Question

Calculate the atomic mass for zinc given the following data for its natural isotopes: $$ \begin{array}{rlr} { }^{64} \mathrm{Zn} & 63.929 \mathrm{amu} & 48.89 \% \\ { }^{66} \mathrm{Zn} & 65.926 \mathrm{amu} & 27.81 \% \\ { }^{67} \mathrm{Zn} & 66.927 \mathrm{amu} & 4.11 \% \\ { }^{68} \mathrm{Zn} & 67.925 \mathrm{amu} & 18.57 \% \\ { }^{70} \mathrm{Zn} & 69.925 \mathrm{amu} & 0.62 \% \end{array} $$

Short answer

The atomic mass of zinc is 65.39 amu.

Step-by-step solution

Understand the Concept of Atomic Mass

Atomic mass is the weighted average mass of an element’s isotopes based on their natural abundance. It accounts for the fact that different isotopes have different masses and occur in different proportions.

Write the Formula for Average Atomic Mass

The formula to calculate the average atomic mass is:\[\text{Average Atomic Mass} = \sum (\text{isotope mass} \times \text{fractional abundance})\]Each isotope's contribution to the atomic mass is calculated by multiplying its mass by its percentage abundance expressed as a fraction (not as a percentage).

Convert Percentage Abundance to Fractional Abundance

Convert the percentage abundance of each isotope into a decimal (fractional abundance) by dividing each percentage by 100.- \({ }^{64} \mathrm{Zn}: 48.89\%\to 0.4889\)- \({ }^{66} \mathrm{Zn}: 27.81\%\to 0.2781\)- \({ }^{67} \mathrm{Zn}: 4.11\%\to 0.0411\)- \({ }^{68} \mathrm{Zn}: 18.57\%\to 0.1857\)- \({ }^{70} \mathrm{Zn}: 0.62\%\to 0.0062\)

Calculate the Contribution of Each Isotope

Multiply the atomic mass of each isotope by its fractional abundance to find its contribution to the atomic mass.\[\begin{align*}{ }^{64} \mathrm{Zn}: & \ 63.929 \times 0.4889 = 31.235 \ \{ }^{66} \mathrm{Zn}: & \ 65.926 \times 0.2781 = 18.333 \ \{ }^{67} \mathrm{Zn}: & \ 66.927 \times 0.0411 = 2.749 \ \{ }^{68} \mathrm{Zn}: & \ 67.925 \times 0.1857 = 12.615 \ \{ }^{70} \mathrm{Zn}: & \ 69.925 \times 0.0062 = 0.434 \\end{align*}\]

Sum the Contributions to Find the Atomic Mass

Add all the contributions from each isotope to find the average atomic mass.\[31.235 + 18.333 + 2.749 + 12.615 + 0.434 = 65.39 \, \text{amu}\]

Conclusion

The calculated average atomic mass of zinc, based on its isotopes and their natural abundances, is approximately 65.39 amu.

Key concepts

Isotopes

Isotopes are atoms of the same element that have the same number of protons but different numbers of neutrons. This means they have different atomic masses. Although chemically similar, isotopes have slight variations in physical properties due to the difference in mass. In the case of zinc, the isotopes include \({ }^{64} \text{Zn}\), \({ }^{66} \text{Zn}\), \({ }^{67} \text{Zn}\), \({ }^{68} \text{Zn}\), and \({ }^{70} \text{Zn}\). Each isotope of zinc contributes to the element's overall atomic mass based on its specific mass and how commonly it occurs in nature.
This is important because isotopes with the same element can vary in their presence across different sources. Natural sampling of these isotopes from materials can inform us of their collective contribution to an element's atomic mass.

Natural Abundance

Natural abundance refers to the relative proportion of each isotope found in a naturally occurring sample of an element. These proportions can vary from one source to another or remain the same. For any given element, natural abundance is expressed as a percentage, showing how much of each isotope is present relative to the total amount of the element.
For zinc, each isotope shows a certain percentage of natural abundance, like \(48.89\%\) for \({ }^{64} \text{Zn}\) or \(0.62\%\) for \({ }^{70} \text{Zn}\). These percentages give insight into how the isotopes affect the calculation of the average atomic mass. The more abundant an isotope is, the more it contributes to the overall atomic weight of the element.

Average Atomic Mass

The average atomic mass is a weighted average of all the isotopes of an element. It reflects the isotopes' masses and their natural abundances. Instead of simply averaging the masses, average atomic mass considers how frequently each isotope occurs.
The average atomic mass can be calculated using the formula:

• \[ \text{Average Atomic Mass} = \sum(\text{isotope mass} \times \text{fractional abundance}) \]

This formula shows that each isotope's mass is multiplied by a fractional abundance—a converted form of the percentage abundance—before summing them for the total. This considers both heavy and light isotopes and their abundance in nature to deliver a more accurate depiction of an element's mass.

Fractional Abundance

Fractional abundance is a way to express the natural abundance of an isotope as a dimensionless number between 0 and 1, rather than as a percentage. This conversion is necessary for mathematical calculations, such as finding the average atomic mass. To get fractional abundance, divide the percentage abundance by 100.
For zinc's isotopes, the conversions are as follows:

• \({ }^{64} \text{Zn}: 48.89\% \to 0.4889\)
• \({ }^{66} \text{Zn}: 27.81\% \to 0.2781\)
• \({ }^{67} \text{Zn}: 4.11\% \to 0.0411\)
• \({ }^{68} \text{Zn}: 18.57\% \to 0.1857\)
• \({ }^{70} \text{Zn}: 0.62\% \to 0.0062\)

These fractional numbers contribute to calculating the weighted average, as they help identify the portion of each isotope's contribution adjusted to 1. This method allows each contribution to be computed accurately and combined to understand the total atomic mass.