Find out the ratio `C_p/C_v` of polyatomic gas for 2 vibrational degree of freedom

Answer

To find the ratio \( \frac{C_p}{C_v} \) for a polyatomic gas with 2 vibrational degrees of freedom, we can follow these steps:

### Step 1: Identify the degrees of freedom for a polyatomic gas
For a polyatomic gas, the degrees of freedom can be categorized as:
- Translational degrees of freedom: 3 (movement in x, y, and z directions)
- Rotational degrees of freedom: 3 (rotation about three axes)
- Vibrational degrees of freedom: Given as 2 in this case

### Step 2: Calculate the total degrees of freedom (F)
The total degrees of freedom \( F \) for the gas can be calculated as:
\[ F = \text{Translational} + \text{Rotational} + \text{Vibrational} = 3 + 3 + 2 = 8 \]

### Step 3: Use the formula for the ratio \( \frac{C_p}{C_v} \)
The ratio \( \frac{C_p}{C_v} \) (denoted as \( \gamma \)) can be calculated using the formula:
\[ \gamma = \frac{C_p}{C_v} = 1 + \frac{2}{F} \]

### Step 4: Substitute the value of F into the formula
Substituting \( F = 8 \) into the formula gives:
\[ \gamma = 1 + \frac{2}{8} = 1 + \frac{1}{4} = 1 + 0.25 = 1.25 \]

### Step 5: Conclusion
Thus, the ratio \( \frac{C_p}{C_v} \) for the polyatomic gas with 2 vibrational degrees of freedom is:
\[ \boxed{1.25} \]