Understanding the Concept
To solve this problem, we need to bridge the gap between mass (grams) and the number of particles (molecules). We use the Mole Concept, which acts as a conversion factor between the macroscopic scale (what we can weigh) and the microscopic scale (atoms and molecules).
Key Constants and Formulae
- Avogadro's Number ($N_A$): $6.022 \times 10^{23}$ molecules per mole.
- Molar Mass of Methane ($CH_4$):
- Atomic mass of Carbon (C) = 12 g/mol
- Atomic mass of Hydrogen (H) = 1 g/mol
- Molar mass of $CH_4$ = $12 + (4 \times 1) = 16$ g/mol.
Step-by-Step Calculation
Step 1: Calculate the number of moles of Methane ($CH_4$)
Using the formula: $\text{Moles} = \frac{\text{Given Mass}}{\text{Molar Mass}}$
$$\text{Moles of } CH_4 = \frac{4 \text{ g}}{16 \text{ g/mol}} = 0.25 \text{ moles}$$
Step 2: Calculate the number of molecules of Methane
Every mole contains $N_A$ particles:
$$\text{Total molecules of } CH_4 = 0.25 \times 6.022 \times 10^{23} = 1.5055 \times 10^{23} \text{ molecules}$$
Step 3: Determining Hydrogen and Carbon atoms
The question asks for the "number of molecules of hydrogen and carbon." In chemistry, carbon and hydrogen in methane exist as atoms within the molecule. Let's break down the composition of one molecule of $CH_4$:
- 1 molecule of $CH_4$ contains 1 atom of Carbon.
- 1 molecule of $CH_4$ contains 4 atoms of Hydrogen.
For Carbon: Since there is 1 Carbon atom per molecule: $$\text{Number of Carbon atoms} = 1.5055 \times 10^{23} \times 1 = 1.5055 \times 10^{23} \text{ atoms}$$
For Hydrogen: Since there are 4 Hydrogen atoms per molecule: $$\text{Number of Hydrogen atoms} = 1.5055 \times 10^{23} \times 4 = 6.022 \times 10^{23} \text{ atoms}$$
Final Answer Summary
- Carbon atoms: $1.5055 \times 10^{23}$
- Hydrogen atoms: $6.022 \times 10^{23}$
Note: While the question asks for "molecules of hydrogen and carbon," it is chemically standard to refer to these as individual atoms since they are covalently bonded in the methane structure.