The rate of diffusion of a gas is proportional to

Options

A. `(P)/(sqrt(d))`
B. `(P)/(d)`
C. `sqrt((P)/(d))`
D. `(sqrt(P))/(d)`

Explanation

To solve the question regarding the rate of diffusion of a gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Graham's Law of Diffusion**:
Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. This can be mathematically expressed as:
\[ \frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}} \]
where \( r_1 \) and \( r_2 \) are the rates of diffusion of gas 1 and gas 2, and \( M_1 \) and \( M_2 \) are their respective molar masses.

2. **Relating Molar Mass to Density**:
At constant temperature and pressure, the molar mass of a gas is related to its density (\( D \)). The relationship can be expressed as:
\[ M = D \cdot R \cdot T \]
where \( R \) is the universal gas constant and \( T \) is the temperature. Thus, we can say that the molar mass is proportional to the density of the gas.

3. **Expressing Rate of Diffusion in Terms of Density**:
Since molar mass is proportional to density, we can rewrite Graham's law in terms of density:
\[ r \propto \frac{1}{\sqrt{D}} \]
This indicates that the rate of diffusion is inversely proportional to the square root of the density of the gas.

4. **Including Pressure in the Relationship**:
The rate of diffusion can also be affected by the pressure of the gas. At constant temperature, the density of a gas is directly proportional to its pressure (from the ideal gas law). Therefore, we can express the rate of diffusion as:
\[ r \propto \frac{P}{\sqrt{D}} \]
where \( P \) is the pressure of the gas.

5. **Final Expression**:
Combining the relationships, we can conclude that the rate of diffusion of a gas is proportional to the pressure divided by the square root of its density:
\[ r \propto \frac{P}{\sqrt{D}} \]

### Conclusion:
The rate of diffusion of a gas is proportional to \( \frac{P}{\sqrt{D}} \).