# Real gas, unable to reach the correct expression

1. Sep 25, 2014

### fluidistic

1. The problem statement, all variables and given/known data
Consider a system of N particles contained in a volume V. The Hamiltonian of the system is $H=\sum{i=1}^N \frac{\vec p_i}{2m}+\sum _{i<j}u(|\vec r_i - \vec r_j|)$ where p_i and r_i are the momentum and position of the i-th's molecule.
1)Show that the state equation of the system is $\frac{Pv}{kT}=1+v\frac{\partial Z(v,T)}{\partial v }$ where v=V/N and $Z(v,T)=\frac{1}{N}\ln \left [ \frac{1}{V^N} \int d^3r_1... d^3 r _N \Pi _{i<j}(1+f_{ij}) \right ]$

Also $f_{ij}=f(|\vec r_i -\vec r_j|)$ with $f(r)=e^{-\beta u(r)}-1$.

2. Relevant equations
Relation between P and Z: $P=-\left ( \frac{\partial A}{\partial V} \right )_{\beta,N}$
Where $A=-\frac{1}{\beta}Z_N(\beta, V)$

3. The attempt at a solution
I used the relevant equations to get $A(\beta,V,N)=-\frac{1}{\beta} \ln [Z_N(\beta,V)]$ so that $P=\frac{1}{\beta}\frac{\partial}{\partial V} \{ \ln [Z_N(\beta,V)] \}$.
Hence $\frac{PV}{kT}=V\frac{\partial}{\partial V} \{ \ln [Z_N(\beta,V)] \}$.
Dividing by N I reach $\frac{Pv}{kT}=v\frac{\partial}{\partial V} \{ \ln [Z_N(\beta, V)] \}$
Now I believe that $\frac{\partial}{\partial V} \{ \ln [Z_N(\beta,V)] \}=N\frac{\partial}{\partial v} \{ \ln [Z_N(v,\beta)] \}$.
Which yields $\frac{Pv}{kT}=Nv \frac{\partial}{\partial v} \{ \ln Z_N (\beta ,v) \}=Nv\frac{1}{Z_N(v,\beta)}\cdot\frac{\partial}{\partial v}[Z_N(v,\beta)]$.
This differs from what I should have reached and I see no way to rewrite my expression into the desired one...
Any help on what's going on is appreciated.

Last edited: Sep 25, 2014