Introduction to Mathematical Physics/N body problems and statistical equilibrium/Van der Waals gas

Van der Waals\footnote{Johannes Diderik van der Waals received the physics Nobel price in 1910.} gas model\index{Van der Walls gas} is a model that also relies on classical approximation, that is on the repartition function that can be written:

${\displaystyle Z={\frac {1}{N!}}{\frac {1}{h^{3N}}}\int \int d^{3}p_{1}\dots d^{3}p_{N}d^{3}r_{1}\dots d^{3}r_{N}e^{-{\frac {H}{k_{B}T}}}}$

where the function ${\displaystyle H(p_{1},\dots ,p_{N},r_{1},\dots ,r_{N})}$ is the hamiltonian of the system.

${\displaystyle H(p_{1},\dots ,p_{N},r_{1},\dots ,r_{N})={\frac {1}{2m}}\sum p_{i}^{2}+U(r_{1},\dots ,r_{N})}$

As for the perfect gas, integration over the ${\displaystyle p_{i}}$'s is immediate:

${\displaystyle Z={\frac {1}{N!}}\left({\frac {2\pi mk_{B}T}{h^{2}}}\right)^{\frac {3N}{2}}Y}$

with

eqYint

${\displaystyle Y=\int d^{3}q_{1}\dots d^{3}q_{N}e^{-{\frac {U(r_{1},\dots ,r_{N})}{k_{B}T}}}}$

We can simplify the evaluation of previuos integral in neglecting correlations between particles. A particle number ${\displaystyle i}$ doesn't feel each particle but rather the average influence of the particles cloud surrounding the considered particle; this approximation is called mean field approximation \index{mean field}. It leads, as if the particles were actually independant to a factorization of the repartition function. Interaction potential ${\displaystyle U(r_{1},\dots ,r_{N})}$ becomes:

${\displaystyle U(r_{1},\dots ,r_{N})={\frac {1}{2}}\sum U_{e}(r_{i})}$

where ${\displaystyle U_{e}(r_{i})}$ is the "effective potential" that describes the mean interaction between particle ${\displaystyle i}$ and all the other particles. It depends only on position of particle ${\displaystyle i}$. Function ${\displaystyle Y}$ introduced at equation ---eqYint--- can be factorized:

${\displaystyle Y=\left(\int dre^{-{\frac {U_{e}(r)}{2k_{B}T}}}\right)}$

In a mean field approximation framework, it is considered that particle distribution is uniform in the volume. Effective potential has thus to traduce an mean attraction. Indeed, it can be shown that at large distances two molecules are attracted, potential varying like ${\displaystyle {\frac {1}{r^{6}}}}$. This can be proofed using quantum mechanics but this is out of the frame of this book. At small distances however, molecules repel themselves strongly. Effective potential undergone by a particle at position ${\displaystyle r=0}$ can thus be modelled by function:

${\displaystyle {\begin{matrix}U_{e}(r)&=&\infty {\mbox{ if }}rr_{0}\\\end{matrix}}}$

Quantity ${\displaystyle Y}$ is

${\displaystyle Y=\left((V-bN)e^{\frac {aN}{k_{B}TV}}\right)^{N}}$

Quantity ${\displaystyle bN}$ represents the excluded volume of the particle. It is proportional to ${\displaystyle N}$ because ${\displaystyle N}$ particles occupying each a volume ${\displaystyle b}$ occupy a ${\displaystyle Nb}$. On another hand mean potential ${\displaystyle U_{0}}$ felt by the test particle depends on ratio ${\displaystyle N/V}$. Usually, one sets:

${\displaystyle U_{0}=-2a{\frac {N}{V}}}$

Introducing

${\displaystyle F=-k_{B}T\log Z}$

and using

${\displaystyle p=-{\frac {\partial F}{\partial V}}}$

one finally obtains Van der Waals state equation:

${\displaystyle \left(p+a{\frac {n^{2}}{V^{2}}}\right)(V-bN)=Nk_{B}T}$

Van der Waals model allows to describe liquid--vapour phase transition.

Over a critical temperature, energy doesnt present any local minimum.}
figvanavant
Below the critical temperatre, energy presents a local minimum. Two phases then appear, characteriszed by points ${\displaystyle A_{0}}$ et ${\displaystyle B_{0}}$.}
figvanapres

When temperature is lower than critical temperature ${\displaystyle T_{c}}$, the energy of the system for a given volume ${\displaystyle V_{M}}$ with ${\displaystyle V_{A_{0}} is not ${\displaystyle F_{v}(V_{M})}$. Indeed, system evolves to a state of lower energy (see figure figvanapres) with energy:

${\displaystyle F(V_{M})=F_{A_{0}}+{\frac {V_{M}-V_{A_{0}}}{V_{B_{0}}-V_{A_{0}}}}(F_{B_{0}}-F_{A_{0}})}$

The apparition of a local minimum of ${\displaystyle F}$ corresponds to the apparition of two phases.\index{phase transition}