When an ideal gas undergoes an adiabatic change causing a temperature change `DeltaT`
(i) there is no heat ganied or lost by the gas
(ii) the work done by the gas is equal to change in internal eenrgy
(iii) the change in internal energy per mole of the gas is `C_(V)DeltaT`, where `C_(V)` is the molar heat capacity at constant volume.

Options

A. (i),(ii),(iii) correct
B. (i),(ii) correct
C. (i),(iii) correct
D. (i)correct

Answer

To solve the question regarding the behavior of an ideal gas undergoing an adiabatic change, we will analyze each of the statements provided.

### Step-by-Step Solution:

1. **Understanding Adiabatic Process**:
- An adiabatic process is defined as one in which no heat is exchanged with the surroundings. This means that the heat transfer, \( \Delta Q \), is equal to zero.
- **Conclusion**: The first statement, which states that "there is no heat gained or lost by the gas," is correct.

2. **Applying the First Law of Thermodynamics**:
- The First Law of Thermodynamics states:
\[ \Delta Q = \Delta U + W \]
where \( \Delta Q \) is the heat added to the system, \( \Delta U \) is the change in internal energy, and \( W \) is the work done by the system.
- Since \( \Delta Q = 0 \) for an adiabatic process, we can rewrite the equation as:
\[ 0 = \Delta U + W \implies \Delta U = -W \]
- This indicates that the work done by the gas is equal to the negative change in internal energy. Therefore, the work done by the gas is not equal to the change in internal energy; rather, it is the opposite.
- **Conclusion**: The second statement, which claims that "the work done by the gas is equal to the change in internal energy," is incorrect.

3. **Change in Internal Energy**:
- The change in internal energy for an ideal gas can be expressed as:
\[ \Delta U = n C_V \Delta T \]
where \( n \) is the number of moles, \( C_V \) is the molar heat capacity at constant volume, and \( \Delta T \) is the change in temperature.
- For one mole of gas, this simplifies to:
\[ \Delta U = C_V \Delta T \]
- Thus, the statement that "the change in internal energy per mole of the gas is \( C_V \Delta T \)" is indeed correct.
- **Conclusion**: The third statement is correct.

### Final Answer:
- The correct statements are:
- (i) There is no heat gained or lost by the gas (Correct)
- (ii) The work done by the gas is equal to the change in internal energy (Incorrect)
- (iii) The change in internal energy per mole of the gas is \( C_V \Delta T \) (Correct)

### Summary:
- Statements (i) and (iii) are correct, while statement (ii) is incorrect.