Question

In a monatomic ideal gas the pressure and volume relation follows the equation $P^6{V}^5=$ constant. If the gas contains $n$ number of moles, find out amount of heat required to increase the temperature of gas by $5K.$

• A. $15nR$
• B. $7.5nR$
• C. $75nR$
• D. $37.5nR$

Answer 1

The correct option is D $37.5nR$

We have, $P{V}^{\frac{5}{6}}=k$,
which is a polytropic process of the form $P{V}^x=k$.
The work done for a polytropic process is $W=\frac{nR\Delta T}{1-x}$
Since, here $x=\frac{5}{6},$
$W=\frac{nR\left(5\right)}{1-\frac{5}{6}}(\because \Delta T=+5)$
$W=30nR$

Also, $\Delta U=n{C}_v\Delta T$
$C_v=\frac{3}{2}R$ , since the gas is monatomic

$=n\left(\frac{3}{2}R\right)\left(5\right)$
$=\frac{15nR}{2}$

From 1st law of thermodynamics:
$\Delta Q=\Delta U+W$
$\Delta Q=30nR+\frac{15nR}{2}$
$\Delta Q=37.5nR$