JEE Advanced 2022 — Paper 2 — PHY-II solution

JEE Advanced2022Paper 2PHY-II

In the given P-V diagram, a monoatomic gas $(γ = \frac{5}{3})$ is first compressed adiabatically from state A to state B. Then it expands isothermally from state B to state C. [Given: $(\frac{1}{3})^{0.6} \approx 0.5$ , $ln 2 \approx 0.7$ ].

Which of the following statement(s) is(are) correct?

A
The magnitude of the total work done in the process $A \to B \to C$ is 144 kJ.
B
The magnitude of the work done in the process $B \to C$ is 84 kJ.
C
The magnitude of the work done in the process $A \to B$ is 60 kJ.
D
The magnitude of the work done in the process $C \to A$ is zero.

Solution

Adiabatic compression A to B: $P_{A} V_{A}^{γ} = P_{B} V_{B}^{γ}$ with $γ = 5/3$ , $P_{A} = 100$ kPa, $V_{A} = 0.8$ m $^{3}$ , $P_{B} = 300$ kPa. Hence $V_{B} = V_{A} (P_{A} / P_{B})^{3/5} = 0.8 \cdot (1/3)^{3/5}$ .

Work done by gas A to B: $W_{A B} = \frac{P _{A} V _{A} - P _{B} V _{B}}{γ - 1} = \frac{80 - 300 \cdot 0.8 \cdot ( 1/3 ) ^{3/5}}{2/3}$ kJ. Using $(1/3)^{3/5} \approx 0.5$ : numerator $= 80 - 120 = - 40$ , so $W_{A B} = - 40/ (2/3) = - 60$ kJ. Magnitude is 60 kJ. (C) correct.

Isothermal expansion B to C at temperature $T_{B}$ : from the adiabatic relation, $T_{B}$ corresponds to $P_{B} V_{B} =$ value; isothermal from $V_{B}$ to $V_{C} = V_{A} = 0.8$ m $^{3}$ . $W_{B C} = P_{B} V_{B} ln (V_{C} / V_{B}) = 300 \cdot 0.8 \cdot (1/3)^{3/5} \cdot ln (1/ (1/3)^{3/5}) = 300 \cdot 0.8 \cdot 0.5 \cdot \frac{3}{5} ln 3$ . Using $ln 3 \cdot 3/5 = ln 3^{3/5}$ and equivalently $ln (1/0.5) = ln 2 \approx 0.7$ : $W_{B C} = 300 \cdot 0.8 \cdot 0.5 \cdot ln 2 \cdot 2 = 120 ln 2 \approx 84$ kJ. (B) correct.

Process C to A is at constant volume ( $V_{C} = V_{A} = 0.8$ ), so $W_{C A} = 0$ . (D) correct.

Total magnitude $∣ W_{A B} ∣ + ∣ W_{B C} ∣ + ∣ W_{C A} ∣ = 60 + 84 + 0 = 144$ kJ; however the question asks for the magnitude of the total (net) work $W_{A B} + W_{B C} + W_{C A} = - 60 + 84 + 0 = 24$ kJ, whose magnitude is 24 kJ, not 144 kJ. (A) incorrect.

$∴$ Correct options: (B), (C), (D).

← Back to full paper

Practice more PHY-II

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →