Two moles of an ideal gas undergo the following process `:`
`(a)` a reversible isobaric expansion from `( P` atm, `V L)` to `( P` atm , `2V L)`.
`(b)` a reversible isochoric change of state from `( P` atm , `2V,L)` to `(P//2` atm, `2V L)`
`(c )` a reversible isothermal compression from `(P//2` atm, `2V L)` to `(P ` atm, `V L)`.
Sketch with labels each of the processes on the same `P-V` diagram.

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the three processes undergone by the ideal gas and sketch them on a P-V diagram. Let's break down each step:

### Step 1: Identify the Processes
1. **Isobaric Expansion**: The gas expands at constant pressure from (P atm, V L) to (P atm, 2V L).
2. **Isochoric Change**: The gas undergoes a change at constant volume from (P atm, 2V L) to (P/2 atm, 2V L).
3. **Isothermal Compression**: The gas is compressed at constant temperature from (P/2 atm, 2V L) to (P atm, V L).

### Step 2: Sketch the P-V Diagram
1. **Draw the Axes**:
- Draw a vertical line for pressure (P) and a horizontal line for volume (V).

2. **Process (a) - Isobaric Expansion**:
- Start at point A (P atm, V L).
- Move horizontally to the right to point B (P atm, 2V L) since pressure is constant (isobaric).

3. **Process (b) - Isochoric Change**:
- From point B (P atm, 2V L), move vertically downwards to point C (P/2 atm, 2V L) since volume is constant (isochoric).

4. **Process (c) - Isothermal Compression**:
- From point C (P/2 atm, 2V L), draw a curve back to point A (P atm, V L). This curve should be hyperbolic since it represents an isothermal process where pressure and volume are inversely related.

### Step 3: Label the Points
- Label point A as (P atm, V L).
- Label point B as (P atm, 2V L).
- Label point C as (P/2 atm, 2V L).

### Final Sketch
The final P-V diagram will show:
- A horizontal line from A to B for the isobaric expansion.
- A vertical line from B to C for the isochoric change.
- A hyperbolic curve from C back to A for the isothermal compression.