A system performs work `DeltaW` when an amount of heat is `DeltaQ` added to the system, the corresponding change in the internal energy is `DeltaU` . A unique function of the initial and final states (irrespective of the mode of change) is

Options

A. `DeltaQ`
B. `DeltaW`
C. `DeltaU`and `DeltaU`
D. `DeltaU`

Answer

To solve the problem, we need to identify a unique function of the initial and final states of a thermodynamic system, which is independent of the path taken to change from the initial state to the final state.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We are given that a system performs work \( \Delta W \) when an amount of heat \( \Delta Q \) is added to it, resulting in a change in internal energy \( \Delta U \).
- We need to find a unique function that depends only on the initial and final states of the system, not on how the system transitions from one state to another.

2. **Applying the First Law of Thermodynamics**:
- The First Law of Thermodynamics states:
\[ \Delta U = \Delta Q - \Delta W \]
- Here, \( \Delta U \) is the change in internal energy, \( \Delta Q \) is the heat added to the system, and \( \Delta W \) is the work done by the system.

3. **Identifying the Unique Function**:
- The change in internal energy \( \Delta U \) is a state function. This means it depends only on the initial and final states of the system, regardless of the path taken to get from one state to the other.
- In contrast, \( \Delta Q \) and \( \Delta W \) are not state functions; they depend on the specific process or path taken.

4. **Conclusion**:
- Therefore, the unique function of the initial and final states, irrespective of the mode of change, is:
\[ \Delta U \]
- This indicates that the change in internal energy is the only quantity that is a function of the state of the system, independent of how the transition occurs.

### Final Answer:
The unique function of the initial and final states is \( \Delta U \) (the change in internal energy).

---