Chapter VII

Chapter VII

MOLECULES AND ATOMS. THE LAWS OF GAY-LUSSAC AND AVOGADRO-GERHARDT

Hydrogen combines with oxygen in the proportion of two volumes to one. The composition by volume of nitrous oxide is exactly similar--it is composed of two volumes of nitrogen and one volume of oxygen. By decomposing ammonia by the action of an electric spark it is easy to prove that it contains one volume of nitrogen to three volumes of hydrogen. So, similarly, it is found, whenever a compound is decomposed and the volumes of the gases proceeding from it are measured, that the volumes of the gases or vapours entering into combination are in a very simple proportion to one another. With water, nitrous oxide, &c., this may be proved by direct observation; but in the majority of cases, and especially with substances which, although volatile--that is, capable of passing into a gaseous (or vaporous) state--are liquid at the ordinary temperature, such a direct method of observation presents many difficulties. But, then, if the densities of the vapours and gases be known, the same simplicity in their ratio is shown by calculation. The volume of a substance is proportional to its weight, and inversely proportional to its density, and therefore by dividing the amount by weight of each substance entering into the composition of a compound by its density in the gaseous or vaporous state we shall obtain factors which will be in the same proportion as the volumes of the substances entering into the composition of the compound.[1] So, for example, water contains eight parts by weight of oxygen to one part by weight of hydrogen, and their densities are 16 and 1, consequently their volumes (or the above-mentioned factors) are 1 and 1/2, and therefore it is seen without direct experiment that water contains two volumes of hydrogen for every one volume of oxygen. So also, knowing that nitric oxide contains fourteen parts of nitrogen and sixteen parts of oxygen, and knowing that the specific gravities of these last two gases are fourteen and sixteen, we find that the volumes in which nitrogen and oxygen combine for the formation of nitric oxide are in the proportion of 1 : 1. We will cite another example. In the last chapter we saw that the density of NO_{2} only becomes constant and equal to twenty-three (referred to hydrogen) above 135°, and as a matter of fact a method of direct observation of the volumetric composition of this substance would be very difficult at so high a temperature. But it may be easily calculated. NO_{2}, as is seen from its formula and analysis, contains thirty-two parts by weight of oxygen to fourteen parts by weight of nitrogen, forming forty-six parts by weight of NO_{2}, and knowing the densities of these gases we find that one volume of nitrogen with two volumes of oxygen gives two volumes of nitrogen peroxide. Therefore, knowing the amounts by weight of the substances participating in a reaction or forming a given substance, and knowing the density of the gas or vapour,[2] the volumetric relations of the substances acting in a reaction or entering into the composition of a compound, may be also determined.

[1] If the weight be indicated by P, the density by D, and the volume by V, then

P/D = _K_V

where K is a coefficient depending on the system of the expressions P, D, and V. If D be the weight of a cubic measure of a substance referred to the weight of the same measure of water--if, as in the metrical system (Chapter I., Note 9), the cubic measure of one part by weight of water be taken as a unit of volume--then K = 1. But, whatever it be, it is cancelled in dealing with the comparison of volumes, because comparative and not absolute measures of volumes are taken. In this chapter, as throughout the book, the weight P is given in grams in dealing with absolute weights; and if comparative, as in the expression of chemical composition, then the weight of an atom is taken as unity. The density of gases, D, is also taken in reference to the density of hydrogen, and the volume V in metrical units (cubic centimetres), if it be a matter of absolute magnitudes of volumes, and if it be a matter of chemical transformations--that is, of relative volumes--then the volume of an atom of hydrogen, or of one part by weight of hydrogen, is taken as unity, and all volumes are expressed according to these units.

[2] As the volumetric relations of vapours and gases, next to the relations of substances by weight, form the most important province of chemistry, and a most important means for the attainment of chemical conclusions, and inasmuch as these volumetric relations are determined by the densities of gases and vapours, necessarily the methods of determining the densities of vapours (and also of gases) are important factors in chemical research. These methods are described in detail in works on physics and physical and analytical chemistry, and therefore we here only touch on the general principles of the subject.

If we know the weight p and volume v, occupied by the vapour of a given substance at a temperature t and pressure h, then its density may be directly obtained by dividing p by the weight of a volume v of hydrogen (if the density be expressed according to hydrogen, see Chapter II., Note 23) at t and h. Hence, the methods of determining the density of vapours and gases are based on the determination of p, v, t, and h. The two last data (the temperature t and pressure h) are given by the thermometer and barometer and the heights of mercury or other liquid confining the gas, and therefore do not require further explanation. It need only be remarked that: (1) In the case of easily volatile liquids there is no difficulty in procuring a bath with a constant temperature, but that it is nevertheless best (especially considering the inaccuracy of thermometers) to have a medium of absolutely constant temperature, and therefore to take either a bath in which some substance is melting--such as melting ice at 0° or crystals of sodium acetate, melting at +56°--or, as is more generally practised, to place the vessel containing the substance to be experimented with in the vapour of a liquid boiling at a definite temperature, and knowing the pressure under which it is boiling, to determine the temperature of the vapour. For this purpose the boiling points of water at different pressures are given in Chapter I., Note 11, and the boiling points of certain easily procurable liquids at various pressures are given in Chapter II., Note 27. (2) With respect to temperatures above 300° (below which mercurial thermometers may be conveniently employed), they are most simply obtained constant (to give time for the weight and volume of a substance being observed in a given space, and to allow that space to attain the calculated temperature t) by means of substances boiling at a high temperature. Thus, for instance, at the ordinary atmospheric pressure the temperature t of the vapour of sulphur is about 445°, of phosphorus pentasulphide 518°, of tin chloride 606°, of cadmium 770°, of zinc 930° (according to Violle and others), or 1040° (according to Deville), &c. (3) The indications of the hydrogen thermometer must be considered as the most exact (but as hydrogen diffuses through incandescent platinum, nitrogen is usually employed). (4) The temperature of the vapours used as the bath should in every case be several degrees higher than the boiling point of the liquid whose density is to be determined, in order that no portion should remain in a liquid state. But even in this case, as is seen from the example of nitric peroxide (Chapter VI.), the vapour density does not always remain constant with a change of t, as it should were the law of the expansion of gases and vapours absolutely exact (Chapter II., Note 26). If variations of a chemical and physical nature similar to that which we saw in nitric peroxide take place in the vapours, the main interest is centred in constant densities, which do not vary with t, and therefore the possible effect of t on the density must always be kept in mind in having recourse to this means of investigation. (5) Usually, for the sake of convenience of observation, the vapour density is determined at the atmospheric pressure which is read on the barometer; but in the case of substances which are volatilised with difficulty, and also of substances which decompose, or, in general, vary at temperatures near their boiling points, it is best or even indispensable to conduct the determination at low pressures, whilst for substances which decompose at low pressures the observations have to be conducted under a more or less considerably increased pressure. (6) In many cases it is convenient to determine the vapour density of a substance in admixture with other gases, and consequently under the partial pressure, which may be calculated from the volume of the mixture and that of the intermixed gas (see Chapter I., Note 1). This method is especially important for substances which are easily decomposable, because, as shown by the phenomena of dissociation, a substance is able to remain unchanged in the atmosphere of one of its products of decomposition. Thus, Wurtz determined the density of phosphoric chloride, PCl_{5}, in admixture with the vapour of phosphorous chloride, PCl_{3}. (7) It is evident, from the example of nitric peroxide, that a change of pressure may alter the density and aid decomposition, and therefore identical results are sometimes obtained (if the density be variable) by raising t and lowering h; but if the density does not vary under these variable conditions (at least, to an extent appreciably exceeding the limits of experimental error), then this constant density indicates the gaseous and invariable state of a substance. The laws hereafter laid down refer only to such vapour densities. But the majority of volatile substances show such a constant density at a certain degree above their boiling points up to the starting point of decomposition. Thus, the density of aqueous vapour does not vary for t between the ordinary temperature and 1000° (there are no trustworthy determinations beyond this) and for pressures varying from fractions of an atmosphere up to several atmospheres. If, however, the density does vary considerably with a variation of h and t, the fact may serve as a guide for the investigation of the chemical changes which are undergone by the substance in a state of vapour, or at least as an indication of a deviation from the laws of Boyle, Mariotte, and Gay-Lussac (for the expansion of gases with t). In certain cases the separation of one form of deviation from the other may be explained by special hypotheses.

With respect to the means of determining p and v, with a view to finding the vapour density, we may distinguish three chief methods: (a) by weight, by ascertaining the weight of a definite volume of vapour; (b) by volume, by measuring the volume occupied by the vapour of a definite weight of a substance; and (c) by displacement. The last-mentioned is essentially volumetric, because a known weight of a substance is taken, and the volume of the air displaced by the vapour at a given t and h is determined.

The method by weight (a) is the most trustworthy and historically important. Dumas' method is typical. An ordinary spherical glass or porcelain vessel, like those shown respectively in figs. 52 and 53, is taken, and an excess of the substance to be experimented upon is introduced into it. The vessel is heated to a temperature t higher than the boiling point of the liquid: this gives a vapour which displaces the air, and fills the spherical space. When the air and vapour cease escaping from the sphere, it is fused up or closed by some means; and when cool, the weight of the vapour remaining in the sphere is determined (either by direct weighing of the vessel with the vapour and introducing the necessary corrections for the weight of the air and of the vapour itself, or the weight of the volatilised substance is determined by chemical methods), and the volume of the vapour at t and the barometric pressure h are then calculated.

The volumetric method (b) originally employed by Gay-Lussac and then modified by Hofmann and others is based on the principle that a weighed quantity of the liquid to be experimented with (placed in a small closed vessel, which is sometimes fused up before weighing, and, if quite full of the liquid, breaks when heated in a vacuum) is introduced into a graduated cylinder heated to t, or simply into a Torricellian vacuum, as shown in fig. 54, and the number of volumes occupied by the vapour noted when the space holding it is heated to the desired temperature t.

The method of displacement (c) proposed by Victor Meyer is based on the fact that a space b is heated to a constant temperature t (by the surrounding vapours of a liquid of constant boiling point), and the air (or other gas enclosed in this space) is allowed to attain this temperature, and when it has done so a glass bulb containing a weighed quantity of the substance to be experimented with is dropped into the space. The substance is immediately converted into vapour, and displaces the air into the graduated cylinder e. The amount of this air is calculated from its volume, and hence the volume at t, and therefore also the volume occupied by the vapour, is found. The general arrangement of the apparatus is given in fig. 55.

Such an investigation (either direct, or by calculation from the densities and composition) of every chemical reaction, resulting in the formation of definite chemical compounds, shows that the volumes of the reacting substances in a gaseous or vaporous state are either equal or are in simple multiple proportion.[3] This forms the first law of those discovered by Gay-Lussac. It may be formulated as follows: The amounts of substances entering into chemical reaction occupy under similar physical conditions, in a gaseous or vaporous state, equal or simple multiple volumes. This law refers not only to elements, but also to compounds entering into mutual chemical combination; thus, for example, one volume of ammonia gas combines with one volume of hydrogen chloride. For in the formation of sal-ammoniac, NH_{4}Cl, there enter into reaction 17 parts by weight of ammonia, NH_{3}, which is 8·5 times denser than hydrogen, and 36·5 parts by weight of hydrogen chloride, whose vapour density is 18·25 times that of hydrogen, as has been proved by direct experiment. By dividing the weights by the respective densities we find that the volume of ammonia, NH_{3}, is equal to two, and so also the volume of hydrogen chloride. Hence the volumes of the compounds which here combine together are equal to each other. Taking into consideration that the law of Gay-Lussac holds good, not only for elements, but also for compounds, it should be expressed as follows: Substances interact with one another in commensurable volumes of their vapours.[4]

[3] Vapours and gases, as already explained in the second chapter, are subject to the same laws, which are, however, only approximate. It is evident that for the deduction of the laws which will presently be enunciated it is only possible to take into consideration a perfect gaseous state (far removed from the liquid state) and chemical invariability in which the _vapour density is constant_--that is, the volume of a given gas or vapour varies like a volume of hydrogen, air, or other gas, with the pressure and temperature.

