Question

$1$ mole of a rigid diatomic gas performs a work of $Q/5$ when heat Q is supplied to it. The molar heat capacity of the gas during this transformation is $\frac{\mathrm{x}}{8}R$. The value of x is _________.

Answer 1

Step 1: Given data

Mole of diatomic gas$=1$

Work$=\frac{\mathrm{Q}}{5}$

Molar heat capacity$=\frac{\mathrm{x}}{8}\mathrm{R}$

Step 2: Calculating the value of “x”

From the first law of thermodynamics we have

$Q=∆U+W\dots ..\left(1\right)$

Where,

$\mathrm{Q}=$Change in heat

$\mathrm{U}=$Change in internal energy

$\mathrm{W}=$work done

Substituting the value of "W" we get,

$\mathrm{Q}=∆\mathrm{U}+\frac{\mathrm{Q}}{5}$

So, $∆\mathrm{U}=\frac{4\mathrm{Q}}{5}$

Also,

$\mathrm{Q}=\mathrm{nC}∆\mathrm{T}$ and $∆\mathrm{U}={\mathrm{nC}}_{\mathrm{v}}∆\mathrm{T}$

Where ${\mathrm{C}}_{\mathrm{V}}$ is the change in specific heat,

Substituting the values in the above equation we get,

${\mathrm{nC}}_{\mathrm{v}}∆\mathrm{T}=\frac{4}{5}\mathrm{nC}∆\mathrm{T}$

$\frac{5}{4}{C}_v=C$

Also, molar heat capacity at constant volume is given as ${\mathrm{C}}_{\mathrm{v}}=\frac{\mathrm{f}}{2}\mathrm{R}$ where f is degree of freedom

For the monoatomic gas degree of freedom$=5$

Substituting value we get, $\mathrm{C}=\frac{5}{4}\times \frac{5}{2}\times \mathrm{R}$, $\mathrm{C}=\frac{25}{8}\mathrm{R}$

Comparing $\mathrm{C}=\frac{25}{8}\mathrm{R}$ with $\mathrm{C}=\frac{\mathrm{x}}{8}\mathrm{R}$, $\mathrm{x}=25$

Therefore, the value of x is $25$