A gas undergoes a change at constant temperature. Which of the following quantities remain fixed ?

Options

A. Pressure
B. Entropy
C. Heat exchanged with the system
D. All the above may change

Answer

To solve the problem, we need to analyze the behavior of a gas undergoing a change at constant temperature. We will use the ideal gas law and the principles of thermodynamics to determine which quantities remain fixed.

### Step-by-Step Solution:

1. **Understanding the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature.

2. **Constant Temperature Condition**:
Since the temperature \( T \) is constant, we can denote it as \( T_0 \). This implies that any changes in volume \( V \) will affect the pressure \( P \) inversely, according to the equation:
\[ P \propto \frac{1}{V} \]
Therefore, if the volume changes, the pressure must also change to maintain the equality.

3. **Analyzing Pressure**:
From the relationship established, if the volume changes, the pressure cannot remain constant. Thus, pressure \( P \) is **not fixed**.

4. **Analyzing Volume**:
The volume \( V \) is also changing as per the problem statement. Therefore, volume \( V \) is **not fixed**.

5. **Work Done**:
When a gas expands or compresses at constant temperature, work is done on or by the gas. The work done \( W \) is given by:
\[ W \neq 0 \]
Hence, work done is **not fixed**.

6. **Change in Internal Energy**:
The change in internal energy \( \Delta U \) for an ideal gas depends only on the temperature. Since the temperature is constant, we have:
\[ \Delta U = 0 \]
Thus, the change in internal energy is **fixed** at zero.

7. **Heat Exchange**:
According to the first law of thermodynamics:
\[ Q = W + \Delta U \]
Since \( \Delta U = 0 \), we have:
\[ Q = W \]
Since work is being done, heat exchange \( Q \) is also **not fixed**.

8. **Entropy Change**:
The change in entropy \( \Delta S \) at constant temperature can be expressed as:
\[ \Delta S = \frac{\Delta Q}{T} \]
Since \( \Delta Q \) is not zero, entropy change \( S \) is also **not fixed**.

### Conclusion:
In summary, during a change at constant temperature:
- Pressure \( P \) is not fixed.
- Volume \( V \) is not fixed.
- Work done \( W \) is not fixed.
- Heat exchange \( Q \) is not fixed.
- Change in internal energy \( \Delta U \) is fixed at zero.
- Entropy \( S \) is not fixed.

Thus, the only quantity that remains fixed is the change in internal energy, which is zero.

### Final Answer:
The only quantity that remains fixed is the change in internal energy \( \Delta U = 0 \).