In a process, temperature and volume of one mole of an ideal monoatomic gas are varied according to the…
In a process, temperature and volume of one mole of an ideal monoatomic gas are varied according to the relation $\mathrm{VT}=\mathrm{K},$ where $\mathrm{K}$ is a constant. In this process the temperature of the gas is increased by $\Delta \mathrm{T}$. The amount of heat absorbed by gas is (R is gas constant):
$\frac{1}{2} \mathrm{R} \Delta \mathrm{T}$
$\frac{1}{2} \mathrm{KR} \Delta \mathrm{T}$
$\frac{3}{2} \mathrm{R} \Delta \mathrm{T}$
$\frac{2 \mathrm{~K}}{3} \Delta \mathrm{T}$
Solution
According to question $\mathrm{VT}=\mathrm{K}$
we also know that $P V=n R T$
$\Rightarrow \mathrm{T}=\left(\frac{\mathrm{PV}}{\mathrm{nR}}\right)$
$\Rightarrow \mathrm{V}\left(\frac{\mathrm{PV}}{\mathrm{nR}}\right)=\mathrm{k} \Rightarrow \mathrm{PV}^{2}=\mathrm{K}$
$\because \mathrm{C}=\frac{\mathrm{R}}{1-\mathrm{x}}+\mathrm{C}_{\mathrm{v}}$ (For polytropic process)
$\mathrm{C}=\frac{\mathrm{R}}{1-2}+\frac{3 \mathrm{R}}{2}=\frac{\mathrm{R}}{2}$
$\therefore \Delta \mathrm{Q}=\mathrm{nC} \Delta \mathrm{T}$
$=\frac{R}{2} \times \Delta \mathrm{T}$ [here, $\left.\mathrm{n}=1 \mathrm{~mole}\right]$