Isotopes: one element, different masses
Isotopes of an element have the same number of protons and different numbers of neutrons. The mass number A counts protons and neutrons: it is an integer, not a mass measurement. B-10 identifies boron with 10 nucleons, but its isotope mass is not exactly 10 u.
The mass of a neutral atom includes its electrons and reflects binding energies. It cannot be obtained simply by counting nucleons. Carbon-12 has a mass of exactly 12 u by definition of the unit; other isotopes do not follow this rule.
Use isotopes of the same element. The label identifies each row; check that the mass corresponds to the isotope named.
Building the weighted average
If there are N atoms and a fraction xᵢ belongs to isotope i, its contribution to the mean mass is xᵢmᵢ. Adding all contributions gives the mass per atom for that composition. Abundances are fractions of the number of atoms, not mass fractions.
The expected total is 100%. A deviation of up to 0.1 percentage points is accepted to accommodate rounded abundances: each percentage is divided by the entered total and the adjustment is reported. This tolerance does not express measurement uncertainty. Outside this range, check the percentages or use “Normalize to 100%”.
Direct example: boron
The OpenStax exercise uses 10.0129 u at 19.9% and 11.0093 u at 80.1%:
The extra digits in the calculator let you check the arithmetic; they do not increase experimental precision. An individual atom retains the mass of its isotope, not the average mass.
Two abundances from an average
For two isotopes, x₁ + x₂ = 1. Substituting x₂ into the average equation leaves one equation with one unknown:
For two hypothetical masses of 10 and 11 u and an average of 10.8 u:
The abundances are 20% and 80%. An average at either endpoint gives fractions of 1 and 0. Outside the interval, no physical composition exists. Equal masses cannot distinguish the abundances. For three or more unknown abundances, the total and average provide only two equations and generally do not determine a unique solution.
Reading the distribution
Each line is positioned at its isotope mass and its height shows normalized abundance. Distances between masses remain proportional. The dashed line marks the average, which always lies between the smallest and largest masses because all fractions are nonnegative and add to one.
In a binary system, the average is closer to the more abundant isotope. With more components, their combined contributions matter; identifying the tallest line is not enough. A zero abundance stays on the baseline.
This chart represents the declared composition. It is neither an instrumental spectrum nor a molecular isotope pattern: it does not include charge, adducts, resolution, fragmentation or the combined probabilities of a molecule.
A sample and a standard atomic weight
The calculated average describes the entered composition. A standard atomic weight is a CIAAW recommendation for normal materials and may be an interval when composition varies naturally. Interval endpoints should not be converted into single percentages without a model of the sample.
The neon and copper examples use central values from CIAAW abundances 2024 and masses from element pages citing Review 2000 (published in 2003). The calculation does not estimate uncertainty in the average from the uncertainties in those sources.
u, relative mass and molar mass
The unit u, equivalent to the dalton (Da), is one twelfth of the mass of a free neutral carbon-12 atom at rest in its ground state. Mean mass in u has the same numerical value as relative atomic mass Aᵣ, but Aᵣ is dimensionless.
Molar mass M refers to one mole and is commonly expressed in g/mol. At ordinary chemical precision, its numerical correspondence with Aᵣ is used as an approximation. In the current SI, Mᵤ = Nₐmᵤ is very close to 1 g/mol but is no longer exactly that constant. This tool reports mass per atom in u and does not convert it to molar mass.
References and versions
Sources accessed October 31, 2026. Example data correspond to the versions cited.