Why Atomic Masses Are Not Whole Numbers

Introduction

Look at a periodic table and you will see that chlorine has a relative atomic mass of 35.5 and copper 63.5. Yet no single chlorine atom has a mass of 35.5 on the carbon-12 scale. How can the average be a value that no atom actually has? The answer lies in isotopes: most elements are natural mixtures of atoms with slightly different masses. This page gives a first look at isotopes and shows how their average produces non-whole numbers.

Core explanation

Isotopes. Dalton assumed that all atoms of one element are identical in mass. We now know this is not quite true. Many elements exist as two or more isotopes : atoms of the same element, with the same chemical behaviour, but different masses. Each isotope is labelled with a whole number called its mass number , for example chlorine-35 and chlorine-37. The mass of each isotope in u is very close to its mass number.

Chlorine as an example. Natural chlorine contains two isotopes:

Isotope Relative mass (approx.) Abundance --- --- --- Chlorine-35 35 about 75% Chlorine-37 37 about 25%

A weighted average. If the two isotopes were equally common, the average would be 36. But chlorine-35 is three times as common as chlorine-37, so the average is pulled towards 35. We calculate a weighted average , in which each mass counts according to its abundance:

Ar = (35 × 75 + 37 × 25) ÷ 100 = (2625 + 925) ÷ 100 = 35.5

So 35.5 is the mass of an "average" chlorine atom, even though every real chlorine atom is either about 35 or about 37.

Other elements. Copper is about 69% copper-63 and 31% copper-65, giving Ar ≈ 63.6 (often rounded to 63.5). Boron is about 20% boron-10 and 80% boron-11, giving Ar ≈ 10.8. Elements with one dominant isotope, such as fluorine (all fluorine-19) or sodium (all sodium-23), have Ar values very close to whole numbers.

Why the mixture is steady. In ordinary chemical samples, the proportions of isotopes are almost the same wherever the element comes from, because isotopes of an element react in essentially the same way. That is why a single Ar value is useful for everyday calculations.

Formulae

Ar = Σ(isotope mass × percentage abundance) ÷ 100

Step-by-step reasoning

To calculate Ar from isotope data:

1. List each isotope's relative mass and its percentage abundance. 2. Multiply each mass by its abundance. 3. Add the products together. 4. Divide by 100 (the total percentage). 5. Check that the answer lies between the lightest and heaviest isotope masses and nearer the most abundant one.

Visual explanation

Imagine a bag of 100 marbles representing chlorine atoms: 75 small marbles labelled 35 and 25 larger ones labelled 37. Pick marbles at random and the typical mass you get is dominated by the 35s. The average over the whole bag is 35.5. The simulation lets you build atoms of different isotopes and compare their masses.

Real-world analogy

Your average mark across a course is a weighted average if exams count for 75% and coursework for 25%. Score 35 in exams and 37 in coursework and your overall mark is 35.5, closer to the exam mark because it carries more weight.

Real-world example

Mass spectrometers in forensic and environmental laboratories measure the exact isotope pattern of samples. Chlorine-containing pollutants show a distinctive pair of peaks roughly three to one in height, from chlorine-35 and chlorine-37, which helps chemists identify them.

Why?

Why do isotopes of one element have the same chemistry? Chemical behaviour depends on the arrangement of electrons, which is the same for all atoms of an element. Isotopes differ only in the number of neutral particles in the nucleus, which changes the mass but not the way the atom bonds.

Common misconception

"Some chlorine atoms have a mass of 35.5." No individual chlorine atom has this mass. 35.5 is an average over a very large number of atoms of mass 35 and 37.

Worked example

Question: Element X has two isotopes: X-69 (60%) and X-71 (40%). Calculate Ar and identify X.

Reasoning: Ar = (69 × 60 + 71 × 40) ÷ 100 = (4140 + 2840) ÷ 100 = 6980 ÷ 100 = 69.8. The element with Ar close to 69.7 is gallium.

Answer: Ar ≈ 69.8; X is gallium.

Quick check

1. Why is the Ar of chlorine closer to 35 than to 37? Answer: Chlorine-35 is about three times as abundant as chlorine-37, so it contributes more to the average.

Exam focus

Show your working clearly: write the multiplication for each isotope, the sum, and the division by 100. Give the answer to the number of decimal places asked for, and check it lies between the isotope masses.

Advanced insight

Even isotope masses are not exactly whole numbers except carbon-12. Chlorine-35 is 34.969 u and chlorine-37 is 36.966 u, because the particles in a nucleus lose a little mass when they bind together. Precise Ar calculations therefore use exact isotope masses, giving chlorine 35.45.

Summary

Many elements are natural mixtures of isotopes: atoms of the same element with different masses. The relative atomic mass is a weighted average of the isotope masses, based on their abundances. That is why values such as chlorine 35.5 and copper 63.5 are not whole numbers, even though no single atom has that mass.

Practice questions

1. What are isotopes? Answer: Atoms of the same element that have different masses but the same chemical behaviour. 2. Boron is 20% boron-10 and 80% boron-11. Calculate its Ar. Answer: (10 × 20 + 11 × 80) ÷ 100 = (200 + 880) ÷ 100 = 10.8. 3. Copper is 69% copper-63 and 31% copper-65. Calculate its Ar to one decimal place. Answer: (63 × 69 + 65 × 31) ÷ 100 = (4347 + 2015) ÷ 100 = 63.62, so about 63.6. 4. Fluorine has only one natural isotope, fluorine-19. Predict its Ar and explain. Answer: About 19, because with only one isotope there is nothing to average, so Ar is close to a whole number.