First slide

Heat ;work and internal energy

Question

1 mole of a monoatomic gas at temperature $T_{0}$ expand slowly according to the law $P^{2}$ is proportional to T . Its final temperature is $2 T_{0},$ then heat supplied to the gas is

Moderate

A

$\frac{R}{T_{0} / 2}$
B

$R T_{0}$
C

$\frac{3}{2} R T_{0}$
D

$2 R T_{0}$

Solution

Given ,Number of moles=1 mole
Temperature $= T_{0}$
From Question
$P^{2} \propto T \Rightarrow P = c \sqrt{T}$ , c is constant
$∴ \frac{d P}{P} = \frac{1}{2} \frac{d T}{T} .... (1)$
$\begin{array}{l} P V = R T \dots . (2) \\ \Rightarrow P d V + V d P = R d T \\ \Rightarrow P d V = R d T - V d P \\ = R d T - R T \cdot \frac{d P}{P} \\ = R d T - \frac{R T}{2} \frac{d T}{T} (by using (1)) \\ = \frac{R d T}{2} \\ ∴ \int_{T_{0}}^{2 T_{0}} P d V = \int_{T_{0}}^{2 T_{0}} \frac{R d T}{2} = \frac{R T_{0}}{2} \end{array}$
From first law of thermodynamics
$Q = Δ U + W = n C_{V} Δ T + \int P d V = \frac{f n R T_{0}}{2} + \frac{R T_{0}}{2} = \frac{3 (1) R}{2} T_{0} + \frac{R T_{0}}{2} = 2 R T_{0}$

Therefore, the correct answer is (D).

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