Ratio of `(/_\T_(b))/(K_(b))` of 10 g `AB_(2)` and 14 g `A_(2)B` per 100 g of solvent in their respective, solution (`AB_(2)and A_(2)B`both are non-electrolytes ) is 1 mole/ kg in both cases. Hence, atomic wt. of A and B are respectively :

Options

A. 100, 40
B. 60, 20
C. 20, 60
D. None of these

Answer

To solve the problem, we need to determine the atomic weights of elements A and B based on the given information about the solutions of non-electrolytes AB₂ and A₂B.

### Step-by-Step Solution:

1. **Understanding the Given Data**:
- We have two compounds: AB₂ and A₂B.
- For AB₂, we have 10 g dissolved in 100 g of solvent.
- For A₂B, we have 14 g dissolved in 100 g of solvent.
- The molality (m) of both solutions is given as 1 mole/kg.

2. **Using the Formula for Molality**:
- The formula for molality (m) is:
\[ m = \frac{\text{number of moles of solute}}{\text{mass of solvent (kg)}} \]
- Since we have 100 g of solvent, this is equivalent to 0.1 kg.

3. **Calculating Moles of Solute**:
- For AB₂:
\[ m = \frac{10 \, \text{g}}{M_{AB2} \, \text{(g/mol)}} \div 0.1 \, \text{kg} = 1 \, \text{mol/kg} \]
Rearranging gives:
\[ 1 = \frac{10}{M_{AB2} \times 0.1} \implies M_{AB2} = 100 \, \text{g/mol} \]

- For A₂B:
\[ m = \frac{14 \, \text{g}}{M_{A2B} \, \text{(g/mol)}} \div 0.1 \, \text{kg} = 1 \, \text{mol/kg} \]
Rearranging gives:
\[ 1 = \frac{14}{M_{A2B} \times 0.1} \implies M_{A2B} = 140 \, \text{g/mol} \]

4. **Setting Up the Equations**:
- For AB₂, the molecular weight can be expressed as:
\[ M_{AB2} = A + 2B = 100 \quad \text{(Equation 1)} \]
- For A₂B, the molecular weight can be expressed as:
\[ M_{A2B} = 2A + B = 140 \quad \text{(Equation 2)} \]

5. **Solving the Equations**:
- From Equation 1:
\[ A + 2B = 100 \quad \text{(1)} \]
- From Equation 2:
\[ 2A + B = 140 \quad \text{(2)} \]
- Multiply Equation (1) by 2:
\[ 2A + 4B = 200 \quad \text{(3)} \]
- Now subtract Equation (2) from Equation (3):
\[ (2A + 4B) - (2A + B) = 200 - 140 \]
This simplifies to:
\[ 3B = 60 \implies B = 20 \]

6. **Finding A**:
- Substitute B back into Equation (1):
\[ A + 2(20) = 100 \implies A + 40 = 100 \implies A = 60 \]

7. **Final Result**:
- The atomic weights of A and B are:
\[ A = 60 \quad \text{and} \quad B = 20 \]

### Conclusion:
The atomic weights of A and B are respectively 60 and 20.