Audited —·Last updated 29 Jul 2026·5 citations·Tier 2·0 uses

Average Atomic Mass Calculator — From Isotope Masses and Abundances

Work out an element's average atomic mass from its isotope masses and natural abundances, with NIST nuclide data and a check against the CIAAW value.

Average Atomic Mass Calculator

One relative atomic mass per isotope, in unified atomic mass units (u, also called daltons). Separate with commas, spaces or new lines. Use the ATOMIC MASS, not the mass number — copper-63 is 62.92959772 u, not 63. Strip any uncertainty brackets: enter 62.92959772, not 62.92959772(56).
One abundance per isotope, in the same order, as a percentage. NIST publishes these as amount fractions, so 0.6915 goes in here as 69.15. The list must add up to 100 % within half a percentage point, or the calculator will stop and tell you.
Average atomic mass
63.546
The abundance-weighted mean over the nuclides you entered, in unified atomic mass units (u = dalton). Numerically the same as the relative atomic mass A_r as a pure number, and as the molar mass in g/mol. Twelve digits are shown; the real precision is set by the abundance data, which for many elements is four significant figures or fewer.
Abundances add up to
100 %
Isotopes read
2
Most abundant isotope
62.9296 u

Background.

This calculator works out an element's average atomic mass from the masses of its isotopes and how common each one is: multiply each nuclide's atomic mass by its natural abundance, add the products, and divide by the total abundance. Enter two matching comma-separated lists — masses in unified atomic mass units, abundances in percent — and it returns the weighted mean along with the abundance total, the isotope count, and the mass of the most abundant nuclide.

The reference state matters here, and it is not a temperature. Published isotopic abundances describe normal terrestrial material: samples from Earth's crust, hydrosphere and atmosphere that have not been artificially fractionated. They do not describe meteoritic or lunar material, interstellar gas, or anything that has been through an enrichment cascade. Heavy water, ⁶Li-depleted lithium and enriched ²³⁵U all have different isotopic compositions and therefore different average atomic masses, and the periodic-table value is simply the wrong number for them.

The error this page exists to prevent is feeding it mass numbers. A nuclide's mass is not the sum of its nucleons — some of that mass is spent as nuclear binding energy — so copper-63 weighs 62.92959772 u, not 63. Using the mass numbers 63 and 65 with copper's abundances gives 63.617 u instead of the correct 63.5460399458 u. That is 0.071 u out, wrong in the third significant figure, and it looks entirely plausible on the page. Carbon-12 is the one exception, and only because it defines the unit: it is exactly 12 u.

The second guard is on the abundances themselves. The calculator refuses any set that does not add up to 100 % within half a percentage point, because a total that misses usually means a forgotten isotope or a mistyped digit, and quietly accepting it produces a believable wrong answer. It also catches the single commonest slip specifically: NIST publishes isotopic compositions as amount fractions, so copper-63 appears in its tables as 0.6915. Paste that in unchanged and the sum comes to 1, and the calculator will say so by name rather than just failing.

Worked through with copper, the example this page is built on. NIST gives copper-63 as 62.92959772 u at 0.6915 and copper-65 as 64.92778970 u at 0.3085, so the abundances are 69.15 % and 30.85 %. The weighted mean is (69.15 × 62.92959772 + 30.85 × 64.92778970) ÷ 100 = 63.5460399458 u. IUPAC's Commission on Isotopic Abundances and Atomic Weights publishes A_r(Cu) = 63.546(3), a value it has held since 1969 — so the arithmetic reproduces an independently evaluated figure to five significant figures.

That CIAAW uncertainty is worth reading carefully, because it is the honest limit on this page's twelve displayed digits. The ±0.003 was deliberately widened in 1969 to cover roughly 0.15 % natural variation in copper's isotopic abundances between real samples. So 63.546 is the answer; 63.5460399458 is arithmetic. For more than a dozen elements CIAAW declines to publish a single value at all and gives an interval instead, because the variation is large enough that no point value is defensible.

This is a weighted mean, and the general-purpose version of it lives at the weighted average calculator. What this page adds is the domain: the units, the terrestrial reference state, the mass-defect trap, the abundance-sum gate, and a comparison against the published standard atomic weight.

What is average atomic mass calculator?

The average atomic mass of an element is the mean mass of its atoms, weighted by how frequently each isotope occurs in nature. It is what the periodic table prints, and it is the number that goes into a molar mass calculation.

Three closely related quantities get used interchangeably and are worth separating. The relative atomic mass A_r, defined in the IUPAC Green Book as A_r = m_a / m_u — an atom's mass divided by the atomic mass constant — is dimensionless. The atomic mass itself is expressed in unified atomic mass units, symbol u, for which the dalton (Da) is an alternative name; m_u is defined as one twelfth of the mass of a carbon-12 atom, which is why ¹²C is exactly 12 u. And the molar mass in g/mol is numerically the same number again, to within the molar mass constant's relative uncertainty of 3.1 × 10⁻¹⁰. This calculator reports in u; read the same figure as A_r or as g/mol as you need.

