# Gas of hard-core spheres

Consider a system of hard spheres of diameter $d_0$ at temperature $T$ and $V/N=v$. Discuss briefly the existence of the thermodynamic limit for the Helmholtz free energy on the basis of the theorem of van Hove. Write the equation (approximated) by van der Waals for the system. Tell whether possibly applying the rule of Maxwell, it preaches or not the liquid-vapor transition. Show that the first order in the amount $d_0^3/v$ equation involves the same correction to $Pv/kT$ can be obtained with the virial expansion, this well in his first order.

So, for an Hard-Core system of spheres the potential is $\varphi=\varphi_{\text{HC}}+\varphi_{\text{attractive}}$ and the canonical partition function is $$Q_{N}(V,T)=\frac{1}{\Lambda^{3N}N!}\int_{q\in V}\text{d}q\exp{(-\beta\varphi(q))}$$ where $\text{d}q=\prod_i^N\text{d}^3q_i$. For now we take $\varphi_{\text{attractive}}=0$, away from the sufrace i assume thatthe total volume for each spheres decrease with the numbers of spheres we take into account, in other words $$V(V-v_0)(V-2v_0)\cdots=V^N\prod_{i=1}^{N-1}(1-iv_0)=\bar{V},$$ where $v_0=\frac{4\pi d_0^3}{3}$.

So for $V\gg Nv_0$ we have $$Q_{N}^{\text{HC}}(V,T)\simeq\frac{1}{\Lambda^{3N}N!}\Big(V-\frac{Nv_0}{2}\Big)^N.$$

If we take into account the attractive part of the potential with the prescription of $$\bar{\varphi}_{ATT}\simeq-\frac{\alpha}{V},$$ the total partition function became $$Q_N^{VdW}(V,T)=\frac{\Big(V-\frac{Nv_0}{2}\Big)^N}{\Lambda^{3N}N!}\exp{\Big(\frac{\alpha N^2}{2k_BTV}\Big)}.$$ From this we can take out the law of that kind of gas, that read $$\Big(P+\frac{\alpha N^2}{2V^2}\Big)(V-\frac{Nv_0}{2})=Nk_BT$$

My questions are, how to discuss the existence of the thermodynamical limit and the use of Maxwell's rule?

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