Why atomic mass is almost never a whole number
Open a periodic table and a question jumps out: why is chlorine’s atomic mass 35.45 u and not a tidy whole number? The answer is isotopes: atoms of the same element with the same number of protons but different numbers of neutrons. Chlorine comes in two natural varieties, chlorine-35 and chlorine-37, and the number on the table is the abundance-weighted average of the two.
That phrase is the whole topic, and it is worth stating precisely:
Average atomic mass is the weighted average of the masses of an element’s naturally occurring isotopes, weighted by how common each one is.
No individual chlorine atom weighs 35.45 u. Every one of them weighs either about 35 u or about 37 u. The 35.45 describes the mixture.
How to calculate average atomic mass
Three steps, and the second is the one people skip.
- Write down each isotope’s mass and its natural abundance. Masses come in atomic mass units (u); abundances are percentages that must add up to 100.
- Convert each percentage to a decimal fraction by dividing by 100. This is the weighting step. Skip it and your answer comes out about a hundred times too big.
- Multiply each mass by its fraction and add the results.
atomic mass = (m₁ × f₁) + (m₂ × f₂) + …
Worked example, chlorine. Chlorine-35 weighs 34.969 u and makes up 75.77% of natural chlorine; chlorine-37 weighs 36.966 u and makes up 24.23%.
(34.969 × 0.7577) + (36.966 × 0.2423) = 26.496 + 8.957 = 35.45 u
Notice where the answer landed. Chlorine-35 is about three-quarters of the mixture, so the average sits about three-quarters of the way from 37 down to 35, nowhere near the middle. A weighted average always leans toward whatever is most common, which is the difference between it and an ordinary average: the plain mean of 34.969 and 36.966 would be 35.97, and that is wrong.
The calculator above does this for as many isotopes as you like. Some elements have three that matter: magnesium is 79% magnesium-24, 10% magnesium-25 and 11% magnesium-26, which is why its atomic mass is 24.305 rather than nearly 24.
Working backwards: finding the abundances
The harder homework question runs the other way. You are told the element’s average atomic mass and the masses of its two isotopes, and asked for the percentages. With two isotopes there is exactly one answer, and you can solve it in one line.
Call the lighter isotope’s fraction x. The heavier one is then 1 − x, because between them they are all of it. So:
m₁x + m₂(1 − x) = average
Rearranged for x:
x = (average − m₂) ÷ (m₁ − m₂)
Worked example, copper. Copper’s average atomic mass is 63.55 u, and its two isotopes weigh 62.93 u and 64.93 u.
x = (63.55 − 64.93) ÷ (62.93 − 64.93) = (−1.38) ÷ (−2.00) = 0.69
So copper is 69% copper-63 and 31% copper-65, which is exactly what the published abundances say. Both differences come out negative and the minus signs cancel; if you get a percentage above 100 or below zero, the average you were given does not lie between the two isotope masses, and an average never can.
Mass number and atomic mass are different numbers
These two get confused constantly, and one particular mix-up is worth naming.
- Mass number (A) counts protons + neutrons in one atom. Always a whole number. Chlorine-35 has 17 protons and 18 neutrons, so A = 35.
- Atomic mass is the average over the whole mixture of isotopes, in unified atomic mass units. Almost never whole.
So when a question asks how many neutrons chlorine has, do not subtract the atomic number from 35.45. That gives 35.45 − 17 = 18.45, and there is no such thing as forty-five hundredths of a neutron. The subtraction only works on a mass number, so round first: 35 − 17 = 18 neutrons in chlorine-35, and 37 − 17 = 20 in chlorine-37. The decimal on the periodic table is not a property of any atom you could point at.
The protons set the element’s identity, its atomic number, and they anchor the Bohr model and the valence electrons that drive chemistry. Neutrons change only the mass.
