In the relation `dQ=dU+dW`, the quantity which remains the same for all process is

Options

A. `dQ`
B. `aW`
C. `dU`
D. all of the above

Explanation

To solve the question regarding the relation \( dQ = dU + dW \) and identify which quantity remains the same for all processes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relation**:
The equation \( dQ = dU + dW \) is known as the first law of thermodynamics. Here, \( dQ \) represents the heat added to the system, \( dU \) represents the change in internal energy, and \( dW \) represents the work done by the system.

2. **Identifying the Quantities**:
- \( dQ \): Change in heat energy.
- \( dU \): Change in internal energy.
- \( dW \): Work done by the system.

3. **Analyzing Each Quantity**:
- The quantity \( dQ \) can vary depending on the process (e.g., isothermal, adiabatic, etc.).
- The quantity \( dW \) also varies depending on the type of work done (e.g., expansion work, pressure-volume work).
- The quantity \( dU \) represents the change in internal energy, which is dependent on the state of the system and the path taken.

4. **Determining Which Quantity Remains the Same**:
- In different thermodynamic processes, while \( dQ \) and \( dW \) can change, the change in internal energy \( dU \) for a given state of the system is determined solely by the initial and final states of the system, not the path taken between them.
- Therefore, \( dU \) remains the same for all processes that connect the same initial and final states.

5. **Conclusion**:
The quantity that remains the same for all processes is the change in internal energy \( dU \).

### Final Answer:
The quantity that remains the same for all processes is \( dU \) (change in internal energy).

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