A gas X diffuses three times faster than another gas Y, the ratio of their densities i.e., `D_(x) : D_(y)` is

Options

A. `1//3`
B. `1//9`
C. `1//6`
D. `1//12`

Explanation

To solve the problem of finding the ratio of the densities of two gases X and Y, given that gas X diffuses three times faster than gas Y, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Diffusion Rate and Density**:
The rate of diffusion of a gas is inversely proportional to the square root of its density. This relationship can be expressed mathematically as:
\[ \text{Rate} \propto \frac{1}{\sqrt{D}} \]
Therefore, we can write:
\[ \frac{\text{Rate of } X}{\text{Rate of } Y} = \frac{\sqrt{D_Y}}{\sqrt{D_X}} \]

2. **Define the Rates of Diffusion**:
Let the rate of diffusion of gas Y be \( R_Y \) and the rate of diffusion of gas X be \( R_X \). According to the problem, gas X diffuses three times faster than gas Y:
\[ R_X = 3R_Y \]

3. **Set Up the Equation**:
Substituting the rates into the equation from step 1 gives:
\[ \frac{3R_Y}{R_Y} = \frac{\sqrt{D_Y}}{\sqrt{D_X}} \]
This simplifies to:
\[ 3 = \frac{\sqrt{D_Y}}{\sqrt{D_X}} \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ 3^2 = \left(\frac{\sqrt{D_Y}}{\sqrt{D_X}}\right)^2 \]
This results in:
\[ 9 = \frac{D_Y}{D_X} \]

5. **Find the Ratio of Densities**:
Rearranging the equation gives us the ratio of the densities:
\[ \frac{D_X}{D_Y} = \frac{1}{9} \]

### Final Result:
The ratio of the densities of gas X to gas Y is:
\[ D_X : D_Y = 1 : 9 \]

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