An ideal gas has molecules with 5 degrees of freedom. The ratio of specific heats at constant pressure `(C_(p))` and at constant volume `(C_(v))` is

Options

A. `7/5`
B. 6
C. `7/2`
D. `5/2`

Answer

To find the ratio of specific heats at constant pressure \( C_p \) and at constant volume \( C_v \) for an ideal gas with 5 degrees of freedom, we can follow these steps:

### Step 1: Understand the relationship between \( C_p \) and \( C_v \)
The specific heat at constant pressure \( C_p \) is related to the specific heat at constant volume \( C_v \) by the equation:
\[ C_p = C_v + R \]
where \( R \) is the universal gas constant.

### Step 2: Use the degrees of freedom to find \( C_v \)
For an ideal gas, the specific heat at constant volume \( C_v \) can be expressed in terms of the degrees of freedom \( F \) of the gas molecules:
\[ C_v = \frac{F}{2} R \]
Given that the gas has 5 degrees of freedom, we can substitute \( F = 5 \):
\[ C_v = \frac{5}{2} R \]

### Step 3: Substitute \( C_v \) into the equation for \( C_p \)
Now we can substitute \( C_v \) into the equation for \( C_p \):
\[ C_p = C_v + R = \frac{5}{2} R + R = \frac{5}{2} R + \frac{2}{2} R = \frac{7}{2} R \]

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

### Conclusion
Thus, the ratio of specific heats at constant pressure and constant volume for the ideal gas with 5 degrees of freedom is:
\[ \frac{C_p}{C_v} = \frac{7}{5} \]