The rates of diffusion of two gases `A` and `B` are in the the ratio `1:4` A mixture contains these gases `A` and `B` in the ratio `2:3` The ratio of mole fraction of the gases `A` and `B` in the mixture is (assume that `P_(A) =P_(B))` .

Options

A. `1:6`
B. `1:12`
C. `1:18`
D. `1:24`

Answer

To solve the problem, we need to find the ratio of the mole fractions of gases A and B in a mixture, given the rates of diffusion and the composition of the mixture.

### Step-by-Step Solution:

1. **Understand the Given Ratios:**
- The rate of diffusion of gases A and B is given as \( R_A : R_B = 1 : 4 \).
- The mixture contains gases A and B in the ratio \( 2 : 3 \).

2. **Use Graham's Law of Effusion:**
- According to Graham's law, the rate of diffusion of a gas is inversely proportional to the square root of its molar mass:
\[ \frac{R_A}{R_B} = \sqrt{\frac{M_B}{M_A}} \]
- Given \( \frac{R_A}{R_B} = \frac{1}{4} \), we can write:
\[ \frac{1}{4} = \sqrt{\frac{M_B}{M_A}} \]

3. **Square Both Sides:**
- Squaring both sides gives:
\[ \left(\frac{1}{4}\right)^2 = \frac{M_B}{M_A} \]
\[ \frac{1}{16} = \frac{M_B}{M_A} \]
- This implies:
\[ M_B = \frac{M_A}{16} \]

4. **Determine the Masses of Gases A and B:**
- Let the mass of gas A be \( 2x \) and the mass of gas B be \( 3x \) (since the ratio of A to B in the mixture is \( 2 : 3 \)).
- The number of moles of gas A (\( N_A \)) and gas B (\( N_B \)) can be calculated using the formula:
\[ N = \frac{mass}{molar\ mass} \]
- Thus,
\[ N_A = \frac{2x}{M_A} \]
\[ N_B = \frac{3x}{M_B} = \frac{3x}{\frac{M_A}{16}} = \frac{48x}{M_A} \]

5. **Calculate the Ratio of Moles:**
- The ratio of moles \( \frac{N_A}{N_B} \) is:
\[ \frac{N_A}{N_B} = \frac{\frac{2x}{M_A}}{\frac{48x}{M_A}} = \frac{2}{48} = \frac{1}{24} \]

6. **Find the Mole Fractions:**
- The mole fraction of A (\( X_A \)) and B (\( X_B \)) can be expressed as:
\[ X_A = \frac{N_A}{N_A + N_B} \quad \text{and} \quad X_B = \frac{N_B}{N_A + N_B} \]
- Since \( N_A : N_B = 1 : 24 \), we can express:
\[ N_A + N_B = N_A + 24N_A = 25N_A \]
- Therefore:
\[ X_A = \frac{N_A}{25N_A} = \frac{1}{25} \]
\[ X_B = \frac{N_B}{25N_A} = \frac{24N_A}{25N_A} = \frac{24}{25} \]

7. **Calculate the Ratio of Mole Fractions:**
- The ratio of mole fractions \( \frac{X_A}{X_B} \) is:
\[ \frac{X_A}{X_B} = \frac{\frac{1}{25}}{\frac{24}{25}} = \frac{1}{24} \]

### Final Result:
The ratio of the mole fractions of gases A and B in the mixture is \( 1 : 24 \).