Understanding the Mole Concept
In chemistry, the mole is a fundamental unit used to bridge the gap between the microscopic world of atoms and the macroscopic world of laboratory measurements (grams).
Key constants to remember:
- Avogadro's Number ($N_A$): $6.022 \times 10^{23}$ atoms/mole
The Problem
We are given:
- Number of atoms ($N$) = $2 \times 10^{21}$
- Mass of these atoms = $0.4\text{ g}$
- Find: Mass of $0.5\text{ mole}$ of the same element.
Step-by-Step Solution
Step 1: Find the Molar Mass ($M$)
First, we determine the mass of one mole of this element. We know that:
$\text{Number of moles } (n) = \frac{\text{Number of atoms } (N)}{\text{Avogadro's number } (N_A)}$
Substituting the given values:
$n = \frac{2 \times 10^{21}}{6.022 \times 10^{23}} \approx 3.321 \times 10^{-3} \text{ moles}$
Now, use the formula $\text{Mass} = n \times M$ to find $M$:
$0.4\text{ g} = (3.321 \times 10^{-3} \text{ mol}) \times M$
$M = \frac{0.4}{3.321 \times 10^{-3}} \approx 120.44 \text{ g/mol}$
Step 2: Calculate the mass of 0.5 mole
Now that we have the molar mass ($M \approx 120.44 \text{ g/mol}$), we can find the mass of $0.5\text{ moles}$:
$\text{Mass} = n \times M$ $\text{Mass} = 0.5 \text{ mol} \times 120.44 \text{ g/mol}$ $\text{Mass} = 60.22 \text{ g}$
Summary
By first determining the molar mass of the element using the provided atom count and mass, we were able to scale up to the mass of $0.5$ moles. This exercise highlights the importance of the relationship between count (atoms), amount (moles), and weight (grams).