Statistical Mechanics Ideal Gas in Harmonic Trap Equiparti

Statistical Mechanics - Ideal Gas in Harmonic Trap - Equipartition Theorem

In class we computed the entropy of an ideal gas using the micro canonical ensemble method. Consider an ideal gas in an external potential trap with Calculate the phase-space volume Q(E) for energy within the shell [E, E + Delta E?]. Calculate the temperature T as a function of E and w from Confirm the equipartition theorem. Consider the parameter w as an externally controlled parameter, and by making the trap strong by increasing w the system receives energy via macroscopic work. Use the first law of thermodynamics dE = TdS - dW, to express dW in T, w, dT and dw.

Solution

Ans for Part (a)

Density of States) = V^N C_3N (2mE)^{3(N/2) - 1} . 3(N/2) . 2m ; where C_3N - volume of the 3N dim sphere with radius 1.

Phase Space Volume = (Density of States) . (del E) ; del E is differenence in the energy levels, such as E &dE

Ans for Part (b)

1 / T = ((del S) / (del E))_w = Nk_B . (1/E) (carloric equation of state)