For polyatomic molecules having 'f' vibrational modes, the ratio of two specific heats, `C_(P)/C_(V)` is

Options

A. `(1+f)/(2+f)`
B. `(2+f)/(3+f)`
C. `(4+f)/(3+f)`
D. `(5+f)/(4+f)`

Explanation

To find the ratio of the specific heats \( \frac{C_P}{C_V} \) for polyatomic molecules with \( f \) vibrational modes, we can follow these steps:

### Step 1: Understand the Degrees of Freedom
For polyatomic gases, the degrees of freedom can be categorized into translational, rotational, and vibrational modes.
- Translational degrees of freedom: 3 (movement in x, y, z directions)
- Rotational degrees of freedom: 3 (rotation about three axes)
- Vibrational degrees of freedom: \( f \) (given in the problem)

### Step 2: Calculate \( C_V \)
Using the equipartition theorem, the molar heat capacity at constant volume \( C_V \) can be expressed as:
\[ C_V = \left( \text{Translational} + \text{Rotational} + \text{Vibrational} \right) \cdot \frac{1}{2} R \]
For polyatomic gases:
\[ C_V = \left( 3 + 3 + f \right) \cdot \frac{1}{2} R = (6 + f) \cdot \frac{1}{2} R \]
Thus,
\[ C_V = \left( 3 + \frac{f}{2} \right) R \]

### Step 3: Calculate \( C_P \)
The molar heat capacity at constant pressure \( C_P \) is related to \( C_V \) by:
\[ C_P = C_V + R \]
Substituting the expression for \( C_V \):
\[ C_P = \left( 3 + \frac{f}{2} \right) R + R = \left( 4 + \frac{f}{2} \right) R \]

### Step 4: Find the Ratio \( \frac{C_P}{C_V} \)
Now we can find the ratio of the specific heats:
\[ \frac{C_P}{C_V} = \frac{4 + \frac{f}{2}}{3 + \frac{f}{2}} \]
To simplify, we can multiply the numerator and denominator by 2:
\[ \frac{C_P}{C_V} = \frac{8 + f}{6 + f} \]

### Final Answer
Thus, the ratio of the specific heats \( \frac{C_P}{C_V} \) for polyatomic molecules having \( f \) vibrational modes is:
\[ \frac{C_P}{C_V} = \frac{4 + f}{3 + f} \]