`214.2` gram of sugar syrup contains `34.2` gram of sugar. Calculate (i) molality of the solution and (ii) mole fraction of the sugar in the syrup

Options

A. `0.555` m, `0.0099`
B. `0.4565` m, `0.0110`
C. `0.355` m, `0.0199`
D. None of these

Explanation

To solve the problem, we need to calculate the molality of the sugar syrup and the mole fraction of sugar in the syrup. Let's break it down step by step.

### Given Data:
- Mass of sugar syrup = 214.2 g
- Mass of sugar = 34.2 g

### Step 1: Calculate the mass of water in the syrup
To find the mass of water, we subtract the mass of sugar from the total mass of the syrup.

\[ \text{Mass of water} = \text{Mass of sugar syrup} - \text{Mass of sugar} \]
\[ \text{Mass of water} = 214.2 \, \text{g} - 34.2 \, \text{g} = 180.0 \, \text{g} \]

### Step 2: Convert the mass of water to kilograms
Since molality is defined in terms of kilograms of solvent, we convert the mass of water from grams to kilograms.

\[ \text{Mass of water in kg} = \frac{180.0 \, \text{g}}{1000} = 0.180 \, \text{kg} \]

### Step 3: Calculate the number of moles of sugar
To find the number of moles of sugar, we use the molar mass of sugar (sucrose, C₁₂H₂₂O₁₁), which is approximately 342 g/mol.

\[ \text{Moles of sugar} = \frac{\text{Mass of sugar}}{\text{Molar mass of sugar}} = \frac{34.2 \, \text{g}}{342 \, \text{g/mol}} \approx 0.100 \, \text{mol} \]

### Step 4: Calculate the molality of the solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent.

\[ \text{Molality} = \frac{\text{Moles of sugar}}{\text{Mass of water in kg}} = \frac{0.100 \, \text{mol}}{0.180 \, \text{kg}} \approx 0.555 \, \text{mol/kg} \]

### Step 5: Calculate the number of moles of water
To find the number of moles of water, we use the molar mass of water (H₂O), which is approximately 18 g/mol.

\[ \text{Moles of water} = \frac{\text{Mass of water}}{\text{Molar mass of water}} = \frac{180.0 \, \text{g}}{18 \, \text{g/mol}} = 10.0 \, \text{mol} \]

### Step 6: Calculate the total number of moles in the syrup
Now we can find the total number of moles in the syrup, which is the sum of the moles of sugar and the moles of water.

\[ \text{Total moles} = \text{Moles of sugar} + \text{Moles of water} = 0.100 \, \text{mol} + 10.0 \, \text{mol} = 10.1 \, \text{mol} \]

### Step 7: Calculate the mole fraction of sugar
The mole fraction (X) of sugar is defined as the number of moles of sugar divided by the total number of moles in the solution.

\[ \text{Mole fraction of sugar} = \frac{\text{Moles of sugar}}{\text{Total moles}} = \frac{0.100 \, \text{mol}}{10.1 \, \text{mol}} \approx 0.00990 \]

### Summary of Results:
(i) Molality of the solution = **0.555 mol/kg**
(ii) Mole fraction of sugar in the syrup = **0.00990**

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