If the density of a mixture of `O_(2)&N_(2)` at N.T.P. is `1.258 g//l.` The
partial volume of `O_(2)` in 3 mole of mixture is

Options

A. `1L`
B. `2L`
C. `3L`
D. `4L`

Explanation

To solve the problem of finding the partial volume of \( O_2 \) in a mixture of \( O_2 \) and \( N_2 \) at N.T.P. with a given density, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Density of the mixture \( = 1.258 \, \text{g/L} \)
- At N.T.P., the volume of 1 mole of gas \( = 22.4 \, \text{L} \)

### Step 2: Calculate the Molar Mass of the Mixture
Using the formula for density:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
We can rearrange this to find the molar mass of the mixture:
\[ \text{Molar Mass} = \text{Density} \times \text{Volume of 1 mole} \]
Substituting the values:
\[ \text{Molar Mass} = 1.258 \, \text{g/L} \times 22.4 \, \text{L} = 28.18 \, \text{g/mol} \]

### Step 3: Set Up the Equation for Molar Mass
Let \( X \) be the moles of \( O_2 \) in 1 mole of the mixture. Then the moles of \( N_2 \) will be \( 1 - X \). The molar masses are:
- Molar mass of \( O_2 = 32 \, \text{g/mol} \)
- Molar mass of \( N_2 = 28 \, \text{g/mol} \)

The average molar mass can be expressed as:
\[ \text{Molar Mass} = (X \times 32) + ((1 - X) \times 28) \]
Setting this equal to the calculated average molar mass:
\[ 28.18 = 32X + 28 - 28X \]

### Step 4: Solve for \( X \)
Rearranging the equation:
\[ 28.18 = 4X + 28 \]
\[ 4X = 28.18 - 28 \]
\[ 4X = 0.18 \]
\[ X = \frac{0.18}{4} = 0.045 \]

### Step 5: Calculate the Volume of \( O_2 \) in 3 Moles of the Mixture
Now, to find the volume of \( O_2 \) in 3 moles of the mixture:
\[ \text{Volume of } O_2 = \text{Moles of } O_2 \times \text{Volume of 1 mole} \]
The moles of \( O_2 \) in 3 moles of the mixture:
\[ \text{Moles of } O_2 = 0.045 \times 3 = 0.135 \]
Calculating the volume:
\[ \text{Volume of } O_2 = 0.135 \times 22.4 = 3.024 \, \text{L} \]

### Final Answer
The partial volume of \( O_2 \) in 3 moles of the mixture is approximately \( 3.02 \, \text{L} \).

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