If a system undergoes an adiabatic change from state 1 to state 2, the work done by the gas is:

Options

A. `U_1 - U_2`
B. `U_2 - U_1`
C. `C_V (T_2 - T_2)`
D. `1/2 (P_1 V_2 - P_1 V_1)`

Explanation

To determine the work done by the gas during an adiabatic change from state 1 to state 2, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Adiabatic Process**:
- In an adiabatic process, there is no heat exchange between the system and its surroundings. This means that the heat (q) is equal to zero.

2. **Apply the First Law of Thermodynamics**:
- The first law of thermodynamics states:
\[ q = \Delta U + W \]
where \( q \) is the heat added to the system, \( \Delta U \) is the change in internal energy, and \( W \) is the work done by the system.

3. **Substitute for Adiabatic Process**:
- Since \( q = 0 \) for an adiabatic process, we can rewrite the equation as:
\[ 0 = \Delta U + W \]

4. **Rearrange the Equation**:
- Rearranging the equation gives us:
\[ W = -\Delta U \]

5. **Define Change in Internal Energy**:
- The change in internal energy (\( \Delta U \)) can be expressed as:
\[ \Delta U = U_2 - U_1 \]
where \( U_1 \) is the internal energy at state 1 and \( U_2 \) is the internal energy at state 2.

6. **Substitute \( \Delta U \) into the Work Equation**:
- Now, substituting \( \Delta U \) into the work equation gives:
\[ W = - (U_2 - U_1) = U_1 - U_2 \]

7. **Conclusion**:
- Therefore, the work done by the gas during the adiabatic process from state 1 to state 2 is:
\[ W = U_1 - U_2 \]