One mole of a gas in the state `A(P_1, V_1, T_1)` is subjected to adiabatic expansion to attain state `B(P_2,V_2,T_2)`. The work done by the gas is:

Options

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

Explanation

To calculate the work done by one mole of a gas during an adiabatic expansion from state A to state B, 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 work done by the gas results in a change in its internal energy.

2. **Identify Given Parameters**:
- We have one mole of gas (n = 1).
- The initial state A has parameters: \( P_1, V_1, T_1 \).
- The final state B has parameters: \( P_2, V_2, T_2 \).

3. **Use the Formula for Work Done in Adiabatic Process**:
- The work done \( W \) by the gas during an adiabatic expansion can be calculated using the formula:
\[ W = \frac{nR(T_2 - T_1)}{\gamma - 1} \]
- Where:
- \( n \) = number of moles (1 mole in this case)
- \( R \) = universal gas constant
- \( T_1 \) = initial temperature
- \( T_2 \) = final temperature
- \( \gamma \) = heat capacity ratio (C_p/C_v)

4. **Substituting the Values**:
- Since \( n = 1 \), the equation simplifies to:
\[ W = \frac{R(T_2 - T_1)}{\gamma - 1} \]

5. **Conclusion**:
- The work done by the gas during the adiabatic expansion from state A to state B is:
\[ W = \frac{R(T_2 - T_1)}{\gamma - 1} \]

### Final Answer:
The work done by the gas in the adiabatic expansion is \( \frac{R(T_2 - T_1)}{\gamma - 1} \).

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