Understanding the Concept
In chemistry, the mole is a fundamental unit used to express the amount of a substance. To find the cost per mole when given the cost per kilogram, we need to bridge the gap between mass (grams/kilograms) and quantity (moles). This requires two key steps:
- Calculate the Molar Mass: Determine the mass of one mole of the substance using atomic masses from the periodic table.
- Conversion: Use the molar mass to convert the price from 'per kilogram' to 'per mole'.
The Problem
Given that the cost of table sugar ($C_{12}H_{22}O_{11}$) is Rs. 24 per kg, what is the cost per mole?
Step-by-Step Solution
1. Calculate the Molar Mass of Sugar ($C_{12}H_{22}O_{11}$)
We use the atomic masses: Carbon ($C) \approx 12.01$, Hydrogen ($H) \approx 1.01$, and Oxygen ($O) \approx 16.00$. (Standard approximate values are $C=12, H=1, O=16$ for simpler calculations).
$$M = (12 \times 12) + (22 \times 1) + (11 \times 16)$$ $$M = 144 + 22 + 176$$ $$M = 342 \text{ g/mol}$$
2. Determine Cost per Gram
We know 1 kg = 1000 grams. If 1000 g costs Rs. 24, then:
$$\text{Cost per gram} = \frac{24}{1000} = 0.024 \text{ Rs/g}$$
3. Calculate Cost per Mole
Since 1 mole of sugar weighs 342 grams, we multiply the cost per gram by the molar mass:
$$\text{Cost per mole} = \text{Cost per gram} \times \text{Molar mass}$$ $$\text{Cost per mole} = 0.024 \text{ Rs/g} \times 342 \text{ g/mol}$$ $$\text{Cost per mole} = 8.208 \text{ Rs/mol}$$
Conclusion
The cost of one mole of table sugar is approximately Rs. 8.21.