Every isotope in the calculator
These are the elements the presets load, with real NIST masses and abundances. The last column is not copied from the periodic table: it is the weighted average computed from the two columns before it, which is the point.
| Element | Isotope | Protons | Neutrons | Mass (u) | Abundance | Weighted average |
|---|---|---|---|---|---|---|
| Hydrogen (H) | ¹₁H H-1 | 1 | 0 | 1.007825 | 99.9885% | 1.008 u |
| ²₁H H-2 | 1 | 1 | 2.014102 | 0.0115% | ||
| Lithium (Li) | ⁶₃Li Li-6 | 3 | 3 | 6.015123 | 7.59% | 6.940 u |
| ⁷₃Li Li-7 | 3 | 4 | 7.016003 | 92.41% | ||
| Boron (B) | ¹⁰₅B B-10 | 5 | 5 | 10.012937 | 19.9% | 10.811 u |
| ¹¹₅B B-11 | 5 | 6 | 11.009305 | 80.1% | ||
| Carbon (C) | ¹²₆C C-12 | 6 | 6 | 12 | 98.93% | 12.011 u |
| ¹³₆C C-13 | 6 | 7 | 13.003355 | 1.07% | ||
| Nitrogen (N) | ¹⁴₇N N-14 | 7 | 7 | 14.003074 | 99.636% | 14.007 u |
| ¹⁵₇N N-15 | 7 | 8 | 15.000109 | 0.364% | ||
| Oxygen (O) | ¹⁶₈O O-16 | 8 | 8 | 15.994915 | 99.757% | 15.999 u |
| ¹⁷₈O O-17 | 8 | 9 | 16.999132 | 0.038% | ||
| ¹⁸₈O O-18 | 8 | 10 | 17.99916 | 0.205% | ||
| Neon (Ne) | ²⁰₁₀Ne Ne-20 | 10 | 10 | 19.99244 | 90.48% | 20.180 u |
| ²¹₁₀Ne Ne-21 | 10 | 11 | 20.993847 | 0.27% | ||
| ²²₁₀Ne Ne-22 | 10 | 12 | 21.991385 | 9.25% | ||
| Magnesium (Mg) | ²⁴₁₂Mg Mg-24 | 12 | 12 | 23.985042 | 78.99% | 24.305 u |
| ²⁵₁₂Mg Mg-25 | 12 | 13 | 24.985837 | 10% | ||
| ²⁶₁₂Mg Mg-26 | 12 | 14 | 25.982593 | 11.01% | ||
| Silicon (Si) | ²⁸₁₄Si Si-28 | 14 | 14 | 27.976927 | 92.223% | 28.085 u |
| ²⁹₁₄Si Si-29 | 14 | 15 | 28.976495 | 4.685% | ||
| ³⁰₁₄Si Si-30 | 14 | 16 | 29.97377 | 3.092% | ||
| Chlorine (Cl) | ³⁵₁₇Cl Cl-35 | 17 | 18 | 34.968853 | 75.77% | 35.453 u |
| ³⁷₁₇Cl Cl-37 | 17 | 20 | 36.965903 | 24.23% | ||
| Copper (Cu) | ⁶³₂₉Cu Cu-63 | 29 | 34 | 62.929598 | 69.15% | 63.546 u |
| ⁶⁵₂₉Cu Cu-65 | 29 | 36 | 64.92779 | 30.85% | ||
| Bromine (Br) | ⁷⁹₃₅Br Br-79 | 35 | 44 | 78.918338 | 50.69% | 79.904 u |
| ⁸¹₃₅Br Br-81 | 35 | 46 | 80.916291 | 49.31% | ||
| Silver (Ag) | ¹⁰⁷₄₇Ag Ag-107 | 47 | 60 | 106.905092 | 51.839% | 107.868 u |
| ¹⁰⁹₄₇Ag Ag-109 | 47 | 62 | 108.904756 | 48.161% |
Read a row like this: ³⁵₁₇Cl carries the mass number above and the atomic number below, so it has 17 protons and 35 − 17 = 18 neutrons. The same isotope is written Cl-35 or chlorine-35 when superscripts are awkward.
Using this with a class
Ask students to predict whether the average will land closer to the lighter or the heavier isotope before they calculate, then check. Then give them the answer and ask for the abundances, which is the same relationship read backwards and much harder to guess. The calculator is free to embed on your site or LMS with the snippet below.