# Statistical distributions for non-ideal gases

Motivated by another question which implicitly suggests calculating the pressure of a gas on the walls of the container, while taking account for molecular scattering. We all know how this calculation is done for ideal gases, using Maxwell-Boltzmann distribution. For non-ideal gases however the corrections are either phenomenological (van der Waals equation) or the pressure/equation-of-state is obtained via the thermodynamic identities.

Question : are there any equivalents of the Maxwell-Boltzmann distribution for the non-ideal gases? (With which a direct calculation of the pressure could be attempted.)

• @YvanVelenik this is also my impression - it is the hard way to get the result. So the context is more historical - if it was tried, or if there are some semi-phenomenological models. I also wonder whether van der Waals equation can be derived or if it must be postulated phenomenologically. Apr 22 '20 at 7:44
• @YvanVelenik Thanks for the recommendations! Apr 22 '20 at 11:06

The Hamiltonian of a real gas is usually taken of the form $$H(p_1,\dots,p_N,q_1,\dots,q_N) = \sum_{i=1}^N \frac{p_i^2}{2m} + \sum_{i Since it is the sum of a momenta-dependent term and a position-dependent term, the canonical probability density (at temperature $$T$$) factorizes: $$f^{\rm can}_T(p_1,\dots,p_N,q_1,\dots,q_N) = f^{{\rm can},p}_T(p_1,\dots,p_N)f^{{\rm can},q}_T(q_1,\dots,q_N),$$ where $$f^p_T(p_1,\dots,p_N)$$ is the marginal density for the momenta and is simply given by a Gaussian density, independently of the interaction $$V$$: $$f_T^{{\rm can},p}(p_1,\dots,p_N) = (2\pi m/\beta)^{-N/2} e^{-\beta\sum_{i}p_i^2/2m}.$$ In particular, the distribution of the momenta is the same as for an ideal gas, which implies that the Maxwellian distribution applies regardless of the interaction term.

The distribution of the positions is however quite complicated. This makes an approach along the lines you want very intricate. The usual way to compute the pressure is thus via the virial expansion, which leads to the virial equation $$PV=Nk_BT\left( 1 + \frac{N}{V}B_2(T) + \frac{N^2}{V^2}B_3(T) + \frac{N^3}{V^3}B_4(T)+ \cdots \right),$$ where the virial coefficients $$B_i(T)$$ are given by explicit expressions and can be (in principle) computed.

Some references:

• You also asked whether the van der Waals equation can be deduced from theoretical considerations or whether it is of a purely phenomenological nature. This equation can indeed be deduced, as an approximation by comparing it to the (exact) virial expansion described above. Alternatively, it is possible to derive it as a type of mean-field limit. A possible reference is our book, where the latter is discussed in Chapter 4 and the former in Chapter 5), although we only discuss the lattice gas.
• A very interesting (although rather old) book where the kind of derivation you are after can be found is this one.

Question : are there any equivalents of the Maxwell-Boltzmann distribution for the non-ideal gases? (With which a direct calculation of the pressure could be attempted.)

No.

The theory is defined by the Maxwell-Boltzmann distribution. That equation leads to the ideal gas equation $$pV_m=RT$$ where $$V_m$$ is the molar volume of the gas.

Unfortunately as more high quality gas data was collected it became obvious that all gases under all conditions didn't follow the ideal gas law.

I'm not sure where a sharp line could be draw for a model being phenomenological versus purely theoretical. The van der Waals equation was supposed to theoretically adjust the ideal gas model. Molecules in real gases interact with one another so the $$\mathrm{a}$$ factor is needed to correct pressure and molecules really do occupy space so the $$\mathrm{b}$$ factor is needed to adjust the volume. Unfortunately the model is a scant improvement over the ideal gas model.

$$\left( p + \mathrm{a}\dfrac{1}{V_m^2}\right) \left( V_m - \mathrm{b}\right)= RT$$

As far as the viral model, it is perturbative treatment of statistical mechanics. The gist is that you add as many terms as it takes to fit the data set.

$$pV_m = RT\left[ 1 + \dfrac{B(T)}{V_m} + \dfrac{C(T)}{V_m^2}+ \dfrac{D(T)}{V_m^3} + ... \right]$$

The van der Waals equation and the viral model are just two of many quasi-theoretical equations that have been tried to explain the behavior of a real gas.

The calculation of the molecular behavior of molecules in a liquid and molecules in gas are somewhat linked. If a 3 body problem is impossible to calculate exactly for gravity imagine roughly $$2.7\times 10^{22}$$ molecules in one liter of a gas at STP.

• I diasagree with this. First, in the canonical ensemble, the distribution of the momenta is independent from the distribution of the positions. You thus get a Maxwellian distribution independently of the interaction term (as long as the latter is only a function of the positions, not the momenta). Second, the coefficients of the virial expansion are completely determined by the interaction. You can in principle compute them. Of course, this is hard (except for the first few). Nevertheless, these are not empirical. Apr 22 '20 at 9:21
• Additionally, the Maxwell distribution by itself does not lead to the ideal gas law: once interactions are present, correlations in positions become very complicated. This is precisely why you get the corrections in the virial expansion. Apr 22 '20 at 9:23
• Finally, you say If a 3 body problem is impossible to calculate exactly for gravity imagine roughly 2.7×1022 molecules in one liter of a gas at STP. Sure, but that's precisely why we use statistical mechanics! The large number of molecules is not a problem, but a good thing: it's what makes the probabilistic analysis reliable. Apr 22 '20 at 9:26