A gas have 3 translation and 2 Rotational degree of freedom. Find `(C_p)/(C_v)`?

Options

A. `5/7`
B. `7/5`
C. `2/5`
D. `5/2`

Answer

To find the ratio \( \frac{C_p}{C_v} \) for a gas with 3 translational and 2 rotational degrees of freedom, we can follow these steps:

### Step 1: Determine the total degrees of freedom (f)
The total degrees of freedom \( f \) for the gas is the sum of the translational and rotational degrees of freedom:
\[ f = f_{\text{translational}} + f_{\text{rotational}} = 3 + 2 = 5 \]

### Step 2: Calculate \( C_v \)
The heat capacity at constant volume \( C_v \) can be expressed in terms of the degrees of freedom:
\[ C_v = \frac{f}{2} R \]
Substituting \( f = 5 \):
\[ C_v = \frac{5}{2} R \]

### Step 3: Calculate \( C_p \)
Using the relation between \( C_p \) and \( C_v \):
\[ C_p = C_v + R \]
Substituting the value of \( C_v \):
\[ C_p = \frac{5}{2} R + R = \frac{5}{2} R + \frac{2}{2} R = \frac{7}{2} R \]

### Step 4: Calculate the ratio \( \frac{C_p}{C_v} \)
Now, we can find the ratio:
\[ \frac{C_p}{C_v} = \frac{\frac{7}{2} R}{\frac{5}{2} R} = \frac{7}{5} \]

### Final Answer
Thus, the ratio \( \frac{C_p}{C_v} \) is:
\[ \frac{C_p}{C_v} = \frac{7}{5} \]

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