Which one of the following is correct when an ideal gas is expanded adaibatically and reversibly ?

Options

A. `q=0,W=nC_(v) dT, DeltaU= nC_(v)dT`
B. `q=0,W=0,DeltaU = nC_(v)dT`
C. `q=0,W= nC_(v)d,DeltaU =0`
D. None of the above

Explanation

To solve the question regarding the behavior of an ideal gas during an adiabatic and reversible expansion, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
- In an adiabatic process, there is no heat exchange with the surroundings. Therefore, the heat energy \( Q = 0 \).

2. **Apply the First Law of Thermodynamics**:
- The first law of thermodynamics states:
\[ \Delta U = Q + W \]
- Since \( Q = 0 \) for an adiabatic process, the equation simplifies to:
\[ \Delta U = W \]

3. **Relate Change in Internal Energy to Work Done**:
- The change in internal energy \( \Delta U \) for an ideal gas can be expressed as:
\[ \Delta U = n C_V \Delta T \]
- Here, \( n \) is the number of moles, \( C_V \) is the molar heat capacity at constant volume, and \( \Delta T \) is the change in temperature.

4. **Express Work Done in Terms of Internal Energy**:
- From the first law, we established that \( W = \Delta U \). Therefore, we can write:
\[ W = n C_V \Delta T \]

5. **Summarize the Relationships**:
- For an ideal gas expanded adiabatically and reversibly:
- Heat exchanged \( Q = 0 \)
- Work done \( W = n C_V \Delta T \)
- Change in internal energy \( \Delta U = n C_V \Delta T \)

6. **Identify the Correct Statement**:
- The correct statement regarding the adiabatic expansion of an ideal gas is:
- \( Q = 0 \), \( W = n C_V \Delta T \), and \( \Delta U = n C_V \Delta T \).

### Conclusion:
Thus, the correct option is that during an adiabatic and reversible expansion of an ideal gas, the heat exchange is zero, and the work done equals the change in internal energy.