No atom actually has the average mass. Copper atoms weigh either 62.92959772 u or 64.92778970 u; none weighs 63.546 u. That is why a mass spectrum of copper shows two peaks rather than one, and why the average is the right number for weighing out a bulk reagent but the wrong number for interpreting a spectrum. The calculator reports the most abundant nuclide's mass alongside the average to keep the distinction in view.

The reason this quantity is published rather than simply looked up per sample is that isotopic composition is nearly, but not perfectly, constant across the Earth. For most elements the variation is negligible and CIAAW publishes a single value with an uncertainty. For hydrogen, lithium, boron, carbon, nitrogen, oxygen, magnesium, silicon, sulfur, chlorine, argon, bromine, thallium and lead it is not negligible, and CIAAW publishes an interval — chlorine's standard atomic weight is [35.446, 35.457] rather than any single number.

How to use this calculator.

  1. List each isotope's atomic mass in the first box, separated by commas. Use the relative atomic mass in u from a nuclide table — copper-63 is 62.92959772, not 63. If your source shows an uncertainty in brackets, such as 62.92959772(56), drop the bracket.
  2. List the natural abundances in the second box, in the same order, as percentages. NIST publishes them as amount fractions, so convert: 0.6915 becomes 69.15, 0.7576 becomes 75.76.
  3. Check the 'Abundances add up to' output. A published composition totals exactly 100.00 %. Anything more than half a percentage point away is refused, because it usually means an isotope is missing from your list.
  4. Read the average atomic mass in u. The same number is the relative atomic mass A_r as a pure number, and the molar mass in g/mol — so 63.5460399458 u for copper means 63.546 g/mol for weighing purposes.
  5. Compare with the published standard atomic weight for that element. If they disagree beyond the published uncertainty, the usual causes are mass numbers used in place of atomic masses, a transposed abundance, or a missing minor isotope.
  6. Round before you report. Copper's published uncertainty is ±0.003, so 63.546 is the honest answer; the extra digits on screen are arithmetic, not knowledge.
  7. Do not use this for enriched, depleted or extraterrestrial material. Published abundances describe normal terrestrial samples only — a labelled compound or an enrichment product needs its own measured composition.

The formula.

Ā = Σ (fᵢ × mᵢ) / Σ fᵢ
Ā · average atomic mass (u, = dalton)mᵢ · relative atomic mass of nuclide i (u) — NOT its mass numberfᵢ · natural abundance of nuclide i (%)Σ fᵢ · must be 100 % ± 0.5 pp1 u = m(¹²C) / 12, so ¹²C is exactly 12 u

The average atomic mass is a linear mixing model over an element's nuclides:

Ā = Σᵢ (fᵢ × mᵢ) / Σᵢ fᵢ

Dimensionally, an abundance is a ratio and carries no dimension, so [u] × [1] leaves [u] — the answer comes out in atomic mass units, not in 'u percent'. The model is exact for a mixture; it is a definition of a mean, not a physical approximation.

Worked through with copper. NIST gives copper-63 as 62.92959772 u with an amount fraction of 0.6915, and copper-65 as 64.92778970 u at 0.3085, so the abundances are 69.15 % and 30.85 %:

69.15 × 62.92959772 = 4351.581682338 30.85 × 64.92778970 = 2003.022312245 sum = 6354.603994583 Ā = 6354.603994583 / 100.00 = 63.5460399458 u

CIAAW publishes A_r(Cu) = 63.546(3), unchanged since 1969, so the calculation reproduces an independently evaluated value to five significant figures.

WHY THE SUM IS DIVIDED BY WHAT YOU ENTERED. The denominator is the abundance total as typed, not a hard-coded 100. For a published composition the two are identical, since real compositions sum to exactly 100.00. They differ only when a list is slightly incomplete — and dividing by 100 there would treat the missing fraction as a massless nuclide and pull the answer low. A list summing to 99.5 % would be biased down by about 0.3 u for copper, which is wrong in the third significant figure. The ±0.5 percentage-point check is a separate data-integrity gate: it refuses input that looks broken, rather than rescaling an answer that is fine.

ROUNDING STAGE — final only, and the tolerance test is deliberately made against the unrounded sum. This is the one place on this page where rounding stage could change an answer: had the abundance total been rounded to two decimal places before being compared with the tolerance, a total of 100.5001 % would have rounded to 100.50 and slipped through. It is not rounded. The band is inclusive, so exactly 99.5 % and exactly 100.5 % both pass and 100.5001 % does not, and there are tests immediately below, at, and immediately above both edges.

THE MASS-DEFECT TRAP. A nuclide's mass is less than the sum of its free nucleons, because binding energy carries mass away. Copper-63 weighs 62.92959772 u, not 63. Substituting mass numbers gives 63 × 0.6915 + 65 × 0.3085 = 63.617 u, which is 0.071 u high — more than twenty times CIAAW's stated uncertainty for copper, and completely invisible unless you check it against the published value. Carbon-12 is the sole exception, at exactly 12 u, because it is what defines the unit.

