Question

An ideal gas undergoes a cyclic process ABCA wherein AB is an isothermal expansion, BC is an isochoric pressure drop, and CA is an adiabatic compression. The work done by the gas in the complete cycle is:

• A. Zero
• B. Positive
• C. Negative
• D. Cannot be determined

Hint:

Key Exam Tip:
For any closed cycle, $\Delta U = 0$, hence $W = Q$. The net work done is the area enclosed by the P-V diagram, which is usually non-zero.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In thermodynamics, a cyclic process is one where the system starts and ends at the same state.
The net work done during a cycle is the area enclosed by the path on a Pressure-Volume (\(P-V\)) diagram.
According to the sign convention:
- Clockwise cycles represent net positive work done by the gas (typical of heat engines).
- Counter-clockwise cycles represent net negative work done (work done on the gas, typical of refrigerators).
Step 2: Detailed Explanation:
Let's analyze each stage of the cycle ABCA:
1. Process AB (Isothermal Expansion): The gas expands at a constant temperature. Expansion means volume increases (\(V_B>V_A\)). On the \(P-V\) graph, this is represented by a curve sloping downwards. Work done during this part is positive as the gas pushes against its surroundings.
2. Process BC (Isochoric Pressure Drop): Isochoric means the volume is kept constant (\(V_B = V_C\)). A pressure drop at constant volume implies the gas is being cooled. Since there is no change in volume (\(dV = 0\)), the work done \(W = \int P \, dV\) is exactly zero.
3. Process CA (Adiabatic Compression): The gas is compressed back to its original state A without any heat exchange with the surroundings. Compression means volume decreases (\(V_A<V_C\)). Work is done on the gas, so this work is negative.
On a \(P-V\) diagram, an adiabatic curve is steeper than an isothermal curve.
- From A to B, we follow a less steep isothermal curve.
- At B, we drop vertically to C.
- From C to A, we return via a steeper adiabatic curve.
Because the adiabatic return path CA is steeper and starts from a lower pressure point C, it stays "underneath" the isothermal expansion path AB.
Visualizing this: The path moves from A (left) to B (right), then down to C, then back up and left to A. This traces a clockwise loop.
In a clockwise loop on a \(P-V\) diagram, the area under the expansion curve (positive work) is greater than the area under the compression curve (negative work).
Step 3: Final Answer:
Since the cycle is clockwise, the net area is positive, meaning the total work done by the gas in the complete cycle is positive.