We consider a thermodynamic system. If `DeltaU` represents the increase in its internal energy and W the work done by the system, which of the following statements is true?

Options

A. `Delta U = - W` in an adiabatic process
B. `Delta U = W` in an isothermal process
C. `Delta U = - W` in an isothernal process
D. `Delta U = W` in an adiabatic process

Answer

To solve the problem, we need to apply the first law of thermodynamics, which states:

\[ \Delta Q = \Delta U + W \]

Where:
- \(\Delta Q\) is the heat added to the system,
- \(\Delta U\) is the change in internal energy,
- \(W\) is the work done by the system.

### Step-by-Step Solution:

1. **Understanding the First Law of Thermodynamics**:
- The first law of thermodynamics relates the change in internal energy of a system to the heat added to the system and the work done by the system.

2. **Consider an Adiabatic Process**:
- In an adiabatic process, there is no heat exchange with the surroundings, which means \(\Delta Q = 0\).

3. **Applying the First Law in an Adiabatic Process**:
- Since \(\Delta Q = 0\), we can substitute this into the first law equation:
\[ 0 = \Delta U + W \]

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

5. **Conclusion**:
- This means that the change in internal energy (\(\Delta U\)) is equal to the negative of the work done by the system (\(W\)). Therefore, if the system does work on the surroundings, its internal energy decreases.

### Final Statement:
Thus, the correct statement is that in an adiabatic process, the change in internal energy is equal to the negative of the work done by the system:
\[ \Delta U = -W \]