To balance the chemical equation [tex]\( \text{BaCl}_2 + \text{Na}_2\text{SO}_4 \rightarrow \text{NaCl} + \text{BaSO}_4 \)[/tex], we need to ensure that the number of atoms of each element is equal on both the reactant and product sides.
Let's denote the coefficients for each compound as follows:
- [tex]\( b \)[/tex] for [tex]\( \text{BaCl}_2 \)[/tex]
- [tex]\( n \)[/tex] for [tex]\( \text{Na}_2\text{SO}_4 \)[/tex]
- [tex]\( c \)[/tex] for [tex]\( \text{NaCl} \)[/tex]
- [tex]\( s \)[/tex] for [tex]\( \text{BaSO}_4 \)[/tex]
We write the following equations by balancing each element:
1. Barium (Ba):
[tex]\( b = s \)[/tex]
2. Chlorine (Cl):
[tex]\( 2b = c \)[/tex]
3. Sodium (Na):
[tex]\( 2n = c \)[/tex]
4. Sulfur (S):
[tex]\( n = s \)[/tex]
Now we will solve these equations.
First, we choose [tex]\( b = 1 \)[/tex]. Then, from the first equation:
[tex]\[ b = s \][/tex]
[tex]\[ 1 = s \implies s = 1 \][/tex]
Using the value of [tex]\( s \)[/tex] in the equation [tex]\( n = s \)[/tex]:
[tex]\[ n = 1 \][/tex]
Next, we use the value of [tex]\( b \)[/tex] in the equation [tex]\( 2b = c \)[/tex]:
[tex]\[ 2 \cdot 1 = c \implies c = 2 \][/tex]
Now we verify that all elements are balanced with these coefficients:
1. [tex]\( \text{Ba} \)[/tex]: [tex]\( 1 \, \text{BaCl}_2 \)[/tex] on the left and [tex]\( 1 \, \text{BaSO}_4 \)[/tex] on the right.
2. [tex]\( \text{Cl} \)[/tex]: [tex]\( 2 \, \text{Cl} \)[/tex] in [tex]\( 1 \, \text{BaCl}_2 \)[/tex] on the left and [tex]\( 2 \, \text{Cl} \)[/tex] in [tex]\( 2 \, \text{NaCl} \)[/tex] on the right.
3. [tex]\( \text{Na} \)[/tex]: [tex]\( 2 \, \text{Na} \)[/tex] in [tex]\( 1 \, \text{Na}_2\text{SO}_4 \)[/tex] on the left and [tex]\( 2 \, \text{Na} \)[/tex] in [tex]\( 2 \, \text{NaCl} \)[/tex] on the right.
4. [tex]\( \text{S} \)[/tex]: [tex]\( 1 \, \text{S} \)[/tex] in [tex]\( 1 \, \text{Na}_2\text{SO}_4 \)[/tex] on the left and [tex]\( 1 \, \text{S} \)[/tex] in [tex]\( 1 \, \text{BaSO}_4 \)[/tex] on the right.
Thus, the coefficients that balance the equation are [tex]\( [1, 1, 2, 1] \)[/tex].
The correct option from the given multiple choices is:
[tex]\[ \boxed{1, 1, 2, 1} \][/tex]
