Page:Elementary Principles in Statistical Mechanics (1902).djvu/208

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184

THERMODYNAMIC ANALOGIES.

be convenient for physical purposes. But when these units have been chosen, the numerical values of $\Theta$ , $d\epsilon /d\log V$ , $d\epsilon /d\phi$ , ${\overline {\eta }}$ , $\log V$ , $\phi$ , are entirely determined,^[1] and in order to compare them with temperature and entropy, the numerical values of which depend upon an arbitrary unit, we must multiply all values of $\Theta$ , $d\epsilon /d\log V$ , $d\epsilon /d\phi$ by a constant ( $K$ ), and divide all values of ${\overline {\eta }}$ , $\log V$ , and $\phi$ by the same constant. This constant is the same for all bodies, and depends only on the units of temperature and energy which we employ. For ordinary units it is of the same order of magnitude as the numbers of atoms in ordinary bodies.

We are not able to determine the numerical value of $K$ as it depends on the number of molecules in the bodies with which we experiment. To fix our ideas, however, we may seek an expression for this value, based upon very probable assumptions, which will show how we would naturally proceed to its evaluation, if our powers of observation were fine enough to take cognizance of individual molecules.

If the unit of mass of a monatomic gas contains $\nu$ atoms, and it may be treated as a system of $3\nu$ degrees of freedom, which seems to be the case, we have for canonical distribution

{\overline {\epsilon }}_{p}={\tfrac {3}{2}}\nu \Theta ,

{\frac {d{\overline {\epsilon }}_{p}}{d\Theta }}={\tfrac {3}{2}}\nu .

(491)

If we write

T

for temperature, and

c_{v}

for the specific heat of the gas for constant volume (or rather the limit toward which this specific heat tends, as rarefaction is indefinitely increased), we have

{\frac {d\epsilon _{p}}{dT}}=c_{v},

(492)

since we may regard the energy as entirely kinetic. We may set the

\epsilon _{p}

of this equation equal to the

{\overline {\epsilon }}_{p}

of the preceding,

↑ The unit of time only affects the last three quantities, and these only by an additive constant, which disappears (with the additive constant of entropy), when differences of entropy are compared with their statistical analogues. See page 19.

[1] The unit of time only affects the last three quantities, and these only by an additive constant, which disappears (with the additive constant of entropy), when differences of entropy are compared with their statistical analogues. See page 19.

[1]