According to Graham's law, at a given temperature the ratio of diffusion `r_A/r_B` of gases A and B is given by
(where P and M are pressures and molecular weights of gases A and B respectively)

Options

A. `(P_A/P_B)(M_A/M_B)^(1//2)`
B. `(M_A/M_B)(P_A/P_B)^(1//2)`
C. `(P_A/P_B)(M_B/M_A)^(1//2)`
D. `(M_A/M_B)(P_B/P_A)^(1//2)`

Explanation

To solve the question regarding Graham's law of diffusion, we can follow these steps:

### Step 1: Understand Graham's Law
Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its density. Mathematically, this can be expressed as:
\[ R \propto \frac{1}{\sqrt{D}} \]
where \( R \) is the rate of diffusion and \( D \) is the density of the gas.

### Step 2: Relate Density to Pressure and Molecular Weight
Density (\( D \)) of a gas can be related to its pressure (\( P \)) and molecular weight (\( M \)) using the ideal gas law:
\[ D = \frac{PM}{RT} \]
where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin.

### Step 3: Write the Ratios for Two Gases
For two gases A and B, we can express their densities as:
\[ D_A = \frac{P_A M_A}{RT} \quad \text{and} \quad D_B = \frac{P_B M_B}{RT} \]
Now, substituting these into the expression for the rates of diffusion, we have:
\[ R_A \propto \frac{1}{\sqrt{D_A}} \quad \text{and} \quad R_B \propto \frac{1}{\sqrt{D_B}} \]

### Step 4: Form the Ratio of Rates of Diffusion
Taking the ratio of the rates of diffusion of gases A and B:
\[ \frac{R_A}{R_B} = \frac{\sqrt{D_B}}{\sqrt{D_A}} \]

### Step 5: Substitute the Densities
Substituting the expressions for \( D_A \) and \( D_B \):
\[ \frac{R_A}{R_B} = \frac{\sqrt{\frac{P_B M_B}{RT}}}{\sqrt{\frac{P_A M_A}{RT}}} \]
The \( RT \) cancels out:
\[ \frac{R_A}{R_B} = \sqrt{\frac{P_B M_B}{P_A M_A}} \]

### Step 6: Final Expression
Thus, we can express the final ratio of the rates of diffusion of gases A and B as:
\[ \frac{R_A}{R_B} = \frac{P_B}{P_A} \cdot \sqrt{\frac{M_B}{M_A}} \]

### Summary
The ratio of the rates of diffusion of gases A and B according to Graham's law is given by:
\[ \frac{R_A}{R_B} = \frac{P_B}{P_A} \cdot \sqrt{\frac{M_B}{M_A}} \]

---