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_1)`
D. `1/2 (P_1 V_2 - P_1 V_1)`

Explanation

To solve the problem of finding 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 with the surroundings. This means that the heat transfer \( q \) 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 an Adiabatic Process**:
Since \( q = 0 \) for an adiabatic process, we can rewrite the equation as:
\[ 0 = \Delta U + W \]
Rearranging this gives us:
\[ W = -\Delta U \]

4. **Express 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.

5. **Substitute Change in Internal Energy into Work Equation**:
Substituting \( \Delta U \) into the work equation gives:
\[ W = -(U_2 - U_1) = U_1 - U_2 \]

6. **Final Expression for Work Done**:
Thus, the work done by the gas during the adiabatic process from state 1 to state 2 is:
\[ W = U_1 - U_2 \]

### Conclusion:
The work done by the gas during an adiabatic change from state 1 to state 2 is equal to the difference in internal energy, expressed as \( W = U_1 - U_2 \).

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