It is necessary to make this statement in order that it may be clearly seen that the laws of gaseous volumes, which we shall describe presently, are in the most intimate connection with the laws of the variations of volumes with pressure and temperature. And as these latter laws (Chapter II.) are not infallible, but only approximately exact, the same, therefore, applies to the laws about to be described. And as it is possible to find more exact laws (a second approximation) for the variation of v with p and t (for example, van der Waals' formula, Chapter II., Note 33), so also a more exact expression of the relation between the composition and the density of vapours and gases is also possible. But to prevent any doubt arising at the very beginning as to the breadth and general application of the laws of volumes, it will be sufficient to mention that the density of such gases as oxygen, nitrogen, and carbonic anhydride is already known to remain constant (within the limits of experimental error) between the ordinary temperature and a white heat; whilst, judging from what is said in my work on the 'Tension of Gases' (vol. i. p. 9), it may be said that, as regards pressure, the relative density remains very constant, even when the deviations from Mariotte's law are very considerable. However, in this respect the number of data is as yet too small to arrive at an exact conclusion.

[4] We must recollect that this law is only approximate, like Boyle and Mariotte's law, and that, therefore, like the latter, a more exact expression may be found for the exceptions.

The law of combining volumes and the law of multiple proportion were discovered independently of each other--the one in France by Gay-Lussac, the other in England by Dalton--almost simultaneously. In the language of the atomic hypothesis it may be said that atomic quantities of elements occupy equal or multiple volumes.

The first law of Gay-Lussac expresses the relation between the volumes of the component parts of a compound. Let us now consider the relation existing between the volumes of the component parts and of the compounds which proceed from them. This may sometimes be determined by direct observation. Thus the volume occupied by water, formed by two volumes of hydrogen and one volume of oxygen, may be determined by the aid of the apparatus shown in fig. 56. The long glass tube is closed at the top and open at the bottom, which is immersed in a cylinder containing mercury. The closed end is furnished with wires like a eudiometer. The tube is filled with mercury, and then a certain volume of detonating gas is introduced. This gas is obtained from the decomposition of water, and therefore in every three volumes contains two volumes of hydrogen and one volume of oxygen. The tube is surrounded by a second and wider glass tube, and the vapour of a substance boiling above 100°--that is, whose boiling point is higher than that of water--is passed through the annular space between them. Amyl alcohol, whose boiling point is 132°, may be taken for this purpose. The amyl alcohol is boiled in the vessel to the right hand and its vapour passed between the walls of the two tubes. In the case of amyl alcohol the outer glass tube should be connected with a condenser to prevent the escape into the air of the unpleasant-smelling vapour. The detonating gas is thus heated up to a temperature of 132°. When its volume becomes constant it is measured, the height of the column of mercury in the tube above the level of the mercury in the cylinder being noted. Let this volume equal v; it will therefore contain 1/3 v of oxygen and 2/3 v of hydrogen. The current of vapour is then stopped, and the gas exploded; water is formed, which condenses into a liquid. The volume occupied by the vapour of the water formed has now to be determined. For this purpose the vapour of the amyl alcohol is again passed between the tubes, and thus the whole of the water formed is converted into vapour at the same temperature as that at which the detonating gas was measured; and the cylinder of mercury being raised until the column of mercury in the tube stands at the same height above the surface of the mercury in the cylinder as it did before the explosion, it is found that the volume of the water formed is equal to 2/3 v--that is, it is equal to the volume of the hydrogen contained in it. Consequently the volumetric composition of water is expressed in the following terms: Two volumes of hydrogen combine with one volume of oxygen to form two volumes of aqueous vapour. For substances which are gaseous at the ordinary temperature, this direct method of observation is sometimes very easily conducted; for instance, with ammonia, nitric and nitrous oxides. Thus to determine the composition by volume of nitrous oxide, the above-described apparatus may be employed. Nitrous oxide is introduced into the tube, and after measuring its volume electric sparks are passed through the gas; it is then found that two volumes of nitrous oxide have given three volumes of gases--namely, two volumes of nitrogen and one volume of oxygen. Consequently the composition of nitrous oxide is similar to that of water; two volumes of nitrogen and one volume of oxygen give two volumes of nitrous oxide. By decomposing ammonia it is found to be composed in such a manner that two volumes give one volume of nitrogen and three volumes of hydrogen; also two volumes of nitric oxide are formed by the union of one volume of oxygen with one volume of nitrogen. The same relations may be proved by calculation from the vapour densities, as was described above.

Comparisons of various results made by the aid of direct observations or calculation, an example of which has just been cited, led Gay-Lussac to the conclusion that the volume of a compound in a gaseous or vaporous state is always in simple multiple proportion to the volume of each of the component parts of which it is formed (and consequently to the sum of the volumes of the elements of which it is formed). This is the second law of Gay-Lussac; it extends the simplicity of the volumetric relations to compounds, and is of the same nature as that presented by the elements entering into mutual combination. Hence not only the substances forming a given compound, but also the substances formed, exhibit a simple relation of volume when measured as vapour or gas.[5]

[5] This second law of volumes may be considered as a consequence of the first law. The first law requires simple ratios between the volumes of the combining substances A and B. A substance AB is produced by their combination. It may, according to the law of multiple proportion, combine, not only with substances C, D, &c., but also with A and with B. In this new combination the volume of AB, combining with the volume of A, should be in simple multiple proportion with the volume of A; hence the volume of the compound AB is in simple proportion to the volume of its component parts. Therefore only one law of volumes need be accepted. We shall afterwards see that there is a third law of volumes embracing also the two first laws.

When a compound is formed from two or more components, there may or may not be a contraction; the volume of the reacting substances is in this case either equal to or greater than the volume of the resultant compound. The reverse is naturally observed in the case of decompositions, when from one substance there are produced several of simpler nature. Therefore in the future we shall term combination a reaction in which a contraction is observed--that is, a diminution in the volume of the component bodies in a state of vapour or gas; and we shall term decomposition a reaction in which an expansion is produced; while those reactions in which the volumes in a gaseous or vaporous state remain constant (the volumes being naturally compared at the same temperature and pressure) we shall term reactions of substitution or of double decomposition. Thus the transition of oxygen into ozone is a reaction of combination, the formation of nitrous oxide from oxygen and nitrogen will also be a combination, the formation of nitric oxide from the same will be a reaction of substitution, the action of oxygen on nitric oxide a combination, and so on.

The degree of contraction produced in the formation of chemical compounds not unfrequently leads to the possibility of distinguishing the degree of change which takes place in the chemical character of the components when combined. In those cases in which a contraction occurs, the properties of the resultant compound are very different from the properties of the substances of which it is composed. Thus ammonia bears no resemblance in its physical or chemical properties to the elements from which it is derived; a contraction takes place in a state of vapour, indicating a proximation of the elements--the distance between the atoms is diminished, and from gaseous substances there is formed a liquid substance, or at any rate one which is easily liquefied. For this reason nitrous oxide formed by the condensation of two permanent gases is a substance which is somewhat easily converted into a liquid; again, nitric acid, which is formed from elements which are permanent gases, is a liquid, whilst, on the contrary, nitric oxide, which is formed without contraction and is decomposed without expansion, remains a gas which is as difficult to liquefy as nitrogen and oxygen. In order to obtain a still more complete idea of the dependence of the properties of a compound on the properties of the component substances, it is further necessary to know the quantity of heat which is developed in the formation of the compound. If this quantity be large--as, for example, in the formation of water--then the amount of energy in the resultant compound will be considerably less than the energy of the elements entering into its composition; whilst, on the contrary, if the amount of heat evolved in the formation of a compound be small, or if there even be an absorption of heat, as in the formation of nitrous oxide, then the energy of the elements is not destroyed, or is only altered to a slight extent; hence, notwithstanding the contraction (compression) involved in its formation, nitrous oxide supports combustion.

The preceding laws were deduced from purely experimental and empirical data and as such evoke further consequences, as the law of multiple proportions gave rise to the atomic theory and the law of equivalents (Chapter IV.) In view of the atomic conception of the constitution of substances, the question naturally arises as to what, then, are the relative volumes proper to those physically indivisible molecules which chemically react on each other and consist of the atoms of elements. The simplest possible hypothesis in this respect would be that the volumes of the molecules of substances are equal; or, what is the same thing, to suppose that equal volumes of vapours and gases contain an equal number of molecules. This proposition was first enunciated by the Italian savant Avogadro in 1810. It was also admitted by the French physico-mathematician Ampère (1815) for the sake of simplifying all kinds of physico-mathematical conceptions respecting gases. But Avogadro and Ampère's propositions were not generally received in science until Gerhardt in the forties had applied them to the generalisation of chemical reactions, and had demonstrated, by aid of a series of phenomena, that the reactions of substances actually take place with the greatest simplicity, and more especially that such reactions take place between those quantities of substances which occupy equal volumes, and until he had stated the hypothesis in an exact manner and deduced the consequences that necessarily follow from it. Following Gerhardt, Clausius, in the fifties, placed this hypothesis of the equality of the number of molecules in equal volumes of gases and vapours on the basis of the kinetic theory of gases. At the present day the hypothesis of Avogadro and Gerhardt lies at the basis of contemporary physical, mechanical, and chemical conceptions; the consequences arising from it have often been subject to doubt, but in the end have been verified by the most diverse methods; and now, when all efforts to refute those consequences have proved fruitless, the hypothesis must be considered as verified,[6] and the law of Avogadro-Gerhardt must be spoken of as fundamental, and as of great importance for the comprehension of the phenomena of nature. The law may now be formulated from two points of view. In the first place, from a physical aspect: equal volumes of gases (or vapours) at equal temperatures and pressures contain the same number of molecules--or of particles of matter which are neither mechanically nor physically divisible--previous to chemical change. In the second place, from a chemical aspect, the same law may be expressed thus: the quantities of substances entering into chemical reactions occupy, in a state of vapour, equal volumes. For our purpose the chemical aspect is the most important, and therefore, before developing the law and its consequences, we will consider the chemical phenomena from which the law is deduced or which it serves to explain.

[6] It must not be forgotten that Newton's law of gravity was first a hypothesis, but it became a trustworthy, perfect theory, and acquired the qualities of a fundamental law owing to the concord between its deductions and actual facts. All laws, all theories, of natural phenomena, are at first hypotheses. Some are rapidly established by their consequences exactly agreeing with facts; others only take root by slow degrees; and there are many which are destined to be refuted owing to their consequences being found to be at variance with facts.

When two isolated substances interact with each other directly and easily--as, for instance, an alkali and an acid--then it is found that the reaction is accomplished between quantities which in a gaseous state occupy equal volumes. Thus ammonia, NH_{3}, reacts directly with hydrochloric acid, HCl, forming sal-ammoniac, NH_{4}Cl, and in this case the 17 parts by weight of ammonia occupy the same volume as the 36·5 parts by weight of hydrochloric acid.[7] Ethylene, C_{2}H_{4}, combines with chlorine, Cl_{2}, in only one proportion, forming ethylene dichloride, C_{2}H_{4}Cl_{2}, and this combination proceeds directly and with great facility, the reacting quantities occupying equal volumes. Chlorine reacts with hydrogen in only one proportion, forming hydrochloric acid, HCl, and in this case equal volumes interact with each other. If an equality of volumes is observed in cases of combination, it should be even more frequently encountered in cases of decomposition, taking place in substances which split up into two others. Indeed, acetic acid breaks up into marsh gas, CH_{4}, and carbonic anhydride, CO_{2}, and in the proportions in which they are formed from acetic acid they occupy equal volumes. Also from phthalic acid, C_{8}H_{6}O_{4}, there may be obtained benzoic acid, C_{7}H_{6}O_{2}, and carbonic anhydride, CO_{2}, and as all the elements of phthalic acid enter into the composition of these substances, it follows that, although they cannot re-form it by their direct action on each other (the reaction is not reversible), still they form the direct products of its decomposition, and they occupy equal volumes. But benzoic acid, C_{7}H_{6}O_{2}, is itself composed of benzene, C_{6}H_{6}, and carbonic anhydride, CO_{2}, which also occupy equal volumes.[8] There is an immense number of similar examples among those organic substances to whose study Gerhardt consecrated his whole life and work, and he did not allow such facts as these to escape his attention. Still more frequently in the phenomena of substitution, when two substances react on one another, and two are produced without a change of volume, it is found that the two substances acting on each other occupy equal volumes as well as each of the two resultant substances. Thus, in general, reactions of substitution take place between volatile acids, HX, and volatile alcohols, R(OH), with the formation of ethereal salts, RX, and water, H(OH), and the volume of the vapour of the reacting quantities, HX, R(OH), and RX, is the same as that of water H(OH), whose weight, corresponding with the formula, 18, occupies 2 volumes, if 1 part by weight of hydrogen occupy 1 volume and the density of aqueous vapour referred to hydrogen is 9. Such general examples, of which there are many,[9] show that the reaction of equal volumes forms a chemical phenomenon of frequent occurrence, indicating the necessity for acknowledging the law of Avogadro-Gerhardt.

