JEE Physics 2025 Question with Solution

Given: The process $ABC$ is a cyclic process on a $P-V$ diagram.

Find: The magnitude of heat exchanged by the system in SI units.

For a cyclic process,

$$\Delta U = 0$$

so by the first law of thermodynamics,

$$Q = W$$

Thus, the magnitude of heat exchanged is equal to the area enclosed by the cycle on the $P-V$ diagram.

From the figure, the path is a circle with horizontal diameter from $V = 200 \, \text{cc}$ to $V = 400 \, \text{cc}$ and vertical diameter from $P = 200 \, \text{kPa}$ to $P = 400 \, \text{kPa}$.

Hence the semi-axes are:

$$r_V = 100 \, \text{cc} = 100 \times 10^{-6} \, \text{m}^3$$ $$r_P = 100 \, \text{kPa} = 100 \times 10^3 \, \text{Pa}$$

Area enclosed by the ellipse-like plot in $P-V$ units is

$$W = \pi r_P r_V$$

Substituting,

$$W = \pi \left(100 \times 10^3\right)\left(100 \times 10^{-6}\right)$$ $$W = 10\pi \; \text{J}$$

Therefore, the magnitude of heat exchanged is $10\pi \, \text{J}$.

The correct option is C.

Given: A closed cyclic process is shown on the $P-V$ diagram.

Find: Magnitude of heat exchanged.

The solution states that the correct option is C. For a cyclic process, internal energy returns to its initial value, so

$$\Delta U = 0$$

and therefore

$$Q = W$$

The required magnitude is the area enclosed by the closed curve.

A common misread is to treat the enclosed shape as a rectangle. However, the figure shows a circular curve, so the enclosed area is not rectangular. Its horizontal and vertical diameters are each $200$ units in the plotted scales, hence each semi-axis is $100$.

So,

$$\text{Area} = \pi ab = \pi \times 100 \, \text{kPa} \times 100 \, \text{cc}$$

Convert units:

$$100 \, \text{kPa} = 100 \times 10^3 \, \text{Pa}$$ $$100 \, \text{cc} = 100 \times 10^{-6} \, \text{m}^3$$

Therefore,

$$W = \pi \times 100 \times 10^3 \times 100 \times 10^{-6}$$ $$W = 10\pi \; \text{J}$$

Hence, the magnitude of heat exchanged is $10\pi \, \text{J}$.

Therefore, the correct option is C.

Note: The solution text contains inconsistent wording such as calling the figure a rectangle, but its final conclusion and option label support $10\pi$.