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If heat is added at constant volume, $6300\,\,J$ of heat are required to raise the temperature of an ideal gas by $150\,\,K$. If instead, heat is added at constant pressure, $8800$ joules are required for the same temperature change. When the temperature of the gas changes by $300\,\,K$, the internal energy of the gas changes by ..... $J$
$5000$
$12600$
$17600$
$22600$
Solution
Heat added at constant volume is equal to the change in internal energy of the system. Change in internal energy $\quad \Delta U=n C_{v} \Delta T$
For $\Delta T=150 K, \quad \Delta U=Q_{v}=6300 \mathrm{J}$
$\therefore \quad 6300=n C_{v}(150)$ $\ldots(1)$
Let the change in internal energy be $\Delta U$ for $\Delta T=300 K$
$\therefore \Delta U=n C_{v}(300)$ $…(2)$
Dividing $( 2 )$ and $( 1 )$ we get $\frac{\Delta U}{6300}=\frac{300}{150}$
$\Longrightarrow \Delta U=12600 \mathrm{J}$
