Which expression is correct for the work done in adiabatic reversible expansion of an ideal gas

Options

A. `W=nRTln.(V_(2))/(V_(1))`
B. `W=n_(e)ln.(T_(2))/(T_(1))`
C. `W=PDeltaV`
D. `W=-int_(1)^(2)PdV`

Explanation

To determine the correct expression for the work done in the adiabatic reversible expansion of an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the First Law of Thermodynamics**:
The first law states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (Q) plus the work done on the system (W):
\[ \Delta U = Q + W \]

2. **Identify the Conditions for an Adiabatic Process**:
In an adiabatic process, there is no heat exchange with the surroundings, which means:
\[ Q = 0 \]
Therefore, the equation simplifies to:
\[ \Delta U = W \]

3. **Relate Internal Energy Change to Temperature Change**:
For an ideal gas, the change in internal energy 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.

4. **Relate Work Done to Internal Energy**:
Since we established that \( \Delta U = W \), we can substitute:
\[ W = n C_v \Delta T \]

5. **Use the Relationship Between Heat Capacities**:
The relationship between the heat capacities at constant pressure (C_p) and constant volume (C_v) is given by:
\[ C_p - C_v = R \]
where \( R \) is the ideal gas constant.

6. **Express Work Done in Terms of Pressure and Volume**:
For a reversible process, we can also express the work done as:
\[ W = \int P \, dV \]
For an ideal gas, we can relate pressure and volume using the ideal gas law:
\[ PV = nRT \]
Thus, we can express changes in volume and pressure in terms of temperature changes.

7. **Final Expression for Work Done**:
From the above relationships, we can derive that:
\[ W = nR \Delta T \]
This expression indicates the work done in an adiabatic reversible expansion of an ideal gas.

### Conclusion:
The correct expression for the work done in the adiabatic reversible expansion of an ideal gas is:
\[ W = nR \Delta T \]