Which sample contains the largest number of atoms?

Options

A. `1 mg of C_(4)H_(10)`
B. `1 mg of N_(2)`
C. `1 mg of Na`
D. `1 mL` of water

Explanation

To determine which sample contains the largest number of atoms, we need to calculate the number of moles of each substance based on their given mass (1 mg) and their molar masses. Then, we can find the total number of atoms in each sample.

### Step-by-Step Solution:

1. **Convert mass from mg to g**:
- Since 1 mg = 0.001 g, we will use 0.001 g for our calculations.

2. **Determine the molar mass of each substance**:
- **Methanol (CH₃OH)**:
- C: 12 g/mol
- H: 1 g/mol (3 H atoms)
- O: 16 g/mol
- Molar mass = 12 + (3 × 1) + 16 = 12 + 3 + 16 = 31 g/mol
- **Nitrogen (N₂)**:
- N: 14 g/mol (2 N atoms)
- Molar mass = 14 × 2 = 28 g/mol
- **Sodium (Na)**:
- Na: 23 g/mol
- Molar mass = 23 g/mol
- **Water (H₂O)**:
- H: 1 g/mol (2 H atoms)
- O: 16 g/mol
- Molar mass = (2 × 1) + 16 = 2 + 16 = 18 g/mol
- **Hydrogen (H₂)**:
- H: 1 g/mol (2 H atoms)
- Molar mass = 1 × 2 = 2 g/mol

3. **Calculate the number of moles for each substance**:
- Number of moles = mass (g) / molar mass (g/mol)
- **Methanol (CH₃OH)**:
- Moles = 0.001 g / 31 g/mol = 0.00003226 mol
- **Nitrogen (N₂)**:
- Moles = 0.001 g / 28 g/mol = 0.00003571 mol
- **Sodium (Na)**:
- Moles = 0.001 g / 23 g/mol = 0.00004348 mol
- **Water (H₂O)**:
- Moles = 0.001 g / 18 g/mol = 0.00005556 mol
- **Hydrogen (H₂)**:
- Moles = 0.001 g / 2 g/mol = 0.0005 mol

4. **Calculate the total number of atoms in each sample**:
- Total atoms = moles × Avogadro's number (6.022 × 10²³ atoms/mol)
- **Methanol (CH₃OH)**:
- Total atoms = 0.00003226 mol × 6.022 × 10²³ = 1.943 × 10²¹ atoms
- Total atoms = 5 atoms per molecule → 1.943 × 10²¹ × 5 = 9.715 × 10²¹ atoms
- **Nitrogen (N₂)**:
- Total atoms = 0.00003571 mol × 6.022 × 10²³ = 2.146 × 10²¹ atoms
- Total atoms = 2 atoms per molecule → 2.146 × 10²¹ × 2 = 4.292 × 10²¹ atoms
- **Sodium (Na)**:
- Total atoms = 0.00004348 mol × 6.022 × 10²³ = 2.617 × 10²¹ atoms
- Total atoms = 1 atom per molecule → 2.617 × 10²¹ × 1 = 2.617 × 10²¹ atoms
- **Water (H₂O)**:
- Total atoms = 0.00005556 mol × 6.022 × 10²³ = 3.344 × 10²¹ atoms
- Total atoms = 3 atoms per molecule → 3.344 × 10²¹ × 3 = 1.003 × 10²² atoms
- **Hydrogen (H₂)**:
- Total atoms = 0.0005 mol × 6.022 × 10²³ = 3.011 × 10²² atoms
- Total atoms = 2 atoms per molecule → 3.011 × 10²² × 2 = 6.022 × 10²² atoms

5. **Compare the total number of atoms**:
- Methanol: 9.715 × 10²¹ atoms
- Nitrogen: 4.292 × 10²¹ atoms
- Sodium: 2.617 × 10²¹ atoms
- Water: 1.003 × 10²² atoms
- Hydrogen: 6.022 × 10²² atoms

### Conclusion:
The sample that contains the largest number of atoms is **1 mg of Hydrogen (H₂)**, with a total of 6.022 × 10²² atoms.