An ideal monatomic gas at the temperature T is confined in a spherical container of radius R. There are N molecules of mass m in the gas. Molecules move in a spherically symmetric potential V(r) where r is the distance from the center of the container. The potential V(r) is given by a piecewise function:
V(r) = -Uo*r/R for r<Ro
= 0 for Ro<r<R
= ∞ for r>R
U0 is a positive constant (i.e., “attractive well”). Thus, there is a potential well (a trap) of radius R0 inside the container. Notice also that the interaction between the molecule and the cavity wall is hard-core like.
a) Calculate the partition function Z.
b) What is the probability distribution function (probability per unit volume), P(r), that a particle is located at distance r from the center of spherical container?
c) What is the probability, Ptrap(r), that a particle is located inside the potential well (i.e. at a distance r smaller than R0 from the center of the spherical container).
d) What is the average number of molecules inside the trap, Ntrap?
e) Calculate the internal energy E=<H(r,p)> and the heat capacity Cv of this gas.
f) Calculate the pressure p exerted by the molecules on the wall of the spherical container.
H = p^2/(2*m) + V(r)
The Attempt at a Solution
ℤ =1/(N!*h^(3N)) ∫ exp(-β*p^2/(2*m)∫exp(-β*V(r))
I get stuck with the integration of the potential term. I don't know if you would need to integrate each part of the piecewise separately?
If someone could help me out with this, I would be extremely grateful![/B]