[7] This is not only seen from the above calculations, but may be proved by experiment. A glass tube, divided in the middle by a stopcock, is taken and one portion filled with dry hydrogen chloride (the dryness of the gases is very necessary, because ammonia and hydrogen chloride are both very soluble in water, so that a small trace of water may contain a large amount of these gases in solution) and the other with dry ammonia, under the atmospheric pressure. One orifice (for instance, of that portion which contains the ammonia) is firmly closed, and the other is immersed under mercury, and the cock is then opened. Solid sal-ammoniac is formed, but if the volume of one gas be greater than that of the other, some of the first gas will remain. By immersing the tube in the mercury in order that the internal pressure shall equal the atmospheric pressure, it may easily be shown that the volume of the remaining gas is equal to the difference between the volumes of the two portions of the tube, and that this remaining gas is part of that whose volume was the greater.

[8] Let us demonstrate this by figures. From 122 grams of benzoic acid there are obtained (a) 78 grams of benzene, whose density referred to hydrogen = 39, hence the relative volume = 2; and (b) 44 grams of carbonic anhydride, whose density = 22, and hence the volume = 2. It is the same in other cases.

[9] A large number of such generalised reactions, showing reaction by equal volumes, occur in the case of the hydrocarbon derivatives, because many of these compounds are volatile. The reactions of alkalis on acids, or anhydrides on water, &c., which are so frequent between mineral substances, present but few such examples, because many of these substances are not volatile and their vapour densities are unknown. But essentially the same is seen in these cases also; for instance, sulphuric acid, H_{2}SO_{4}, breaks up into the anhydride, SO_{3}, and water, H_{2}O, which exhibit an equality of volumes. Let us take another example where three substances combine in equal volumes: carbonic anhydride, CO_{2}, ammonia, NH_{3}, and water, H_{2}O (the volumes of all are equal to 2), form acid ammonium carbonate, (NH_{4})HCO_{3}.

But the question arises, What is the relation of volumes if the reaction of two substances takes place in more than one proportion, according to the law of multiple proportions? A definite answer can only be given in cases which have been very thoroughly studied. Thus chlorine, in acting on marsh gas, CH_{4}, forms four compounds, CH_{3}Cl, CH_{2}Cl_{2}, CHCl_{3}, and CCl_{4}, and it may be established by direct experiment that the substance CH_{3}Cl (methylic chloride) precedes the remainder, and that the latter proceed from it by the further action of chlorine. And this substance, CH_{3}Cl, is formed by the reaction of equal volumes of marsh gas, CH_{4}, and chlorine, Cl_{2}, according to the equation CH_{4} + Cl_{2} = CH_{3}Cl + HCl. A great number of similar cases are met with amongst organic--that is, carbon--compounds. Gerhardt was led to the discovery of his law by investigating many such reactions, and by observing that in them the reaction of equal volumes precedes all others.

But if nitrogen or hydrogen give several compounds with oxygen, the question proposed above cannot be answered with complete clearness, because the successive formations of the different combinations cannot be so strictly defined. It may be supposed, but neither definitely affirmed nor experimentally confirmed, that nitrogen and oxygen first give nitric oxide, NO, and only subsequently the brown vapours N_{2}O_{3} and NO_{2}. Such a sequence in the combination of nitrogen with oxygen can only be supposed on the basis of the fact that NO forms N_{2}O_{3} and NO_{2} directly with oxygen. If it be admitted that NO (and not N_{2}O or NO_{2}) be first formed, then this instance would also confirm the law of Avogadro-Gerhardt, because nitric oxide contains equal volumes of nitrogen and oxygen. So, also, it may be admitted that, in the combination of hydrogen with oxygen, hydrogen peroxide is first formed (equal volumes of hydrogen and oxygen), which is decomposed by the heat evolved into water and oxygen. This explains the presence of traces of hydrogen peroxide (Chapter IV.) in almost all cases of the combustion or oxidation of hydrogenous substances; for it cannot be supposed that water is first formed and then the peroxide of hydrogen, because up to now such a reaction has not been observed, whilst the formation of H_{2}O from H_{2}O_{2} is very easily reproduced.[10]

[10] This opinion which I have always held (since the first editions of this work), as to the primary origin of hydrogen peroxide and of the formation of water by means of its decomposition, has in latter days become more generally accepted, thanks more especially to the work of Traube. Probably it explains most simply the necessity for the presence of traces of water in many reactions, as, for instance, in the explosion of carbonic oxide with oxygen, and perhaps the theory of the explosion of detonating gas itself and of the combustion of hydrogen will gain in clearness and truth if we take into consideration the preliminary formation of hydrogen peroxide and its decomposition. We may here point out the fact that Ettingen (at Dorpat, 1888) observed the existence of currents and waves in the explosion of detonating gas by taking photographs, which showed the periods of combustion and the waves of explosion, which should be taken into consideration in the theory of this subject. As the formation of H_{2}O_{2} from O_{2} and H_{2} corresponds with a less amount of heat than the formation of water from H_{2} and O, it may be that the temperature of the flame of detonating gas depends on the pre-formation of hydrogen peroxide.

Thus a whole series of phenomena show that the chemical reaction of substances actually takes place, as a rule, between equal volumes, but this does not preclude the possibility of the frequent reaction of unequal volumes, although, in this case, it is often possible to discover a preceding reaction between equal volumes.[11]

[11] The possibility of reactions between unequal volumes, notwithstanding the general application of the law of Avogadro-Gerhardt, may, in addition to what has been said above, depend on the fact that the participating substances, at the moment of reaction, undergo a preliminary modification, decomposition, isomeric (polymeric) transformation, &c. Thus, if NO_{2}, seems to proceed from N_{2}O_{4}, if O_{2} is formed from O_{3}, and the converse, then it cannot be denied that the production of molecules containing only one atom is also possible--for instance, of oxygen--as also of higher polymeric forms--as the molecule N from N_{2}, or H_{3} from H_{2}. In this manner it is obviously possible, by means of a series of hypotheses, to explain the cases of the formation of ammonia, NH_{3}, from 3 vols. of hydrogen and 1 vol. of nitrogen. But it must be observed that perhaps our information in similar instances is, as yet, far from being complete. If hydrazine or diamide N_{2}H_{4} (Chapter VI. Note 20 bis) is formed and the imide N_{2}H_{2} in which 2 vols. of hydrogen are combined with 2 vols. of nitrogen, then the reaction here perhaps first takes place between equal volumes. If it be shown that diamide gives nitrogen and ammonia (3N_{2}H_{4} = N_{2} + 4NH_{3}) under the action of sparks, heat, or the silent discharge, &c., then it will be possible to admit that it is formed before ammonia. And perhaps the still less stable imide N_{2}H_{2}, which may also decompose with the formation of ammonia, is produced before the amide N_{2}H_{4}.

I mention this to show that the fact of apparent exceptions existing to the law of reactions between equal volumes does not prove the impossibility of their being included under the law on further study of the subject. Having put forward a certain law or hypothesis, consequences must be deduced from it, and if by their means clearness and consistency are attained--and especially, if by their means that which could not otherwise be known can be predicted--then the consequences verify the hypothesis. This was the case with the law now under discussion. The mere simplicity of the deduction of the weights proper to the atoms of the elements, or the mere fact that having admitted the law it follows (as will afterwards be shown) that the vis viva of the molecules of all gases is a constant quantity, is quite sufficient reason for retaining the hypothesis, if not for believing in it as a fact beyond doubt. And such is the whole doctrine of atoms. And since by the acceptance of the law it became possible to foretell even the properties and atomic weights of elements which had not yet been discovered, and these predictions afterwards proved to be in agreement with the actual facts, it is evident that the law of Avogadro-Gerhardt penetrates deeply into the nature of the chemical relation of substances. This being granted, it is possible at the present time to exhibit and deduce the truth under consideration in many ways, and in every case, like all that is highest in science (for example, the laws of the indestructibility of matter, of the conservation of energy, of gravity, &c.), it proves to be not an empirical conclusion from direct observation and experiment, not a direct result of analysis, but a creation, or instinctive penetration, of the inquiring mind, guided and directed by experiment and observation--a synthesis of which the exact sciences are capable equally with the highest forms of art. Without such a synthetical process of reasoning, science would only be a mass of disconnected results of arduous labour, and would not be distinguished by that vitality with which it is really endowed when once it succeeds in attaining a synthesis, or concordance of outward form with the inner nature of things, without losing sight of the diversities of individual parts; in short, when it discovers by means of outward phenomena, which are apparent to the sense of touch, to observation, and to the common mind, the internal signification of things--discovering simplicity in complexity and uniformity in diversity. And this is the highest problem of science.

The law of Avogadro-Gerhardt may also be easily expressed in an algebraical form. If the weight of a molecule, or of that quantity of a substance which enters into chemical reaction and occupies in a state of vapour, according to the law, a volume equal to that occupied by the molecules of other bodies, be indicated by the letters M_{1}, M_{2} ... or, in general, M, and if the letters D_{1}, D_{2}, ... or, in general, D, stand for the density or weight of a given volume of the gases or vapours of the corresponding substances under certain definite conditions of temperature and pressure, then the law requires that

M_{1}/D_{1} = M_{2}/D_{2} ... = M/D = C

where C is a certain constant. This expression shows directly that the volumes corresponding with the weights M_{1}, M_{2} ... M, are equal to a certain constant, because the volume is proportional to the weight and inversely proportional to the density. The magnitude of C is naturally conditioned by and dependent on the units taken for the expression of the weights of the molecules and the densities. The weight of a molecule (equal to the sum of the atomic weights of the elements forming it) is usually expressed by taking the weight of an atom of hydrogen as unity, and hydrogen is now also chosen as the unit for the expression of the densities of gases and vapours; it is therefore only necessary to find the magnitude of the constant for any one compound, as it will be the same for all others. Let us take water. Its reacting mass is expressed (conditionally and relatively) by the formula or molecule H_{2}O, for which M = 18, if H = 1, as we already know from the composition of water. Its vapour density, or D, compared to hydrogen = 9, and consequently for water C = 2, and therefore and in general for the molecules of all substances M/D = 2.

Consequently the weight of a molecule is equal to twice its vapour density expressed in relation to hydrogen, and conversely the density of a gas is equal to half the molecular weight referred to hydrogen.

The truth of this may be seen from a very large number of observed vapour densities by comparing them with the results obtained by calculation. As an illustration, we may point out that for ammonia, NH_{3}, the weight of the molecule or quantity of the reacting substance, as well as the composition and weight corresponding with the formula, is expressed by the figures 14 + 3 = 17. Consequently M = 17. Hence, according to the law, D = 8·5. And this result is also obtained by experiment. The density, according to both formula and experiment, of nitrous oxide, N_{2}O, is 22, of nitric acid 15, and of nitric peroxide 23. In the case of nitrous anhydride, N_{2}O_{3}, as a substance which dissociates into NO + NO_{2}, the density should vary between 38 (so long as the N_{2}O_{3} remains unchanged) and 19 (when NO + NO_{2} is obtained). There are no figures of constant density for H_{2}O_{2}, NHO_{3}, N_{2}O_{4}, and many similar compounds which are either wholly or partially decomposed in passing into vapour. Salts and similar substances either have no vapour density because they do not pass into vapour (for instance, potassium nitrate, KNO_{3}) without decomposition, or, if they pass into vapour without decomposing, their vapour density is observed with difficulty only at very high temperatures. The practical determination of the vapour density at these high temperatures (for example, for sodium chloride, ferrous chloride, stannous chloride, &c.) requires special methods which have been worked out by Sainte-Claire Deville, Crafts, Nilson and Pettersson, Meyer, Scott, and others. Having overcome the difficulties of experiment, it is found that the law of Avogadro-Gerhardt holds good for such salts as potassium iodide, beryllium chloride, aluminium chloride, ferrous chloride, &c.--that is, the density obtained by experiment proves to be equal to half the molecular weight--naturally within the limits of experimental error or of possible deviation from the law.

Gerhardt deduced his law from a great number of examples of volatile carbon compounds. We shall become acquainted with certain of them in the following chapters; their entire study, from the complexity of the subject, and from long-established custom, forms the subject of a special branch of chemistry termed 'organic' chemistry. With all these substances the observed and calculated densities are very similar.

