Understanding the Mole Concept
To determine which sample contains more molecules, we need to convert the given masses into the number of molecules. The bridge between mass and the number of particles is the mole.
Key formulas:
- Number of moles ($n$) = $\frac{\text{Given mass } (m)}{\text{Molar mass } (M)}$
- Number of molecules = $n \times N_A$ (where $N_A$ is Avogadro's constant, approx $6.022 \times 10^{23}$).
Since $N_A$ is constant for both, we simply need to compare the number of moles ($n$).
Step-by-Step Calculation
1. Analyzing Nitrogen ($N_2$)
- Given mass ($m$): 7 g
- Molar mass of Nitrogen atom ($N$): 14 g/mol
- Molar mass of Nitrogen gas ($N_2$): $2 \times 14 = 28$ g/mol
- Number of moles ($n_1$): $$n_1 = \frac{7}{28} = 0.25 \text{ moles}$$
2. Analyzing Hydrogen ($H_2$)
- Given mass ($m$): 1 g
- Molar mass of Hydrogen atom ($H$): 1 g/mol
- Molar mass of Hydrogen gas ($H_2$): $2 \times 1 = 2$ g/mol
- Number of moles ($n_2$): $$n_2 = \frac{1}{2} = 0.50 \text{ moles}$$
Comparison and Conclusion
Comparing the results:
- Moles of $N_2 = 0.25$
- Moles of $H_2 = 0.50$
Since $0.50 > 0.25$, the sample of 1 g of Hydrogen contains a larger number of molecules.
Why does this happen?
Even though nitrogen atoms are heavier than hydrogen atoms, the key factor is the mass per mole. Hydrogen molecules ($H_2$) are significantly lighter (2 g/mol) than nitrogen molecules ($N_2$, 28 g/mol). Therefore, for any fixed mass, you will always have more individual molecules of the lighter substance because each molecule "costs" less mass.