Which of the following expressions is true for an ideal gas ?

Options

A. `((delV)/(delT))_(P) = 0`
B. `((delP)/(delT))_(V) = 0`
C. `((delU)/(delP))_(T) = 0`
D. `((delU)/(delT))_(T) = 0`

Explanation

To determine which expression is true for an ideal gas, we will analyze each option step by step using the ideal gas law and relevant thermodynamic equations.

### Step-by-Step Solution:

1. **Understanding the Ideal Gas Law**:
The ideal gas law is given by the equation:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature.

2. **Analyzing Option 1**:
The first option states:
\[ \frac{\Delta V}{\Delta T} \text{ at constant pressure} = 0 \]
- From the ideal gas law, at constant pressure, we can differentiate:
\[ P \frac{\Delta V}{\Delta T} = nR \]
- This implies:
\[ \frac{\Delta V}{\Delta T} = \frac{nR}{P} \]
- Since \( nR/P \) is a constant (not zero), this option is **incorrect**.

3. **Analyzing Option 2**:
The second option states:
\[ \frac{\Delta P}{\Delta T} \text{ at constant volume} = 0 \]
- Again, using the ideal gas law:
\[ V \frac{\Delta P}{\Delta T} = nR \]
- Thus:
\[ \frac{\Delta P}{\Delta T} = \frac{nR}{V} \]
- Since \( nR/V \) is also a constant (not zero), this option is **incorrect**.

4. **Analyzing Option 3**:
The third option states:
\[ \frac{\Delta U}{\Delta P} \text{ at constant temperature} = 0 \]
- The change in internal energy \( \Delta U \) for an ideal gas is given by:
\[ \Delta U = nC_v \Delta T \]
- At constant temperature, \( \Delta T = 0 \), hence:
\[ \Delta U = 0 \]
- Therefore:
\[ \frac{\Delta U}{\Delta P} = \frac{0}{\Delta P} = 0 \]
- This option is **correct**.

5. **Analyzing Option 4**:
The fourth option states:
\[ \frac{\Delta U}{\Delta T} \text{ at constant temperature} = 0 \]
- Here, we have:
\[ \Delta U = nC_v \Delta T \]
- Thus:
\[ \frac{\Delta U}{\Delta T} = nC_v \]
- Since \( nC_v \) is not zero, this option is **incorrect**.

### Conclusion:
The correct answer is **Option 3**: \(\frac{\Delta U}{\Delta P} \text{ at constant temperature} = 0\).