When the consequences of a law are verified by a great number of observations, it should be considered as confirmed by experiment. But this does not exclude the possibility of apparent deviations. They may evidently be of two kinds: the fraction M/D may be found to be either greater or less than 2--that is, the calculated density may be either greater or less than the observed density. When the difference between the results of experiment and calculation falls within the possible errors of experiment (for example, equal to hundredths of the density), or within a possible error owing to the laws of gases having an only approximate application (as is seen from the deviations, for instance, from the law of Boyle and Mariotte), then the fraction M/D proves but slightly different from 2 (between 1·9 and 2·2), and such cases as these may be classed among those which ought to be expected from the nature of the subject. It is a different matter if the quotient of M/D be several times, and in general a multiple, greater or less than 2. The application of the law must then be explained or it must be laid aside, because the laws of nature admit of no exceptions. We will therefore take two such cases, and first one in which the quotient M/D is greater than 2, or the density obtained by experiment is less than is in accordance with the law.

It must be admitted, as a consequence of the law of Avogadro-Gerhardt, that there is a decomposition in those cases where the volume of the vapour corresponding with the weight of the amount of a substance entering into reaction is greater than the volume of two parts by weight of hydrogen. Suppose the density of the vapour of water to be determined at a temperature above that at which it is decomposed, then, if not all, at any rate a large proportion of the water will be decomposed into hydrogen and oxygen. The density of such a mixture of gases, or of detonating gas, will be less than that of aqueous vapour; it will be equal to 6 (compared with hydrogen), because 1 volume of oxygen weighs 16, and 2 volumes of hydrogen 2; and, consequently, 3 volumes of detonating gas weigh 18 and 1 volume 6, while the density of aqueous vapour = 9. Hence, if the density of aqueous vapour be determined after its decomposition, the quotient M/D would be found to be 3 and not 2. This phenomenon might be considered as a deviation from Gerhardt's law, but this would not be correct, because it may be shown by means of diffusion through porous substances, as described in Chapter II., that water is decomposed at such high temperatures. In the case of water itself there can naturally be no doubt, because its vapour density agrees with the law at all temperatures at which it has been determined.[12] But there are many substances which decompose with great ease directly they are volatilised, and therefore only exist as solids or liquids, and not in a state of vapour. There are, for example, many salts of this kind, besides all definite solutions having a constant boiling point, all the compounds of ammonia for example, all ammonium salts--&c. Their vapour densities, determined by Bineau, Deville, and others, show that they do not agree with Gerhardt's law. Thus the vapour density of sal-ammoniac, NH_{4}Cl, is nearly 14 (compared with hydrogen), whilst its molecular weight is not less than 53·5, whence the vapour density should be nearly 27, according to the law. The molecule of sal-ammoniac cannot be less than NH_{4}Cl, because it is formed from the molecules NH_{3} and HCl, and contains single atoms of nitrogen and chlorine, and therefore cannot be divided; it further never enters into reactions with the molecules of other substances (for instance, potassium hydroxide, or nitric acid) in quantities of less than 53·5 parts by weight, &c. The calculated density (about 27) is here double the observed density (about 13·4); hence M/D = 4 and not 2. For this reason the vapour density of sal-ammoniac for a long time served as an argument for doubting the truth of the law. But it proved otherwise, after the matter had been fully investigated. The low density depends on the decomposition of sal-ammoniac, on volatilising, into ammonia and hydrogen chloride. The observed density is not that of sal-ammoniac, but of a mixture of NH_{3} and HCl, which should be nearly 14, because the density of NH_{3} = 8·5 and of HCl = 18·2, and therefore the density of their mixture (in equal volumes) should be about 13·4.[13] The actual decomposition of the vapours of sal-ammoniac was demonstrated by Pebal and Than by the same method as the decomposition of water, by passing the vapour of sal-ammoniac through a porous substance. The experiment demonstrating the decomposition during volatilisation of sal-ammoniac may be made very easily, and is a very instructive point in the history of the law of Avogadro-Gerhardt, because without its aid it would never have been imagined that sal-ammoniac decomposed in volatilising, as this decomposition bears all the signs of simple sublimation; consequently the knowledge of the decomposition itself was forestalled by the law. The whole aim and practical use of the discovery of the laws of nature consists in, and is shown by, the fact that they enable the unknown to be foretold, the unobserved to be foreseen. The arrangement of the experiment is based on the following reasoning.[14] According to the law and to experiment, the density of ammonia, NH_{3}, is 8-1/2, and of hydrochloric acid, HCl, 18-1/4, if the density of hydrogen = 1. Consequently, in a mixture of NH_{3} and HCl, the ammonia will penetrate much more rapidly through a porous mass, or a fine orifice, than the heavier hydrochloric acid, just as in a former experiment the hydrogen penetrated more rapidly than the oxygen. Therefore, if the vapour of sal-ammoniac comes into contact with a porous mass, the ammonia will pass through it in greater quantities than the hydrochloric acid, and this excess of ammonia may be detected by means of moist red litmus paper, which should be turned blue. If the vapour of sal-ammoniac were not decomposed, it would pass through the porous mass as a whole, and the colour of the litmus paper would not be altered, because sal-ammoniac is a neutral salt. Thus, by testing with litmus the substances passing through the porous mass, it may be decided whether the sal-ammoniac is decomposed or not when passing into vapour. Sal-ammoniac volatilises at so moderate a temperature that the experiment may be conducted in a glass tube heated by means of a lamp, an asbestos plug being placed near the centre of the tube.[15] The asbestos forms a porous mass, which is unaltered at a high temperature. A piece of dry sal-ammoniac is placed at one side of the asbestos plug, and is heated by a Bunsen burner. The vapours formed are driven by a current of air forced from a gasometer or bag through two tubes containing pieces of moist litmus paper, one blue and one red paper in each. If the sal-ammoniac be heated, then the ammonia appears on the opposite side of the asbestos plug, and the litmus there turns blue. And as an excess of hydrochloric acid remains on the side where the sal-ammoniac is heated, it turns the litmus at that end red. This proves that the sal-ammoniac, when converted into vapour, splits up into ammonia and hydrochloric acid, and at the same time gives an instance of the possibility of correctly conjecturing a fact on the basis of the law of Avogadro-Gerhardt.[15 bis]

[12] As the density of aqueous vapour remains constant within the limits of experimental accuracy, even at 1,000°, when dissociation has certainly commenced, it would appear that only a very small amount of water is decomposed at these temperatures. If even 10 p.c. of water were decomposed, the density would be 8·57 and the quotient M/D = 2·1, but at the high temperatures here concerned the error of experiment is not greater than the difference between this quantity and 2. And probably at 1,000° the dissociation is far from being equal to 10 p.c. Hence the variation in the vapour density of water does not give us the means of ascertaining the amount of its dissociation.

[13] This explanation of the vapour density of sal-ammoniac, sulphuric acid, and similar substances which decompose in being distilled was the most natural to resort to as soon as the application of the law of Avogadro-Gerhardt to chemical relations was begun; it was, for instance, given in my work on Specific Volumes, 1856, p. 99. The formula, M/D = 2, which was applied later by many other investigators, had already been made use of in that work.

[14] The beginner must remember that an experiment and the mode in which it is carried out must be determined by the principle or fact which it is intended to illustrate, and not vice versa, as some suppose. The idea which determines the necessity of an experiment is the chief consideration.

[15] It is important that the tubes, asbestos, and sal-ammoniac should be dry, as otherwise the moisture retains the ammonia and hydrogen chloride.

[15 bis] Baker (1894) showed that the decomposition of NH_{4}Cl in the act of volatilising only takes place in the presence of water, traces of which are amply sufficient, but that in the total absence of moisture (attained by carefully drying with P_{2}O_{5}) there is no decomposition, and the vapour density of the sal-ammoniac is found to be normal, i.e., nearly 27. It is not yet quite clear what part the trace of moisture plays here, and it must be presumed that the phenomenon belongs to the category of electrical and contact phenomena, which have not yet been fully explained (see Chapter IX., Note 29).

So also the fact of a decomposition may be proved in the other instances where M/D proved greater than 2, and hence the apparent deviations appear in reality as an excellent proof of the general application and significance of the law of Avogadro-Gerhardt.

In those cases where the quotient M/D proves to be less than 2, or the observed density greater than that calculated, by a multiple number of times, the matter is evidently more simple, and the fact observed only indicates that the weight of the molecule is as many times greater as that taken as the quotient obtained is less than 2. So, for instance, in the case of ethylene, whose composition is expressed by CH_{2}, the density was found by experiment to be 14, and in the case of amylene, whose composition is also CH_{2}, the density proved to be 35, and consequently the quotient for ethylene = 1, and for amylene = 2/5. If the molecular weight of ethylene be taken, not as 14, as might be imagined from its composition, but as twice as great--namely, as 28--and for amylene as five times greater--that is as 70--then the molecular composition of the first will be C_{2}H_{4}, and of the second C_{5}H_{10}, and for both of them M/D will be equal to 2. This application of the law, which at first sight may appear perfectly arbitrary, is nevertheless strictly correct, because the amount of ethylene which reacts--for example, with sulphuric and other acids--is not equal to 14, but to 28 parts by weight. Thus with H_{2}SO_{4}, Br_{2}, or HI, &c., ethylene combines in a quantity C_{2}H_{4}, and amylene in a quantity C_{5}H_{10}, and not CH_{2}. On the other hand, ethylene is a gas which liquefies with difficulty (absolute boiling point = +10°), whilst amylene is a liquid boiling at 35° (absolute boiling point = +192°), and by admitting the greater density of the molecules of amylene (M = 70) its difference from the lighter molecules of ethylene (M = 28) becomes clear. Thus, the smaller quotient M/D is an indication of polymerisation, as the larger quotient is of decomposition. The difference between the densities of oxygen and ozone is a case in point.

On turning to the elements, it is found in certain cases, especially with metals--for instance, mercury, zinc, and cadmium--that that weight of the atoms which must be acknowledged in their compounds (of which mention will be afterwards made) appears to be also the molecular weight. Thus, the atomic weight of mercury must be taken as = 200, but the vapour density = 100, and the quotient = 2. Consequently the molecule of mercury contains one atom, Hg. It is the same with sodium, cadmium, and zinc. This is the simplest possible molecule, which necessarily is only possible in the case of elements, as the molecule of a compound must contain at least two atoms. However, the molecules of many of the elements prove to be complex--for instance, the weight of an atom of oxygen = 16, and its density = 16, so that its molecule must contain two atoms, O_{2}, which might already be concluded by comparing its density with that of ozone, whose molecule contains O_{3} (Chapter IV.) So also the molecule of hydrogen equals H_{2}, of chlorine Cl_{2}, of nitrogen N_{2}, &c. If chlorine react with hydrogen, the volume remains unaltered after the formation of hydrochloric acid, H_{2} + Cl_{2} = HCl + HCl. It is a case of substitution between the one and the other, and therefore the volumes remain constant. There are elements whose molecules are much more complex--for instance, sulphur, S_{6}--although, by heating, the density is reduced to a third, and S_{2} is formed. Judging from the vapour density of phosphorus (D = 62) the molecule contains four atoms P_{4}. Hence many elements when polymerised appear in molecules which are more complex than the simplest possible. In carbon, as we shall afterwards find, a very complex molecule must be admitted, as otherwise its non-volatility and other properties cannot be understood. And if compounds are decomposed by a more or less powerful heat, and if polymeric substances are depolymerised (that is, the weight of the molecule diminishes) by a rise of temperature, as N_{2}O_{4} passes into NO_{2}, or ozone, O_{3}, into ordinary oxygen, O_{2}, then we might expect to find the splitting-up of the complex molecules of elements into the simplest molecule containing a single atom only--that is to say, if O_{2} be obtained from O_{3}, then the formation of O might also be looked for. The possibility but not proof of such a proposition is indicated by the vapour of iodine. Its normal density = 127 (Dumas, Deville, and others), which corresponds with the molecule I_{2}. At temperatures above 800° (up to which the density remains almost constant), this density distinctly decreases, as is seen from the verified results obtained by Victor Meyer, Crafts, and Troost. At the ordinary pressure and 1,000° it is about 100, at 1,250° about 80, at 1,400° about 75, and apparently it strives to reduce itself to one-half--that is, to 63. Under a reduced pressure this splitting-up, or depolymerisation, of iodine vapour actually reaches a density[16] of 66, as Crafts demonstrated by reducing the pressure to 100 mm. and raising the temperature to 1,500°. From this it may be concluded that at high temperatures and low pressures the molecule I_{2} gradually passes into the molecule I containing one atom like mercury, and that something similar occurs with other elements at a considerable rise of temperature, which tends to bring about the disunion of compounds and the decomposition of complex molecules.[17]

[16] Just as we saw (Chapter VI. Note 46) an increase of the dissociation of N_{2}O_{4} and the formation of a large proportion of NO_{2}, with a decrease of pressure. The decomposition of I_{2} into I + I is a similar dissociation.

