To balance the chemical equation involving the reaction of boron trifluoride ([tex]\(BF_3\)[/tex]) and lithium sulfite ([tex]\(Li_2SO_3\)[/tex]), we need to ensure that the number of each type of atom on the reactant side is equal to the number on the product side.
Given unbalanced reaction:
[tex]\[ ?\, BF_3 + Li_2SO_3 \longrightarrow ?\, B_2(SO_3)_3 + ?\, LiF \][/tex]
Let's proceed step-by-step:
### Step 1: Determine the Atoms Involved
- Boron (B)
- Fluorine (F)
- Lithium (Li)
- Sulfur (S)
- Oxygen (O)
### Step 2: Set up the Unbalanced Equation with Coefficients
Assume the coefficients are [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] for [tex]\(BF_3\)[/tex], [tex]\(Li_2SO_3\)[/tex], [tex]\(B_2(SO_3)_3\)[/tex], and [tex]\(LiF\)[/tex] respectively:
[tex]\[ a\, BF_3 + b\, Li_2SO_3 \longrightarrow c\, B_2(SO_3)_3 + d\, LiF \][/tex]
### Step 3: Balance Each Element
1. Boron (B):
Each [tex]\(BF_3\)[/tex] has 1 Boron, and each [tex]\(B_2(SO_3)_3\)[/tex] has 2 Borons:
[tex]\[ a = 2c \][/tex]
2. Fluorine (F):
Each [tex]\(BF_3\)[/tex] has 3 Fluorines, and each [tex]\(LiF\)[/tex] has 1 Fluorine:
[tex]\[ 3a = d \][/tex]
3. Lithium (Li):
Each [tex]\(Li_2SO_3\)[/tex] has 2 Lithiums, and each [tex]\(LiF\)[/tex] has 1 Lithium:
[tex]\[ 2b = d \][/tex]
4. Sulfur (S):
Each [tex]\(Li_2SO_3\)[/tex] has 1 Sulfur, and each [tex]\(B_2(SO_3)_3\)[/tex] has 3 Sulfurs:
[tex]\[ b = 3c \][/tex]
5. Oxygen (O):
Each [tex]\(Li_2SO_3\)[/tex] has 3 Oxygens, and each [tex]\(B_2(SO_3)_3\)[/tex] has 9 Oxygens:
(This is inherently balanced if [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are balanced as determined from sulfur)
### Step 4: Solve the System of Equations
From our constraints, we can express all coefficients in terms of [tex]\(c\)[/tex]:
1. Boron: [tex]\(a = 2c\)[/tex]
2. Fluorine: [tex]\(3a = d\)[/tex] => [tex]\(3(2c) = d\)[/tex] => [tex]\(d = 6c\)[/tex]
3. Lithium: [tex]\(2b = d\)[/tex] => [tex]\(2b = 6c\)[/tex] => [tex]\(b = 3c\)[/tex]
### Step 5: Determine the Smallest Integer Coefficients
Let [tex]\(c = 1\)[/tex]:
1. [tex]\(a = 2c = 2(1) = 2\)[/tex]
2. [tex]\(b = 3c = 3(1) = 3\)[/tex]
3. [tex]\(d = 6c = 6(1) = 6\)[/tex]
### Step 6: Write the Balanced Equation
[tex]\[ 2\, BF_3 + 3\, Li_2SO_3 \longrightarrow 1\, B_2(SO_3)_3 + 6\, LiF \][/tex]
Therefore, the coefficient of lithium sulfite ([tex]\(Li_2SO_3\)[/tex]) is [tex]\(\boxed{3}\)[/tex].
