An ideal gas undergoes a reversible isothermal expansion from state I to state II followed by a reversible adiabatic expansion from state II to state III. The correct plot(s) representing the changes from state I to state III is(are)

Options

A.
B.
C.
D.

Answer

To solve the problem of an ideal gas undergoing a reversible isothermal expansion followed by a reversible adiabatic expansion, we need to analyze the processes step by step and determine the correct plots that represent the changes from state I to state III.

### Step 1: Understand the Processes
1. **Reversible Isothermal Expansion (State I to State II)**:
- In this process, the temperature remains constant (T = constant).
- As the gas expands, its volume increases (V increases).
- Since the process is isothermal for an ideal gas, the pressure decreases (P decreases) according to the ideal gas law (PV = nRT).

2. **Reversible Adiabatic Expansion (State II to State III)**:
- In this process, there is no heat exchange with the surroundings (Q = 0).
- The gas does work on the surroundings as it expands, which results in a decrease in internal energy (U decreases).
- Consequently, the temperature of the gas decreases (T decreases) during this expansion.

### Step 2: Analyze the Graphs
1. **Graph A (P vs. V)**:
- For the isothermal process (I to II), the curve is hyperbolic, showing a decrease in pressure with an increase in volume.
- For the adiabatic process (II to III), the curve is steeper than the isothermal curve, indicating a more significant decrease in pressure with volume increase.
- This graph is correct.

2. **Graph B (P vs. T)**:
- From I to II, since the temperature is constant, the line is horizontal.
- From II to III, as the gas expands adiabatically, the temperature decreases, resulting in a downward slope.
- This graph is also correct.

3. **Graph C (H vs. S)**:
- For the isothermal process, enthalpy (H) remains constant, but the entropy (S) increases.
- For the adiabatic process, the temperature decreases, which should lead to a decrease in enthalpy as well.
- If the graph shows an increase in enthalpy while the temperature is decreasing, this is incorrect.

4. **Graph D (T vs. S)**:
- During the isothermal expansion, the temperature remains constant, leading to a horizontal line.
- During the adiabatic expansion, the temperature decreases while entropy increases.
- This graph is correct.

### Conclusion
The correct plots representing the changes from state I to state III are:
- **Graph A (P vs. V)**
- **Graph B (P vs. T)**
- **Graph D (T vs. S)**

**Incorrect Plot**:
- **Graph C (H vs. S)** is incorrect due to the inconsistency in the expected behavior of enthalpy during the adiabatic process.

### Final Answer:
The correct plots are A, B, and D. The incorrect plot is C.