`9.7xx10^(17)` atoms of iron weigh as much as 1 cc of `H_(2)` at S.T.P. What is the atomic mass of iron?

Explanation

To find the atomic mass of iron based on the information given, we can follow these steps:

### Step 1: Determine the mass of 1 cc (1 mL) of H₂ at STP
At standard temperature and pressure (STP), 1 mole of any gas occupies 22.4 liters. Since 1 mole of H₂ weighs approximately 2 grams, we can calculate the mass of 1 mL of H₂.

1. **Convert liters to mL**:
\[ 1 \text{ L} = 1000 \text{ mL} \]
Therefore, 22.4 L = 22400 mL.

2. **Calculate the mass of 1 mL of H₂**:
\[ \text{Mass of 1 mL of H₂} = \frac{2 \text{ g}}{22400 \text{ mL}} = \frac{2}{22400} \text{ g} = 9 \times 10^{-5} \text{ g} \]

### Step 2: Set up the relationship between the mass of iron and the number of atoms
We know that the mass of \(9.7 \times 10^{17}\) atoms of iron is equal to the mass of 1 mL of H₂, which we calculated to be \(9 \times 10^{-5}\) g.

### Step 3: Calculate the atomic mass of iron
The atomic mass of iron can be calculated using Avogadro's number (\(N_A\)), which is approximately \(6 \times 10^{23}\) atoms/mole.

1. **Set up the equation**:
\[ \text{Mass of } N_A \text{ atoms of iron} = \left(\frac{9 \times 10^{-5} \text{ g}}{9.7 \times 10^{17} \text{ atoms}}\right) \times 6 \times 10^{23} \text{ atoms} \]

2. **Calculate**:
\[ \text{Mass of } N_A \text{ atoms of iron} = \frac{9 \times 10^{-5}}{9.7 \times 10^{17}} \times 6 \times 10^{23} \]
\[ = \frac{9 \times 6}{9.7} \times 10^{-5 + 23 - 17} \]
\[ = \frac{54}{9.7} \times 10^{1} \]
\[ \approx 5.57 \times 10^{1} \text{ g} \approx 55.9 \text{ g} \]

### Conclusion
The atomic mass of iron is approximately 56 g/mol.

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