A mixture of gasses consists of $16 \mathrm{~g}$ of Helium and $16 \mathrm{~g}$ of Oxygen. The ratio of…
A mixture of gasses consists of $16 \mathrm{~g}$ of Helium and $16 \mathrm{~g}$ of Oxygen. The ratio of specific heats of the mixture is nearly
$1.33$
$1.4$
$1.56$
$1.62$
Solution
Helium mass, $m_H=16 g$
oxygen mass, $m_o=16 \mathrm{~g}$
let,
$\mathrm{M}_{\mathrm{o}}=$ molar mass of oxygen
$\mathrm{M}_{\mathrm{H}}=$ molar mass of Helium
Number of moles of oxygen,
$\mathrm{n}_{\mathrm{o}}=\frac{\mathrm{m}_{\mathrm{o}}}{\mathrm{M}_{\mathrm{o}}}=\frac{16}{32}=\frac{1}{2}$
No. of moles of Helium,
$\mathrm{n}_{\mathrm{h}}=\frac{\mathrm{M}_{\mathrm{h}}}{\mathrm{M}_{\mathrm{h}}}=\frac{16}{4}=4$
$\begin{aligned} & \mathrm{C}_{\mathrm{v}_1}=\text { Specific heat of oxygen at constant volume, } \\ & \mathrm{C}_{\mathrm{v}_1}=\frac{5}{2} \mathrm{R} \\ & \mathrm{C}_{\mathrm{v}_2}=\text { specific heat of Helium at constant volume, } \\ & \mathrm{C}_{\mathrm{v}_2}=\frac{3}{2} \mathrm{R}\end{aligned}$
Specific heat of mixture at constant volume,
$\begin{aligned} & \mathrm{C}_{\mathrm{v}}=\frac{\mathrm{n}_{\mathrm{o}} \mathrm{C}_{\mathrm{v}_1}+\mathrm{n}_{\mathrm{h}} \mathrm{C}_{\mathrm{v}_2}}{\mathrm{n}_{\mathrm{o}}+\mathrm{n}_{\mathrm{h}}}=\frac{\frac{1}{2} \times \frac{5}{2} \mathrm{R}+4 \times \frac{3}{2} \mathrm{R}}{\frac{1}{2}+4} \\ & \mathrm{C}_{\mathrm{v}}=\frac{29 \mathrm{R}}{18}\end{aligned}$
Specific heat of mixture at constent pressure,
$\mathrm{C}_{\mathrm{P}}=\mathrm{C}_{\mathrm{V}}+\mathrm{R}=\frac{29}{18} \mathrm{R}+\mathrm{R}=\frac{47 \mathrm{R}}{18}$
Ratio of specific heats is given by;
$\begin{aligned} & \frac{C_P}{C_V}=\frac{47}{18} R \times \frac{18}{29 R} \\ & =1.62\end{aligned}$