In Wikipedia's ideal gas article there is a derivation for the thermodynamic entropy that results in $$S = Nk\ln\left[\frac{V}{N} \left(\frac{U}{\hat{c}_V Nk}\right)^{\hat{c}_V} \frac{1}{\Phi}\right],$$ and the article states:

[...] $\Phi$ may vary for different gases, but will be independent of the thermodynamic state of the gas.

I have a notion that $\Phi$ should look something like the volume of the individual gas molecules multiplied by the latent heat of vaporization per molecule divided by $k$ to the power of $\hat{c}_V$, which means it would have the following (approximate) values for the named gases

  1. $\mathrm{N}_2$: $\Phi = 2\times \frac{4}{3}\pi(56\operatorname{pm})^3\times(58 \operatorname{meV} / k)^{5/2}= 1.7\times10^{-23} \operatorname{K}^{5/2} \operatorname{m}^3$
  2. $\mathrm{O}_2$: $\Phi = 2\times \frac{4}{3}\pi(48\operatorname{pm})^3\times (71 \operatorname{meV} / k)^{5/2}= 1.8\times10^{-23} \operatorname{K}^{5/2} \operatorname{m}^3$
  3. $\mathrm{H}_2$: $\Phi = 2\times \frac{4}{3}\pi(53\operatorname{pm})^3 \times (9.4 \operatorname{meV} / k)^{5/2}= 1.5\times10^{-25} \operatorname{K}^{5/2} \operatorname{m}^3$
  4. $\mathrm{He}$: $\Phi = \frac{4}{3}\pi(31\operatorname{pm})^3\times (0.86 \operatorname{meV} / k)^{5/2}= 3.9\times10^{-30} \operatorname{K}^{3/2} \operatorname{m}^3$

Basically, $\Phi$ should give some information of where the ideal gas model breaks down for the gases in question. At $P=1\operatorname{atm}$ these values of $\Phi$ imply that entropy hits zero at $T = 29,\ 29,\ 7.4,\ \mathrm{and}\ 0.25 \operatorname{K}$ for each of these gases, which doesn't seem too far off from their boiling points of $77$, $90$, $20$, and $4.2$ Kelvin. Even so, this is about as well as you'd expect to do from unit analysis. What are the actual factors that go in to determining $\Phi$?


For the entropy of monatomic ideal gas there is Sackur-Tetrode equation. $\Phi$ can be explicitly found from this equation for inert gases, for example. As one can see in this case $\Phi$ is determined by atomic mass, Boltzmann's constant, Planck's constant and mathematical constants.

For diatomic gases Sackur-Tetrode equation is modified by addition of terms related to inner motions of molecules. Besides temperature and constants this terms are determined by microscopic details of molecules such as moment of inertia and frequency of vibration.

Phase transitions are due to interaction between molecules. This interaction is not taken into account in ideal gas approximation. Hence I think, that $\Phi$ is fully determined by mentioned microscopic characteristics of molecules.


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