The atomic mass of boron is 10.81100 amu. Boron has two isotopes: Boron-10 has a mass of 10.01294 amu: Boron-11 has a mass of 11.00931 amu What is the percentage of the boron-10 isotope
The Correct Answer and Explanation is:
To determine the percentage of the boron-10 isotope, we use the weighted average formula for atomic mass. Let xx be the fraction of boron-10, while the fraction of boron-11 is 1−x1 – x. The atomic mass of boron is the sum of the contributions from both isotopes:
10.81100=(10.01294×x)+(11.00931×(1−x))10.81100 = (10.01294 \times x) + (11.00931 \times (1 – x))
Expanding the equation:
10.81100=10.01294x+11.00931−11.00931×10.81100 = 10.01294x + 11.00931 – 11.00931x
10.81100=11.00931−0.99637×10.81100 = 11.00931 – 0.99637x
Rearranging to solve for xx:
0.99637x=11.00931−10.811000.99637x = 11.00931 – 10.81100
0.99637x=0.198310.99637x = 0.19831
x=0.198310.99637x = \frac{0.19831}{0.99637}
x≈0.199x \approx 0.199
Multiplying by 100 gives the percentage:
Percentage of boron-10≈19.9%\text{Percentage of boron-10} \approx 19.9\%
Explanation:
The atomic mass of an element represents the weighted average of its isotopic masses based on their relative abundances. Boron exists in nature with two stable isotopes: boron-10 and boron-11. Their individual masses influence the overall atomic mass of boron, given as 10.81100 amu.
To determine the fraction of boron-10 in a natural sample, the weighted average equation is applied. The atomic mass is expressed as the sum of isotope masses, weighted by their relative proportions. By setting up an algebraic equation, we isolate the variable representing boron-10’s fraction. Solving for this fraction allows us to determine that approximately 19.9% of boron atoms in nature are boron-10.
This calculation is crucial in various fields such as nuclear chemistry, environmental science, and materials engineering. The isotopic distribution of boron affects its chemical behavior, its application in nuclear reactors, and its role in biological processes. Understanding isotope percentages helps scientists in isotope enrichment, radiochemical analysis, and geochemical dating. Boron’s isotopic composition also influences its reaction pathways and interaction with other elements, making this calculation a key aspect of boron-related research.
