Show how internal energy U varies with T in isochoric, isobaric and adiabatic process?

Explanation

To show how internal energy \( U \) varies with temperature \( T \) in isochoric, isobaric, and adiabatic processes, we can follow these steps:

### Step 1: Understand the Processes
- **Isochoric Process**: This is a process where the volume \( V \) remains constant. Therefore, any heat added to the system increases the internal energy.
- **Isobaric Process**: This is a process where the pressure \( P \) remains constant. Heat added to the system can do work and also increase internal energy.
- **Adiabatic Process**: In this process, there is no heat exchange with the surroundings (\( \Delta Q = 0 \)). Any change in internal energy is due to work done on or by the system.

### Step 2: Use the Formula for Change in Internal Energy
The change in internal energy \( \Delta U \) is given by the formula:
\[ \Delta U = N \cdot C_v \cdot \Delta T \]
where:
- \( N \) is the number of moles,
- \( C_v \) is the molar specific heat at constant volume,
- \( \Delta T \) is the change in temperature.

This formula applies to all three processes.

### Step 3: Analyze Each Process
1. **Isochoric Process**:
- Since the volume is constant, all the heat added goes into increasing the internal energy.
- Thus, \( \Delta U = N \cdot C_v \cdot \Delta T \).
- Internal energy \( U \) increases linearly with temperature \( T \).

2. **Isobaric Process**:
- Here, heat added does work as well as increases internal energy.
- The relationship still holds: \( \Delta U = N \cdot C_v \cdot \Delta T \).
- Again, internal energy \( U \) increases linearly with temperature \( T \).

3. **Adiabatic Process**:
- In this case, since there is no heat exchange, the change in internal energy is equal to the work done on or by the system.
- The relationship \( \Delta U = N \cdot C_v \cdot \Delta T \) still applies.
- Internal energy \( U \) increases linearly with temperature \( T \).

### Conclusion
In all three processes (isochoric, isobaric, and adiabatic), the internal energy \( U \) varies linearly with temperature \( T \). This means that as the temperature increases, the internal energy also increases proportionally.