Calulate the molesules mass of a substance whose 7.0% by mass solution in water freezes at `-0.93^(@)C`. the cryosctopic constant of water is `1.86^(@)C kg mol^(-1)` :
(a)`140 g mol ^(-1)`
(b)`150.5 g mol ^(-1)`
(c)`160 g mol ^(-1)`
(d)`155 g mol ^(-1)`

Options

A. `140 g mol ^(-1)`
B. `150.5 g mol ^(-1)`
C. `160 g mol ^(-1)`
D. `155 g mol ^(-1)`

Answer

To solve the problem, we need to calculate the molecular mass of a substance based on the freezing point depression of a 7% by mass solution in water. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
- We have a 7% by mass solution of a solute in water.
- The freezing point of the solution is -0.93°C.
- The cryoscopic constant (Kf) of water is 1.86°C kg/mol.

### Step 2: Calculate the Mass of Solute and Solvent
Since the solution is 7% by mass, we can assume:
- Mass of the solute (solute) = 7 g
- Mass of the solution = 100 g
- Therefore, mass of the solvent (water) = 100 g - 7 g = 93 g

### Step 3: Convert Mass of Solvent to Kilograms
To use the cryoscopic constant, we need the mass of the solvent in kilograms:
- Mass of solvent = 93 g = 93/1000 kg = 0.093 kg

### Step 4: Calculate the Freezing Point Depression (ΔTf)
The change in freezing point (ΔTf) is given by:
- ΔTf = Kf × molality
Where:
- ΔTf = 0.93°C (since the freezing point is lowered)
- Kf = 1.86°C kg/mol

### Step 5: Rearranging the Formula to Find Molality
We can rearrange the formula to find molality (m):
- m = ΔTf / Kf
- m = 0.93°C / 1.86°C kg/mol
- m = 0.5 mol/kg

### Step 6: Use the Definition of Molality
Molality (m) is defined as:
- m = (moles of solute) / (mass of solvent in kg)

Let’s denote the molecular mass of the solute as M. The number of moles of solute can be expressed as:
- Moles of solute = mass of solute / M = 7 g / M

Substituting into the molality equation:
- 0.5 mol/kg = (7 g / M) / 0.093 kg

### Step 7: Solve for Molecular Mass (M)
Now we can solve for M:
- 0.5 = (7 / M) / 0.093
- 0.5 × 0.093 = 7 / M
- 0.0465 = 7 / M
- M = 7 / 0.0465
- M ≈ 150.5 g/mol

### Step 8: Conclusion
The molecular mass of the substance is approximately 150.5 g/mol.

### Final Answer
(b) 150.5 g/mol

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