If the process carried out on one mole of monatomic ideal gas is shown in the figure $P V$ -diagram with $P_{0} V_{0} = \frac{1}{3} R T_{0},$ the correct match is, [2019]

Question

Smart Achievers · Accepted Answer

(1)

$I. W_{1 \to 2 \to 3} = W_{1 \to 2} + W_{2 \to 3}$ $[W_{2 \to 3} = 0, Δ V = 0]$

$= P_{0} \times V_{0} + 0$

$= P_{0} V_{0} = \frac{R T_{0}}{3}$ $[∵ P_{0} V_{0} = \frac{1}{3} R T_{0} given]$

$∴$ $I \to Q$

$II. Δ U_{1 \to 2 \to 3} = Δ U_{1 \to 2} + Δ U_{2 \to 3}$

$= n C_{v} Δ T_{1 \to 2} + n C_{v} Δ T_{2 \to 3}$

$= 1 \times \frac{3 R}{2} {(T_{f} - T_{i})}_{1 \to 2} + 1 \times \frac{3 R}{2} {(T_{f} - T_{i})}_{2 \to 3}$

$= \frac{3}{2} [2 P_{0} V_{0} - P_{0} V_{0}] + \frac{3}{2} [\frac{3 P_{0}}{2} \times 2 V_{0} - P_{0} \times 2 V_{0}]$

$= 3 P_{0} V_{0} = 3 \times \frac{1}{3} R T_{0} = R T_{0}$

$II \to R$

$III. Q_{1 \to 2 \to 3} = Q_{1 \to 2} + Q_{2 \to 3} = n C_{p} Δ T_{1 \to 2} + n C_{v} Δ T_{2 \to 3}$

$= \frac{5}{2} P_{0} V_{0} + 1 \times \frac{3}{2} [\frac{3 P_{0}}{2} \times 2 V_{0} - P_{0} (2 V_{0})]$

$= \frac{8}{2} P_{0} V_{0} = \frac{8}{2} \times \frac{R T_{0}}{3} = \frac{4}{3} R T_{0}$

$III \to S$

$IV. Q_{1 \to 2} = n C_{p} Δ T_{1 \to 2} = n C_{p} (T_{f} - T_{i})$

$= 1 \times \frac{5 R}{2} [\frac{P_{0} (2 V_{0})}{R} - \frac{P_{0} V_{0}}{R}]$ $[∵ P V = n R T]$

$= \frac{5}{2} P_{0} V_{0} = \frac{5}{2} (\frac{R T_{0}}{3}) = \frac{5 R T_{0}}{6}$

$∴$ $IV \to U$

Solution

Q.
If the process carried out on one mole of monatomic ideal gas is shown in the figure $P V$ -diagram with $P_{0} V_{0} = \frac{1}{3} R T_{0},$ the correct match is, [2019]

1 I → Q, II → R, III → S, IV → U

2 I → S, II → R, III → Q, IV → T

3 I → Q, II → R, III → P, IV → U

4 I → Q, II → S, III → R, IV → U

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