Question

A sample of an ideal gas has diatomic molecules at a temperature at which effective degree of freedom is $5$. Under the action of radiation, each molecule splits into two atoms. If the ratio of the number of dissociated molecules to the total number of molecules is $\alpha$, then find the ratio of molar specific heat capacities $(\gamma =\frac{C_P}{C_V})$.

• A. $\frac{2}{\alpha +5}$
• B. $\frac{5}{\alpha +2}$
• C. $1+\frac{2}{\alpha +5}$
• D. $1+\frac{5}{\alpha +2}$

Answer 1

The correct option is C $1+\frac{2}{\alpha +5}$

Let us consider one mole of a sample of diatomic gas. Due to radiation, $\alpha$ moles of diatomic gas split into $2\alpha$ moles of monoatomic gas and $1-\alpha$ moles remain as diatomic gas.
Specific heat capacity at constant volume for the mixture will be
$C_V=\frac{2\alpha .\frac{3}{2}R+(1-\alpha )\frac{5}{2}R}{(1+\alpha )}=\frac{(\alpha +5)}{2(1+\alpha )}R=\frac{R\alpha +5R}{2+2\alpha }$
$\because C_P=C_V+R$, we get
$C_P=\frac{R\alpha +5R}{2+2\alpha }+R=\frac{(3\alpha +7)R}{(2+2\alpha )}$
$\therefore \gamma =\frac{C_P}{C_V}=\frac{\left[\frac{(3\alpha +7)R}{(2+2\alpha )}\right]}{\left[\frac{(\alpha +5)R}{(2+2\alpha )}\right]}=\frac{(3\alpha +7)}{(\alpha +5)}=[1+\frac{2(\alpha +1)}{(\alpha +5)}]$
Thus, option (c) is the correct answer.