Carbon dioxide and helium are kept in a container at partial pressure of `P_(CO_2)` and `P_(He)` at temperature T. A small aperture is made in the wall of the container. It is observed that both the gases effuse at the same rate. Ratio of `P_(CO_2) : P_(He)` in the container is

Options

A. `1:3:33`
B. `2:1 `
C. `3.33:1`
D. `1:2`

Answer

To solve the problem, we need to find the ratio of the partial pressures of carbon dioxide (CO₂) and helium (He) in a container where both gases are effusing at the same rate. We will apply Graham's law of effusion, which states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have two gases: Carbon Dioxide (CO₂) and Helium (He).
- Their partial pressures are denoted as \( P_{CO_2} \) and \( P_{He} \).
- The temperature is constant (T).
- Both gases are effusing at the same rate.

2. **Apply Graham's Law of Effusion**:
- According to Graham's law, the rate of effusion of a gas is given by:
\[ \text{Rate} \propto \frac{P}{\sqrt{M}} \]
where \( P \) is the partial pressure and \( M \) is the molar mass of the gas.

3. **Set Up the Equation**:
- Since both gases effuse at the same rate, we can write:
\[ \frac{\text{Rate}_{CO_2}}{\text{Rate}_{He}} = 1 \]
- This leads to:
\[ \frac{P_{CO_2}}{\sqrt{M_{CO_2}}} = \frac{P_{He}}{\sqrt{M_{He}}} \]

4. **Rearranging the Equation**:
- Rearranging gives us:
\[ \frac{P_{CO_2}}{P_{He}} = \frac{\sqrt{M_{CO_2}}}{\sqrt{M_{He}}} \]

5. **Substituting Molar Mass Values**:
- The molar mass of CO₂ is approximately 44 g/mol (12 for Carbon and 16 for each of the two Oxygen atoms).
- The molar mass of He is approximately 4 g/mol.
- Therefore:
\[ \frac{P_{CO_2}}{P_{He}} = \frac{\sqrt{44}}{\sqrt{4}} = \frac{\sqrt{44}}{2} \]

6. **Calculating the Ratio**:
- Simplifying further:
\[ \frac{P_{CO_2}}{P_{He}} = \frac{\sqrt{44}}{2} = \frac{2\sqrt{11}}{2} = \sqrt{11} \]
- Since \( \sqrt{11} \) is approximately 3.32, we can express the ratio as:
\[ P_{CO_2} : P_{He} \approx 3.32 : 1 \]

### Final Answer:
The ratio of \( P_{CO_2} : P_{He} \) in the container is approximately \( 3.32 : 1 \).