Question

Two moles of an ideal gas with $\frac{C_{P}}{C_{V}} = \frac{5}{3}$ are mixed with $3$ moles of another ideal gas with $\frac{C_{P}}{C_{V}} = \frac{4}{3}$. The value of $\frac{C_{P}}{C_{V}}$ for the mixture is :

• A. 1.50
• B. 1.45
• C. 1.47
• D. 1.42

Answer 1

Answer: (D)

$\gamma_{mixture}=\frac{n_{1}C_{P_1}+n_{2}C_{P_2}}{n_{1}C_{V_1}+n_{2}C_{V_2}}=\frac{n_{1}\frac{\gamma_{1}R}{\gamma_{1}-1}+n_{2} \frac{\gamma_{2}R}{\gamma_{2}-1}}{\frac{n_{1}R}{\gamma_{1}-1}+\frac{n_{2}R}{\gamma_{2}-1}}$
on rearranging we get,
$\frac{n_{1}+n_{2}}{\gamma_{mix}-1}=\frac{n_{1}}{\gamma_{1}-1}+\frac{n_{2}}{\gamma_{2}-1}$
$\frac{5}{\gamma_{mic}-1}=\frac{3}{1/3}+\frac{3}{2/3}$
$\frac{5}{\gamma _{mic}-1}=9+3=12$
$\Rightarrow \gamma_{mixture}=\frac{17}{12}=1+\frac{5}{12}$
$\gamma_{mix}=1.42$