Which of the following P-V diagram represent the graph of isometric process?

Options

A.
B.
C.
D.

Explanation

To determine which P-V diagram represents an isometric process, we need to understand what an isometric process is. An isometric process, also known as an isochoric process, is one in which the volume remains constant.

### Step-by-Step Solution:

1. **Understand the Isometric Process**:
- An isometric process is characterized by constant volume. This means that during the process, the volume of the gas does not change.

2. **Ideal Gas Law**:
- The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is temperature.
- For an isometric process, since the volume \( V \) is constant, we can rearrange the ideal gas law to express pressure in terms of temperature:
\[ P = \frac{nR}{V} T \]
- Here, \( \frac{nR}{V} \) is a constant (let's call it \( k \)), so we can write:
\[ P = kT \]
- This indicates that pressure \( P \) is directly proportional to temperature \( T \).

3. **Identify the Graph**:
- In a P-V diagram, if the volume is constant, the graph will be a vertical line. This is because, for a given volume, pressure can change with temperature, but the volume does not change.
- Therefore, we need to look for a graph that shows a vertical line, indicating that as temperature increases, pressure increases while the volume remains constant.

4. **Analyze the Options**:
- **Option A**: If \( T \) increases and \( P \) decreases, this is not an isometric process.
- **Option B**: Similar to option A, if \( T \) increases and \( P \) decreases, this is also not an isometric process.
- **Option C**: If \( T \) increases and \( P \) increases, this indicates a constant volume process, which is characteristic of an isometric process.
- **Option D**: If \( P \) is constant while \( T \) increases, this represents an isobaric process, not an isometric process.

5. **Conclusion**:
- Based on the analysis, **Option C** is the correct representation of an isometric process.

### Final Answer:
The correct P-V diagram that represents an isometric process is **Option C**.