Understanding the Problem
In this problem, we are given the cost of one mole of sugar ($C_{12}H_{22}O_{11}$) and asked to find the cost of a 2 kg packet. To solve this, we must bridge the gap between chemical units (moles) and physical units (kilograms).
Step 1: Calculate the Molar Mass of Sugar
First, we need to find the mass of one mole of sucrose ($C_{12}H_{22}O_{11}$). We use the atomic masses of Carbon (C = 12), Hydrogen (H = 1), and Oxygen (O = 16).
- C: $12 \times 12 = 144$
- H: $22 \times 1 = 22$
- O: $11 \times 16 = 176$
Total Molar Mass = $144 + 22 + 176 = 342 \text{ g/mol}$
Step 2: Convert Mass from Grams to Kilograms
Since the price is per mole, and we need to know the price for a specific weight, let's establish that 1 mole of sugar weighs 342 grams, which is equal to 0.342 kg.
Step 3: Determine the Number of Moles in 2 kg
We want to find out how many moles are present in a 2 kg packet of sugar. We use the formula: $\text{Number of moles} = \frac{\text{Given Mass}}{\text{Molar Mass}}$
$\text{Moles} = \frac{2000 \text{ g}}{342 \text{ g/mol}} \approx 5.848 \text{ moles}$
Step 4: Calculate the Total Cost
Now that we know the 2 kg packet contains approximately 5.848 moles of sugar, and we know the cost is Rs. 20 per mole:
$\text{Total Cost} = \text{Number of moles} \times \text{Price per mole}$ $\text{Total Cost} = 5.848 \times 20$ $\text{Total Cost} \approx 116.96$
Summary
A 2 kg packet of sugar would cost approximately Rs. 116.96.