Let's carefully analyze the chemical formula [tex]\(3 \left( NH_4 \right)_2 SO_4\)[/tex] and Elena's calculations to identify the mistake:
1. Understanding the chemical formula:
- The expression [tex]\(3 \left( NH_4 \right)_2 SO_4\)[/tex] means we have 3 units of the compound [tex]\(\left( NH_4 \right)_2 SO_4\)[/tex].
- Within each unit of [tex]\(\left( NH_4 \right)_2 SO_4\)[/tex]:
- [tex]\(\left( NH_4 \right)_2\)[/tex]:
- 2 Nitrogen atoms because [tex]\(N\)[/tex] appears once and the subscript 2 indicates there are 2 of them.
- 2 groups of [tex]\(H_4\)[/tex], so we have [tex]\(4 \times 2 = 8\)[/tex] Hydrogen atoms.
- [tex]\(SO_4\)[/tex]:
- 1 Sulfur atom.
- 4 Oxygen atoms.
- Multiplying these by 3 because we have 3 such units in the entire formula.
2. Elena's Calculations:
- Nitrogen:
- Calculation: [tex]\(1 \times 2 \times 3 = 6\)[/tex]
- Elena's result is correct: there are 6 Nitrogen atoms.
- Hydrogen:
- Calculation: [tex]\(4 \times 2 \times 3 = 24\)[/tex]
- Elena's result is correct: there are 24 Hydrogen atoms.
- Sulfur:
- Calculation: [tex]\(1 \times 4 \times 3 = 12\)[/tex]
- Elena's result here is incorrect.
- Correct calculation: Sulfur does not have any subscript following it (other than implicitly 1), so it should be [tex]\(1 \times 3 = 3\)[/tex]. Elena mistakenly multiplied by the subscript 4 of [tex]\(O_4\)[/tex].
- Oxygen:
- Calculation: [tex]\(4 \times 3 = 12\)[/tex]
- Elena's result is correct: there are 12 Oxygen atoms.
3. Identifying the mistake:
- Elena erroneously included the subscript 4 in the calculation for the total number of sulfur atoms. She incorrectly multiplied 1 sulfur atom by 4 and then by 3, leading to [tex]\(1 \times 4 \times 3 = 12\)[/tex].
Therefore, the mistake Elena made is:
- She should not have multiplied the sulfur atoms by the subscript 4.
