n moles of an ideal triatomic linear gas undergoes a process in which the temperature changes with volume as `k_(1)V ^ 2` where `k_(1)` is a constant. Choose incorrect alternative.

Options

A. At normal temperature `C_(v)=(5)/(2)R`
B. At any temperature `C_(p)-C_(v)=R`
C. At normal temperature molar heat capacity C=3R
D. At any temperature molar heat capacity C=3R

Explanation

To solve the problem, we need to analyze the given statements about the ideal triatomic linear gas undergoing a process where the temperature varies with volume as \( T = k_1 V^2 \). We will evaluate each statement to identify the incorrect one.

### Step-by-Step Solution:

1. **Understanding the Degrees of Freedom**:
- For a triatomic linear gas, the degrees of freedom \( F \) is given by \( F = 5 \) (3 translational + 2 rotational).
- Therefore, the molar heat capacity at constant volume \( C_v \) can be calculated as:
\[ C_v = \frac{F}{2} R = \frac{5}{2} R \]

2. **Evaluating the First Statement**:
- The first statement claims that at normal temperature, \( C_v = \frac{5}{2} R \).
- Since we have calculated \( C_v \) as \( \frac{5}{2} R \), this statement is **correct**.

3. **Evaluating the Second Statement**:
- The second statement states that \( C_p - C_v = R \).
- We know that for any ideal gas, this relation holds true. Thus:
\[ C_p = C_v + R = \frac{5}{2} R + R = \frac{7}{2} R \]
- Therefore, \( C_p - C_v = R \) is also **correct**.

4. **Evaluating the Third Statement**:
- The third statement claims that at normal temperature, the molar heat capacity \( C = 3R \).
- We need to check if the molar heat capacity \( C \) can be derived from the process \( T = k_1 V^2 \) and the ideal gas equation \( PV = nRT \).
- Using the relation derived from the process and the ideal gas law, we find:
\[ C = C_v + R = \frac{5}{2} R + R = \frac{7}{2} R \]
- This means that the statement claiming \( C = 3R \) is **incorrect**.

5. **Evaluating the Fourth Statement**:
- The fourth statement claims that at any temperature, the molar heat capacity \( C = 3R \).
- Since we have established that \( C = \frac{7}{2} R \) at normal conditions and it does not hold for all temperatures, this statement is also **incorrect**.

### Conclusion:
The incorrect statement is the third one, which states that at normal temperature, the molar heat capacity \( C = 3R \).