If `alpha` moles of a monoatomic gas are mixed with `beta` moles of a polyatomic gas and mixture behaves like diatomic gas, then [neglect the vibration mode of freedom]

Options

A. `2 alpha = beta`
B. ` alpha = 2beta`
C. `alpha = -3 beta`
D. `3 alpha = - beta`

Answer

To solve the problem of mixing `alpha` moles of a monoatomic gas with `beta` moles of a polyatomic gas such that the mixture behaves like a diatomic gas, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Degrees of Freedom**:
- For a monoatomic gas, the degree of freedom (F) is 3.
- For a diatomic gas, F is 5.
- For a polyatomic gas, F is 6 (neglecting vibrational modes).

2. **Use the Formula for Specific Heat Capacity**:
- The specific heat capacity at constant volume (C_V) for a gas can be expressed as:
\[ C_V = \frac{F}{2} R \]
- Where R is the universal gas constant.

3. **Calculate Specific Heat Capacities**:
- For the monoatomic gas (C1):
\[ C_1 = \frac{3}{2} R \]
- For the polyatomic gas (C2):
\[ C_2 = \frac{6}{2} R = 3R \]

4. **Calculate the Specific Heat Capacity of the Mixture**:
- The specific heat capacity of the mixture (C_mixture) behaves like a diatomic gas:
\[ C_{mixture} = \frac{5}{2} R \]

5. **Set Up the Equation for the Mixture**:
- The specific heat capacity of the mixture can also be expressed as:
\[ C_{mixture} = \frac{n_1 C_1 + n_2 C_2}{n_1 + n_2} \]
- Substituting the values:
\[ \frac{C_{mixture}}{R} = \frac{\alpha \cdot \frac{3}{2} + \beta \cdot 3}{\alpha + \beta} \]

6. **Equate the Two Expressions for C_mixture**:
- Setting the two expressions for C_mixture equal:
\[ \frac{5}{2} = \frac{\alpha \cdot \frac{3}{2} + \beta \cdot 3}{\alpha + \beta} \]

7. **Cross Multiply to Eliminate the Denominator**:
- Cross-multiplying gives:
\[ 5(\alpha + \beta) = 3\alpha + 6\beta \]

8. **Rearranging the Equation**:
- Expanding and rearranging:
\[ 5\alpha + 5\beta = 3\alpha + 6\beta \]
\[ 5\alpha - 3\alpha = 6\beta - 5\beta \]
\[ 2\alpha = \beta \]

### Final Relation:
The final relation between the moles of the gases is:
\[ \beta = 2\alpha \]