To balance the chemical equation:
[tex]\[ a \, \text{HNO}_3 + b \, \text{Mg(OH)}_2 \rightarrow c \, \text{H}_2\text{O} + d \, \text{Mg(NO}_3\text{)}_2 \][/tex]
We need to find the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] that will balance the number of atoms of each element on both sides of the equation.
1. Balancing the Nitrogen (N) atoms:
- On the left side, in HNO[tex]\(_3\)[/tex], we have 1 nitrogen per molecule.
- On the right side, in Mg(NO[tex]\(_3\)[/tex])[tex]\(_2\)[/tex], each molecule has 2 nitrogen atoms.
- Thus, we need 2 HNO[tex]\(_3\)[/tex] molecules to balance these nitrogen atoms.
Therefore, [tex]\(a = 2\)[/tex].
2. Balancing the Magnesium (Mg) atoms:
- On the left side, in Mg(OH)[tex]\(_2\)[/tex], there is 1 magnesium atom per molecule.
- On the right side, in Mg(NO[tex]\(_3\)[/tex])[tex]\(_2\)[/tex], there is 1 magnesium atom per molecule.
- Thus, 1 molecule of Mg(OH)[tex]\(_2\)[/tex] is required to balance the magnesium atoms.
Therefore, [tex]\(b = 1\)[/tex].
3. Balancing the Oxygen (O) atoms:
- On the left side:
- 2 HNO[tex]\(_3\)[/tex]: each has 3 oxygen atoms, totaling 6 oxygen atoms.
- 1 Mg(OH)[tex]\(_2\)[/tex]: each has 2 oxygen atoms.
- Total oxygen atoms on left = 6 + 2 = 8.
- On the right side:
- Each H[tex]\(_2\)[/tex]O molecule has 1 oxygen atom.
- Each Mg(NO[tex]\(_3\)[/tex])[tex]\(_2\)[/tex] molecule has 6 oxygen atoms (3 from each NO[tex]\(_3\)[/tex] and there are 2 NO[tex]\(_3\)[/tex] groups).
- Therefore: [tex]\(6\)[/tex] (from Mg(NO[tex]\(_3\)[/tex])[tex]\(_2\)[/tex]) + [tex]\(c \times 1\)[/tex] (1 oxygen per H[tex]\(_2\)[/tex]O molecule).
According to the balanced total, we have [tex]\(c\)[/tex] as 2 H[tex]\(_2\)[/tex]O molecules resulting in [tex]\(2\)[/tex] oxygen atoms. So, these add up to [tex]\(6 + 2 = 8\)[/tex] oxygen atoms on the right side as well.
Therefore, [tex]\(c = 2\)[/tex].
4. Balancing the Hydrogen (H) atoms:
- On the left side:
- 2 HNO[tex]\(_3\)[/tex]: each has 1 hydrogen atom, totaling 2 hydrogen atoms.
- 1 Mg(OH)[tex]\(_2\)[/tex]: each has 2 hydrogen atoms.
- Total hydrogen atoms on left = 2 + 2 = 4.
- On the right side, in H[tex]\(_2\)[/tex]O, each molecule has 2 hydrogen atoms.
Therefore, [tex]\(c = 2\)[/tex] for aligning the hydrogen atoms resulting in [tex]\(2 \times 2 = 4\)[/tex] hydrogen atoms on the right side.
Hence the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] that balance the equation are:
[tex]\[
a = 2, \quad b = 1, \quad c = 2, \quad d = 1
\][/tex]
