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So it dawned on me the other day that the ideal gas constant $$R = \pu{8.31 J mol-1 K-1}$$ has the same units as molar entropy. Is there some deeper meaning behind this or is it just a coincidence that this occurs?

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  • $\begingroup$ Well, R is in the equation for the entropy change of an ideal gas at constant temperature. Is that what you were asking? $\endgroup$ – Chet Miller Apr 8 at 19:40
  • $\begingroup$ What is that, equation as it may answer my question. But, i was asking why does R have the units for entropy - Is it representing entropy? $\endgroup$ – H.Linkhorn Apr 8 at 19:42
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    $\begingroup$ Both entropy and the ideal gas constant are in some sense related to the Boltzmann constant. $\endgroup$ – Zhe Apr 8 at 20:12
  • $\begingroup$ Could you elaborate on that please. How are they related? $\endgroup$ – H.Linkhorn Apr 8 at 20:13
  • $\begingroup$ $\Delta S=R\ln{(V_f/V_i)}=-R\ln{(P_f/P_i)}$ $\endgroup$ – Chet Miller Apr 8 at 20:14
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The gas constant is equal to Avogadro's constant times Boltzmann's constant, the latter serving as a proportionality constant between the average thermal (kinetic) energy of the particles in an ideal gas and the temperature:

$$\left(\frac{\partial \bar U}{\partial T}\right)_p=\frac{3}{2}k_\mathrm{B}$$

The entropy can be regarded as a proportionality constant between the change in free energy $G$ with change in temperature of a system at constant pressure, since

$$\left(\frac{\partial G}{\partial T}\right)_p=-S$$

Note also that the average entropy of a particle can be written as

$$S=k_\mathrm{B} \log\Omega$$

So both $R$ and $S$ can be regarded as proportionality constants between energy and temperature.

The Wikipedia page on the equipartition theorem may provide some enlightenment on the origin of these proportionalities.

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The fundamental equation is Boltzmann's for the entropy $S =k_B\ln(\Omega)$ where $k_B$, Boltzmann's constant, has units J/molecule/K. When we use $R$ instead of $k_B$ it is trivially because molar units are used to define $S$.

In the equation $\Omega$ is the number of arrangements or configurations (or 'complexions' to use an old word) of distinguishable 'particles' among all the available energy levels. In that sense entropy is a measure of the uniformity of population in these levels.

(If there are $N$ distinguishable particles then $\displaystyle \Omega =\frac{N!}{n_1!n_2!n_3!\cdots}$ where there are $n_i$ particles in level $i$. The log is evaluated by assuming all the numbers $N,\, n_i$ etc. are large numbers so that Stirling's approximation for $\ln(N!) $ etc. is valid.)

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