(4)
where M is the molecular weight of the salt in solution, θ the absolute temperature, and R a constant which has the value 8.32 joules, or nearly 2 calories, per degree C. It is necessary to consider two cases, corresponding to the curves CB and AB in fig. 1, in which the solution is saturated with respect to salt and water respectively. To facilitate description we take the case of a salt dissolved in water, but similar results apply to solutions in other liquids and alloys of metals.
(a) If unit mass of salt is separated in the solid state from a saturated solution of salt (curve CB) by forcing out through a semi-permeable membrane against the osmotic pressure P the corresponding volume of water V in which it is dissolved, the heat evolved is the latent heat of saturated solution of the salt Q together with the work done PV. Writing (Q + PV) for L, and V for (v″ − v′) in equation (1), and substituting P for p, we obtain
Q + PV = VθdP / dθ,
(5)
which is equivalent to equation (1), and may be established by similar reasoning. Substituting for P and V in terms of C from equation (4), if Q is measured in calories, R = 2, and we obtain
QC = 2θ²dC / dθ,
(6)
which may be integrated, assuming Q constant, with the result
2logeC″ / C′ = Q / θ′ − Q / θ″,