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, and determining the relationship between `alpha` and `beta` when 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 degrees of freedom (f) is 3 (only translational motion).
- For a polyatomic gas (neglecting vibrational modes), the degrees of freedom is 6 (3 translational + 3 rotational).

2. **Calculate Molar Heat Capacities:**
- The molar heat capacity at constant volume (Cv) for a monoatomic gas is given by:
\[ C_{v,\text{mono}} = \frac{f}{2} R = \frac{3}{2} R \]
- The molar heat capacity at constant volume (Cv) for a polyatomic gas is:
\[ C_{v,\text{poly}} = \frac{f}{2} R = \frac{6}{2} R = 3R \]

3. **Mixture Behaving Like a Diatomic Gas:**
- For a diatomic gas, the molar heat capacity at constant volume is:
\[ C_{v,\text{diatomic}} = \frac{5}{2} R \]

4. **Set Up the Equation for the Mixture:**
- The total heat capacity of the mixture can be expressed as:
\[ C_{v,\text{mixture}} = \frac{n_1 C_{v,\text{mono}} + n_2 C_{v,\text{poly}}}{n_1 + n_2} \]
- Substituting the values:
\[ \frac{\alpha \cdot \frac{3}{2} R + \beta \cdot 3R}{\alpha + \beta} = \frac{5}{2} R \]

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

6. **Simplify the Equation:**
- Dividing through by R (assuming R ≠ 0):
\[ 5(\alpha + \beta) = 3\alpha + 6\beta \]
- Expanding the left side:
\[ 5\alpha + 5\beta = 3\alpha + 6\beta \]

7. **Rearranging Terms:**
- Rearranging gives:
\[ 5\alpha - 3\alpha = 6\beta - 5\beta \]
- Simplifying further:
\[ 2\alpha = \beta \]

8. **Final Result:**
- The relationship between `alpha` and `beta` is:
\[ \beta = 2\alpha \]