[17] Although at first there appeared to be a similar phenomenon in the case of chlorine, it was afterwards proved that if there is a decrease of density it is only a small one. In the case of bromine it is not much greater, and is far from being equal to that for iodine.

As in general we very often involuntarily confuse chemical processes with physical, it may be that a physical process of change in the coefficient of expansion with a change of temperature participates with a change in molecular weight, and partially, if not wholly, accounts for the decrease of the density of chlorine, bromine, and iodine. Thus, I have remarked (Comptes Rendus, 1876) that the coefficient of expansion of gases increases with their molecular weight, and (Chapter II., Note 26) the results of direct experiment show the coefficient of expansion of hydrobromic acid (M = 81) to be 0·00386 instead of 0·00367, which is that of hydrogen (M = 2). Hence, in the case of the vapour of iodine (M = 254) a very large coefficient of expansion is to be expected, and from this cause alone the relative density would fall. As the molecule of chlorine Cl_{2} is lighter (= 71) than that of bromine (= 160), which is lighter than that of iodine (= 254), we see that the order in which the decomposability of the vapours of these haloids is observed corresponds with the expected rise in the coefficient of expansion. Taking the coefficient of expansion of iodine vapour as 0·004, then at 1,000° its density would be 116. Therefore the dissociation of iodine may be only an apparent phenomenon. However, on the other hand, the heavy vapour of mercury (M = 200, D = 100) scarcely decreases in density at a temperature of 1,500° (D = 98, according to Victor Meyer); but it must not be forgotten that the molecule of mercury contains only one atom, whilst that of iodine contains two, and this is very important. Questions of this kind which are difficult to decide by experimental methods must long remain without a certain explanation, owing to the difficulty, and sometimes impossibility, of distinguishing between physical and chemical changes.

Besides these cases of apparent discrepancy from the law of Avogadro-Gerhardt there is yet a third, which is the last, and is very instructive. In the investigation of separate substances they have to be isolated in the purest possible form, and their chemical and physical properties, and among them the vapour density, then determined. If it be normal--that is, if D = M/2--it often serves as a proof of the purity of the substance, i.e. of its freedom from all foreign matter. If it be abnormal--that is, if D be not equal to M/2--then for those who do not believe in the law it appears as a new argument against it and nothing more; but to those who have already grasped the important significance of the law it becomes clear that there is some error in the observation, or that the density was determined under conditions in which the vapour does not follow the laws of Boyle or Gay-Lussac, or else that the substance has not been sufficiently purified, and contains other substances. The law of Avogadro-Gerhardt in that case furnishes convincing evidence of the necessity of a fresh and more exact research. And as yet the causes of error have always been found. There are not a few examples in point in the recent history of chemistry. We will cite one instance. In the case of pyrosulphuryl chloride, S_{2}O_{5}Cl_{2}, M = 215, and consequently D should = 107·5, instead of which Ogier and others obtained 53·8--that is, a density half as great; and further, Ogier (1882) demonstrated clearly that the substance is not dissociated by distillation into SO_{3} and SO_{2}Cl_{2}, or any other two products, and thus the abnormal density of S_{2}O_{5}Cl_{2} remained unexplained until D. P. Konovaloff (1885) showed that the previous investigators were working with a mixture (containing SO_{3}HCl), and that pyrosulphuryl chloride has a normal density of approximately 107. Had not the law of Avogadro-Gerhardt served as a guide, the impure liquid would have still passed as pure; the more so since the determination of the amount of chlorine could not aid in the discovery of the impurity. Thus, by following a true law of nature we are led to true deductions.

All cases which have been studied confirm the law of Avogadro-Gerhardt, and as by it a deduction is obtained, from the determination of the vapour density (a purely physical property), as to the weight of the molecule or quantity of a substance entering into chemical reaction, this law links together the two provinces of learning--physics and chemistry--in the most intimate manner. Besides which, the law of Avogadro-Gerhardt places the conceptions of molecules and atoms on a firm foundation, which was previously wanting. Although since the days of Dalton it had become evident that it was necessary to admit the existence of the elementary atom (the chemical individual indivisible by chemical or other forces), and of the groups of atoms (or molecules) of compounds, indivisible by mechanical and physical forces; still the relative magnitude of the molecule and atom was not defined with sufficient clearness. Thus, for instance, the atomic weight of oxygen might be taken as 8 or 16, or any multiple of these numbers, and nothing indicated a reason for the acceptation of one rather than another of these magnitudes;[18] whilst as regards the weights of the molecules of elements and compounds there was no trustworthy knowledge whatever. With the establishment of Gerhardt's law the idea of the molecule was fully defined, as well as the relative magnitude of the elementary atom.

[18] And so it was in the fifties. Some took O = 8, others O = 16. Water in the first case would be HO and hydrogen peroxide HO_{2}, and in the second case, as is now generally accepted, water H_{2}O and hydrogen peroxide H_{2}O_{2} or HO. Disagreement and confusion reigned. In 1860 the chemists of the whole world met at Carlsruhe for the purpose of arriving at some agreement and uniformity of opinion. I was present at this Congress, and well remember how great was the difference of opinion, and how a compromise was advocated with great acumen by many scientific men, and with what warmth the followers of Gerhardt, at whose head stood the Italian professor, Canizzaro, followed up the consequences of the law of Avogadro. In the spirit of scientific freedom, without which science would make no progress, and would remain petrified as in the middle ages, and with the simultaneous necessity of scientific conservatism, without which the roots of past study could give no fruit, a compromise was not arrived at, nor ought it to have been, but instead of it truth, in the form of the law of Avogadro-Gerhardt, received by means of the Congress a wider development, and soon afterwards conquered all minds. Then the new so-called Gerhardt atomic weights established themselves, and in the seventies they were already in general use.

The chemical particle or molecule must be considered as the quantity of a substance which enters into chemical reaction with other molecules, and occupies in a state of vapour the same volume as two parts by weight of hydrogen.

The molecular weight (which has been indicated by M) of a substance is determined by its composition, transformations, and vapour density.

The molecule is not divisible by the mechanical and physical changes of substances, but in chemical reaction it is either altered in its properties, or quantity, or structure, or in the nature of the motion of its parts.

An agglomeration of molecules, which are alike in all chemical respects, makes up the masses of homogeneous substances in all states.[19]

[19] A bubble of gas, a drop of liquid, or the smallest crystal, presents an agglomeration of a number of molecules, in a state of continual motion (like the stars of the Milky Way), distributing themselves evenly or forming new systems. If the aggregation of all kinds of heterogeneous molecules be possible in a gaseous state, where the molecules are considerably removed from each other, then in a liquid state, where they are already close together, such an aggregation becomes possible only in the sense of the mutual reaction between them which results from their chemical attraction, and especially in the aptitude of heterogeneous molecules for combining together. Solutions and other so-called indefinite chemical compounds should be regarded in this light. According to the principles developed in this work we should regard them as containing both the compounds of the heterogeneous molecules themselves and the products of their decomposition, as in peroxide of nitrogen, N_{2}O_{4} and NO_{2}. And we must consider that those molecules A, which at a given moment are combined with B in AB, will in the following moment become free in order to again enter into a combined form. The laws of chemical equilibrium proper to dissociated systems cannot be regarded in any other light.

Molecules consist of atoms in a certain state of distribution and motion, just as the solar system[20] is made up of inseparable parts (the sun, planets, satellites, comets, &c.) The greater the number of atoms in a molecule, the more complex is the resultant substance. The equilibrium between the dissimilar atoms may be more or less stable, and may for this reason give more or less stable substances. Physical and mechanical transformations alter the velocity of the motion and the distances between the individual molecules, or of the atoms in the molecules, or of their sum total, but they do not alter the original equilibrium of the system; whilst chemical changes, on the other hand, alter the molecules themselves, that is, the velocity of motion, the relative distribution, and the quality and quantity of the atoms in the molecules.

[20] This strengthens the fundamental idea of the unity and harmony of type of all creation and is one of those ideas which impress themselves on man in all ages, and give rise to a hope of arriving in time, by means of a laborious series of discoveries, observations, experiments, laws, hypotheses, and theories, at a comprehension of the internal and invisible structure of concrete substances with that same degree of clearness and exactitude which has been attained in the visible structure of the heavenly bodies. It is not many years ago since the law of Avogadro-Gerhardt took root in science. It is within the memory of many living scientific men, and of mine amongst others. It is not surprising, therefore, that as yet little progress has been made in the province of molecular mechanics; but the theory of gases alone, which is intimately connected with the conception of molecules, shows by its success that the time is approaching when our knowledge of the internal structure of matter will be defined and established.

Atoms are the smallest quantities or chemically indivisible masses of the elements forming the molecules of elements and compounds.

Atoms have weight, the sum of their weights forms the weight of the molecule, and the sum of the weights of the molecules forms the weight of masses, and is the cause of gravity, and of all the phenomena which depend on the mass of a substance.

The elements are characterised, not only by their independent existence, their incapacity of being converted into each other, &c., but also by the weight of their atoms.

Chemical and physical properties depend on the weight, composition, and properties of the molecules forming a substance, and on the weight and properties of the atoms forming the molecules.

This is the substance of those principles of molecular mechanics which lie at the basis of all contemporary physical and chemical constructions since the establishment of the law of Avogadro-Gerhardt. The fecundity of the principles enunciated is seen at every step in all the particular cases forming the present store of chemical data. We will here cite a few examples of the application of the law.

As the weight of an atom must be understood as the minimum quantity of an element entering into the composition of all the molecules formed by it, therefore, in order to find the weight of an atom of oxygen, let us take the molecules of those of its compounds which have already been described, together with the molecules of certain of those carbon compounds which will be described in the following chapter:

Molecular Amount of Molecular Amount of Weight Oxygen Weight Oxygen

H_{2}O 18 16 HNO_{3} 63 48 N_{2}O 44 16 CO 28 16 NO 30 16 CO_{2} 44 32 NO_{2} 46 32

The number of substances taken might be considerably increased, but the result would be the same--that is, the molecules of the compounds of oxygen would never be found to contain less than 16 parts by weight of this element, but always _n_16, where n is a whole number. The molecular weights of the above compounds are found either directly from the density of their vapour or gas, or from their reactions. Thus, the vapour density of nitric acid (as a substance which easily decomposes above its boiling point) cannot be accurately determined, but the fact of its containing one part by weight of hydrogen, and all its properties and reactions, indicate the above molecular composition and no other. In this manner it is very easy to find the atomic weight of all the elements, knowing the molecular weight and composition of their compounds. It may, for instance, be easily proved that less than _n_12 parts of carbon never enters into the molecules of carbon compounds, and therefore C must be taken as 12, and not as 6 which was the number in use before Gerhardt. In a similar manner the atomic weights now accepted for the elements oxygen, nitrogen, carbon, chlorine, sulphur, &c., were found and indubitably established, and they are even now termed the Gerhardt atomic weights. As regards the metals, many of which do not give a single volatile compound, we shall afterwards see that there are also methods by which their atomic weights may be established, but nevertheless the law of Avogadro-Gerhardt is here also ultimately resorted to, in order to remove any doubt which may be encountered. Thus, for instance, although much that was known concerning the compounds of beryllium necessitated its atomic weight being taken as Be = 9--that is, the oxide as BeO and the chloride BeCl_{2}--still certain analogies gave reason for considering its atomic weight to be Be = 13·5, in which case its oxide would be expressed by the composition Be_{2}O_{3}, and the chloride by BeCl_{3}.[21] It was then found that the vapour density of beryllium chloride was approximately 40, when it became quite clear that its molecular weight was 80, and as this satisfies the formula BeCl_{2}, but does not suit the formula BeCl_{3}, it therefore became necessary to regard the atomic weight of Be as 9 and not as 13-1/2.

