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.

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Tags: Entropy, Friday Physics, Ideal Gas, Ideal Gas Law, Maxwell Relations, physics, Sackur-Tetrode Equation

This entry was posted on July 10, 2009 at 12:02 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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July 17, 2009 at 12:42 am |

[…] and f is the number of available degrees of freedom for a molecule of the gas (see here). We previously found that (for high enough temperatures) the entropy of an ideal gas can be expressed as for some […]