First slide

Kinetic theory of ideal gases

Question

At a certain absolute temperature T, an ideal diatomic gas has total energy U. Energy E is supplied to the gas and its temperature increases by t on Kelvin scale. If an additional energy E is supplied to the gas, find the final temperature of the gas on Kelvin scale.

Moderate

A

$T + \frac{3 t}{2}$
B

$T + \frac{5 t}{4}$
C

$T + \frac{7 t}{4}$
D

$T + 2 t$

Solution

$\begin{array}{l} U = \frac{5 nRT}{2} . . . . . (1) \\ U + E = \frac{5 nR (T + t)}{2} = \frac{5 nRT}{2} + \frac{5 nRt}{2} . . . . . . (2) \\ \Rightarrow E = \frac{5 nRt}{2} . . . . . . (3) \\ U + 2 E = \frac{5 nR (T + t + t^{'})}{2} = \frac{5 nR (T + t)}{2} + \frac{5 n {Rt}^{'}}{2} = U + E + \frac{5 {nRt}^{'}}{2} . . . . . . (4) \\ \Rightarrow E = \frac{5 {nRt}^{'}}{2} . . . . (5) \end{array}$
From (3) and (5),
$t' = t$
Hence, final temperature of the gas is $T + t + t' = T + 2 t$ .

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