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

$Z_{c}(T,V,N)={\frac {1}{h^{3N}N!}}\int \!\!\!\int _{q,p}e^{-\beta H({p,q})}d{q}d{p}$

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

Calculate the integral:

$Z_{c}(T,V,N)={\frac {V^{N}}{h^{3N}N!}}{\Big [}\int _{\infty }^{\infty }e^{-\beta {\frac {p^{2}}{2m}}}{\Big ]}^{N}={\frac {V^{N}}{h^{3N}N!}}{\Big (}{\frac {2\pi m}{\beta }}{\Big )}^{3N/2}$

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:

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