A sample of `H_(2)O_(2)` solution labelled as "28 volume" has density of 265 g/L. Mark the correct option(s) representing concentration of same solution in other units :

Options

A. `M_(H_(2)O_(2)) = 2.5`
B. `%(w)/(v) = 17`
C. `"mole fraction of" H_(2)O_(2)=0.2`
D. `m_(H_(2)O_(2))=13.88`

Explanation

To solve the problem, we need to determine the concentration of the hydrogen peroxide (H₂O₂) solution labeled as "28 volume" in different units. We will calculate the molarity, percentage by weight/volume, mole fraction, and molality step by step.

### Step 1: Understanding Volume Strength
Volume strength of a solution indicates the volume of gas that one volume of the solution can produce at standard temperature and pressure (STP). For H₂O₂, "28 volume" means that 1 liter of this solution can produce 28 liters of oxygen gas (O₂) at STP.

### Step 2: Calculate Molarity (M)
The formula to convert volume strength to molarity is:
\[ M = \frac{VS}{11.2} \]
Where:
- \( VS \) = volume strength (28 in this case)

Substituting the values:
\[ M = \frac{28}{11.2} = 2.5 \, \text{mol/L} \]

### Step 3: Calculate Percentage by Weight/Volume (W/V)
The formula to convert volume strength to percentage W/V is:
\[ \text{Percentage W/V} = \frac{VS \times 68}{22.4} \]
Where:
- \( 68 \) is the molar mass of H₂O₂,
- \( 22.4 \) is the molar volume of gas at STP.

Substituting the values:
\[ \text{Percentage W/V} = \frac{28 \times 68}{22.4} = 8.5\% \]

### Step 4: Calculate the Number of Moles of H₂O₂ in 1 L of Solution
Given the density of the solution is 265 g/L, we can find the mass of H₂O₂ in 1 liter:
\[ \text{Mass of solution} = 265 \, \text{g} \]
Since we have already calculated the molarity (2.5 mol/L), we can find the number of moles of H₂O₂:
\[ \text{Number of moles of H₂O₂} = 2.5 \, \text{mol} \]

### Step 5: Calculate the Number of Moles of Water (H₂O)
The molar mass of water (H₂O) is 18 g/mol. The mass of H₂O in 1 L of solution can be calculated as follows:
Using the density and the molarity:
\[ \text{Mass of H₂O} = \text{Total mass of solution} - \text{Mass of H₂O₂} \]
\[ \text{Mass of H₂O₂} = 2.5 \, \text{mol} \times 34 \, \text{g/mol} = 85 \, \text{g} \]
\[ \text{Mass of H₂O} = 265 \, \text{g} - 85 \, \text{g} = 180 \, \text{g} \]
Now, calculate the number of moles of H₂O:
\[ \text{Number of moles of H₂O} = \frac{180 \, \text{g}}{18 \, \text{g/mol}} = 10 \, \text{mol} \]

### Step 6: Calculate Mole Fraction of H₂O₂
The mole fraction of H₂O₂ can be calculated as:
\[ \text{Mole fraction of H₂O₂} = \frac{\text{Moles of H₂O₂}}{\text{Moles of H₂O₂} + \text{Moles of H₂O}} \]
\[ \text{Mole fraction of H₂O₂} = \frac{2.5}{2.5 + 10} = \frac{2.5}{12.5} = 0.2 \]

### Step 7: Calculate Molality (m)
Molality is calculated using the formula:
\[ m = \frac{\text{Moles of solute}}{\text{Mass of solvent (kg)}} \]
The mass of the solvent (water) is 180 g, which is 0.180 kg:
\[ m = \frac{2.5 \, \text{mol}}{0.180 \, \text{kg}} \approx 13.89 \, \text{mol/kg} \]

### Summary of Results
- Molarity (M) = 2.5 mol/L
- Percentage W/V = 8.5%
- Mole Fraction of H₂O₂ = 0.2
- Molality (m) = 13.89 mol/kg