JEE Main · PhysicsHard

$A mixture of two ideal gases with adiabatic exponents $\gamma_1$ and $\gamma_2$ and mole fractions $x_1$ and $x_2$ is kept in a container. What is the effective adiabatic exponent of the mixture?$

A. $\gamma_{\text{eff}} = \frac{x_1\gamma_1 + x_2\gamma_2}{x_1 + x_2}$
B. $\frac{1}{\gamma_{\text{eff}} - 1} = \frac{x_1}{\gamma_1 - 1} + \frac{x_2}{\gamma_2 - 1}$
C. $\gamma_{\text{eff}} = \frac{x_1\gamma_1 C_{v1} + x_2\gamma_2 C_{v2}}{x_1 C_{v1} + x_2 C_{v2}}$
D. $\gamma_{\text{eff}} = \frac{x_1\gamma_1 + x_2\gamma_2}{x_1 C_{v1} + x_2 C_{v2}}$

Show correct answer & step-by-step solution

Correct answer: B — $\frac{1}{\gamma_{\text{eff}} - 1} = \frac{x_1}{\gamma_1 - 1} + \frac{x_2}{\gamma_2 - 1}$

Solution

The effective molar heat capacity at constant volume is

C_{v, mix} = x_{1} C_{v 1} + x_{2} C_{v 2}

. Substituting

C_{v} = \frac{R}{γ - 1}

and using

C_{p, mix} - C_{v, mix} = R

yields the result.