Question

In a certain gas, the ratio of the speed of sound and root mean square speed is $\sqrt{\frac{5}{9}}$. The molar heat capacity of the gas in a process given by $PT=constant$ is
(Take $R=2cal/moleK)$. Treat the gas as an ideal gas

• A. $\frac{R}{2}$
• B. $\frac{3R}{2}$
• C. $\frac{5R}{2}$
• D. $\frac{7R}{2}$

Answer 1

The correct option is D $\frac{7R}{2}$

Given process $PT=$ constant $\Rightarrow PT=K\to \left(1\right)$

Ideal gas equation $\to PV=nRT\Rightarrow P=\frac{nRT}{V}$

$\Rightarrow \frac{{nRT}^2}{V}=$Constant/K $\to \left(2\right)$

Differentiating w.r.t $V$ we get

$\Rightarrow 2nRT△T=K△V$ $[PT=K]$

$2nRT△T=PT△V$

$2nR△T=P△V\to \left(3\right)$

Now, for $r$ i.e, degree of freedom,

The ratio of speed of sound and root mean square speed $=\sqrt{\frac{r}{3}}$ and given $=\sqrt{\frac{5}{9}}$

So, $\sqrt{\frac{r}{3}}=\sqrt{\frac{5}{9}}\Rightarrow \frac{5}{3}$

First law of thermodynamic says.

$\Rightarrow dQ=dV+PdV$

$\Rightarrow C△T=C_V△T+2nR△T$

$\Rightarrow C=\frac{R}{r-1}+2R$ $[n-1,C_V=\frac{R}{r-1},r=\frac{5}{3}]$

$C=\frac{R}{\frac{5}{3}-1}+2R$

$C=\frac{3R}{2}+2R$

$C=\frac{7R}{2}.$

Hence, the answer is $\frac{7R}{2}.$