Partition Function Analysis

The given molecular partition function is:

$q(T) = \left(\frac{k_B T}{hc}\right)^{\frac{3}{2}} \left(\frac{8\pi^2 mk_B T}{h^2}\right)^{\frac{3}{2}}$

Let's simplify this expression. We can combine the terms and group the temperature dependence:

$q(T) = \left( \frac{k_B}{hc} \cdot \frac{8\pi^2 m k_B}{h^2} \right)^{\frac{3}{2}} \cdot (T)^{\frac{3}{2} + \frac{3}{2}}$

$q(T) = \left( \frac{8\pi^2 m k_B^2}{h^3 c} \right)^{\frac{3}{2}} \cdot T^3$

This simplifies to the form $q(T) = C \cdot T^3$, where $C$ is a constant.

Average Energy Calculation

The average internal energy ($U$) of the system is related to the partition function by:

$U = k_B T^2 \left( \frac{\partial \ln q}{\partial T} \right)_V$

First, find the natural logarithm of the partition function:

$\ln q(T) = \ln C + \ln(T^3) = \ln C + 3 \ln T$

Next, calculate the derivative of $\ln q(T)$ with respect to temperature ($T$):

$\frac{\partial \ln q}{\partial T} = \frac{3}{T}$

Now substitute this into the formula for average energy:

$U = k_B T^2 \left( \frac{3}{T} \right)$

$U = 3 k_B T$

Heat Capacity Derivation

The heat capacity at constant volume ($C_V$) is the derivative of the average energy with respect to temperature:

$C_V = \left( \frac{\partial U}{\partial T} \right)_V$

Differentiating the expression for $U$:

$C_V = \frac{\partial}{\partial T} (3 k_B T)$

$C_V = 3 k_B$

Relating to Ideal Gas Constant (R)

The result $C_V = 3 k_B$ is the heat capacity per particle. To express this in terms of the ideal gas constant ($R$) for one mole of substance, we use the relationship $R = N_A k_B$, where $N_A$ is Avogadro's number.

Molar Heat Capacity $= N_A \times C_V = N_A \times (3 k_B)$

Molar Heat Capacity $= 3 (N_A k_B) = 3R$

Therefore, the heat capacity at constant volume for this system is $3R$.