For polyatomic molecules having 'f' vibrational modes, the ratio of two specific heat, `(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 two specific heats, \( \frac{C_P}{C_V} \), for polyatomic molecules having \( f \) vibrational modes, we can follow these steps:

### Step 1: Determine the degrees of freedom for polyatomic molecules
For a polyatomic molecule, the total degrees of freedom \( f_{total} \) can be calculated as follows:
- Translational degrees of freedom: 3 (movement in x, y, and z directions)
- Rotational degrees of freedom: 3 (rotation about x, y, and z axes)
- Vibrational degrees of freedom: \( f \) (given in the problem)

Thus, the total degrees of freedom is:
\[ f_{total} = 3 + 3 + f = 6 + f \]

### Step 2: Calculate the internal energy \( U \)
According to the law of equipartition of energy, the internal energy \( U \) for one mole of gas can be expressed as:
\[ U = \frac{f_{total}}{2} RT \]
Substituting \( f_{total} \):
\[ U = \frac{6 + f}{2} RT \]

### Step 3: Determine \( C_V \)
The molar specific heat at constant volume \( C_V \) is defined as the change in internal energy with respect to temperature:
\[ C_V = \frac{dU}{dT} = \frac{d}{dT}\left(\frac{6 + f}{2} RT\right) = \frac{6 + f}{2} R \]

### Step 4: Determine \( C_P \)
Using the relation \( C_P = C_V + R \):
\[ C_P = \frac{6 + f}{2} R + R = \left(\frac{6 + f}{2} + 1\right) R = \frac{6 + f + 2}{2} R = \frac{8 + f}{2} R \]

### Step 5: Calculate the ratio \( \frac{C_P}{C_V} \)
Now, we can find the ratio:
\[ \frac{C_P}{C_V} = \frac{\frac{8 + f}{2} R}{\frac{6 + f}{2} R} = \frac{8 + f}{6 + f} \]

### Final Result
Thus, the ratio of the specific heats for polyatomic molecules having \( f \) vibrational modes is:
\[ \frac{C_P}{C_V} = \frac{8 + f}{6 + f} \]

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