Let's examine and solve the system of given chemical equations step-by-step:
### Given Equations:
1. [tex]\( \text{Equation 1: } Ca + \text{CO}_2 + \frac{1}{2} O_2 \rightarrow CaCO_3 \)[/tex]
2. [tex]\( \text{Equation 2: } 2 Ca + O_2 \rightarrow 2 CaO \)[/tex]
We are given directions to modify these equations:
- Multiply equation (1) by 3.
- Multiply equation (2) by [tex]\( \frac{1}{2} \)[/tex].
### Step-by-step modification:
1. Multiply Equation 1 by 3:
[tex]\[
3 \times \left( Ca + \text{CO}_2 + \frac{1}{2} O_2 \rightarrow CaCO_3 \right)
\][/tex]
Let's distribute the 3 to each term in the reaction:
[tex]\[
3 Ca + 3 \text{CO}_2 + 3 \left(\frac{1}{2} O_2 \right) \rightarrow 3 CaCO_3
\][/tex]
Simplifying the terms, this becomes:
[tex]\[
3 Ca + 3 \text{CO}_2 + \frac{3}{2} O_2 \rightarrow 3 CaCO_3
\][/tex]
2. Multiply Equation 2 by [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[
\frac{1}{2} \times \left( 2 Ca + O_2 \rightarrow 2 CaO \right)
\][/tex]
Let's distribute the [tex]\( \frac{1}{2} \)[/tex] to each term in the reaction:
[tex]\[
\frac{1}{2} \times 2 Ca + \frac{1}{2} \times O_2 \rightarrow \frac{1}{2} \times 2 CaO
\][/tex]
Simplifying the terms, this becomes:
[tex]\[
1 Ca + \frac{1}{2} O_2 \rightarrow 1 CaO
\][/tex]
### Resulting Modified Equations:
1. Modified Equation 1:
[tex]\[
3 Ca + 3 \text{CO}_2 + \frac{3}{2} O_2 \rightarrow 3 CaCO_3
\][/tex]
2. Modified Equation 2:
[tex]\[
1 Ca + \frac{1}{2} O_2 \rightarrow 1 CaO
\][/tex]
### Summary:
The detailed, step-by-step modified coefficients for each term of the equations based on the directives provided are as follows:
- Modified Equation 1:
- [tex]\( Ca \)[/tex] coefficient: 3
- [tex]\( \text{CO}_2 \)[/tex] coefficient: 3
- [tex]\( O_2 \)[/tex] coefficient: [tex]\( \frac{3}{2} = 1.5 \)[/tex]
- [tex]\( CaCO_3 \)[/tex] coefficient: 3
- Modified Equation 2:
- [tex]\( Ca \)[/tex] coefficient: 1.0
- [tex]\( O_2 \)[/tex] coefficient: 0.5
- [tex]\( CaO \)[/tex] coefficient: 1.0
Thus, the final coefficients are:
[tex]\[
((3, 3, 1.5, 3), (1.0, 0.5, 1.0))
\][/tex]
