■272 
mi. J. LAiUiOR ON A DYNAMICAL THEORY OF 
ihe dissolved molecules. The solvent will traiisjjire across the partition into the 
solution, unless a definite osmotic pressure acts against it, when there will be equili¬ 
brium. Let us examine the change of free energy involved in the very slow trans¬ 
piration of a certain volume; all essential that has happened has been an expansion 
of the molecules of the contained gas, each with its fluid environment, into a larger 
space. We may compare the two states of the gas, as it would exist free with these 
two different volumes, and then suppose that by an ideal process the fluid environ¬ 
ment of the molecules is directly brought about in each case : that process will, as 
regards change of intimate molecular configuration, be essentially the same for both 
states of the gas, therefore the change of free energy due to the dilution of the solu¬ 
tion is simply that which corresponds to the free gaseous expansion of the dissolved 
gas.'" This conclusion carries with it, by the thermodynamic principle of free or 
available energy, a theoretical proof of vax’t Hoff’s generalization that the osmotic 
pressure of a very dilute solution is equal to the gaseous pressure of the dissolved 
molecules when they are supposed to occupy the same volume in the gaseous state. 
The extension of this proof to dissolved liquids and solids, which form the practically 
important case, is at first sight barred (unless it is formulated as in the footnote) by 
the fact that we cannot then actually have the molecules existing free at the same 
volume as they occupy in the dilute solution. But when the Andrews isothermal 
for the dissolved substance is made into a continuous curve by inserting a super¬ 
saturated wavy part, there will always be a real point on it corresponding to the 
volume occupied by the substance thus existing in a homogeneous condition, and also 
a corresponding pressure which at the small density under consideration would prac¬ 
tically be that of the gaseous state : thus there would be no difficulty in the exten¬ 
sion to dissolved solids and liquids, were it not that this point on the isothermal 
might be on the thoroughly unstable reach, along which rise of density corresponds 
to fall of pressure, so that any slight accidental inequality of density would be spon¬ 
taneously increased. The successful use made of the Andrews diagram for numerical 
calculation of the properties of substances by Van der Waals shows however that 
its physical reality is not destroyed by this instabdity ; and when it is remembered 
that instability can be theoretically removed by slight constraint which does not 
sensibly afiect the material transformations and does not affect the energy relations 
at all, it will appear that there is good reason for generalizing the law of osmotic 
pressure above demonstrated for gases. As before stated, what is most desirable to 
* The circumstance which makes this purely imaginary pi’ocess legitimate is that the available 
energy is a function of the constitution of the matter in hulk, not depending on the accidental charac¬ 
teristics of state or motion of the individual molecules: now the only change that has occurred as 
regards the constitution of the substance in bulk, that can affect either the available or the total energy, 
is the change of volume of the solution by transpiration of the pure solvent across the partition, whicli 
by the above affects it in a manner absolutely independent of the nature of the homogeneous solvent, 
and therefore of the existence of the solvent at all, because the relation of each molecule to Ihe joortion 
of the solvent within its sphere of influence is not changed. 
