Problem 98 - Chemistry and Chemical Reactivity

Question

Potassium has three naturally occurring isotopes \(\left(^{39} \mathrm{K}\right.\) \(\left.^{40} \mathrm{K}, \text { and }^{41} \mathrm{K}\right),\) but \(^{40} \mathrm{K}\) has a very low natural abundance. Which of the other two isotopes is more abundant? Briefly explain your answer.

Step-by-Step Solution

Answer

The isotope \(^{39}\mathrm{K}\) is more abundant because its atomic weight is closer to the average atomic weight of potassium.

Step 1: Identify Atomic Weights

The approximate atomic weights for the potassium isotopes are as follows: \(^{39}\mathrm{K}\) has an atomic weight of about 39 u (unified atomic mass units), and \(^{41}\mathrm{K}\) has an atomic weight of about 41 u.

Step 2: Consider Average Atomic Weight

The average atomic weight of potassium that you would find on the periodic table is about 39.10 u. This average should closely reflect the relative abundance of each isotope.

Step 3: Compare Isotope Weights to Average

Compare the isotopic atomic weights to the average atomic weight of potassium. \(^{39}\mathrm{K}\) is closer to the average atomic weight of 39.10 u than \(^{41}\mathrm{K}\) is.

Step 4: Conclusion on Abundance

Given that \(^{39}\mathrm{K}\) is closer to the average atomic weight of potassium, it indicates that \(^{39}\mathrm{K}\) is more abundant than \(^{41}\mathrm{K}\), because the average atomic weight is skewed towards the more abundant isotope.

Key Concepts

Average Atomic Weight

The average atomic weight of an element gives us a clue about the relative abundance of its isotopes. This value is a weighted average, not simply a mean. It considers both the mass of each isotope and its relative abundance. For potassium, the average atomic weight is approximately 39.10 u.
This means that isotopes closer to this number are more prevalent. The average atomic weight reflects the natural distribution of isotopes in a sample.
Here’s how it works:

• If an isotope has a higher percentage in nature, the average atomic weight will be closer to its atomic weight.
• This concept helps us predict which isotopes are more common based on their contributions to the average value found on the periodic table.

Potassium Isotopes

Potassium is an element with three naturally occurring isotopes:

• \(^{39}\mathrm{K}\) - Atomic weight of about 39 u.
• \(^{40}\mathrm{K}\) - Present in low abundance, not a focus for common calculations.
• \(^{41}\mathrm{K}\) - Atomic weight of about 41 u.

These isotopes differ in the number of neutrons they have, but they all share the same number of protons. In nature, these isotopes occur in varying amounts.
Since \(^{40}\mathrm{K}\) has a negligible natural abundance, between \(^{39}\mathrm{K}\) and \(^{41}\mathrm{K}\), it is \(^{39}\mathrm{K}\) that is more abundant because its atomic weight is closer to the average atomic weight of potassium, 39.10 u. This proximity suggests it is the dominant isotope by mass.

Atomic Weights

Atomic weights are a critical part of understanding isotopes and their behavior. An atomic weight is essentially the mass of an atom, expressed in unified atomic mass units (u).
For isotopes, individual atomic weights are not integers because they take into consideration the binding energy of nuclei and other quantum effects. Each isotope of an element has its unique atomic weight:

• \(^{39}\mathrm{K}\) - approximately 39 u, which is why it strongly influences the average atomic weight of potassium.
• \(^{41}\mathrm{K}\) - about 41 u, contributes less to the average due to lower abundance.

Understanding atomic weights helps chemists and scientists predict chemical behaviors and interactions of elements, determine molecular weights, and perform quantitative chemical analyses.