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

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

($$\frac{1}{N!}$$ 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

$$F(T,N,V)=-kT ln(Z_c)$$

From F we can calculate the pressure:

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