The work done in adiabatic process is given by

Options

A. `(nR(T_(1))-T_(2))/(gamma)`
B. `(nR(T_(1))-T_(2))/(gamma-1)`
C. `(nR(T_(1))-T_(2))R`
D. `(gamma(T_(1))-T_(2)R)/(n)`

Explanation

To find the work done in an adiabatic process, 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. The relationship between pressure (P), volume (V), and temperature (T) for an ideal gas during an adiabatic process is given by the equation \( PV^\gamma = \text{constant} \), where \( \gamma \) (gamma) is the heat capacity ratio \( C_p/C_v \).

2. **Work Done in an Adiabatic Process**: The work done (W) during an adiabatic process can be calculated using the integral:
\[ W = \int_{V_1}^{V_2} P \, dV \]

3. **Substituting for Pressure**: From the adiabatic condition, we can express pressure in terms of volume:
\[ P = \frac{k}{V^\gamma} \]
where \( k \) is a constant.

4. **Setting Up the Integral**: Substitute \( P \) into the work integral:
\[ W = \int_{V_1}^{V_2} \frac{k}{V^\gamma} \, dV \]

5. **Integrating**: The integral becomes:
\[ W = k \int_{V_1}^{V_2} V^{-\gamma} \, dV = k \left[ \frac{V^{1 - \gamma}}{1 - \gamma} \right]_{V_1}^{V_2} \]

6. **Evaluating the Integral**: Substitute the limits \( V_1 \) and \( V_2 \):
\[ W = k \left( \frac{V_2^{1 - \gamma} - V_1^{1 - \gamma}}{1 - \gamma} \right) \]

7. **Relating to Temperature**: Using the ideal gas law \( PV = nRT \), we can express \( k \) in terms of temperature:
\[ k = P V^\gamma = nRT \cdot V^{\gamma - 1} \]

8. **Final Expression for Work Done**: After substituting and simplifying, we arrive at the expression for work done in an adiabatic process:
\[ W = \frac{nR(T_1 - T_2)}{\gamma - 1} \]

### Conclusion:
The work done in an adiabatic process is given by:
\[ W = \frac{nR(T_1 - T_2)}{\gamma - 1} \]
Thus, the correct option is the second one: \( \frac{nR(T_1 - T_2)}{\gamma - 1} \).