1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Real gas, unable to reach the correct expression

  1. Sep 25, 2014 #1

    fluidistic

    User Avatar
    Gold Member

    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
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Real gas, unable to reach the correct expression
Loading...