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Calculate the number of iron atoms in a [tex][tex]$100.0 \text{ g}$[/tex][/tex] sample of iron(III) oxide [tex]\left( Fe_2O_3 \right)[/tex].

Reserve your answer with a unit symbol if necessary, and round it to 4 significant digits.

[tex]
7.540 \times 10^{23}
[/tex]



Answer :

Certainly! Let’s calculate the number of iron (Fe) atoms in a 100.0 g sample of iron(III) oxide ([tex]\( Fe_2O_3 \)[/tex]) in a step-by-step manner.

1. Determine the molar mass of iron(III) oxide ([tex]\( Fe_2O_3 \)[/tex]):
- The molar mass ([tex]\(M\)[/tex]) of iron(III) oxide ([tex]\( Fe_2O_3 \)[/tex]) is given as 159.687 g/mol.

2. Calculate the number of moles of [tex]\( Fe_2O_3 \)[/tex] in the sample:
- Mass of the sample ([tex]\(m\)[/tex]) provided is 100.0 g.
- Use the formula to find the number of moles ([tex]\(n\)[/tex]):
[tex]\[ n = \frac{m}{M} \][/tex]
[tex]\[ n = \frac{100.0 \text{ g}}{159.687 \text{ g/mol}} \approx 0.6262 \text{ mol} \quad \text{(to four significant digits)} \][/tex]

3. Determine the number of molecules of [tex]\( Fe_2O_3 \)[/tex] in the sample:
- Avogadro's number ([tex]\(N_A\)[/tex]) is [tex]\(6.022 \times 10^{23}\)[/tex] molecules/mol.
- Using Avogadro's number, calculate the number of molecules:
[tex]\[ \text{Number of molecules} = n \times N_A \][/tex]
[tex]\[ \text{Number of molecules} = 0.6262 \text{ mol} \times 6.022 \times 10^{23} \text{ molecules/mol} \approx 3.7702 \times 10^{23} \text{ molecules} \][/tex]

4. Determine the number of iron atoms in the sample:
- Each molecule of [tex]\( Fe_2O_3 \)[/tex] contains 2 atoms of iron (Fe).
- Therefore, the number of iron atoms:
[tex]\[ \text{Number of iron (Fe) atoms} = \text{Number of molecules} \times 2 \][/tex]
[tex]\[ \text{Number of iron (Fe) atoms} = 3.7702 \times 10^{23} \text{ molecules} \times 2 = 7.5404 \times 10^{23} \text{ atoms} \][/tex]

5. Round the answer to 4 significant digits:
- The rounded result is:
[tex]\[ 7.540 \times 10^{23} \][/tex]

So, the number of iron ([tex]\(Fe\)[/tex]) atoms in a 100.0 g sample of iron(III) oxide ([tex]\( Fe_2O_3 \)[/tex]) is [tex]\(7.540 \times 10^{23}\)[/tex] atoms.