What is the relationship between entropy and heat capacity?
The entropy depends on the mass. The translational part is given by the Sakur-Tetrode eqn., the rotational part depends on the moment of inertia, and the vibrational part on the vibrational frequency. In each case the partition function $z$ has to be calculated then the entropy via $\displaystyle S=R\ln(z)+RT\left(\frac{\partial z}{\partial T} \right)_V -R\ln(N)+R$ (for a perfect gas). The physical reason is that for the heavier molecule there are more energy levels up to a given energy that can be occupied and so more ways of populating these levels and so more entropy.

The heat capacity is the slope of internal energy $U$ with temperature (at constant volume) $\displaystyle C_V=\left({\partial U}{\partial T}\right)_V$ and the internal energy is related to the partition function as $\displaystyle \frac{U}{T}=RT\left(\frac{\partial z}{\partial T}\right)_V$ and so also to the entropy. The classical thermodynamic relationship is $\displaystyle S=S_0+\int_0^T \frac{C_p}{T}dT$ where $C_p$ is a function of temperature.