Atomic Mass

Atomic Mass

In the previous lecture on isotopes, we saw that atoms of the same element can have different numbers of neutrons, giving them different masses. This raises a practical question: when the periodic table lists a single mass value for each element, which mass is it showing? The answer is the weighted average atomic mass — a single number that accounts for all naturally occurring isotopes and the abundance of each.

Percent Abundance

Different isotopes of the same element exist in nature in different proportions. The fraction of a naturally occurring sample of an element that consists of a particular isotope is called its percent abundance. It is simply the answer to the question: out of every 100 atoms of this element found in nature, how many are this specific isotope?

A quick way to spot elements that have only one naturally occurring isotope: look at the atomic mass listed on the periodic table. If the value is very close to a whole number — for example, fluorine is listed as 18.998 amu — it may have only one isotope (or one overwhelmingly dominant one). If the atomic mass falls notably between two whole numbers, multiple isotopes of significant abundance are almost certainly present. Fluorine (F) and phosphorus (P) are among the elements with only a single stable isotope.

What Is a Weighted Average?

A simple (unweighted) average treats every value equally. A weighted average gives each value a weight proportional to how often it occurs. The atomic mass in the periodic table is a weighted average because the more abundant an isotope is, the more it contributes to the element’s average mass.

A familiar analogy: suppose you want to find the average mass of a group of 25 people, 13 women and 12 men. If women average 130 lbs and men average 180 lbs, a simple average of the two masses (155 lbs) would be slightly misleading because there are slightly more women in the group. A weighted average accounts for this:

(13/25 × 130 lbs) +(12/25 × 180 lbs)

The answer (154 lbs) is pulled slightly toward the women’s mass because there are more of them. This is exactly how atomic mass works: the weighted average is pulled toward the massof whichever isotope is most abundant.

Calculating Weighted Average Atomic Mass

The weighted average atomic massof an element is calculated by multiplying each isotope’s mass by its fractional abundance (the percent abundance divided by 100), then summing all the contributions:

The formula for weighted atomic mass (average atomic mass) is:

Weighted Atomic Mass=∑(fractional abundance×isotopic mass)\text{Weighted Atomic Mass} = \sum (\text{fractional abundance} \times \text{isotopic mass})

or more explicitly:

Atomic Mass=(f1×m1)+(f2×m2)+(f3×m3)+⋯\text{Atomic Mass} = (f_1 \times m_1) + (f_2 \times m_2) + (f_3 \times m_3) + \cdots

where:

• ff = fractional abundance of each isotope (convert percentages to decimals)

• mm = mass of each isotope (amu)

Worked Example 1: Iron (Fe)

Iron has four naturallyoccurring isotopes. Using the percent abundances and isotope masses below, wecan calculate the weighted average atomic mass:

Table 1. Isotopic data for iron.

Atomic Mass of Fe =3.128 + 51.306 + 1.253 + 0.162

= 55.85 amu

This matches the value shown on the periodic table. Notice that the result is very close to the mass of ⁵⁶Fe(55.9349 amu) because that isotope accounts for 91.72% of all iron atoms in nature. The weighted average is always pulled strongly toward the most abundant isotope.

Worked Example 2: Chlorine (Cl)

Chlorine has two naturally occurring stable isotopes:

Table 2. Isotopic data for chlorine.

Atomic Mass of Cl =26.496 + 8.957

= 35.453 amu ≈ 35.45 amu

Again, this matches the periodic table. The result (35.45) is much closer to 35 than to 37, because ³⁵Cl is about three times more abundant than ³⁷Cl. The average is pulled toward the lighter, more common isotope.

Working Backwards: Finding Percent Abundance

If you know the atomic mass of an element and the masses of its isotopes, you can work backwards to find the percent abundance of each isotope. This is straight forward for elements with exactly two isotopes.

Example: Boron has two stable isotopes: ¹⁰B (mass 10.013 amu) and ¹¹B (mass 11.009 amu). The atomic mass of boron is 10.811 amu. What is the percent abundance of each isotope?

Let x = fraction of ¹⁰B. Then (1− x) = fraction of ¹¹B.

10.013x + 11.009(1 −x) = 10.811

10.013x+ 11.009 − 11.009x = 10.811

−0.996x = − 0.198

x = 0.199 → ¹⁰B is 19.9% abundant; ¹¹Bis 80.1% abundant

The known values are 19.9%(¹⁰B) and 80.1% (¹¹B) — a perfect match. This type of calculation is the reverse of the weighted average and appears frequently on exams.

Summary

The atomic masses listed on the periodic table are weighted average atomic masses. Because most elements exist as mixtures of naturally occurring isotopes, the average atomic mass reflects both the mass of each isotope and its abundance in nature.

More abundant isotopes contribute more strongly to the weighted average, causing the average atomic mass to lie closer to the mass of the dominant isotope. By multiplying isotope masses by their fractional abundances and summing the contributions, chemists can calculate atomic masses. Conversely, if the atomic mass is known, isotope abundances can often be determined through algebraic analysis.

Key Points

• Most elements exist as mixtures of isotopes.

• Percent abundance describes how common an isotope is in nature.

• Atomic masses on the periodic table are weighted averages.

• More abundant isotopes contribute more heavily to the average.

• Fractional abundance equals percent abundance divided by 100.

• Weighted average atomic mass equals the sum of isotope mass × fractional abundance.

• Iron's atomic mass is dominated by ⁵⁶Fe because it is over 91% abundant.

• Chlorine's atomic mass lies closer to 35 because ³⁵Cl is more abundant.

• Isotope abundance problems can be solved algebraically.

• Weighted average calculations are widely used throughout chemistry.

• Atomic mass calculations connect isotopes directly to periodic table values.