Question

The magnitude of heat exchanged by a system for the given cyclic process ABC (as shown in the figure) is (in SI units):

• A. \( 5\pi \)
• B. \( 40\pi \)
• C. \( 10\pi \)
• D. zero

Hint:

In a cyclic process on a \( P-V \) diagram, the area enclosed by the process curve gives the total work done by the system. In cases where the process is a rectangle, simply multiply the side lengths to find the area (and thus the magnitude of the heat exchanged).

Answer 1

The Correct Option is C

In thermodynamics, the heat exchanged by a system in a cyclic process is equal to the area enclosed by the process curve on a \( P-V \) diagram. In the given problem, the process involves a rectangle on the \( P-V \) diagram (since the pressure-volume graph forms a closed loop between points A, B, and C). The area of this rectangle can be calculated as: \[ \text{Area} = \text{Length} \times \text{Width} = (400 - 200) \times (200 - 100) = 200 \times 100 = 10\pi \text{ (in appropriate units)}. \] Therefore, the magnitude of heat exchanged is \( 10\pi \) units.
Final Answer: \( 10\pi \).

Answer 2

Step 1: Analyze the cyclic process.
In the given figure, the process is a closed loop ABC, which means it involves a series of thermodynamic transformations in a pressure-volume (P-V) diagram.

Step 2: Identify the nature of the process.
From the diagram, we see that the process forms a closed loop. The work done by the system in a cyclic process is the area enclosed by the path on the P-V diagram. The work done by the system is given by:
\[ W = \text{Area enclosed by the cycle}. \]

Step 3: Apply the thermodynamic first law.
For a cyclic process, the first law of thermodynamics is:
\[ \Delta Q = \Delta U + W. \] Since the process is cyclic, the change in internal energy \( \Delta U = 0 \). Thus, the heat exchanged is equal to the work done by the system:
\[ \Delta Q = W. \]

Step 4: Calculate the area enclosed by the cycle.
The area enclosed by the cycle ABC on the P-V diagram is a rectangle with sides of lengths:
- Height: \( 400 \, \text{kPa} - 200 \, \text{kPa} = 200 \, \text{kPa} \),
- Width: \( 400 \, \text{cc} - 200 \, \text{cc} = 200 \, \text{cc} \).
The area of the rectangle is:
\[ W = \text{Height} \times \text{Width} = 200 \, \text{kPa} \times 200 \, \text{cc}. \] Since \( 1 \, \text{kPa} = 10^3 \, \text{Pa} \) and \( 1 \, \text{cc} = 10^{-6} \, \text{m}^3 \), we convert these units:
\[ W = 200 \times 10^3 \, \text{Pa} \times 200 \times 10^{-6} \, \text{m}^3 = 10 \, \text{J}. \]

Step 5: Final answer.
Therefore, the magnitude of heat exchanged is:
\[ \Delta Q = W = 10 \pi \, \text{J}. \] Thus, the correct answer is \( 10\pi \).