Heat and Thermodynamics 5 Question 3
3. When heat $Q$ is supplied to a diatomic gas of rigid molecules, at constant volume, its temperature increases by $\Delta T$. The heat required to produce the same change in temperature, at a constant pressure is
(2019 Main, 10 April II)
(a) $\frac{2}{3} Q$
(b) $\frac{5}{3} Q$
(c) $\frac{3}{2} Q$
(d) $\frac{7}{5} Q$
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Solution:
- Given, heat supplied at constant volume is $Q$.
$$ \therefore \quad Q=n C _V \Delta T $$
For same change in temperature, if heat supplied at constant pressure is $Q$, then
So, we have
$$ Q^{\prime}=n C _p \Delta T $$
$$ \frac{Q^{\prime}}{Q}=\frac{n C _p \Delta T}{n C _V \Delta T} \Rightarrow \frac{Q^{\prime}}{Q}=\frac{C _p}{C _V} $$
$$ \Rightarrow \quad Q^{\prime}=Q \gamma \quad\left[\because \gamma=\frac{C _p}{C _V}\right] $$
As given gas is diatomic, so $\gamma=\frac{7}{5}$
$$ \therefore \quad Q^{\prime}=\frac{7}{5} Q $$
