The expression `[deltaU//deltaT]_(V)` represents

Options

A. Heat capacity at constant volume
B. Heat capacity at constant pressure
C. Enthalpy change
D. Entropy change

Answer

To solve the question regarding the expression \(\left[\frac{\Delta U}{\Delta T}\right]_{V}\), we will break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Terms**:
- \(\Delta U\) represents the change in internal energy of a system.
- \(\Delta T\) represents the change in temperature of the system.
- The subscript \(V\) indicates that the process is occurring at constant volume.

2. **Definition of Heat Capacity**:
- Heat capacity (\(C\)) is defined as the amount of heat required to raise the temperature of a system by 1 degree Celsius.
- There are two types of heat capacities:
- Heat capacity at constant volume (\(C_V\))
- Heat capacity at constant pressure (\(C_P\))

3. **Expression for Heat Capacity at Constant Volume**:
- At constant volume, the heat capacity can be expressed mathematically as:
\[ C_V = \left(\frac{\Delta Q}{\Delta T}\right)_{V} \]
- According to the first law of thermodynamics:
\[ \Delta Q = \Delta U + P \Delta V \]
- At constant volume, \(\Delta V = 0\), hence:
\[ \Delta Q = \Delta U \]

4. **Relating Heat Capacity to Internal Energy**:
- Substituting \(\Delta Q\) in the heat capacity equation gives:
\[ C_V = \left(\frac{\Delta U}{\Delta T}\right)_{V} \]

5. **Conclusion**:
- Therefore, the expression \(\left[\frac{\Delta U}{\Delta T}\right]_{V}\) represents the heat capacity at constant volume (\(C_V\)).
- Thus, the final answer is that \(\left[\frac{\Delta U}{\Delta T}\right]_{V}\) represents the heat capacity at constant volume.

### Final Answer:
\[ \left[\frac{\Delta U}{\Delta T}\right]_{V} = C_V \quad \text{(Heat Capacity at Constant Volume)} \]

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