The internal energy U is a unique function of any state because change in U

Options

A. does not depends upon path
B. depends upon path
C. corresponds to adiabatic process
D. corresponds to an isothermal process

Answer

To solve the question regarding the internal energy \( U \) and its unique function of any state due to the change in \( U \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Internal Energy**:
- Internal energy \( U \) is a thermodynamic property of a system that reflects the total energy contained within the system due to the kinetic and potential energies of its particles.

2. **First Law of Thermodynamics**:
- The first law of thermodynamics states that the change in internal energy \( dU \) of a system is equal to the heat added to the system \( dQ \) minus the work done by the system \( dW \):
\[ dQ = dU + dW \]
- Rearranging gives:
\[ dU = dQ - dW \]

3. **Path Functions vs. State Functions**:
- In thermodynamics, \( dQ \) (heat) and \( dW \) (work) are path-dependent quantities, meaning their values depend on the specific path taken during a process.
- In contrast, \( dU \) (the change in internal energy) is a state function, meaning it depends only on the initial and final states of the system, not on the path taken to get there.

4. **Exact Differentials**:
- Since \( dU \) is a state function, it is represented as an exact differential. This means that the change in internal energy between two states is the same regardless of the path taken between those states.

5. **Conclusion**:
- Therefore, we can conclude that the change in internal energy \( dU \) is a unique function of the state of the system because it does not depend on the path taken to reach that state. This makes it a state function.

### Final Answer:
The internal energy \( U \) is a unique function of any state because the change in \( U \) (denoted as \( dU \)) depends only on the initial and final states of the system and not on the path taken between them.