Ratio of the rate of effusion of oxygen gas at 1.5 atm to that of helium gas at 4.5 atm will be

Options

A. `1 : 6sqrt2`
B. `1 : 12sqrt2`
C. `1 : 2sqrt2`
D. ` 1 : 3`

Answer

To solve the problem of finding the ratio of the rate of effusion of oxygen gas at 1.5 atm to that of helium gas at 4.5 atm, we can follow these steps:

### Step 1: Understand Graham's Law of Effusion
Graham's law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass and directly proportional to its pressure. The formula can be expressed as:

\[ \frac{\text{Rate of Effusion of Gas 1}}{\text{Rate of Effusion of Gas 2}} = \frac{P_1}{P_2} \times \sqrt{\frac{M_2}{M_1}} \]

Where:
- \( P_1 \) and \( P_2 \) are the pressures of gas 1 and gas 2 respectively.
- \( M_1 \) and \( M_2 \) are the molar masses of gas 1 and gas 2 respectively.

### Step 2: Identify the Gases and Their Properties
In this case:
- Gas 1: Oxygen (O₂)
- Gas 2: Helium (He)

From the question:
- Pressure of oxygen gas (\( P_1 \)) = 1.5 atm
- Pressure of helium gas (\( P_2 \)) = 4.5 atm
- Molar mass of oxygen (\( M_1 \)) = 32 g/mol
- Molar mass of helium (\( M_2 \)) = 4 g/mol

### Step 3: Substitute the Values into the Formula
Now, we can substitute the known values into the formula:

\[ \frac{\text{Rate of Effusion of O}_2}{\text{Rate of Effusion of He}} = \frac{1.5}{4.5} \times \sqrt{\frac{4}{32}} \]

### Step 4: Simplify the Expression
First, simplify the pressure ratio:

\[ \frac{1.5}{4.5} = \frac{1}{3} \]

Next, simplify the square root term:

\[ \sqrt{\frac{4}{32}} = \sqrt{\frac{1}{8}} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}} \]

Now, combine these results:

\[ \frac{\text{Rate of Effusion of O}_2}{\text{Rate of Effusion of He}} = \frac{1}{3} \times \frac{1}{2\sqrt{2}} = \frac{1}{6\sqrt{2}} \]

### Step 5: Final Ratio
Thus, the final ratio of the rate of effusion of oxygen gas to that of helium gas is:

\[ \frac{1}{6\sqrt{2}} \text{ or } 1 : 6\sqrt{2} \]

### Conclusion
The ratio of the rate of effusion of oxygen gas at 1.5 atm to that of helium gas at 4.5 atm is \( 1 : 6\sqrt{2} \).

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