Gallium has only two naturally occurring isotopes, Ga69 and Ga-71 . The mass of Ga-69 is 68.9256

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Gallium has only two naturally occurring isotopes, Ga69 and...

Gallium has only two naturally occurring isotopes, Ga69 and $\mathrm{Ga}-71 .$ The mass of $\mathrm{Ga}-69$ is $68.9256 \mathrm{amu},$ and the mass of Ga-71 is 70.9247 amu. Use the atomic mass of gallium to determine the relative abundance of each isotope in a naturally occurring sample.

Key Concepts

Isotopic Abundance

Isotopic abundance refers to the relative percentage with which each isotope of an element occurs naturally. It is a measure of the proportion of each isotope present in a sample and is critical for determining the average atomic mass of an element. In problems like this, knowing the isotopic abundances allows one to relate the distinct masses of isotopes to the overall atomic mass of the element as found in nature.

Weighted Average Calculation

The atomic mass of an element is computed as a weighted average of the masses of its isotopes, where each isotope's contribution is weighted by its relative abundance. This means that the average atomic mass equals the sum of the product of each isotope's mass and its abundance. Understanding this concept is essential in setting up the equation that relates the known atomic mass of the element to the masses and percentages of its isotopes.

System of Equations in Mixture Problems

Solving for the unknown isotopic abundances involves setting up a system of equations. One equation represents the weighted average of the isotopic masses (the atomic mass equation), while a second equation represents the conservation of mass or the sum of the relative abundances, which must equal 100%. This methodical approach using simultaneous equations is a common technique in quantitative chemistry for solving mixture problems.

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