Sodium Bose Gas: Critical Temperature & Law of Diminishing N0

[Schwarzschild90]

Homework Statement

3: When the temperature is increased slightly above T = 0, it is self-consistent to assume to that the chemical potential stays \mu(T) = 0, as long as there is a macroscopic number N0 oscillators in the ground state (You should not prove this!). The N atoms in the system are distributed with N0 in the ground state and the residual N-N0 particles in the excited state. Show that

When the temperature is increased, N0 gradually disappears. Show that this happens at a critical temperature T_C and calculate the value of T_C. Also, show that N0 diminishes according to the law

Homework Equations

The Attempt at a Solution

 

[BigJDubs]

When the temperature is increased, the energy levels of the oscillators become further apart as the particles gain energy. This means that more particles will be able to occupy excited states instead of the ground state. This process continues until all of the particles are in the excited states and N0 becomes zero. At the critical temperature T_C, the energy gap between the ground state and the first excited state is equal to kT_C, where k is the Boltzmann constant. We can find the value of T_C by solving the equation:kT_C = E_1 - E_0 where E_1 and E_0 are the energy levels of the first excited state and the ground state respectively.The number of particles in the ground state N0 diminishes according to the law:N_0 = N_0(0)e^(-E_1/kT)where N0(0) is the initial number of particles in the ground state, and E1 is the energy difference between the first excited state and the ground state.