An ideal gas consists of molecules, each molecule being a linear chain of four atoms. If this gas is heated up to a temperature at which all degrees of freedom, including vibrational, are excited, the molar specific heat of such a gas at constant volume will be (include vibrational degrees also)

Options

A. `(19R)/(2)`
B. `(15R)/(2)`
C. `6R`
D. `9R`

Explanation

To find the molar specific heat of an ideal gas consisting of linear chains of four atoms at constant volume, we need to calculate the degrees of freedom (F) of the gas molecules, which includes translational, rotational, and vibrational degrees of freedom.

### Step-by-Step Solution:

1. **Identify the Degrees of Freedom:**
- For a linear molecule, the degrees of freedom can be categorized as:
- **Translational Degrees of Freedom (T):** Each molecule has 3 translational degrees of freedom (movement along x, y, and z axes).
- **Rotational Degrees of Freedom (R):** A linear molecule can rotate about two axes perpendicular to its length. Therefore, it has 2 rotational degrees of freedom.
- **Vibrational Degrees of Freedom (V):** Each atom in the molecule can vibrate. For a linear molecule with \( n \) atoms, the number of vibrational degrees of freedom is given by \( 2n - 5 \). Here, \( n = 4 \), so:
\[ V = 2(4) - 5 = 3 \]

2. **Calculate Total Degrees of Freedom (F):**
- Now, we can sum up all the degrees of freedom:
\[ F = T + R + V = 3 + 2 + 3 = 8 \]

3. **Use the Formula for Molar Specific Heat at Constant Volume (C_V):**
- The molar specific heat at constant volume is given by the formula:
\[ C_V = \frac{F}{2} R \]
- Substituting the value of \( F \):
\[ C_V = \frac{8}{2} R = 4R \]

4. **Final Answer:**
- The molar specific heat of the gas at constant volume is:
\[ C_V = 4R \]