Question

If one mole of a monatomic gas $(\gamma =\frac{5}{3})$ is mixed with one mole of a diatomic gas $(\gamma =\frac{7}{5})$ the value of $\gamma$ for the mixture is:

Answer 1

For a monoatomic gas,
$C_p=\frac{5R}{2}$ and $C_v=\frac{3R}{2}$
For a diatomic gas,
$C_p=\frac{7R}{2}$ and $C_v=\frac{5R}{2}$

So, for a mixture of 1 mole of each gas in solution
$C_{v_{mix}}$ = $\frac{\frac{3R}{2}+\frac{5R}{2}}{2}$ = $2R$

$C_{p_{mix}}$ = $\frac{\frac{5R}{2}+\frac{7R}{2}}{2}$ = $3R$
${\gamma }_{mix}=\frac{C_{p_{mix}}}{C_{v_{mix}}}$

${\gamma }_{mix}=\frac{3R}{2R}=1.50$