[21] If Be = 9, and beryllium chloride be BeCl_{2}, then for every 9 parts of beryllium there are 71 parts of chlorine, and the molecular weight of BeCl_{2} = 80; hence the vapour density should be 40 or n{4}0. If Be = 13·5, and beryllium chloride be BeCl_{3}, then to 13·5 of beryllium there are 106·5 of chlorine; hence the molecular weight would be 120, and the vapour density 60 or _n_60. The composition is evidently the same in both cases, because 9 : 71 :: 13·5 : 106·5. Thus, if the symbol of an element designate different atomic weights, apparently very different formulæ may equally well express both the percentage composition of compounds, and those properties which are required by the laws of multiple proportions and equivalents. The chemists of former days accurately expressed the composition of substances, and accurately applied Dalton's laws, by taking H = 1, O = 8, C = 6, Si = 14, &c. The Gerhardt equivalents are also satisfied by them, because O = 16, C = 12, Si = 28, &c., are multiples of them. The choice of one or the other multiple quantity for the atomic weight is impossible without a firm and concrete conception of the molecule and atom, and this is only obtained as a consequence of the law of Avogadro-Gerhardt, and hence the modern atomic weights are the results of this law (see Note 28).

With the establishment of a true conception of molecules and atoms, chemical formulæ became direct expressions, not only of composition,[22] but also of molecular weight or vapour density, and consequently of a series of fundamental chemical and physical data, inasmuch as a number of the properties of substances are dependent on their vapour density, or molecular weight and composition. The vapour density D = M/2. For instance, the formula of ethyl ether is C_{4}H_{10}O, corresponding with the molecular weight 74, and the vapour density 37, which is the fact. Therefore, the density of vapours and gases has ceased to be an empirical magnitude obtained by experiment only, and has acquired a rational meaning. It is only necessary to remember that 2 grams of hydrogen, or the molecular weight of this primary gas in grams, occupies, at 0° and 760 mm. pressure, a volume of 22·3 litres (or 22,300 cubic centimetres), in order to directly determine the weights of cubical measures of gases and vapours from their formulæ, because the molecular weights in grams of all other vapours at 0° and 760 mm. occupy the same volume, 22·3 litres. Thus, for example, in the case of carbonic anhydride, CO_{2}, the molecular weight M = 44, hence 44 grams of carbonic anhydride at 0° and 760 mm. occupy a volume of 22·3 litres--consequently, a litre weighs 1·97 gram. By combining the laws of gases--Gay-Lussac's, Mariotte's, and Avogadro-Gerhardt's--we obtain[23] a general formula for gases

6200_s_(273 + t) = M_p_

where s is the weight in grams of a cubic centimetre of a vapour or gas at a temperature t and pressure p (expressed in centimetres of mercury) if the molecular weight of the gas = M. Thus, for instance, at 100° and 760 millimetres pressure (i.e. at the atmospheric pressure) the weight of a cubic centimetre of the vapour of ether (M = 74) is s = 0·0024.[24]

[22] The percentage amounts of the elements contained in a given compound may be calculated from its formula by a simple proportion. Thus, for example, to find the percentage amount of hydrogen in hydrochloric acid we reason as follows:--HCl shows that hydrochloric acid contains 35·5 of chlorine and 1 part of hydrogen. Hence, in 36·5 parts of hydrochloric acid there is 1 part by weight of hydrogen, consequently 100 parts by weight of hydrochloric acid will contain as many more units of hydrogen as 100 is greater than 36·5; therefore, the proportion is as follows--x : 1 :: 100 : 36·5 or x = 100/36·5 = 2·739. Therefore 100 parts of hydrochloric acid contain 2·739 parts of hydrogen. In general, when it is required to transfer a formula into its percentage composition, we must replace the symbols by their corresponding atomic weights and find their sum, and knowing the amount by weight of a given element in it, it is easy by proportion to find the amount of this element in 100 or any other quantity of parts by weight. If, on the contrary, it be required to find the formula from a given percentage composition, we must proceed as follows: Divide the percentage amount of each element entering into the composition of a substance by its atomic weight, and compare the figures thus obtained--they should be in simple multiple proportion to each other. Thus, for instance, from the percentage composition of hydrogen peroxide, 5·88 of hydrogen and 94·12 of oxygen, it is easy to find its formula; it is only necessary to divide the amount of hydrogen by unity and the amount of oxygen by 16. The numbers 5·88 and 5·88 are thus obtained, which are in the ratio 1 : 1, which means that in hydrogen peroxide there is one atom of hydrogen to one atom of oxygen.

The following is a proof of the practical rule given above that to find the ratio of the number of atoms from the percentage composition, it is necessary to divide the percentage amounts by the atomic weights of the corresponding substances, and to find the ratio which these numbers bear to each other. Let us suppose that two radicles (simple or compound), whose symbols and combining weights are A and B, combine together, forming a compound composed of x atoms of A and y atoms of B. The formula of the substance will be A_x_B_y_. From this formula we know that our compound contains _x_A parts by weight of the first element, and _y_B of the second. In 100 parts of our compound there will be (by proportion) (100. _x_A)/(_x_A + _y_B) of the first element, and (100. _y_B)/(_x_A + _y_B) of the second. Let us divide these quantities, expressing the percentage amounts by the corresponding combining weights; we then obtain 100_x_/(_x_A + _y_B) for the first element and 100_y_/(_x_A + _y_B) for the second element. And these numbers are in the ratio x : y--that is, in the ratio of the number of atoms of the two substances.

It may be further observed that even the very language or nomenclature of chemistry acquires a particular clearness and conciseness by means of the conception of molecules, because then the names of substances may directly indicate their composition. Thus the term 'carbon dioxide' tells more about and expresses CO_{2} better than carbonic acid gas, or even carbonic anhydride. Such nomenclature is already employed by many. But expressing the composition without an indication or even hint as to the properties, would be neglecting the advantageous side of the present nomenclature. Sulphur dioxide, SO_{2}, expresses the same as barium dioxide, BaO_{2}, but sulphurous anhydride indicates the acid properties of SO_{2}. Probably in time one harmonious chemical language will succeed in embracing both advantages.

[23] This formula (which is given in my work on 'The Tension of Gases,' and in a somewhat modified form in the 'Comptes Rendus,' Feb. 1876) is deduced in the following manner. According to the law of Avogadro-Gerhardt, M = 2D for all gases, where M is the molecular weight and D the density referred to hydrogen. But it is equal to the weight _s__{0} of a cubic centimetre of a gas in grams at 0° and 76 cm. pressure, divided by 0·0000898, for this is the weight in grams of a cubic centimetre of hydrogen. But the weight s of a cubic centimetre of a gas at a temperature t and under a pressure p (in centimetres) is equal to _s__{0}p/76(1 + at). Therefore, _s__{0} = s.76(1 + at)/p; hence D = 76.s(1 + at)/0·0000898_p_, whence M = 152_s_(1 + at)/0·0000898_p_, which gives the above expression, because 1/a = 273, and 152 multiplied by 273 and divided by 0·0000898 is nearly 6200. In place of s, m/v may be taken, where m is the weight and v the volume of a vapour.

[24] The above formula may be directly applied in order to ascertain the molecular weight from the data; weight of vapour m grms., its volume v c.c., pressure p cm., and temperature t°; for s = the weight of vapour m, divided by the volume v, and consequently M = 6,200_m_(273 + t)/pv. Therefore, instead of the formula (see Chapter II., Note 34), pv = R(273 + t), where R varies with the mass and nature of a gas, we may apply the formula pv = 6,200(m/M)(273 + t). These formulæ simplify the calculations in many cases. For example, required the volume v occupied by 5 grms. of aqueous vapour at a temperature t = 127° and under a pressure p = 76 cm. According to the formula M = 6,200_m_(273 + t)/pv, we find that v = 9,064 c.c., as in the case of water M = 18, m in this instance = 5 grms. (These formulæ, however, like the laws of gases, are only approximate.)

As the molecules of many elements (hydrogen, oxygen, nitrogen, chlorine, bromine, sulphur--at least at high temperatures) are of uniform composition, the formulæ of the compounds formed by them directly indicate the composition by volume. So, for example, the formula HNO_{3} directly shows that in the decomposition of nitric acid there is obtained 1 vol. of hydrogen, 1 vol. of nitrogen, and 3 vols. of oxygen.

And since a great number of mechanical, physical, and chemical properties are directly dependent on the elementary and volumetric composition, and on the vapour density, the accepted system of atoms and molecules gives the possibility of simplifying a number of most complex relations. For instance, it may be easily demonstrated that the vis viva of the molecules of all vapours and gases is alike. For it is proved by mechanics that the vis viva of a moving mass = (1/2) mv^2, where m is the mass and v the velocity. For a molecule, m = M, or the molecular weight, and the velocity of the motion of gaseous molecules = a constant which we will designate by C, divided by the square root of the density of the gas[25] = C/[sqrt]D, and as D = M/2, the vis viva of molecules = C^2--that is, a constant for all molecules. Q.E.D.[26] The specific heat of gases (Chapter XIV.), and many other of their properties, are determined by their density, and consequently by their molecular weight. Gases and vapours in passing into a liquid state evolve the so-called latent heat, which also proves to be in connection with the molecular weight. The observed latent heats of carbon bisulphide, CS_{2} = 90, of ether, C_{4}H_{10}O, = 94, of benzene, C_{6}H_{6}, = 109, of alcohol, C_{2}H_{6}O, = 200, of chloroform, CHCl_{3}, = 67, &c., show the amount of heat expended in converting one part by weight of the above substances into vapour. A great uniformity is observed if the measure of this heat he referred to the weight of the molecule. For carbon bisulphide the formula CS_{2} expresses a weight 76, hence the latent heat of evaporation referred to the molecular quantity CS_{2} = 76 x 90 = 6,840, for ether = 9,656, for benzene = 8,502, for alcohol = 9,200, for chloroform = 8,007, for water = 9,620, &c. That is, for molecular quantities, the latent heat varies comparatively little, from 7,000 to 10,000 heat units, whilst for equal parts by weight it is ten times greater for water than for chloroform and many other substances.[27]

[25] Chapter I., Note 34.

[26] The velocity of the transmission of sound through gases and vapours closely bears on this. It = [sqrt](Kpg)/D(1 + [Greek: a]t), where K is the ratio between the two specific heats (it is approximately 1·4 for gases containing two atoms in a molecule), p the pressure of the gas expressed by weight (that is, the pressure expressed by the height of a column of mercury multiplied by the density of mercury), g the acceleration of gravity, D the weight of a cubic measure of the gas, [Greek: a] = 0·00367, and t the temperature. Hence, if K be known, and as D can he found from the composition of a gas, we can calculate the velocity of the transmission of sound in that gas. Or if this velocity be known, we can find K. The relative velocities of sound in two gases can he easily determined (Kundt).

If a horizontal glass tube (about 1 metre long and closed at both ends) be full of a gas, and be firmly fixed at its middle point, then it is easy to bring the tube and gas into a state of vibration, by rubbing it from centre to end with a damp cloth. The vibration of the gas is easily rendered visible, if the interior of the tube be dusted with lycopodium (the yellow powder-dust or spores of the lycopodium plant is often employed in medicine), before the gas is introduced and the tube fused up. The fine lycopodium powder arranges itself in patches, whose number depends on the velocity of sound in the gas. If there be 10 patches, then the velocity of sound in the gas is ten times slower than in glass. It is evident that this is an easy method of comparing the velocity of sound in gases. It has been demonstrated by experiment that the velocity of sound in oxygen is four times less than in hydrogen, and the square roots of the densities and molecular weights of hydrogen and oxygen stand in this ratio.

[27] If the conception of the molecular weights of substances does not give an exact law when applied to the latent heat of evaporation, at all events it brings to light a certain uniformity in figures, which otherwise only represent the simple result of observation. Molecular quantities of liquids appear to expend almost equal amounts of heat in their evaporation. It may be said that the latent heat of evaporation of molecular quantities is approximately constant, because the vis viva of the motion of the molecules is, as we saw above, a constant quantity. According to thermodynamics the latent heat of evaporation is equal to ((t + 273)/E)(n´-n)dp/_d_T × 13·59, where t is the boiling point, n´ the specific volume (i.e. the volume of a unit of weight) of the vapour, and n the specific volume of the liquid, dp/_d_T the variation of the tension with a rise of temperature per 1°, and 13·89 the density of the mercury according to which the pressure is measured. Thus the latent heat of evaporation increases not only with a decrease in the vapour density (i.e. the molecular weight), but also with an increase in the boiling point, and therefore depends on different factors.

