If a gas has n degrees of freedom ratio of specific heats of gas is

Options

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

Explanation

To find the ratio of specific heats (γ) of a gas with n degrees of freedom, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Degrees of Freedom**:
- The degrees of freedom (n) of a gas refers to the number of independent ways in which the gas molecules can move. For example, in a monatomic gas, n = 3 (translational motion), while for a diatomic gas, n can be 5 (3 translational + 2 rotational).

2. **Internal Energy of the Gas**:
- The internal energy (U) of an ideal gas is entirely kinetic energy. For one mole of an ideal gas with n degrees of freedom, the internal energy can be expressed as:
\[ U = \frac{n}{2}RT \]
- Here, R is the universal gas constant and T is the absolute temperature.

3. **Finding the Heat Capacity at Constant Volume (C_V)**:
- The heat capacity at constant volume (C_V) is defined as the change in internal energy with respect to temperature:
\[ C_V = \frac{dU}{dT} = \frac{n}{2}R \]

4. **Finding the Heat Capacity at Constant Pressure (C_P)**:
- The heat capacity at constant pressure (C_P) can be related to C_V using the equation:
\[ C_P = C_V + R \]
- Substituting the expression for C_V:
\[ C_P = \frac{n}{2}R + R = \left(\frac{n}{2} + 1\right)R \]

5. **Calculating the Ratio of Specific Heats (γ)**:
- The ratio of specific heats (γ) is given by:
\[ \gamma = \frac{C_P}{C_V} \]
- Substituting the expressions for C_P and C_V:
\[ \gamma = \frac{\left(\frac{n}{2} + 1\right)R}{\frac{n}{2}R} \]
- Simplifying this gives:
\[ \gamma = \frac{\frac{n}{2} + 1}{\frac{n}{2}} = 1 + \frac{2}{n} \]

6. **Final Result**:
- Therefore, the ratio of specific heats of the gas is:
\[ \gamma = 1 + \frac{2}{n} \]