The molar heat capacity at constant pressure of nitrogen gas at `STP` is nearly `3.5 R`. Now when the temperature is increased, it gradually increases and approaches `4.5 R`. The most approprite reason for this behaviour is that at high temperatures

Options

A. nitrogen does not behave as an ideal gas
B. nitrogen molecules dissociate in atoms
C. the molecules collides more frequently
D. molecules vibration gradually beome effective

Explanation

To solve the problem regarding the molar heat capacity at constant pressure (Cp) of nitrogen gas and its behavior with temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Molar Heat Capacity**:
- The molar heat capacity at constant pressure (Cp) is related to the degrees of freedom of the gas molecules. For a diatomic gas like nitrogen (N2), at standard temperature and pressure (STP), Cp is approximately 3.5 R.

2. **Degrees of Freedom**:
- The degrees of freedom (F) for a diatomic molecule at STP includes:
- Translational degrees of freedom: 3 (movement in x, y, and z directions)
- Rotational degrees of freedom: 2 (rotation about two axes perpendicular to the bond axis)
- Total at STP: F = 3 (translational) + 2 (rotational) = 5.

3. **Calculating Cp at STP**:
- Using the relation \( Cp = \frac{F}{2} R + R \):
\[ Cp = \frac{5}{2} R + R = \frac{7}{2} R = 3.5 R \]

4. **Effect of Temperature Increase**:
- As the temperature increases, the kinetic energy of the nitrogen molecules also increases. This increase in kinetic energy leads to more vigorous molecular motion.

5. **Activation of Vibrational Degrees of Freedom**:
- At higher temperatures, the vibrational modes of the nitrogen molecules become significant. Each vibrational mode contributes an additional degree of freedom.
- For diatomic gases, there are typically 2 vibrational degrees of freedom.

6. **Total Degrees of Freedom at High Temperature**:
- At high temperatures, the total degrees of freedom becomes:
\[ F = 3 \text{ (translational)} + 2 \text{ (rotational)} + 2 \text{ (vibrational)} = 7 \]

7. **Calculating Cp at High Temperature**:
- Now, using the new total degrees of freedom:
\[ Cp = \frac{7}{2} R + R = \frac{9}{2} R = 4.5 R \]

8. **Conclusion**:
- The increase in Cp from 3.5 R to 4.5 R as the temperature rises is primarily due to the activation of the vibrational degrees of freedom of the nitrogen molecules.

### Final Answer:
The most appropriate reason for the behavior of Cp increasing with temperature is that **molecules' vibrations gradually become active**.

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