when the system consists of an intimate mixture (solution) of masses m1, m2, ... mn of given constituents, which differ physically or chemically but may be partially transformable into each other by chemical or physical action during the changes under consideration, the whole being of volume v and under extraneous pressure p, while W is potential energy arising from physical forces such as those of gravity, capillarity, &c. The variables m1, m2, ... mn may not be all independent; for example, if the system were chloride of ammonium gas existing along with its gaseous products of dissociation, hydrochloric acid and ammonia, only one of the three masses would be independently variable. The sufficient number of these variables (independent components) together with two other variables, which may be v and T, or v and φ, specifies and determines the state of the system, considered as matter in bulk, at each instant. It is usual to include δW in μ1δm1 + ...; in all cases where this is possible the single equation
δE = Tδφ − pδv + μ1δm1 + μ2δm2 + ... + μnδmn
(1)
thus expresses the complete variation of the energy-function E arising from change of state; and when the part involving the n constitutive differentials has been expressed in terms of the number of them that are really independent, this equation by itself becomes the unique expression of all the thermodynamic relations of the system. These are in fact the various relations ensuring that the right-hand side is an exact differential, and are of the type of reciprocal relations such as dμr/dφ = dT/dmr.
The condition that the state of the system be one of stable equilibrium is that δφ, the variation of entropy, be negative for all formally imaginable infinitesimal transformations which make δE vanish; for as δφ cannot actually be negative for any spontaneous variation, none of these transformations can then occur. From the form of the equation, this condition is the same as that δE − Tδφ must be positive for all possible variations of state of the system as above defined in terms of co-ordinates representing its constitution in bulk, without restriction.
We can change one of the independent variables expressing the state of the system from φ to T by subtracting δ(φT) from both sides of the equation of variation: then
δ(E − Tφ) = −φδT − pδv + μ1δm1 + ... + μnδmn.
It follows that for isothermal changes, i.e. those for which δT is maintained null by an environment at constant temperature, the condition of stable equilibrium is that the function E − Tφ shall be a minimum. If the system is subject to an external pressure p, which as well as the temperature is imposed constant from without and thus incapable of variation through internal changes, the condition of stable equilibrium is similarly that E − Tφ + pv shall be a minimum.
A chemical system maintained at constant temperature by communication of heat from its environment may thus have several states of stable equilibrium corresponding to different minima of the function here considered, just as there may be several minima of elevation on a landscape, one at the bottom of each depression; in fact, this analogy, when extended to space of n dimensions, exactly fits the case. If the system is sufficiently disturbed, for example, by electric shock, it may pass over (explosively) from a higher to a lower minimum, but never (without compensation from outside) in the opposite direction. The former passage, moreover, is often effected by introducing a new substance into the system; sometimes that substance is recovered unaltered at the end of the process, and then its action is said to be purely catalytic; its presence modifies the form of the function E − Tφ so as to obliterate the ridge between the two equilibrium states in the graphical representation.
There are systems in which the equilibrium states are but very slightly dependent on temperature and pressure within wide limits, outside which reaction takes place. Thus while there are cases in which a state of mobile dissociation exists in the system which changes continuously as a function of these variables, there are others in which change does not sensibly occur at all until a certain temperature of reaction is attained, after which it proceeds very rapidly owing to the heat developed, and the system soon becomes sensibly permanent in a transformed phase by completion of the reaction. In some cases of this latter type the cause of the delay in starting lies possibly in passive resistance to change, of the nature of viscosity or friction, which is competent to convert an unstable mechanical equilibrium into a moderately stable one; but in most such reactions there seems to be no exact equilibrium at any temperature, short of the ultimate state of dissipated energy in which the reaction is completed, although the velocity of reaction is found to diminish exponentially with change of temperature, and thus becomes insignificant at a small interval from the temperature of pronounced activity.