Question

Potassium has three stable isotopes, \({ }^{39} \mathrm{~K},{ }^{40} \mathrm{~K},\) and \({ }^{41} \mathrm{~K}\) but \({ }^{40} \mathrm{~K}\) has a very low natural abundance. Which of the other two is the more abundant? (No calculation should be necessary.)

Short answer

The more abundant isotope is { }^{39} ext{~K}.

Step-by-step solution

Understanding Isotopes

Isotopes are atoms of the same element that have the same number of protons but different numbers of neutrons. Therefore, they have different mass numbers. The element in question, potassium, has isotopes { }^{39} ext{~K}, { }^{40} ext{~K}, ext{ and } { }^{41} ext{~K}.

Natural Abundance Consideration

Natural abundance refers to how common an isotope is in nature compared to other isotopes of the same element. In this exercise, { }^{40} ext{~K} is stated to have a very low natural abundance, so we need to compare the other two isotopes: { }^{39} ext{~K} and { }^{41} ext{~K}.

Periodic Table Insight

The average atomic mass of potassium found on the periodic table is approximately 39.1. This value gives us insight into the predominant isotopes because it is a weighted average of the isotopic masses.

Compare to Average Atomic Mass

The average atomic mass of potassium (39.1) is very close to the mass number of { }^{39} ext{~K} (39) compared to { }^{41} ext{~K} (41). This suggests that { }^{39} ext{~K} must have a higher natural abundance than { }^{41} ext{~K} as it pulls the average mass closer to 39.

Key concepts

Isotopic Abundance

Isotopic abundance refers to the proportion of a particular isotope among all the isotopes of the same element in nature. For any element, isotopes are variations of the element with the same number of protons but differing numbers of neutrons. Each isotope has a unique mass number, reflecting the combined number of protons and neutrons.
When looking at isotopic abundance, it tells us how common each isotope is in a natural sample of the element. In most cases, one or two isotopes are significantly more abundant than others.
For example, potassium has isotopes \( {}^{39} \text{K}, {}^{40} \text{K}, \text{and} {}^{41} \text{K} \). Given that \( {}^{40} \text{K} \) is known to have very low natural abundance, the focus shifts to \( {}^{39} \text{K} \) and \( {}^{41} \text{K} \). The isotopic abundance of these isotopes allows us to determine which one is more common in nature.
By examining natural abundance, we can identify which isotopes significantly influence an element's average atomic mass, providing insights into the element's properties and behavior.

Average Atomic Mass

Average atomic mass is a value that reflects the weighted average of the masses of an element's isotopes found naturally on Earth. It's calculated based on both the mass and the relative abundance of each isotope of the element. This value is often represented on the periodic table and gives insight into the isotopes' distribution.
The formula to determine average atomic mass is:\[\text{Average Atomic Mass} = (\text{mass of } {}^{39} \text{K}) \times (\text{abundance of } {}^{39} \text{K}) + (\text{mass of } {}^{40} \text{K}) \times (\text{abundance of } {}^{40} \text{K}) + (\text{mass of } {}^{41} \text{K}) \times (\text{abundance of } {}^{41} \text{K})\]For potassium, this average atomic mass is approximately 39.1. Since this number is much closer to 39 than to 41, it indicates that \( {}^{39} \text{K} \) is the most abundant isotope.
It's this concept of average atomic mass that helps chemists and physicists alike to better understand the elemental composition of naturally occurring samples and their isotopic distributions.

Periodic Table

The periodic table is a systematic arrangement of the chemical elements, where elements are ordered by their atomic number, electron configuration, and recurring chemical properties. It serves as a comprehensive reference for understanding the relationships between elements, including trends in properties and electron arrangements.
A key feature of the periodic table is the listing of each element's average atomic mass. This provides a quick snapshot of an element’s isotopic composition and its most common isotopes. For example, potassium, with an average atomic mass of about 39.1, indicates a predominance of its lighter isotope, \( {}^{39} \text{K} \).
The periodic table not only provides atomic masses but also facilitates the prediction of how elements might react chemically with each other. It is a foundational tool in chemistry that allows for the visualization of elements and their isotopic characteristics at a glance, guiding scientists in numerous applications from laboratory research to educational endeavors.