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