fundamental characteristic equation expressing the energy of the
system in terms of its entropy and constitution, or the pressure
in terms of the temperature and the potentials, which includes
them all,—can be readily approximated to by experimental
determinations. Even in the case of the simple system water-vapour,
which is fundamental for the theory of the steam-engine,
this has not yet been completely accomplished. The general
theory is thus largely confined, as above, to defining the restrictions
on the degree of variability of a complex chemical system
which the principle of Carnot imposes. The tracing out of these
general relations of continuity of state is much facilitated by
geometrical diagrams, such as James Thomson first introduced in
order to exhibit and explain Andrews’ results as to the range of
coexistent phases in carbonic acid. Gibbs’s earliest thermodynamic
surface had for its co-ordinates volume, entropy and
energy; it was constructed to scale by Maxwell for water-substance,
and is fully explained in later editions of the Theory of
Heat (1875); it forms a relief map which, by simple inspection,
reveals the course of the transformations of water, with the
corresponding mechanical and thermal changes, in its three
coexistent states of solid, liquid and gas. In the general case,
when the substance has more than one independently variable
constituent, there are more than three variables to be represented;
but Gibbs has shown the utility of surfaces representing,
for instance, the entropy in terms of the constitutive variables
when temperature and pressure are maintained constant. Such
graphical methods are now of fundamental importance in
connexion with the phase rule, for the experimental exploration
of the trend of the changes of constitution of complex mixtures
with interacting components, which arise as the physical conditions
are altered, as, for example in modern metallurgy, in the
theory of alloys. The study of the phenomena of condensation
in a mixture of two gases or vapours, initiated by Andrews and
developed in this manner by van der Waals and his pupils, forms
a case in point (see Condensation of Gases).
Dilute Components: Perfect Gases and Dilute Solutions.—There are, however, two simple limiting cases, in which the theory can be completed by a determination of the functions involved in it, which throw much light on the phenomena of actual systems not far removed from these ideal limits. They are the cases of mixtures of perfect gases, and of very dilute solutions.
If, following Gibbs, we apply his equation (2) expressing the pressure in terms of the temperature and the potentials, to a very dilute solution of substances m2, m3, ... mn in a solvent substance m1, and vary the co-ordinate mr alone, p and T remaining unvaried, we have in the equilibrium state
| mr | dμr | + m1 | dμ1 | + ... + mn | dμn | = 0, |
| dmr | dmr | dmr |
in which every m except m1 is very small, while dμ1/dmr is presumably finite. As the second term is thus finite, this requires that the total potential of each component mr, which is mrdμr/dmr, shall be finite, say kr, in the limit when mr is null. Thus for very small concentrations the potential μr of a dilute component must be of the form krlog mr/v, being proportional to the logarithm of the density of that component; it thus tends logarithmically to an infinite value at evanescent concentrations, showing that removal of the last traces of any impurity would demand infinite proportionate expenditure of available energy, and is therefore practically impossible with finite intensities of force. It should be noted, however, that this argument applies only to fluid phases, for in the case of deposition of a solid mr is not uniformly distributed throughout the phase; thus it remains possible for the growth of a crystal at its surface in aqueous solution to extrude all the water except such as is in some form of chemical combination.
