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:

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):

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

( is a rule of the boltzmann counting)

Calculate the integral:

note that Z is adimansional

Now calculate the Helmotz free energy

From F we can calculate the pressure: