## Physics Friday 80

In several past posts, we explored the properties of a classical ideal gas. Next, we consider the entropy of an ideal gas. Using the differential dS and the chain rule, we can express it in terms of the temperature and volume differentials as

(with volume and temperature both possibly functions of particle number N).
For the first term on the right, we recall our discussion of heat capacity, and that . This means our entropy expression becomes
.
For the latter, we use one of the Maxwell relations, specifically the one (derived from the mixed second partial derivatives of the Helmholtz free energy) which states:
.
Now, solving the ideal gas equation for pressure,
,
we can take the partial derivative to get

Applying these to the entropy differential, we get:
.
Now, recalling our definition of the dimensionless specific heat capacity , we can change the above to

Integrating this, we get:
,
where C(N) is our constant of integration, an as yet unknown function of the particle number. Using our properties of logarithms, this may be restated as
.
where the function f(N) has the same units as .

Now, we need to use the fact that entropy is an extensive property. This means that if the extensive parameters, here V and N, are scaled by a constant, the entropy will be multiplied by that same constant:
. Plugging this into our above, we get:

which solving for f(cN), tells us
. This, in turn, tells us that f(N) must be a constant multiple of N. Thus
,
where φ is a (positive) constant with the same units as .

This is the limits of where classical thermodynamics takes us. However, we note that this formula cannot be valid for lower temperatures: the above, for any value of φ, gives entropy going to negative infinity as the temperature approaches absolute zero, with zero entropy at some positive temperature .

For a monatomic ideal gas (so ), quantum mechanical arguments can be used to give a value for φ which gives an entropy equation which holds for a wide range of states in the classical regime. The result is the Sackur-Tetrode equation.