An ideal monatomic gas at temperature $27°C$ and pressure ${10}^{6}N/m^2$ occupies $10L$ volume. 10,000 cal of heat is added to the system without changing the volume. Calculate the change in temperature of the gas. Given : $R=8.31$ $J/mol-K$ and $J=4.18J/cal$

 For $n$ moles of gas, we have $pV=nRT$

Here, $p={10}^{6}N/m^2, V=10 L={10}^{-2}{m}^a$ and $T=27°C=300K$

$\therefore$ $n=\frac{pV}{RT}=\frac{{10}^6\times {10}^{-2}}{8.31\times 300}=4.0$

For monatomic gas, $C_V=\frac{3}{2}R$

Thus,

$C_V=\frac{3}{2}\times 8.31J/mol-K$

$=\frac{3}{2}\times \frac{8.31}{4.18}=3cal/mol-K$

Let $\Delta T$ be the rise in temperature when $n$ moles of the gas is given $Q$ cal of heat at constant volume. Then,

$Q=n{C}_V\Delta T\text{ or }\Delta T=\frac{Q}{n{C}_V}$

$=\frac{10000cal}{4.0mole\times 3cal/mol-K}$

$=833K$