Page:Elementary Principles in Statistical Mechanics (1902).djvu/230

This is evidently a negligible quantity, since $K$ is of the same order of magnitude as the number of molecules in ordinary bodies. It is to be observed that ${\overline {\eta }}_{\rm {gen}}$ is here the average in the grand ensemble, whereas the quantity which we wish to compare with ${\overline {\mathrm {H} }}$ is the average in a petit ensemble. But as we have seen that in the case considered the grand ensemble would appear to human observation as a petit ensemble, this distinction may be neglected.

The differences therefore, in the case considered, between the quantities which may be represented by the notations^[1]

${\overline {\mathrm {H} _{\rm {gen}}}}|_{\rm {grand}},\quad {\overline {\eta _{\rm {gen}}}}|_{\rm {grand}},\quad {\overline {\eta _{\rm {gen}}}}|_{\rm {petit}}$

are not sensible to human faculties. The difference

${\overline {\eta _{\rm {gen}}}}|_{\rm {petit}}-{\overline {\eta _{\rm {spec}}}}|_{\rm {petit}}={\nu _{1}}!\ldots {\nu _{h}}!,$

and is therefore constant, so long as the numbers $\nu _{1},\ldots \nu _{h}$ are constant. For constant values of these numbers, therefore, it is immaterial whether we use the average of $\eta _{\rm {gen}}$ or of $\eta$ for entropy, since this only affects the arbitrary constant of integration which is added to entropy. But when the numbers $\nu _{1},\ldots \nu _{h}$ are varied, it is no longer possible to use the index for specific phases. For the principle that the entropy of any body has an arbitrary additive constant is subject to limitation, when different quantities of the same substance are concerned. In this case, the constant being determined for one quantity of a substance, is thereby determined for all quantities of the same substance. To fix our ideas, let us suppose that we have two identical fluid masses in contiguous chambers. The entropy of the whole is equal to the sum of the entropies of the parts, and double that of one part. Suppose a valve is now opened, making a communication between the chambers. We do not regard this as making any change in the entropy, although the masses of gas or liquid diffuse into one another, and al-though the same process of diffusion would increase the

1. In this paragraph, for greater distinctness, ${\overline {\mathrm {H} _{\rm {gen}}}}|_{\rm {grand}}$ and ${\overline {\eta _{\rm {spec}}}}|_{\rm {petit}}$ have been written for the quantities which elsewhere are denoted by ${\overline {\mathrm {H} }}$ and ${\overline {\eta }}$.