A football bladder contains equimolar proportions of `H_(2)` and `O_(2)`. The composition by mass of the mixture effusing out of punctured football is in the ration `(H_(2) : O_(2))`

Options

A. `1:4`
B. `2sqrt2 : 1`
C. `1: 2 sqrt 2`
D. `4:1`

Answer

To solve the problem of the composition by mass of the mixture effusing out of a punctured football bladder containing equimolar proportions of \( H_2 \) and \( O_2 \), we can follow these steps:

### Step 1: Understand the Problem
We have a mixture of two gases, hydrogen (\( H_2 \)) and oxygen (\( O_2 \)), in equal molar proportions. We need to find the mass ratio of the gases that effuse out of the punctured football.

### Step 2: Apply Graham's Law of Effusion
According to Graham's law of effusion, the rate of effusion of a gas is inversely proportional to the square root of its molar mass. This can be expressed mathematically as:

\[ \frac{\text{Rate of } H_2}{\text{Rate of } O_2} = \sqrt{\frac{M_{O_2}}{M_{H_2}}} \]

Where \( M_{O_2} \) and \( M_{H_2} \) are the molar masses of oxygen and hydrogen, respectively.

### Step 3: Determine Molar Masses
The molar masses of the gases are:
- Molar mass of \( H_2 \) = 2 g/mol
- Molar mass of \( O_2 \) = 32 g/mol

### Step 4: Calculate the Ratio of Effusion Rates
Substituting the molar masses into Graham's law equation:

\[ \frac{\text{Rate of } H_2}{\text{Rate of } O_2} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4 \]

This means that hydrogen effuses 4 times faster than oxygen.

### Step 5: Relate Effusion Rates to Mass
Since the rates of effusion are proportional to the masses of the gases that effuse, we can express the ratio of the masses of \( H_2 \) and \( O_2 \) that effuse as:

\[ \frac{m_{H_2}}{m_{O_2}} = \frac{\text{Rate of } H_2}{\text{Rate of } O_2} = 4 \]

### Step 6: Finalize the Mass Ratio
Thus, the mass ratio of \( H_2 \) to \( O_2 \) in the effusing mixture is:

\[ H_2 : O_2 = 4 : 1 \]

### Conclusion
The composition by mass of the mixture effusing out of the punctured football is in the ratio \( H_2 : O_2 = 4 : 1 \).

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