Consider a system of N identical particles. Each particle has two energy levels: a ground
state with energy 0, and an upper level with energy �. The upper level is four-fold degenerate
(i.e., there are four excited states with the same energy �).
(a) Write down the partition function for a single particle.
(b) Find an expression for the internal energy of the system of N particles.
(c) Calculate the heat capacity at constant volume of this system, and sketch a graph to
show its temperature dependence.
(d) Find an expression for the Helmholtz free energy of the system.
(e) Find an expression for the entropy of the system, as a function of temperature. Verify
that the entropy goes to zero in the limit T ! 0. What is the entropy in the limit
T ! 1? How many microstates are accessible in the high-temperature limit?
(a) To find the partition function for a single particle, we need to consider the possible configurations of the energy levels. The ground state has energy 0, and the upper level is four-fold degenerate with energy ε.
The partition function, denoted by Z_1, can be calculated as the sum of the Boltzmann factors for each energy level:
Z_1 = e^(-β * 0) + 4 * e^(-β * ε)
where β = 1 / (k_B * T) is the inverse temperature, k_B is the Boltzmann constant, and T is the temperature.
(b) The internal energy of the system, U, can be obtained by taking the derivative of the logarithm of the partition function with respect to β:
U = -d(ln(Z_1)) / dβ
Differentiating the expression for Z_1 and simplifying, we get:
U = 0 + 4ε * e^(-β * ε) / (e^(-β * 0) + 4 * e^(-β * ε))
(c) The heat capacity at constant volume, C_V, can be calculated as the derivative of the internal energy with respect to temperature:
C_V = dU / dT
To evaluate this derivative, we can rewrite U in terms of β and differentiate with respect to β, and then use the relation between β and T.
(d) The Helmholtz free energy, F, is defined as F = -k_B * T * ln(Z). For a system of N identical particles, the total partition function, Z, is given by:
Z = Z_1^N
Therefore, the Helmholtz free energy is:
F = -k_B * T * ln(Z_1^N) = -N * k_B * T * ln(Z_1)
(e) The entropy, S, can be calculated from the Helmholtz free energy:
S = -dF / dT
Using the expression for F from (d), differentiation, and simplification, we can find an expression for the entropy of the system as a function of temperature.
To verify that the entropy goes to zero in the limit T → 0, we can calculate the entropy expression for small values of T and observe the limit.
In the limit T → 1, we can also calculate the entropy expression and observe its value.
The number of microstates accessible in the high-temperature limit can be obtained by considering Boltzmann's entropy formula: S = k_B * ln(W), where W is the number of microstates. The high-temperature limit corresponds to large values of T, for which W can be calculated using the partition function and β.