A sugar syrup of weight `214.2g` contains `34.2 g` of sugar `(C_(12)H_(22)O_(11))`. Calculate (i) molal concentration and (ii) mole fraction of sugar in the syrup

Answer

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

### Given:
- Weight of sugar syrup = 214.2 g
- Weight of sugar (C₁₂H₂₂O₁₁) = 34.2 g

### Step 1: Calculate the number of moles of sugar

To find the number of moles of sugar, we use the formula:

\[ \text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]

The molar mass of sugar (C₁₂H₂₂O₁₁) is calculated as follows:
- Carbon (C): 12.01 g/mol × 12 = 144.12 g/mol
- Hydrogen (H): 1.008 g/mol × 22 = 22.176 g/mol
- Oxygen (O): 16.00 g/mol × 11 = 176.00 g/mol

Total molar mass of sugar:
\[ 144.12 + 22.176 + 176.00 = 342.296 \text{ g/mol} \approx 342 \text{ g/mol} \]

Now, calculate the number of moles of sugar:
\[ \text{Number of moles of sugar} = \frac{34.2 \text{ g}}{342 \text{ g/mol}} \approx 0.100 \text{ mol} \]

### Step 2: Calculate the mass of the solvent (water)

The mass of the solvent (water) can be calculated by subtracting the mass of sugar from the total mass of the syrup:
\[ \text{Mass of water} = \text{Total mass of syrup} - \text{Mass of sugar} = 214.2 \text{ g} - 34.2 \text{ g} = 180.0 \text{ g} \]

### Step 3: Convert the mass of water to kilograms

Since molality is defined as moles of solute per kilogram of solvent, we convert the mass of water to kilograms:
\[ \text{Mass of water in kg} = \frac{180.0 \text{ g}}{1000} = 0.180 \text{ kg} \]

### Step 4: Calculate the molal concentration

Molality (m) is calculated using the formula:
\[ m = \frac{\text{Number of moles of solute}}{\text{Mass of solvent (kg)}} \]
\[ m = \frac{0.100 \text{ mol}}{0.180 \text{ kg}} \approx 0.556 \text{ mol/kg} \]

### 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:
\[ \text{Number of moles of water} = \frac{180.0 \text{ g}}{18 \text{ g/mol}} = 10.0 \text{ mol} \]

### Step 6: Calculate the mole fraction of sugar

The mole fraction (X) of sugar is calculated using the formula:
\[ X_{\text{sugar}} = \frac{\text{Number of moles of sugar}}{\text{Number of moles of sugar} + \text{Number of moles of water}} \]
\[ X_{\text{sugar}} = \frac{0.100 \text{ mol}}{0.100 \text{ mol} + 10.0 \text{ mol}} \approx \frac{0.100}{10.1} \approx 0.00990 \]

### Final Answers:
(i) Molal concentration = 0.556 mol/kg
(ii) Mole fraction of sugar = 0.00990