The pressure p and volume V of an ideal gas both increase in a process.

Options

A. such a process is not possible.
B. The work done by the system is positive.
C. The tempreture of the system must increase.
D. Heat supplied to the gas is equal to the change in internal energy.

Explanation

To analyze the process where both the pressure \( P \) and volume \( V \) of an ideal gas increase, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Work Done by the Gas**:
- The work done \( W \) by the gas during expansion is given by the formula:
\[ W = P \Delta V \]
- Here, \( \Delta V = V_f - V_i \) (final volume - initial volume). Since the volume is increasing, \( \Delta V \) is positive.

**Conclusion**: The work done by the system is positive.

2. **Analyzing the Ideal Gas Law**:
- The ideal gas law is expressed as:
\[ PV = nRT \]
- Given that both \( P \) and \( V \) are increasing, we can infer that the temperature \( T \) must also increase to satisfy the equation, assuming the number of moles \( n \) and the gas constant \( R \) remain constant.

**Conclusion**: The temperature of the system must increase.

3. **Applying the First Law of Thermodynamics**:
- The first law of thermodynamics states:
\[ \Delta Q = \Delta W + \Delta U \]
- Where \( \Delta Q \) is the heat added to the system, \( \Delta W \) is the work done by the system, and \( \Delta U \) is the change in internal energy.
- Since we have established that the work done \( \Delta W \) is positive (not zero), it implies that heat supplied to the gas cannot be equal to the change in internal energy \( \Delta U \) unless the work done is zero.

**Conclusion**: The heat supplied to the gas is not equal to the change in internal energy.

4. **Final Evaluation of Options**:
- Based on our analysis:
- **Option 1**: Incorrect (such a process is possible).
- **Option 2**: Correct (the work done by the system is positive).
- **Option 3**: Correct (the temperature of the system must increase).
- **Option 4**: Incorrect (heat supply to the gas is not equal to the change in internal energy).

### Final Answer:
The correct options are **Option 2** and **Option 3**.