Generalising from the above, the weight of the molecule determines the properties of a substance independently of its composition--i.e. of the number and quality of the atoms entering into the molecule--whenever the substance is in a gaseous state (for instance, the density of gases and vapours, the velocity of sound in them, their specific heat, &c.), or passes into that state, as we see in the latent heat of evaporation. This is intelligible from the point of view of the atomic theory in its present form, for, besides a rapid motion proper to the molecules of gaseous bodies, it is further necessary to postulate that these molecules are dispersed in space (filled throughout with the luminiferous ether) like the heavenly bodies distributed throughout the universe. Here, as there, it is only the degree of removal (the distance) and the masses of substances which take effect, while those peculiarities of a substance which are expressed in chemical transformations, and only come into action on near approach or on contact, are in abeyance by reason of the dispersal. Hence it is at once obvious, in the first place, that in the case of solids and liquids, in which the molecules are closer together than in gases and vapours, a greater complexity is to be expected, i.e. a dependence of all the properties not only upon the weight of the molecule but also upon its composition and quality, or upon the properties of the individual chemical atoms forming the molecule; and, in the second place, that, in the case of a small number of molecules of any substance being disseminated through a mass of another substance--for example, in the formation of weak (dilute) solutions (although in this case there is an act of chemical reaction--i.e. a combination, decomposition, or substitution)--the dispersed molecules will alter the properties of the medium in which they are dissolved, almost in proportion to the molecular weight and almost independently of their composition. The greater the number of molecules disseminated--i.e. the stronger the solution--the more clearly defined will those properties become which depend upon the composition of the dissolved substance and its relation to the molecules of the solvent, for the distribution of one kind of molecules in the sphere of attraction of others cannot but be influenced by their mutual chemical reaction. These general considerations give a starting point for explaining why, since the appearance of Van't Hoff's memoir (1886), 'The Laws of Chemical Equilibrium in a Diffused Gaseous or Liquid State' (see Chapter I., Note 19), it has been found more and more that dilute (weak) solutions exhibit such variations of properties as depend wholly upon the weight and number of the molecules and not upon their composition, and even give the means of determining the weight of molecules by studying the variations of the properties of a solvent on the introduction of a small quantity of a substance passing into solution. Although this subject has been already partially considered in the first chapter (in speaking of solutions), and properly belongs to a special (physical) branch of chemistry, we touch upon it here because the meaning and importance of molecular weights are seen in it in a new and peculiar light, and because it gives a method for determining them whenever it is possible to obtain dilute solutions. Among the numerous properties of dilute solutions which have been investigated (for instance, the osmotic pressure, vapour tension, boiling point, internal friction, capillarity, variation with change of temperature, specific heat, electroconductivity, index of refraction, &c.) we will select one--the 'depression' or fall of the temperature of freezing (Raoult's cryoscopic method), not only because this method has been the most studied, but also because it is the most easily carried out and most frequently applied for determining the weight of the molecules of substances in solution, although here, owing to the novelty of the subject there are also many experimental discrepancies which cannot as yet be explained by theory.[27 bis]

[27 bis] The osmotic pressure, vapour tension of the solvent, and several other means applied like the cryoscopic method to dilute solutions for determining the molecular weight of a substance in solution, are more difficult to carry out in practice, and only the method of determining the rise of the boiling point of dilute solutions can from its facility be placed parallel with the cryoscopic method, to which it bears a strong resemblance, as in both the solvent changes its state and is partially separated. In the boiling point method it passes off in the form of a vapour, while in cryoscopic determinations it separates out in the form of a solid body.

Van't Hoff, starting from the second law of thermodynamics, showed that the dependence of the rise of pressure (dp) upon a rise of temperature (_d_T) is determined by the equation dp = (kmp/2T^2)_d_T, where k is the latent heat of evaporation of the solvent, m its molecular weight, p the tension of the saturated vapour of the solvent at T, and T the absolute temperature (T = 273 + t), while Raoult found that the quantity (p-p´)/p (Chapter I., Note 50) or the measure of the relative fall of tension (p the tension of the solvent or water, and p´ of the solution) is found by the ratio of the number of molecules, n of the substance dissolved, and N of the solvent, so that (p-p´)/p = C_n_/(N + n) where C is a constant. With very dilute solutions p -p´ may be taken as equal to dp, and the fraction n/(N + n) as equal to n/N (because in that case the value of N is very much greater than n), and then, judging from experiment, C is nearly unity--hence: dp/p = n/N or dp = np/N, and on substituting this in the above equation we have (kmp/2T^2)_d_T = np/N. Taking a weight of the solvent m/N = 100, and of the substance dissolved (per 100 of the solvent) q, where q evidently = _n_M, if M be the molecular weight of the substance dissolved, we find that n/N = qm/100M, and hence, according to the preceding equation, we have M = (0·02T^2/k)·(q/_d_T), that is, by taking a solution of q grms. of a substance in 100 grms. of a solvent, and determining by experiment the rise of the boiling point _d_T, we find the molecular weight M of the substance dissolved, because the fraction 0·02T^2/k is (for a given pressure and solvent) a constant; for water at 100° (T = 373°) when k = 534 (Chapter I., Note 11), it is nearly 5·2, for ether nearly 21, for bisulphide of carbon nearly 24, for alcohol nearly 11·5, &c. As an example, we will cite from the determinations made by Professor Sakurai, of Japan (1893), that when water was the solvent and the substance dissolved, corrosive sublimate, HgCl_{2}, was taken in the quantity q = 8·978 and 4·253 grms., the rise in the boiling point _d_T was = O°·179 and 0°·084, whence M = 261 and 263, and when alcohol was the solvent, q = 10·873 and 8·765 and _d_T = 0°·471 and 0°·380, whence M = 266 and 265, whilst the actual molecular weight of corrosive sublimate = 271, which is very near to that given by this method. In the same manner for aqueous solutions of sugar (M = 342), when q varied from 14 to 2·4, and the rise of the boiling point from 0°·21 to 0°·035, M was found to vary between 339 and 364. For solutions of iodine I_{2} in ether, the molecular weight was found by this method to be between 255 and 262, and I_{2} = 254. Sakurai obtained similar results (between 247 and 262) for solutions of iodine in bisulphide of carbon.

We will here remark that in determining M (the molecular weight of the substance dissolved) at small but increasing concentrations (per 100 grms. of water), the results obtained by Julio Baroni (1893) show that the value of M found by the formula may either increase or decrease. An increase, for instance, takes place in aqueous solutions of HgCl_{2} (from 255 to 334 instead of 271), KNO_{3} (57-66 instead of 101), AgNO_{3} (104-107 instead of 170), K_{2}SO_{4} (55-89 instead of 174), sugar (328-348 instead of 342), &c. On the contrary the calculated value of M decreases as the concentration increases, for solutions of KCl (40-39 instead of 74·5), NaCl (33-28 instead of 58·5), NaBr (60-49 instead of 103), &c. In this case (as also for LiCl, NaI, C_{2}H_{3}NaO_{2}, &c.) the value of i (Chapter I., Note 49), or the ratio between the actual molecular weight and that found by the rise of the boiling point, was found to increase with the concentration, i.e. to be greater than 1, and to differ more and more from unity as the strength of the solution becomes greater. For example, according to Schlamp (1894), for LiCl, with a variation of from 1·1 to 6·7 grm. LiCl per 100 of water, i varies from 1·63 to 1·89. But for substances of the first series (HgCl_{2}, &c.), although in very dilute solutions i is greater than 1, it approximates to 1 as the concentration increases, and this is the normal phenomenon for solutions which do not conduct an electric current, as, for instance, of sugar. And with certain electrolytes, such as HgCl_{2}, MgSO_{4}, &c., i exhibits a similar variation; thus, for HgCl_{2} the value of M is found to vary between 255 and 334; that is, i (as the molecular weight = 271) varies between 1·06 and 0·81. Hence I do not believe that the difference between i and unity (for instance, for CaCl_{2}, i is about 3, for KI about 2, and decreases with the concentration) can at present be placed at the basis of any general chemical conclusions, and it requires further experimental research. Among other methods by which the value of i is now determined for dilute solutions is the study of their electroconductivity, admitting that i = 1 + a(k-1), where a = the ratio of the molecular conductivity to the limiting conductivity corresponding to an infinitely large dilution (see Physical Chemistry), and k is the number of ions into which the substance dissolved can split up. Without entering upon a criticism of this method of determining i, I will only remark that it frequently gives values of i very close to those found by the depression of the freezing point and rise of the boiling point; but that this accordance of results is sometimes very doubtful. Thus for a solution containing 5·67 grms. CaCl_{2} per 100 grms. of water, i, according to the vapour tension = 2·52, according to the boiling point = 2·71, according to the electroconductivity = 2·28, while for solutions in propyl alcohol (Schlamp 1894) i is near to 1·33. In a word, although these methods of determining the molecular weight of substances in solution show an undoubted progress in the general chemical principles of the molecular theory, there are still many points which require explanation.

We will add certain general relations which apply to these problems. Isotonic (Chapter I., Note 19) solutions exhibit not only similar osmotic pressures, but also the same vapour tension, boiling point and freezing temperature. The osmotic pressure bears the same relation to the fall of the vapour tension as the specific gravity of a solution does to the specific gravity of the vapour of the solvent. The general formulæ underlying the whole doctrine of the influence of the molecular weight upon the properties of solutions considered above, are: 1. Raoult in 1886-1890 showed that

((p-p´)/p) × (100/a) × (M/m) = a constant C

where p and p´ are the vapour tensions of the solvent and substance dissolved, a the amount in grms. of the substance dissolved per 100 grms. of solvent, M and m the molecular weights of the substance dissolved and solvent. 2. Raoult and Recoura in 1890 showed that the constant above C = the ratio of the actual vapour density d´ of the solvent to the theoretical density d calculated according to the molecular weight. This deduction may now be considered proved, because both the fall of tension and the ratio of the vapour densities d´/d give, for water 1·03, for alcohol 1·02, for ether 1·04, for bisulphide of carbon 1·00, for benzene 1·02, for acetic acid 1·63. 3. By applying the principles of thermodynamics and calling L_{1} the latent heat of fusion and T_{1} the absolute (= t + 273) temperature of fusion of the solvent, and L_{2} and T_{2} the corresponding values for the boiling point, Van't Hoff in 1886-1890 deduced:--

(Depression of freezing point)/(Rise of boiling point) = (L_{2}/L_{1}) × (T_{1}^2/T_{2}^2)

Depression of freezing point = (AT_{1}^{2}a)/(L_{1}M_{1})

Rise of boiling point = (AT_{2}^{2}a)/(L_{2}M_{1})

where A = 0·01988 (or nearly 0·02 as we took it above), a is the weight in grms. of the substance dissolved per 100 grms. of the solvent, M_{1} the molecular weight of the dissolved substance (in the solution), and M the molecular weight of this substance according to its composition and vapour density, then i = M/M_{1}. The experimental data and theoretical considerations upon which these formulæ are based will be found in text-books of physical and theoretical chemistry.

