Calculating Percent Abundance of Isotopes

Core Formula for Average Atomic Mass

• The average atomic mass ($A$) of an element is the weighted average of its naturally occurring isotopes.
• Formula: $A = (m_1 \times p_1) + (m_2 \times p_2) + ...$
• where:   - $m_n$ is the mass of isotope $n$.   - $p_n$ is the fractional (decimal) percent abundance of isotope $n$.

Variables and Algebraic Setup

• To solve for two unknown abundances, express them in terms of a single variable $x$.
• Let $p_1 = x$.
• Since the total abundance must equal $100\%$ (or $1$ as a decimal), let $p_2 = 1 - x$.
• Substitute these into the formula: $A = (m_1 \times x) + (m_2 \times (1 - x))$.

Calculation Steps for Bromine

• Given Data for Bromine ($Br$):   - Average Atomic Mass ($A$): $79.9\,amu$   - Mass of $Br-79$ ($m_1$): $78.918$   - Mass of $Br-81$ ($m_2$): $80.916$
• Algebraic Equation: $79.9 = (78.918 \times x) + (80.916 \times (1 - x))$
• Distribution: $79.9 = 78.918x + 80.916 - 80.916x$
• Combine like terms: $-1.016 = -1.998x$
• Solve for $x$: $x = \frac{-1.016}{-1.998} \approx 0.509$

Final Abundance Results

• Percent Abundance of $Br-79$ ($x$): $50.9\%$
• Percent Abundance of $Br-81$ ($1 - x$): $1 - 0.509 = 0.491$ or $49.1\%$
• Verification: $(78.918 \times 0.509) + (80.916 \times 0.491) \approx 79.9\,amu$.