Let's balance the given chemical equation step-by-step and determine the coefficient for [tex]\( \mathrm{Zn(s)} \)[/tex].
The given unbalanced equation is:
[tex]\[
\mathrm{Al}(s) + 3 \, \mathrm{Zn(NO}_3\mathrm{)_2}(aq) \rightarrow \_\_ \, \mathrm{Al(NO}_3\mathrm{)_3}(aq) + \_\_ \, \mathrm{Zn}(s)
\][/tex]
### Step 1: Identify the number of atoms of each element on both sides.
- On the left side:
- 1 Al
- 3 Zn
- [tex]\(3 \times 2 = 6\)[/tex] N (from [tex]\(\mathrm{Zn(NO}_3\mathrm{)_2}\)[/tex])
- [tex]\(3 \times 6 = 18\)[/tex] O (from [tex]\(\mathrm{Zn(NO}_3\mathrm{)_2}\)[/tex])
- On the right side:
- \_\_ Al
- \_\_ Zn
- \_\_ N (from [tex]\(\mathrm{Al(NO}_3\mathrm{)_3}\)[/tex])
- \_\_ O (from [tex]\(\mathrm{Al(NO}_3\mathrm{)_3}\)[/tex])
### Step 2: Balance the Al atoms.
Since we have 1 Al atom on the left, we need 1 Al atom on the right. Also, [tex]\(\mathrm{Al(NO}_3\mathrm{)_3}\)[/tex] indicates that there are 3 nitrate ([tex]\(\mathrm{NO}_3\)[/tex]) groups per Al atom.
Balanced Al term:
[tex]\[
\mathrm{Al}(s) + 3 \, \mathrm{Zn(NO}_3\mathrm{)_2}(aq) \rightarrow \mathrm{Al(NO}_3\mathrm{)_3}(aq) + \_\_ \, \mathrm{Zn}(s)
\][/tex]
To match the nitrates, let's add another Al on the left:
[tex]\[
2 \, \mathrm{Al}(s) + 3 \, \mathrm{Zn(NO}_3\mathrm{)_2}(aq) \rightarrow 2 \, \mathrm{Al(NO}_3\mathrm{)_3}(aq) + \_\_ \, \mathrm{Zn}(s)
\][/tex]
### Step 3: Balance the Zn atoms.
Now we need to balance the Zn atoms. There are 3 Zn atoms on the left (from [tex]\(3 \, \mathrm{Zn(NO}_3\mathrm{)_2}(aq))\)[/tex], therefore we need 3 Zn atoms on the right:
[tex]\[
2 \, \mathrm{Al}(s) + 3 \, \mathrm{Zn(NO}_3\mathrm{)_2}(aq) \rightarrow 2 \, \mathrm{Al(NO}_3\mathrm{)_3}(aq) + 3 \, \mathrm{Zn}(s)
\][/tex]
### Step 4: Verify the balance of other atoms.
Let's check all the atoms:
- Left side:
- 2 Al
- 3 Zn
- 6 N (from [tex]\(3 \times 2 = 6\)[/tex] [tex]\(\mathrm{NO}_3\)[/tex] groups)
- 18 O (from [tex]\(3 \times 6 = 18\)[/tex] [tex]\(\mathrm{NO}_3\)[/tex] groups)
- Right side:
- 2 Al
- 3 Zn
- 6 N (from [tex]\(2 \times 3 = 6\)[/tex] [tex]\(\mathrm{NO}_3\)[/tex] groups)
- 18 O (from [tex]\(2 \times 9 = 18\)[/tex] [tex]\(\mathrm{NO}_3\)[/tex] [tex]\([]\)[/tex])
The atoms are balanced on both sides of the equation.
### Conclusion
The balanced equation is:
[tex]\[
2 \, \mathrm{Al}(s) + 3 \, \mathrm{Zn(NO}_3\mathrm{)_2}(aq) \rightarrow 2 \, \mathrm{Al(NO}_3\mathrm{)_3}(aq) + 3 \, \mathrm{Zn}(s)
\][/tex]
Therefore, the coefficient for [tex]\(\mathrm{Zn(s)}\)[/tex] in the balanced equation is:
[tex]\[
\boxed{3}
\][/tex]
So, the correct answer is:
C. 3
