The elements A and B form purely covalent compounds having molecular formulae AB2 and AB4. When dissolved in 20g of benzene, 1g of AB2 lowers the freezing point by 2.3K, whereas 1g of AB4 lowers it by 1.3K. the molal depression constant for benzene is 5.1 K kg/mol, calculate the atomic mass of A and atomic mass of B.

To solve the problem of determining the atomic masses of elements A and B from the given information about their compounds AB2 and AB4, we can use the concept of freezing point depression. This phenomenon occurs when a solute is added to a solvent, resulting in a lower freezing point than that of the pure solvent. The relationship can be expressed using the formula:

Freezing Point Depression Formula

The freezing point depression can be calculated using the equation:

ΔTf = Kf × m

Where:

• ΔTf = change in freezing point (in K)
• Kf = molal depression constant (in K kg/mol)
• m = molality of the solution (in mol/kg)

Step 1: Calculate Molality for Each Compound

First, we need to find the molality of the solutions for both compounds AB2 and AB4. Molality (m) is defined as the number of moles of solute per kilogram of solvent. The mass of the solvent (benzene) is given as 20g, which is equivalent to 0.020 kg.

For AB2:

Given that 1g of AB2 lowers the freezing point by 2.3K:

Using the freezing point depression formula:

• ΔTf = 2.3 K
• Kf = 5.1 K kg/mol

We can rearrange the formula to find molality:

m = ΔTf / Kf = 2.3 K / 5.1 K kg/mol = 0.45098 mol/kg

For AB4:

Similarly, for AB4, where 1g lowers the freezing point by 1.3K:

ΔTf = 1.3 K

Calculating molality:

m = ΔTf / Kf = 1.3 K / 5.1 K kg/mol = 0.25490 mol/kg

Step 2: Relate Molality to Moles of Solute

Now, we can relate the molality to the number of moles of solute:

For AB2:

m = moles of AB2 / mass of solvent (kg)

0.45098 mol/kg = moles of AB2 / 0.020 kg

moles of AB2 = 0.45098 mol/kg × 0.020 kg = 0.0090196 mol

For AB4:

0.25490 mol/kg = moles of AB4 / 0.020 kg

moles of AB4 = 0.25490 mol/kg × 0.020 kg = 0.005098 mol

Step 3: Calculate Molar Masses of Compounds

Next, we can find the molar masses of the compounds using the number of moles calculated:

For AB2:

Since we have 1g of AB2:

Molar mass of AB2 = mass of AB2 / moles of AB2 = 1g / 0.0090196 mol = 110.9 g/mol

For AB4:

For 1g of AB4:

Molar mass of AB4 = mass of AB4 / moles of AB4 = 1g / 0.005098 mol = 196.1 g/mol

Step 4: Set Up Equations for Atomic Masses

Now we can express the molar masses in terms of the atomic masses of A and B:

• For AB2: Molar mass = M(A) + 2 × M(B) = 110.9 g/mol
• For AB4: Molar mass = M(A) + 4 × M(B) = 196.1 g/mol

Step 5: Solve the System of Equations

We now have a system of two equations:

• M(A) + 2M(B) = 110.9
• M(A) + 4M(B) = 196.1

Subtract the first equation from the second:

(M(A) + 4M(B)) - (M(A) + 2M(B)) = 196.1 - 110.9

2M(B) = 85.2

M(B) = 42.6 g/mol

Now substitute M(B) back into one of the equations to find M(A):

M(A) + 2(42.6) = 110.9

M(A) + 85.2 = 110.9

M(A) = 110.9 - 85.2 = 25.7 g/mol

Final Results

The atomic masses of the elements are:

• Atomic mass of A: 25.7 g/mol
• Atomic mass of B: 42.6 g/mol