Three moles of an ideal gas `(C_p=7/2R)` at pressure, `P_A` and temperature `T_A` is isothermally expanded to twice its initial volume. It is then compressed at constant pressure to its original volume. Finally gas is compressed at constant volume to its original pressure `P_A`.
(a) Sketch P-V and P-T diagrams for the complete process.
(b) Calculate the net work done by the gas, and net heat supplied to the gas during the complete process.

Options

A.
B.
C.
D.

Explanation

To solve the problem step by step, we will analyze the processes involved, sketch the required diagrams, and calculate the net work done and net heat supplied during the complete process.

### Step 1: Understanding the Processes
1. **Isothermal Expansion (A to B)**: The gas expands isothermally from volume \( V_A \) to \( V_B = 2V_A \) at constant temperature \( T_A \).
2. **Isobaric Compression (B to C)**: The gas is compressed at constant pressure from volume \( V_B \) back to \( V_C = V_A \).
3. **Isochoric Compression (C to A)**: The gas is compressed at constant volume back to its original pressure \( P_A \).

### Step 2: Sketching the P-V Diagram
- **Point A**: \( (P_A, V_A) \)
- **Point B**: \( (P_B, V_B) \) where \( P_B = \frac{P_A}{2} \) and \( V_B = 2V_A \)
- **Point C**: \( (P_C, V_C) \) where \( P_C = \frac{P_A}{2} \) and \( V_C = V_A \)

The P-V diagram will show:
- A hyperbolic curve from A to B (isothermal).
- A straight line from B to C (isobaric).
- A vertical line from C to A (isochoric).

### Step 3: Sketching the P-T Diagram
- **Point A**: \( (P_A, T_A) \)
- **Point B**: \( (P_B, T_A) \) where \( P_B = \frac{P_A}{2} \)
- **Point C**: \( (P_C, T_C) \) where \( T_C = \frac{T_A}{2} \)

The P-T diagram will show:
- A horizontal line from A to B (isothermal).
- A straight line from B to C (isobaric).
- A straight line from C to A (isochoric).

### Step 4: Calculating the Work Done
1. **Work Done in Isothermal Process (A to B)**:
\[ W_{AB} = nRT_A \ln\left(\frac{V_B}{V_A}\right) = 3R T_A \ln(2) \]

2. **Work Done in Isobaric Process (B to C)**:
\[ W_{BC} = P_B (V_C - V_B) = \frac{P_A}{2} (V_A - 2V_A) = -\frac{P_A V_A}{2} = -\frac{3R T_A}{2} \]

3. **Work Done in Isochoric Process (C to A)**:
\[ W_{CA} = 0 \]

4. **Net Work Done**:
\[ W_{net} = W_{AB} + W_{BC} + W_{CA} = 3R T_A \ln(2) - \frac{3R T_A}{2} + 0 \]

### Step 5: Calculating the Heat Supplied
1. **Heat Supplied in Isothermal Process (A to B)**:
\[ Q_{AB} = W_{AB} = 3R T_A \ln(2) \]

2. **Heat Supplied in Isobaric Process (B to C)**:
- Change in Internal Energy:
\[ \Delta U_{BC} = n C_V \Delta T = 3 \cdot \frac{5R}{2} \left(\frac{T_A}{2} - T_A\right) = -\frac{15R T_A}{4} \]
- Heat Supplied:
\[ Q_{BC} = W_{BC} + \Delta U_{BC} = -\frac{3R T_A}{2} - \frac{15R T_A}{4} = -\frac{21R T_A}{4} \]

3. **Heat Supplied in Isochoric Process (C to A)**:
\[ Q_{CA} = \Delta U_{CA} = n C_V \Delta T = 3 \cdot \frac{5R}{2} \left(T_A - \frac{T_A}{2}\right) = \frac{15R T_A}{4} \]

4. **Net Heat Supplied**:
\[ Q_{net} = Q_{AB} + Q_{BC} + Q_{CA} = 3R T_A \ln(2) - \frac{21R T_A}{4} + \frac{15R T_A}{4} \]
\[ Q_{net} = 3R T_A \ln(2) - \frac{6R T_A}{4} = 3R T_A \ln(2) - \frac{3R T_A}{2} \]

### Final Answers
- **Net Work Done**:
\[ W_{net} = 3R T_A \ln(2) - \frac{3R T_A}{2} \]
- **Net Heat Supplied**:
\[ Q_{net} = 3R T_A \ln(2) - \frac{3R T_A}{2} \]