The precise value of this logarithmic expression for the potential can be readily determined for the case of a perfect gas from its characteristic properties, and can be thence extended to other dilute forms of matter. We have pv = R/m·T for unit mass of the gas, where m is the molecular weight, being 2 for hydrogen, and R is a constant equal to 82 × 106 in C.G.S. dynamical units, or 2 calories approximately in thermal energy units, which is the same for all gases because they have all the same number of molecules per unit volume. The increment of heat received by the unit mass of the gas is δH = pδv + κδT, κ being thus the specific heat at constant volume, which can be a function only of the temperature. Thus
and the available energy A per unit mass is E − Tφ + Tφ0 where E = ε + ƒκdT, the integral being for a standard state, and ε being intrinsic energy of chemical constitution; so that
If there are ν molecules in the unit mass, and N per unit volume, we have mν = Nmv, each being 2 ν′, where ν′ is the number of molecules per unit mass in hydrogen; thus the free energy per molecule is a′ + R′T log bN, where b = m/2ν′, R′ = R/2ν′, and a′ is a function of T alone. It is customary to avoid introducing the unknown molecular constant ν′ by working with the available energy per “gramme-molecule,” that is, for a number of grammes expressed by the molecular weight of the substance; this is a constant multiple of the available energy per molecule, and is a + RT logρ, ρ being the density equal to bN where b = m/2ν′. This formula may now be extended by simple summation to a mixture of gases, on the ground of Dalton’s experimental principle that each of the components behaves in presence of the others as it would do in a vacuum. The components are, in fact, actually separable wholly or partially in reversible ways which may be combined into cycles, for example, either (i.) by diffusion through a porous partition, taking account of the work of the pressures, or (ii.) by utilizing the modified constitution towards the top of a long column of the mixture arising from the action of gravity, or (iii.) by reversible absorption of a single component.
If we employ in place of available energy the form of characteristic equation which gives the pressure in terms of the temperature and potentials, the pressure of the mixture is expressed as the sum of those belonging to its components: this equation was made by Gibbs the basis of his analytical theory of gas mixtures, which he tested by its application to the only data then available, those of the equilibrium of dissociation of nitrogen peroxide (2NO2 ⇆ N2O4) vapour.
Van ’t Hoff’s Osmotic Principle: Theoretical Explanation.—We proceed to examine how far the same formulae as hold for gases apply to the available energy of matter in solution which is so dilute that each molecule of the dissolved substance, though possibly the centre of a complex of molecules of the solvent, is for nearly all the time beyond the sphere of direct influence of the other molecules of the dissolved substance. The available energy is a function only of the co-ordinates of the matter in bulk and the temperature; its change on further dilution, with which alone we are concerned in the transformations of dilute solutions, can depend only on the further separation of these molecular complexes in space that is thereby produced, as no one of them is in itself altered. The change is therefore a function only of the number N of the dissolved molecules per unit volume, and of the temperature, and is, per molecule, expressible in a form entirely independent of their constitution and of that of the medium in which they are dissolved. This suggests that the expression for the change on dilution is the same as the known one for a gas, in which the same molecules would exist free and in the main outside each other’s spheres of influence; which confirms and is verified by the experimental principle of van ’t Hoff, that osmotic pressure obeys the laws of gaseous pressure with identically the same physical constants as those of gases. It can be held, in fact, that this suggestion does not fall short of a demonstration, on the basis of Carnot’s principle, and independent of special molecular theory, that in all cases where the molecules of a component, whether it be of a gas or of a solution, are outside each other’s spheres of influence, the available energy, so far as regards dilution, must have a common form, and the physical constants must therefore be the known gas-constants. The customary exposition derives this principle, by an argument involving cycles, from Henry’s law of solution of gases; it is sensibly restricted to such solutes as appear concomitantly in the free gaseous state, but theoretically it becomes general when it is remembered that no solute can be absolutely non-volatile.
Source of the Idea of Temperature.—The single new element that thermodynamics introduces into the ordinary dynamical specification of a material system is temperature. This conception is akin to that of potential, except that it is given to us directly by our sense of heat. But if that were not so, we could still demonstrate, on the basis of Carnot’s principle, that there is a definite function of the state of a body which must be the same for all of a series of connected bodies, when thermal equilibrium has become established so that there is no tendency for heat to flow from one to another. For we can by mere geometrical displacement change the order of the bodies so as to bring different ones into direct contact. If this disturbed the thermal equilibrium, we could construct cyclic processes to take advantage of the resulting flow of heat to do mechanical work, and such processes might be carried on without limit. Thus it is proved
