The ratio of the number of moles of a monoatomic to a polyatomic gas in a mixture of the two, behaving as an diatomic gas is : (vibrational modes of freedom is to be ignored)

Options

A. `2:1`
B. `1:2`
C. `2:3`
D. `3:2`

Explanation

To solve the problem of finding the ratio of the number of moles of a monoatomic gas to a polyatomic gas in a mixture that behaves like a diatomic gas, we will follow these steps:

### Step-by-Step Solution:

1. **Define Degrees of Freedom**:
- For a monoatomic gas, the degrees of freedom (f) is 3.
- For a polyatomic gas, the degrees of freedom (f) is 6.
- For a diatomic gas, the degrees of freedom (f) is 5.

2. **Set Up the Ratio**:
- Let the number of moles of the monoatomic gas be \( x \) and the number of moles of the polyatomic gas be \( y \).
- The mixture behaves like a diatomic gas, so we will equate the effective degrees of freedom of the mixture to that of a diatomic gas.

3. **Write the Equation for Degrees of Freedom**:
- The effective degrees of freedom for the mixture can be expressed as:
\[ \frac{3x + 6y}{x + y} = 5 \]
- Here, \( 3x \) is the contribution from the monoatomic gas, \( 6y \) is from the polyatomic gas, and \( x + y \) is the total number of moles.

4. **Cross Multiply to Eliminate the Denominator**:
- Cross multiplying gives:
\[ 3x + 6y = 5(x + y) \]

5. **Expand and Rearrange the Equation**:
- Expanding the right side:
\[ 3x + 6y = 5x + 5y \]
- Rearranging gives:
\[ 3x + 6y - 5x - 5y = 0 \]
- Simplifying this leads to:
\[ -2x + y = 0 \quad \text{or} \quad y = 2x \]

6. **Find the Ratio**:
- The ratio of the number of moles of monoatomic gas to polyatomic gas is:
\[ \frac{x}{y} = \frac{x}{2x} = \frac{1}{2} \]
- Thus, the ratio \( x : y = 1 : 2 \).

### Final Answer:
The ratio of the number of moles of a monoatomic gas to a polyatomic gas in the mixture is \( 1 : 2 \).

---