[Physics] Gamma of a mixture of multiple gases

Question

Gamma is defined as $$\frac{C_p}{C_v}$$

Now say you have a mixture of gases eg: 1 mol of monoatomic gas and 2 moles of diatomic gas, then how would I find the value of gamma of this new mixture of gases?

Answer 1

Once the gases are mixed and at same temperature,

$$ (n_1 + n_2) \overline{C_v} T = n_1 C_{v_1} T + n_2 C_{v_2} T$$

Cancelling temperatures,

$$(n_1+n_2)\bar{C_v}=n_1C_{v1}+n_2C_{v2}$$ where $\bar{C_v}$ is the molar average heat capacity at constant volume of the mixture. So $$\bar{C_v}=x_1C_{v1}+x_2C_{v2}$$and $$\bar{C_p}=x_1C_{v1}+x_2C_{v2}+R$$So, for the mixture, $$\gamma=\frac{\bar{C_p}}{\bar{C_v}}=1+\frac{R}{x_1C_{v1}+x_2C_{v2}}$$or $$\gamma=1+\frac{1}{\frac{x_1}{(\gamma_1-1)}+\frac{x_2}{(\gamma_2-1)}}$$ I'm sure that this can be simplified some more, but I'll leave that up to you.