# Does DCl have a higher rotational partition function (qrot) than HCl? And if so why?

I know that DCl has a higher moment of inertia than HCl, but I can't really explain how this actually affects the rotational partition function, I know larger molecules have a higher moment of inertia and this means the energy levels are closer, but I don't know how to explain how this has an effect on the partition function of rotation, can someone help?

• Do you know how to obtain an expression for the rotational partition function (specifically, by approximating the sum as an integral and evaluating the subsequent integral)? – orthocresol Nov 22 '16 at 18:26

The partition function is the sum over states, which means over all the energy levels, thus $$Z=\sum _i g_i \exp(-E_i/(k_BT))$$ where $g_i$ is the degeneracy of each level with energy $E_i$. The rigid rotor rotational energy levels of a diatomic molecules are $$E_J= \hbar^2J(J+1)/(2I)$$ where I is the moment of inertia $I=\mu r^2$ for reduced mass $\mu=m_1m_2/(m_1+m_2)u$ (in kg) and bond length r. The degeneracy of rotational levels is $g_J=2J+1$.
As the moment of inertial of DCl is greater than that of HCl (due to reduced mass effect) the energy levels are lower in energy in DCl at the same J than in HCl and so at the same temperature more levels are populated and the partition function is greater. This means that the exponential term in the partition function is greater in DCl at a given J and so this increases the partition function. (recall that $e^0 = 1$ and $e^{-\text{big number}} \rightarrow 0$). The energy levels of DCl and HCl are shown in the left figure and the terms in the partition function on the right as $g_J \exp(-E_J/(k_BT))$. (The curves are plotted for clarity assuming that J is continuous not discrete). The respective partition function is the sum of the values at each J in the right-hand figure.
(To plot the curves I used a mass of Cl = $35$ and H and D, $1$ & $2$ respectively. The unified atomic mass unit $u = 1.660\cdot 10^{-27}$ kg, and assume a temperature of $300$ K. The bond length is $74.15$ pm.)