Portal:Biochemistry/Pressure ideal Boltzmann gas exercise

Calculate the pressure of an ideal Boltzmann gas in a volume V at the temperature T

The Hamiltonian of the system is:

$H(p,q)=\sum _i^N\frac{p_i^2}{2m}$

where N is the total number of particles and m is the mass. The gas is ideal because there are no interaction between particles.

We work in the canonical ensamble, the partition function is (s=state):

$Z_c=\sum _s{e}^{-\beta H(s)}$

we work in a continous state-space, so Z is

$Z_c(T,V,N)=\frac{1}{h^{3N}N!}\int \underset{q,p}{\int }{e}^{-\beta H(p,q)}dqdp$

($\frac{1}{N!}$ is a rule of the boltzmann counting)

Calculate the integral:

$Z_c(T,V,N)=\frac{V^N}{h^{3N}N!}[\underset{\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-\beta \frac{p^2}{2m}}{]}^N=\frac{V^N}{h^{3N}N!}(\frac{2\pi m}{\beta }{)}^{3N/2}$

note that Z is adimansional

Now calculate the Helmotz free energy

$F(T,N,V)=-kTln(Z_c)$

From F we can calculate the pressure:

$p(T,N,V)=-\frac{\partial F}{\partial V}=\frac{kTN}{V}$