Question

Consider a mixture of $\mathrm{n}$ moles of helium gas and $2\mathrm{n}$ moles of oxygen gas (molecules taken to be rigid) as an ideal gas. Its ($\raisebox{1ex}{${\mathrm{C}}_{\mathrm{p}}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{C}}_{\mathrm{v}}$}\right.$) value will be

• A. $\frac{67}{45}$
• B. $\frac{40}{27}$
• C. $\frac{19}{13}$
• D. $\frac{23}{15}$

Answer 1

The correct option is C

$\frac{19}{13}$

Explanation for the correct option:

C) ${\mathrm{C}}_{\mathrm{v}}=\mathrm{Specific}\mathrm{heat}\mathrm{at}\mathrm{constant}\mathrm{volume}$

${\mathrm{C}}_{\mathrm{p}}=\mathrm{Specific}\mathrm{heat}\mathrm{at}\mathrm{constant}\mathrm{pressure}$

Now

${\mathrm{C}}_{\mathrm{v}}\mathrm{of}H\mathrm{elium}=\frac{3}{2}\mathrm{R}$

${\mathrm{C}}_{\mathrm{v}}\mathrm{of}O\mathrm{xygen}=\frac{5}{2}\mathrm{R}$

${\mathrm{C}}_{\mathrm{p}}\mathrm{of}H\mathrm{elium}=\frac{5}{2}\mathrm{R}$

${\mathrm{C}}_{\mathrm{p}}\mathrm{of}O\mathrm{xygen}=\frac{7}{2}\mathrm{R}$

${\mathrm{C}}_{\mathrm{v}\mathrm{mix}}=\frac{{\mathrm{n}}_1{\mathrm{C}}_{\mathrm{v}1}+{\mathrm{n}}_2{\mathrm{C}}_{\mathrm{v}2}}{{\mathrm{n}}_1+{\mathrm{n}}_2}$

Where ${\mathrm{n}}_1,{\mathrm{n}}_2$ are number of moles

${\mathrm{C}}_{\mathrm{v}\mathrm{mix}}=\frac{\mathrm{n}\times {\displaystyle \frac{3}{2}}\mathrm{R}+2\mathrm{n}\times {\displaystyle \frac{5}{2}}\mathrm{R}}{\mathrm{n}+2\mathrm{n}}$

$$\begin{gathered} =\frac{13\mathrm{nR}}{6\mathrm{n}} \\ =\frac{\mathbf{13}\mathbf{R}}{\mathbf{6}} \end{gathered}$$

${\mathrm{C}}_{\mathrm{p}\mathrm{mix}}=\frac{{\mathrm{n}}_1{\mathrm{C}}_{\mathrm{p}1}+{\mathrm{n}}_2{\mathrm{C}}_{\mathrm{p}2}}{{\mathrm{n}}_1+{\mathrm{n}}_2}$

$=\frac{\mathrm{n}\times {\displaystyle \frac{5}{2}}\mathrm{R}+2\mathrm{n}\times {\displaystyle \frac{7}{2}}\mathrm{R}}{\mathrm{n}+2\mathrm{n}}$

$$\begin{gathered} =\frac{19\mathrm{nR}}{6\mathrm{n}} \\ =\mathbf{}\frac{\mathbf{19}\mathbf{R}}{\mathbf{6}} \end{gathered}$$

Now,

$\frac{C_p}{C_v}=\frac{{\displaystyle \frac{19}{6}}R}{{\displaystyle \frac{13}{6}}R}$

$=\mathbf{}\frac{\mathbf{19}}{\mathbf{13}}$

Hence, option C is correct.