Question

Two isotopes of gallium are naturally occurring, with \({ }_{31}^{69} \mathrm{Ga}\) at \(60.11 \%(68.93 \mathrm{amu})\) and \({ }_{31}^{71} \mathrm{Ga}\) at \(39.89 \%(70.92 \mathrm{amu})\). What is the atomic mass of gallium?

Short answer

69.73 amu

Step-by-step solution

Identify the Given Values

The data provided includes two isotopes of gallium. The first isotope \({ }_{31}^{69}\text{Ga}\) has a mass of 68.93 amu and a natural abundance of 60.11%, and the second isotope \({ }_{31}^{71}\text{Ga}\) has a mass of 70.92 amu and a natural abundance of 39.89%.

Convert Percentages to Decimals

Convert the percentage abundances to decimal form by dividing each by 100. For \({ }_{31}^{69}\text{Ga}\) this is \(60.11\% = 0.6011\), and for \({ }_{31}^{71}\text{Ga}\) this is \(39.89\% = 0.3989\).

Multiply Each Isotope's Mass by Its Abundance

Calculate the contribution of each isotope to the atomic mass. For \({ }_{31}^{69}\text{Ga}\), multiply its mass by its abundance: \(68.93 \text{ amu} \times 0.6011 = 41.43 \text{ amu}\). For \({ }_{31}^{71}\text{Ga}\), multiply its mass by its abundance: \(70.92 \text{ amu} \times 0.3989 = 28.30 \text{ amu}\).

Sum the Contributions

Add the contributions from each isotope to find the total atomic mass of gallium. \((41.43 \text{ amu} + 28.30 \text{ amu} = 69.73 \text{ amu})\).

Key concepts

isotopes

An element's isotopes are atoms that have the same number of protons but different numbers of neutrons. Though isotopes of an element share chemical properties, they have different masses. For example, the isotopes of gallium in the exercise are \(_{31}^{69}\mathrm{Ga}\) and \(_{31}^{71}\mathrm{Ga}\). Both isotopes have 31 protons because they are both gallium, but \(_{31}^{69}\mathrm{Ga}\) has 38 neutrons, while \(_{31}^{71}\mathrm{Ga}\) contains 40 neutrons.
Understanding isotopes is crucial for calculating the atomic mass of elements, as each isotope's mass and abundance contribute to the average atomic mass.

natural abundance

Natural abundance refers to the relative proportions of an element's isotopes found naturally on Earth. It is usually expressed as a percentage. For gallium, as highlighted in the exercise, the two naturally occurring isotopes have abundances of 60.11% for \(_{31}^{69}\mathrm{Ga}\) and 39.89% for \(_{31}^{71}\mathrm{Ga}\).
To perform calculations, these percentages should be converted into decimal form by dividing by 100: 60.11% becomes 0.6011 and 39.89% becomes 0.3989.
Natural abundance is essential for determining the average atomic mass of an element, giving insight into how much each isotope contributes to the weighted average.

atomic mass unit (amu)

The atomic mass unit (amu) is a standard unit of mass that quantifies the mass of atoms and molecules. It is defined as one twelfth of the mass of a carbon-12 atom. In this context, amu allows us to express isotopic masses clearly. For instance, the masses of gallium isotopes in the exercise are 68.93 amu for \(_{31}^{69}\mathrm{Ga}\) and 70.92 amu for \(_{31}^{71}\mathrm{Ga}\).
Using amu in calculations makes it easier to work with the small masses of atoms, making the data comprehensible and comparable.

elemental composition

Elemental composition denotes the presence and ratio of various isotopes or elements within a given sample. In terms of the exercise provided, gallium's elemental composition includes a mix of two isotopes. The atomic mass of an element reflects this composition by considering the masses and natural abundances of its isotopes.
To calculate the average atomic mass of gallium, we sum the products of each isotope's mass and its natural abundance:
\(68.93 \text{ amu} \times 0.6011 = 41.43 \text{ amu} \)
\(70.92 \text{ amu} \times 0.3989 = 28.30 \text{ amu} \)
Add these contributions to obtain the average atomic mass: \(41.43 \text{ amu} + 28.30 \text{ amu} = 69.73 \text{ amu}\).