If 100 gram-molecules of water, i.e. 1,800 grms, be taken and n gram-molecules of sugar, C_{12}H_{22}O_{11}, i.e. n 342 grms., be dissolved in them, then the depression d, or fall (counting from 0°) of the temperature of the formation of ice will be (according to Pickering)

n = 0 0·010 0·025 0·100 0·250 1·000 d = 0° 0°·0103 0°·0280 0°·1115 0°·2758 1°·1412

which shows that for high degrees of dilution (up to 0·25_n_) d approximately (estimating the possible errors of experiment at ±0°·005) = _n_1·10, because then d = 0°, 0°·0110, 0°·0275, 0°·1100, 0°·2750, 1°·1000, and the difference between these figures and the results of experiment for very dilute solutions is less than the possible errors of experiment (for n = 1 the difference is already greater) and therefore for dilute solutions of sugar it may be said that n molecules of sugar in dissolving in 100 molecules of water give a depression of about 1°·1_n_. Similar data for acetone (Chapter I., Note 49) give a depression of 1°·006_n_ for n molecules of acetone per 100 molecules of water. And in general, for indifferent substances (the majority of organic bodies) the depression per 100H_{2}O is _nearly n_1°·1 to _n_1°·0 (ether, for instance, gives the last number), and consequently in dissolving in 100 grms. of water it is about 18°·0_n_ to 19°·0_n_, taking this rule to apply to the case of a small number of n (not over 0·2_n_). If instead of water, other liquid or fused solvents (for example, benzene, acetic acid, acetone, nitrobenzene or molten naphthaline, metals, &c.) be taken and in the proportion of 100 molecules of the solvent to n molecules of a dissolved indifferent (neither acid nor saline) substance, then the depression is found to be equal to from 0°·62_n_ to 0°·65_n_ and in general K_n_. If the molecular weight of the solvent = m, then 100 gram-molecules will weigh 100_m_ grms., and the depression will be approximately (taking 0·63_n_) equal to _m_0·63_n_ degrees for n molecules of the substance dissolved in 100 grms. of the solvent, or in general the depression for 100 grms. of a given solvent = kn where k is almost a constant quantity (for water nearly 18, for acetone nearly 37, &c.) for all dilute solutions. Thus, having found a convenient solvent for a given substance and prepared a definite (by weight) solution (i.e. knowing how many grms. r of the solvent there are to q grms. of the substance dissolved) and having determined the depression d--i.e. the fall in temperature of freezing for the solvent--it is possible to determine the molecular weight of the substance dissolved, because d = kn where d is found by experiment and k is determined by the nature of the solvent, and therefore n or the number of molecules of the substance dissolved can be found. But if r grms. of the solvent and q grms. of the substance dissolved are taken, then there are 100_q_/r of the latter per 100 grms. of the former, and this quantity = _n_X, where n is found from the depression and = d/k and X is the molecular weight of the substance dissolved. Hence X = 100_qk_/rd, which gives the molecular weight, naturally only approximately, but still with sufficient accuracy to easily indicate, for instance, whether in peroxide of hydrogen the molecule contains HO or H_{2}O_{2} or H_{3}O_{3}, &c. (H_{2}O_{2} is obtained). Moreover, attention should be drawn to the fact that a great many substances taken as solvents give per 100 molecules a depression of about 0·63_n_, whilst water gives about 1·05_n_, i.e. a larger quantity, as though the molecules of liquid water were more complex than is expressed by the formula H_{2}O.[28] A similar phenomenon which repeats itself in the osmotic pressure, vapour tension of the solvent, &c. (see Chapter I., Notes 19 and 49), i.e. a variation of the constant (k for 100 grms. of the solvent or K for 100 molecules of it), is also observed in passing from indifferent substances to saline (to acids, alkalis and salts) both in aqueous and other solutions as we will show (according to Pickering's data 1892) for solutions of NaCl and CuSO_{4} in water. For

n = 0·01 0·03 0·05 0·1 0·5

molecules of NaCl the depression is

d = 0°·0177 0°·0598 0°·0992 0°·1958 0°·9544

which corresponds to a depression per molecule

K = 1·77 1·96 1·98 1·96 1·91

i.e. here in the most dilute solutions (when n is nearly 0) d is obtained about 1·7_n_, while in the case of sugar it was about 1·1_n_. For CuSO_{4} for the same values of n, experiment gave:

d = 0°·0164 0°·0451 0°·0621 0°·1321 0°·5245 K = 1·64 1·50 1·44 1·32 1·05

i.e. here again d for very dilute solutions is nearly 1·7_n_, but the value of K falls as the solution becomes more concentrated, while for NaCl it at first increased and only fell for the more concentrated solutions. The value of K in the solution of n molecules of a body in 100H_{2}O, when d = K_n_, for very dilute solutions of CaCl_{2} is nearly 2·6, for Ca(NO_{3}){2} nearly 2·5, for HNO{3}, KI and KHO nearly 1·9-2·O, for borax Na_{2}B_{4}O_{7} nearly 3·7, &c., while for sugar and similar substances it is, as has been already mentioned, nearly 1·0-1·1. Although these figures are very different[28 bis] still k and K may be considered constant for analogous substances, and therefore the weight of the molecule of the body in solution can be found from d. And as the vapour tension of solutions and their boiling points (see Note 27 bis and Chapter I., Note 51) vary in the same manner as the freezing point depression, so they also may serve as means for determining the molecular weight of a substance in solution.[29]

[28] A similar conclusion respecting the molecular weight of liquid water (i.e. that its molecule in a liquid state is more complex than in a gaseous state, or polymerized into H_{8}O_{4}, H_{6}O_{3} or in general into _n_H_{2}O) is frequently met in chemico-physical literature, but as yet there is no basis for its being fully admitted, although it is possible that a polymerization or aggregation of several molecules into one takes place in the passage of water into a liquid or solid state, and that there is a converse depolymerization in the act of evaporation. Recently, particular attention has been drawn to this subject owing to the researches of Eötvös (1886) and Ramsay and Shields (1893) on the variation of the surface tension N with the temperature (N = the capillary constant a^2 multiplied by the specific gravity and divided by 2, for example, for water at 0° and 100° the value of a^2 = 15·41 and 12·58 sq. mm., and the surface tension 7·92 and 6·04). Starting from the absolute boiling point (Chapter II., Note 29) and adding 6°, as was necessary from all the data obtained, and calling this temperature T, it is found that AS = _k_T, where S is the surface of a gram-molecule of the liquid (if M is its weight in grams, s its sp. gr., then its sp. volume = M/s, and the surface S = [3root](M/s)^2), A the surface tension (determined by experiment at T), and k a constant which is independent of the composition of the molecule. The equation AS = _k_T is in complete agreement with the well-known equation for gases vp = RT (p. 140) which serves for deducing the molecular weight from the vapour density. Ramsay's researches led him to the conclusion that the liquid molecules of CS_{2}, ether, benzene, and of many other substances, have the same value as in a state of vapour, whilst with other liquids this is not the case, and that to obtain an accordance, that is, that k shall be a constant, it is necessary to assume the molecular weight in the liquid state to be n times as great. For the fatty alcohols and acids n varies from 1-1/2 to 3-1/2, for water from 2-1/4 to 4, according to the temperature (at which the depolymerization takes place). Hence, although this subject offers a great theoretical interest, it cannot be regarded as firmly established, the more so since the fundamental observations are difficult to make and not sufficiently numerous; should, however, further experiments confirm the conclusions arrived at by Professor Ramsay, this will give another method of determining molecular weights.

[28 bis] Their variance is expressed in the same manner as was done by Van't Hoff (Chapter I., Notes 19 and 49) by the quantity i, taking it as = 1 when k = 1·05, in that case for KI, i is nearly 2, for borax about 4, &c.

[29] We will cite one more example, showing the direct dependence of the properties of a substance on the molecular weight. If one molecular part by weight of the various chlorides--for instance, of sodium, calcium, barium, &c.--be dissolved in 200 molecular parts by weight of water (for instance, in 3,600 grams) then it is found that the greater the molecular weight of the salt dissolved, the greater is the specific gravity of the resultant solution.

Molecular Sp. gr. Molecular Sp. gr. weight at 15° weight at 15° HCl 36·5 1·0041 CaCl_{2} 111 1·0236 NaCl 58·5 1·0106 NiCl_{2} 130 1·0328 KCl 74·5 1·0121 ZnCl_{2} 136 1·0331 BeCl_{2} 80 1·0138 BaCl_{2} 208 1·0489 MgCl_{2} 95 1·0203

Thus not only in vapours and gases, but also in dilute solutions of solid and liquid substances, we see that if not all, still many properties are wholly dependent upon the molecular weight and not upon the quality of a substance, and that this gives the possibility of determining the weight of molecules by studying these properties (for instance, the vapour density, depression of the freezing point, &c.) It is apparent from the foregoing that the physical and even more so the chemical properties of homogeneous substances, more especially solid and liquid, do not depend exclusively upon the weights of their molecules, but that many are in definite (see Chapter XV.) dependence upon the weights of the atoms of the elements entering into their composition, and are determined by their quantitative and individual peculiarities. Thus the density of solids and liquids (as will afterwards be shown) is chiefly determined by the weights of the atoms of the elements entering into their composition, inasmuch as dense elements (in a free state) and compounds are only met with among substances containing elements with large atomic weights, such as gold, platinum, and uranium. And these elements themselves, in a free state, are the heaviest of all elements. Substances containing such light elements as hydrogen, carbon, oxygen and nitrogen (like many organic substances) never have a high specific gravity; in the majority of cases it scarcely exceeds that of water. The density generally decreases with the increase of the amount of hydrogen, as the lightest element, and a substance is often obtained lighter than water. The refractive power of substances also entirely depends on the composition and the properties of the component elements.[29 bis] The history of chemistry presents a striking example in point--Newton foresaw from the high refractive index of the diamond that it would contain a combustible substance since so many combustible oils have a high refractive power. We shall afterwards see (Chapter XV.) that many of those properties of substances which are in direct dependence not upon the weight of the molecules but upon their composition, or, in other words, upon the properties and quantities of the elements entering into them, stand in a peculiar (periodic) dependence upon the atomic weight of the elements; that is, the mass (of molecules and atoms), proportional to the weight, determines the properties of substances as it also determines (with the distance) the motions of the heavenly bodies.

[29 bis] With respect to the optical refractive power of substances, it must first be observed that the coefficient of refraction is determined by two methods: (a) either all the data are referred to one definite ray--for instance, to the Fraunhofer (sodium) line D of the solar spectrum--that is, to a ray of definite wave length, and often to that red ray (of the hydrogen spectrum) whose wave length is 656 millionths of a millimetre; (b) or Cauchy's formula is used, showing the relation between the coefficient of refraction and dispersion to the wave length n = A + (B/([Greek: l]^2)), where A and B are two constants varying for every substance but constant for all rays of the spectrum, and [Greek: l] is the wave length of that ray whose coefficient of refraction is n. In the latter method the investigation usually concerns the magnitudes of A, which are independent of dispersion. We shall afterwards cite the data, investigated by the first method, by which Gladstone, Landolt, and others established the conception of the refraction equivalent.

It has long been known that the coefficient of refraction n for a given substance decreases with the density of a substance D, so that the magnitude (n-1) ÷ D = C is almost constant for a given ray (having a definite wave length) and for a given substance. This constant is called the refractive energy, and its product with the atomic or molecular weight of a substance the refraction equivalent. The coefficient of refraction of oxygen is 1·00021, of hydrogen 1·00014, their densities (referred to water) are 0·00143 and 0·00009, and their atomic weights, O = 16, H = 1; hence their refraction equivalents are 3 and 1·5. Water contains H_{2}O, consequently the sum of the equivalents of refraction is (2 × 1·5) + 3 = 6. But as the coefficient of refraction of water = 1·331, its refraction equivalent = 5·958, or nearly 6. Comparison shows that, approximately, the sum of the refraction equivalents of the atoms forming compounds (or mixtures) is equal to the refraction equivalent of the compound. According to the researches of Gladstone, Landolt, Hagen, Brühl and others, the refraction equivalents of the elements are--H = 1·3, Li = 3·8, B = 4·0, C = 5·0, N = 4·1 (in its highest state of oxidation, 5·3), O = 2·9, F = 1·4, Na = 4·8, Mg = 7·0, Al = 8·4, Si = 6·8, P = 18·3, S = 16·0, Cl = 9·9, K = 8·1, Ca = 10·4, Mn = 12·2, Fe = 12·0 (in the salts of its higher oxides, 20·1), Co = 10·8, Cu = 11·6, Zn = 10·2, As = 15·4, Bi = 15·3, Ag = 15·7, Cd = 13·6, I = 24·5, Pt = 26·0, Hg = 20·2, Pb = 24·8, &c. The refraction equivalents of many elements could only be calculated from the solutions of their compounds. The composition of a solution being known it is possible to calculate the refraction equivalent of one of its component parts, those for all its other components being known. The results are founded on the acceptance of a law which cannot be strictly applied. Nevertheless the representation of the refraction equivalents gives an easy means for directly, although only approximately, obtaining the coefficient of refraction from the chemical composition of a substance. For instance, the composition of carbon bisulphide is CS_{2} = 76, and from its density, 1·27, we find its coefficient of refraction to be 1·618 (because the refraction equivalent = 5 + 2 × 16 = 37), which is very near the actual figure. It is evident that in the above representation compounds are looked on as simple mixtures of atoms, and the physical properties of a compound as the sum of the properties present in the elementary atoms forming it. If this representation of the presence of simple atoms in compounds had not existed, the idea of combining by a few figures a whole mass of data relating to the coefficient of refraction of different substances could hardly have arisen. For further details on this subject, see works on Physical Chemistry.