For an ideal gas, the relationship between $C_p$ and $C_v$ is (Mayer's relation):

Mayer's relation: $C_p - C_v = R$, where $R$ is the universal gas constant. Derivation: at constant volume, heat $Q = nC_v\Delta T$ all becomes internal energy. At constant pressure, $Q = nC_p\Delta T$, but some heat does PdV work against the surroundings; that work equals $nR\Delta T$. Setting $nC_p\Delta T = nC_v\Delta T + nR\Delta T$ gives $C_p - C_v = R$. Monoatomic: $C_v = \dfrac{3}{2}R$, $C_p = \dfrac{5}{2}R$, $\gamma = 5/3$. Diatomic: $C_v = \dfrac{5}{2}R$, $C_p = \dfrac{7}{2}R$, $\gamma = 7/5$. The $R$ gap is always there because of that PdV work.