For one mole of a diatomic gas `gamma` is equal to

Options

A. `3/5`
B. `5/3`
C. `7/5`
D. `8/6`

Answer

To find the value of gamma (γ) for one mole of a diatomic gas, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Degrees of Freedom**:
For a diatomic gas, the degrees of freedom (F) include:
- Translational degrees of freedom: 3 (movement in x, y, z directions)
- Rotational degrees of freedom: 2 (rotation about two axes)
- Total degrees of freedom (F) = 3 (translational) + 2 (rotational) = 5.

2. **Use the Formula for Heat Capacities**:
The relationship between the heat capacities at constant pressure (Cₚ) and constant volume (Cᵥ) is given by:
\[ Cₚ - Cᵥ = R \]
where R is the universal gas constant.

3. **Express Cᵥ in Terms of Degrees of Freedom**:
The heat capacity at constant volume (Cᵥ) can be expressed as:
\[ Cᵥ = \frac{F}{2} R \]
Substituting F = 5:
\[ Cᵥ = \frac{5}{2} R \]

4. **Calculate Cₚ**:
Using the relationship from step 2:
\[ Cₚ = Cᵥ + R = \frac{5}{2} R + R = \frac{5}{2} R + \frac{2}{2} R = \frac{7}{2} R \]

5. **Calculate Gamma (γ)**:
Gamma is defined as:
\[ \gamma = \frac{Cₚ}{Cᵥ} \]
Substituting the values we found:
\[ \gamma = \frac{\frac{7}{2} R}{\frac{5}{2} R} = \frac{7}{5} \]

6. **Final Result**:
Therefore, for one mole of a diatomic gas, the value of gamma (γ) is:
\[ \gamma = \frac{7}{5} \]