Question

A gaseous mixture consists of $16\, g$ of helium and $16\, g$ of oxygen. The ratio $\frac{C_{p}}{C_{v}}$ of the mixture is :

• A. $1.59$
• B. $1.62$
• C. $1.4$
• D. $1.54$

Answer 1

Answer: (B)

$C_{v}=\frac{n_{1}\,C_{v_1}+n_{2}\,C_{v_2}}{n_{1}+n_{2}}$
For helium, $n_{1}=\frac{16}{4}=4$ and $\gamma_{1}=\frac{5}{3}$
$n_{2}=\frac{16}{32}=\frac{1}{2} $ and $\gamma _{2}=\frac{7}{5}$
$C_{v_1}=\frac{R}{\gamma_{1}-1}$
$=\frac{R}{\frac{5}{3}-1}=\frac{3}{2}R$
$C_{v_2}=\frac{R}{\gamma_{2}-1}=\frac{R}{\frac{7}{5}-1}=\frac{5}{2}R$
$\therefore C_{v}=\frac{4\times\frac{3}{2}R+\frac{1}{2}. \frac{5}{2}R}{4+\frac{1}{2}}$
$=\frac{6\,R+\frac{5}{4}R}{\frac{9}{2}}$
$=\frac{29\,R\times2}{9\times4}=\frac{29\,R}{18}$
Now, $C_{v}=\frac{R}{\gamma-1}$
$\Rightarrow \gamma-1=\frac{R}{C_{v}}$
$\gamma =\frac{R}{C_{v}}+1=\frac{R}{\frac{29}{18}R}+1$
$\Rightarrow \frac{C_{p}}{C_{v}}=\frac{18}{29}+1$
$=\frac{18+29}{29}=1.62$