Calculate the number of atoms in each of the following (i) 52 moles of Ar (ii) 52 u of He (iii) 52 g of He.

Explanation

To solve the problem, we will calculate the number of atoms in each case step by step.

### Step 1: Calculate the number of atoms in 52 moles of Argon (Ar)

1. **Identify the number of moles**: We have 52 moles of Argon.
2. **Use Avogadro's number**: Avogadro's number (Na) is approximately \(6.022 \times 10^{23}\) atoms/mole.
3. **Calculate the total number of atoms**:
\[ \text{Number of atoms} = \text{Number of moles} \times \text{Avogadro's number} \]
\[ \text{Number of atoms} = 52 \, \text{moles} \times 6.022 \times 10^{23} \, \text{atoms/mole} \]
\[ \text{Number of atoms} = 3.13 \times 10^{25} \, \text{atoms} \]

### Step 2: Calculate the number of atoms in 52 u of Helium (He)

1. **Identify the mass in atomic mass units (u)**: We have 52 u of Helium.
2. **Convert atomic mass units to grams**: Since 1 u = \( \frac{1}{N_a} \) grams, we convert 52 u to grams:
\[ \text{Mass in grams} = \frac{52 \, \text{u}}{N_a} \, \text{grams} \]
3. **Calculate the number of moles**:
- The molar mass of Helium (He) is 4 g/mol.
\[ \text{Number of moles} = \frac{\text{Mass in grams}}{\text{Molar mass}} = \frac{\frac{52}{N_a}}{4} = \frac{52}{4N_a} \, \text{moles} \]
4. **Calculate the total number of atoms**:
\[ \text{Number of atoms} = \text{Number of moles} \times N_a = \left(\frac{52}{4N_a}\right) \times N_a \]
\[ \text{Number of atoms} = \frac{52}{4} = 13 \, \text{atoms} \]

### Step 3: Calculate the number of atoms in 52 g of Helium (He)

1. **Identify the mass**: We have 52 grams of Helium.
2. **Calculate the number of moles**:
\[ \text{Number of moles} = \frac{\text{Mass}}{\text{Molar mass}} = \frac{52 \, \text{g}}{4 \, \text{g/mol}} = 13 \, \text{moles} \]
3. **Calculate the total number of atoms**:
\[ \text{Number of atoms} = \text{Number of moles} \times N_a = 13 \times 6.022 \times 10^{23} \, \text{atoms/mole} \]
\[ \text{Number of atoms} = 7.8 \times 10^{24} \, \text{atoms} \]

### Summary of Results:
- (i) Number of atoms in 52 moles of Ar: \(3.13 \times 10^{25}\) atoms
- (ii) Number of atoms in 52 u of He: 13 atoms
- (iii) Number of atoms in 52 g of He: \(7.8 \times 10^{24}\) atoms