Consider two ideal gases that share the same value of C_p / C_V = γ...

For an ideal gas, the molar heat capacities at constant volume and constant pressure are related to $\gamma$ by $C_V = \dfrac{R}{\gamma - 1}$ and $C_P = \dfrac{\gamma R}{\gamma - 1}$.

Since both gases have the same value of $\gamma$, their molar heat capacities are equal, i.e., $C_{V1} = C_{V2} = C_V$ and $C_{P1} = C_{P2} = C_P$.

For a mixture of two gases with number of moles $n_1$ and $n_2$, the equivalent heat capacities are given by:

$C_{V_{mix}} = \dfrac{n_1 C_{V1} + n_2 C_{V2}}{n_1 + n_2} = \dfrac{(n_1 + n_2) C_V}{n_1 + n_2} = C_V$

$C_{P_{mix}} = \dfrac{n_1 C_{P1} + n_2 C_{P2}}{n_1 + n_2} = \dfrac{(n_1 + n_2) C_P}{n_1 + n_2} = C_P$

The ratio of specific heats for the mixture is:

$\gamma_{mix} = \dfrac{C_{P_{mix}}}{C_{V_{mix}}} = \dfrac{C_P}{C_V} = \gamma$

The value of $\gamma$ for the mixture remains unchanged and is independent of the mixing ratio.