A system can be taken from the initial state `p_(1),V_(1)` to the final state `p_(2,V_(2)` by two different methods, let `DeltaQ and DeltaW` represent the heat given to the system and the work done by the system. Which of the following must be the same in both the method?

Options

A. `DeltaQ`
B. `DeltaW`
C. `DeltaQ+DeltaW`
D. `DeltaQ-DeltaW`.

Explanation

To solve the problem, we need to analyze the relationship between heat (ΔQ), work (ΔW), and internal energy (ΔU) as described by the first law of thermodynamics.

### Step-by-Step Solution:

1. **Understand the First Law of Thermodynamics**:
The first law of thermodynamics states that the change in internal energy (ΔU) of a system is equal to the heat added to the system (ΔQ) minus the work done by the system (ΔW). This can be expressed mathematically as:
\[ \Delta U = \Delta Q - \Delta W \]

2. **Identify Internal Energy as a State Function**:
Internal energy (ΔU) is a state function, meaning it depends only on the initial and final states of the system, not on the path taken to get from one state to another. Therefore, if a system transitions from an initial state (p1, V1) to a final state (p2, V2), the change in internal energy will be the same regardless of the path taken.

3. **Apply the Concept to Both Paths**:
Since the internal energy change (ΔU) is the same for both paths (let's call them Path 1 and Path 2), we can write:
\[ \Delta U_{Path1} = \Delta U_{Path2} \]

4. **Relate ΔQ and ΔW for Both Paths**:
From the first law, we can express the changes in internal energy for both paths:
\[ \Delta U_{Path1} = \Delta Q_{Path1} - \Delta W_{Path1} \]
\[ \Delta U_{Path2} = \Delta Q_{Path2} - \Delta W_{Path2} \]
Since ΔU is the same for both paths, we can set the two equations equal to each other:
\[ \Delta Q_{Path1} - \Delta W_{Path1} = \Delta Q_{Path2} - \Delta W_{Path2} \]

5. **Conclude About ΔQ and ΔW**:
Rearranging the above equation gives us:
\[ \Delta Q_{Path1} - \Delta Q_{Path2} = \Delta W_{Path1} - \Delta W_{Path2} \]
This indicates that ΔQ and ΔW can vary depending on the path taken, but the difference between the heat added and the work done must remain constant.

6. **Final Conclusion**:
Therefore, the only quantity that must be the same for both methods is the change in internal energy (ΔU), which can also be expressed as:
\[ \Delta U = \Delta Q - \Delta W \]
Thus, the correct answer is:
\[ \Delta Q - \Delta W \text{ must be the same for both methods.} \]

### Answer:
The quantity that must be the same in both methods is \( \Delta Q - \Delta W \).

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