Q&A

# Is it possible to derive the Dieterici equation starting from assumptions about microstates?

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I was introduced to a somewhat novel derivation of the ideal gas law that starts by thinking about the number of microstates of an ideal gas, $\Omega$. Say we have a gas with a single particle in a volume $V$. Doubling the volume should double the number of microstates, as it doubles the possible positions. The same goes for increasing the volume by a factor of 3, 4, etc., and so it holds that the number of microstates is proportional to the volume available to the particle; $\Omega\propto V$. If we have two non-interacting particles, each obeys this same proportionality, and the total number of microstates of this two-particle gas is $$\Omega=\Omega_1\Omega_2\propto V^2$$ Using this same logic, the number of microstates of an ideal gas with $N$ particles should scale as $\Omega\propto V^N$. (You can also do this more formally by integrating over the $3N$-dimensional subspace of phase space corresponding to positions, if you'd like - that's a bit more rigorous.)

Now, the entropy obeys Boltzmann's equation $S=k_B\ln\Omega$, so we have $$S(E,V,N)=Nk_B\ln V+S_0(E,N)$$ with $E$ the energy and $S_0$ some arbitrary function encapsulating additional dependence on $E$ and $N$. Finally, recalling that $$\left(\frac{\partial S}{\partial V}\right)_{E,N}=\frac{P}{T}$$ with $T$ the temperature and $P$ the pressure, we can see that $$\frac{P}{T}=\frac{Nk_B}{V}\implies PV=Nk_BT$$ which is the ideal gas law.

It's a nice alternative to some of the other more explicit derivations I've seen, e.g. ones invoking the equipartition theorem, partition functions or kinetic theory. However, I'm interested in whether it can be modified for related non-ideal gases, in particular the Van der Waals equation and the Dieterici equation. I came across what seems to be a nice extension of it applied to the former, but I've been unable to find anything relevant applied to the latter.

Is it possible to extend this reasoning to derive the Dieterici equation of state - to reason out the dependence of $\Omega$ on the extensive parameters $E$, $V$ and $N$ and the two appropriate constants, then invoke Boltzmann's equation and the appropriate thermodynamic relations? My guess is that there isn't, but I'd be happy to be proven wrong.

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