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 of finding the ratio of partial pressures of carbon dioxide (CO₂) and helium (He) when both gases effuse at the same rate, we can follow these steps:

### Step 1: Understand the relationship between effusion rate, pressure, and molar mass.
The rate of effusion of a gas is directly proportional to its partial pressure and inversely proportional to the square root of its molar mass. This can be expressed mathematically as:
\[ \text{Rate} \propto \frac{P}{\sqrt{M}} \]
where \( P \) is the partial pressure and \( M \) is the molar mass of the gas.

### Step 2: Write the equations for the rates of effusion for both gases.
Let \( R_{CO_2} \) be the rate of effusion for carbon dioxide and \( R_{He} \) be the rate of effusion for helium. We can write:
\[ R_{CO_2} = k \frac{P_{CO_2}}{\sqrt{M_{CO_2}}} \]
\[ R_{He} = k \frac{P_{He}}{\sqrt{M_{He}}} \]
where \( k \) is a constant.

### Step 3: Substitute the molar masses.
The molar mass of CO₂ is 44 g/mol and that of He is 4 g/mol. Thus, we can rewrite the equations as:
\[ R_{CO_2} = k \frac{P_{CO_2}}{\sqrt{44}} \]
\[ R_{He} = k \frac{P_{He}}{\sqrt{4}} \]

### Step 4: Set the rates equal to each other.
Since it is given that both gases effuse at the same rate, we can set \( R_{CO_2} = R_{He} \):
\[ k \frac{P_{CO_2}}{\sqrt{44}} = k \frac{P_{He}}{\sqrt{4}} \]
We can cancel \( k \) from both sides (assuming \( k \neq 0 \)):
\[ \frac{P_{CO_2}}{\sqrt{44}} = \frac{P_{He}}{\sqrt{4}} \]

### Step 5: Cross-multiply to find the ratio of pressures.
Cross-multiplying gives us:
\[ P_{CO_2} \cdot \sqrt{4} = P_{He} \cdot \sqrt{44} \]
This simplifies to:
\[ P_{CO_2} \cdot 2 = P_{He} \cdot \sqrt{44} \]

### Step 6: Solve for the ratio \( \frac{P_{CO_2}}{P_{He}} \).
Rearranging the equation gives:
\[ \frac{P_{CO_2}}{P_{He}} = \frac{\sqrt{44}}{2} \]
Calculating \( \sqrt{44} \):
\[ \sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11} \]
Thus,
\[ \frac{P_{CO_2}}{P_{He}} = \frac{2\sqrt{11}}{2} = \sqrt{11} \]

### Step 7: Final ratio.
To express this in a more standard form, we can approximate \( \sqrt{11} \):
\[ \sqrt{11} \approx 3.32 \]
Thus, the ratio of \( P_{CO_2} : P_{He} \) is approximately:
\[ P_{CO_2} : P_{He} \approx 3.32 : 1 \]

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

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