When one mole of monatomic gas is mixed with one mole of a diatomic gas, the equivalent value of `gamma` for the mixture will be (vibration mode neglected)

Options

A. `1.33`
B. `1.40`
C. `1.50`
D. `1.6`

Answer

To find the equivalent value of \( \gamma \) (gamma) for a mixture of one mole of a monatomic gas and one mole of a diatomic gas, we can follow these steps:

### Step 1: Identify the degrees of freedom
- For a monatomic gas, the degrees of freedom \( F \) is 3.
- For a diatomic gas (neglecting vibrational modes), the degrees of freedom \( F \) is 5.

### Step 2: Calculate \( C_V \) for each gas
- The formula for \( C_V \) (specific heat at constant volume) is given by:
\[ C_V = \frac{F}{2} R \]
- For the monatomic gas:
\[ C_{V, \text{monatomic}} = \frac{3}{2} R \]
- For the diatomic gas:
\[ C_{V, \text{diatomic}} = \frac{5}{2} R \]

### Step 3: Calculate the mixture's \( C_V \)
- The total \( C_V \) for the mixture can be calculated using the formula:
\[ C_{V, \text{mix}} = \frac{N_1 C_{V1} + N_2 C_{V2}}{N_1 + N_2} \]
- Here, \( N_1 = 1 \) (monatomic gas), \( C_{V1} = \frac{3}{2} R \), \( N_2 = 1 \) (diatomic gas), \( C_{V2} = \frac{5}{2} R \).
- Plugging in the values:
\[ C_{V, \text{mix}} = \frac{1 \cdot \frac{3}{2} R + 1 \cdot \frac{5}{2} R}{1 + 1} = \frac{\frac{3}{2} R + \frac{5}{2} R}{2} = \frac{8/2 R}{2} = 2R \]

### Step 4: Calculate \( C_P \) for each gas
- The formula for \( C_P \) (specific heat at constant pressure) is given by:
\[ C_P = C_V + R \]
- For the monatomic gas:
\[ C_{P, \text{monatomic}} = \frac{3}{2} R + R = \frac{5}{2} R \]
- For the diatomic gas:
\[ C_{P, \text{diatomic}} = \frac{5}{2} R + R = \frac{7}{2} R \]

### Step 5: Calculate the mixture's \( C_P \)
- Using the same formula for the mixture:
\[ C_{P, \text{mix}} = \frac{N_1 C_{P1} + N_2 C_{P2}}{N_1 + N_2} \]
- Plugging in the values:
\[ C_{P, \text{mix}} = \frac{1 \cdot \frac{5}{2} R + 1 \cdot \frac{7}{2} R}{1 + 1} = \frac{\frac{5}{2} R + \frac{7}{2} R}{2} = \frac{12/2 R}{2} = 3R \]

### Step 6: Calculate \( \gamma \) for the mixture
- Finally, \( \gamma \) is calculated using the formula:
\[ \gamma = \frac{C_P}{C_V} \]
- Substituting the values:
\[ \gamma = \frac{3R}{2R} = \frac{3}{2} = 1.5 \]

### Final Answer
The equivalent value of \( \gamma \) for the mixture is \( 1.5 \).