Heat and Thermodynamics 5 Question 25
26. Two cylinders $A$ and $B$ fitted with pistons contain equal amounts of an ideal diatomic gas at $300 K$. The piston of $A$ is free to move, while that of $B$ is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $30 K$, then the rise in temperature of the gas in $B$ is
(1998, 2M)
(a) $30 K$
(b) $18 K$
(c) $50 K$
(d) $42 K$
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Solution:
- $A$ is free to move, therefore, heat will be supplied at constant pressure
$\therefore \quad d Q _A=n C _p d T _A$
$B$ is held fixed, therefore, heat will be supplied at constant volume.
$\therefore \quad d Q _B=n C _V d T _B$
But $\quad d Q _A=d Q _B$
(given)
$\therefore \quad n C _p d T _A=n C _V d T _B$
$\therefore \quad d T _B=\left(\frac{C _p}{C _V}\right) d T _A$
$=\gamma\left(d T _A\right) \quad[\gamma=1.4($ diatomic $)]$
$\left(d T _A=30 K\right)$
$=(1.4)(30 K)$
$\therefore \quad d T _B=42 K$
