Internal Energy

In addition to simple translation, gas particles can sometimes also rotate or swing which increases the degrees of freedom $f$. The general form of the inner energy of a gas can therefore be written as $$\boxed{U = \frac{1}{2}fNk_BT}$$ The inner energy per mole is then given as $$U = \frac{1}{2}fRT$$ Inserting the heat $\Delta Q$ into the system leads to an increase of the inner energy according to $$\Delta U = \Delta Q = C\Delta T$$ where $C$ is the previously defined heat capacitance. Then it follows for the molar heat capacity for a constant volume: $$\boxed{c_v = \frac{1}{2}fR}$$ As one can see, it only depends on the degree of freedoms in case of an ideal gas. If the volume of the gas can increase, i.e. the pressure remains constant, the increase of the inner energy can be written as $$\Delta U = c_v\Delta T + p\Delta V$$ Using the ideal gas equation again leads to $$\Delta U = c_v\Delta T + R\Delta T$$ Now we can define the heat capacity for a constant pressure according to $$\Delta U = c_p\Delta T$$ where $c_p$ is given as $$\boxed{c_p = c_v +R}$$ which is the sum of the heat capacity for a constant volume and the gas constant.