Sure, let's solve for the percentage abundance of the isotopes Carbon-12 ([tex]\(C^{12}\)[/tex]) and Carbon-13 ([tex]\(C^{13}\)[/tex]) given that the average atomic mass of carbon is [tex]\(12.011 \mu\)[/tex].
1. Define the Variables:
- Let [tex]\(x\)[/tex] be the percentage abundance of [tex]\(C^{12}\)[/tex].
- Let [tex]\(y\)[/tex] be the percentage abundance of [tex]\(C^{13}\)[/tex].
2. Set up the Equations:
- Since the total percentage abundance of the isotopes must add up to 100%, we can write:
[tex]\[
x + y = 100
\][/tex]
- The average atomic mass is given by the weighted average of the isotopic masses:
[tex]\[
\text{Average Atomic Mass} = \frac{x \cdot \text{mass}_{C^{12}} + y \cdot \text{mass}_{C^{13}}}{100}
\][/tex]
Given the values:
[tex]\[
12.011 = \frac{x \cdot 12 + y \cdot 13}{100}
\][/tex]
3. Substitute [tex]\(y\)[/tex]:
- From the equation [tex]\(x + y = 100\)[/tex], we get [tex]\(y = 100 - x\)[/tex].
- Substitute [tex]\(y\)[/tex] in the average atomic mass equation:
[tex]\[
12.011 = \frac{x \cdot 12 + (100 - x) \cdot 13}{100}
\][/tex]
4. Solve the Equation for [tex]\(x\)[/tex]:
[tex]\[
12.011 = \frac{12x + 1300 - 13x}{100}
\][/tex]
[tex]\[
12.011 = \frac{1300 - x}{100}
\][/tex]
[tex]\[
1201.1 = 1300 - x
\][/tex]
[tex]\[
x = 1300 - 1201.1
\][/tex]
[tex]\[
x = 98.9
\][/tex]
5. Determine the Abundance of [tex]\(C^{13}\)[/tex]:
[tex]\[
y = 100 - x
\][/tex]
[tex]\[
y = 100 - 98.9
\][/tex]
[tex]\[
y = 1.1
\][/tex]
Conclusion:
- The percentage abundance of [tex]\(C^{12}\)[/tex] is approximately [tex]\(98.9\%\)[/tex].
- The percentage abundance of [tex]\(C^{13}\)[/tex] is approximately [tex]\(1.1\%\)[/tex].
