For an ideal gas, the value of `((dU)/(dV))_(T)` is:

Options

A. Positive
B. Zero
C. Negative
D. Interchangeable

Answer

To find the value of \(\left(\frac{dU}{dV}\right)_{T}\) for an ideal gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Variables**:
- \(U\) represents the internal energy of the gas.
- \(V\) represents the volume of the gas.
- \(T\) represents the temperature of the gas.

2. **Recognize the Relationship**:
- For an ideal gas, the internal energy \(U\) is primarily a function of temperature \(T\) alone. This means that \(U\) does not depend on the volume \(V\) when the temperature is held constant.

3. **Apply the Concept of Partial Derivatives**:
- When we take the partial derivative of \(U\) with respect to \(V\) while keeping \(T\) constant, we are essentially looking at how \(U\) changes with \(V\) at a fixed temperature.

4. **Determine the Value**:
- Since \(U\) is a function of \(T\) only for an ideal gas, if \(T\) is constant, \(U\) remains constant regardless of changes in \(V\). Therefore, the change in internal energy with respect to volume at constant temperature is zero.
- Mathematically, this can be expressed as:
\[ \left(\frac{dU}{dV}\right)_{T} = 0 \]

5. **Conclusion**:
- The value of \(\left(\frac{dU}{dV}\right)_{T}\) for an ideal gas is \(0\).

### Final Answer:
\[ \left(\frac{dU}{dV}\right)_{T} = 0 \]