The ratio of the specific heats `(C_(P))/(C_(upsilon)) = gamma` in terms of degrees of freedom is given by

Options

A. `(1 + (1)/(n))`
B. `(1 + (n)/(3))`
C. `(1 + (2)/(n))`
D. `(1 + (n)/(2))`

Answer

To find the ratio of specific heats \( \frac{C_P}{C_V} \) (denoted as \( \gamma \)) in terms of degrees of freedom, we can follow these steps:

### Step 1: Understand Degrees of Freedom
Degrees of freedom (denoted as \( n \)) refer to the number of independent ways in which a molecule can move or store energy. For different types of gases:
- **Monoatomic gases** have 3 degrees of freedom (translational motion).
- **Diatomic gases** have 5 degrees of freedom (3 translational and 2 rotational).
- **Polyatomic gases** have more degrees of freedom depending on their structure.

### Step 2: Relate Specific Heats to Degrees of Freedom
The specific heats at constant pressure \( C_P \) and at constant volume \( C_V \) are related to the degrees of freedom of the gas. The relationship can be expressed as:
\[ C_V = \frac{n}{2} R \]
\[ C_P = C_V + R = \frac{n}{2} R + R = \left(\frac{n}{2} + 1\right) R \]
where \( R \) is the universal gas constant.

### Step 3: Calculate the Ratio \( \gamma \)
Now, we can find the ratio \( \gamma \):
\[ \gamma = \frac{C_P}{C_V} = \frac{\left(\frac{n}{2} + 1\right) R}{\frac{n}{2} R} \]
Simplifying this gives:
\[ \gamma = \frac{\frac{n}{2} + 1}{\frac{n}{2}} = \frac{n + 2}{n} \]

### Step 4: Express in Terms of Degrees of Freedom
We can rewrite this ratio in a more simplified form:
\[ \gamma = 1 + \frac{2}{n} \]

### Final Result
Thus, the ratio of specific heats \( \frac{C_P}{C_V} \) in terms of degrees of freedom is given by:
\[ \gamma = 1 + \frac{2}{n} \]