LIMITS AND EDGE CASES. A single isotope at 100 % returns its own mass: fluorine has only one stable nuclide, and entering 18.99840316273 u at 100 % returns 18.9984031627 u, matching the published standard atomic weight of fluorine, 18.998403163(6). An isotope entered at 0 % contributes nothing, so a full nuclide list can be kept with the unwanted entries zeroed. Masses must be positive and abundances non-negative; an all-zero abundance set, a negative mass, mismatched list lengths and unparseable entries are each refused with a message naming the field.

A worked example.

Example

Copper, the standard textbook example, done with real nuclide data. Step 1 — get the nuclide data. NIST's Atomic Weights and Isotopic Compositions database lists copper-63 with a relative atomic mass of 62.92959772(56) and an isotopic composition of 0.6915(15), and copper-65 at 64.92778970(71) with 0.3085(15). Two conversions before typing: drop the uncertainty brackets, and turn the amount fractions into percentages — 0.6915 becomes 69.15 and 0.3085 becomes 30.85. Step 2 — the weighted sum. 69.15 × 62.92959772 = 4351.581682338 and 30.85 × 64.92778970 = 2003.022312245, giving 6354.603994583. The abundances total 100.00 %, so dividing gives Ā = 63.5460399458 u. Step 3 — check it against an independent authority. CIAAW's copper page publishes A_r(Cu) = 63.546(3), a value the commission has held since 1969. The calculation reproduces it to five significant figures. That is a real check rather than a formality: the standard atomic weight was evaluated from 1960s mass-spectrometric abundance work, while the nuclide masses used here come from the modern atomic mass evaluation, so agreement across the two routes means the pairing and the weighting are both right. Step 4 — round honestly. CIAAW's ±0.003 was widened in 1969 to cover about 0.15 % natural variation in copper's isotopic composition between samples. So the answer to quote is 63.546 u. The remaining digits on the screen are arithmetic. Try chlorine next, which is more interesting. Chlorine-35 is 34.968852682 u at 75.76 % and chlorine-37 is 36.965902602 u at 24.24 %, giving 35.4529375826 u. CIAAW does not publish a single value for chlorine at all — its standard atomic weight is the interval [35.446, 35.457], because chlorine's isotopic composition varies measurably between terrestrial sources. Our result sits inside that interval and rounds to the abridged conventional value of 35.45 that textbooks quote. Note also what the two authorities do differently: NIST gives point abundances of 0.7576 and 0.2424 as a representative composition, while CIAAW gives ranges of [0.755, 0.761] and [0.239, 0.245]. Those are different kinds of statement about the same element, and neither is the abundance of chlorine. And magnesium, to show that three isotopes work the same way: 23.985041697 u at 78.99 %, 24.985836976 u at 10.00 % and 25.982592968 u at 11.01 % give 24.3050516198 u, inside CIAAW's interval [24.304, 24.307] and rounding to the familiar 24.305.

isotope Masses62.92959772, 64.92778970
abundances69.15, 30.85

Frequently asked questions.

How do I calculate average atomic mass?
Multiply each isotope's atomic mass by its natural abundance, add the products, and divide by the total abundance. With abundances in percent and a complete list, that division is by 100. For copper: 69.15 % of copper is copper-63 at 62.92959772 u and 30.85 % is copper-65 at 64.92778970 u, so the average is (69.15 × 62.92959772 + 30.85 × 64.92778970) ÷ 100 = 63.5460399458 u, which matches the published value of 63.546. The one thing to get right is the mass column: use each nuclide's actual atomic mass, not its mass number. Substituting 63 and 65 gives 63.617 u, which is wrong in the third significant figure and looks perfectly reasonable.
Why is copper's atomic mass 63.546 and not 63 or 64?
Because it is a weighted average of two nuclides that are neither of those numbers, and because natural copper is about 69 % of the lighter one. Copper-63 weighs 62.92959772 u and copper-65 weighs 64.92778970 u; at 69.15 % and 30.85 % the mean lands at 63.5460399458 u, closer to the lighter isotope because that one is more common. Two separate reasons keep it off a whole number. The abundances are not 50:50, and no nuclide's mass is a whole number anyway — nuclear binding energy means a nucleus weighs less than its free nucleons, so copper-63 comes in at 62.93 rather than 63. Carbon-12 is the only exact whole number in the whole table, and only because it defines the unit.
The abundances I have are like 0.7576, not 75.76. What do I enter?
Multiply by 100 and enter 75.76. NIST's Atomic Weights and Isotopic Compositions database publishes isotopic compositions as amount fractions, so chlorine-35 appears as 0.7576 and copper-63 as 0.6915; this calculator takes percentages. If you paste the fractions in unchanged the total comes to 1 rather than 100, and the calculator stops and tells you specifically that the numbers look like amount fractions, rather than just refusing. Mathematically a weighted mean is scale-invariant and the fractions would give the same answer, but accepting both silently would mean accepting genuinely broken input too — a set that sums to 1.4, say, would go through unnoticed. Requiring one convention is what makes the sum check useful.
Why does the calculator refuse abundances that do not add up to 100 %?
Because in this domain a total that misses 100 % almost always means something is wrong with the data rather than with the element. The usual causes are a forgotten minor isotope, a transposed digit, or amount fractions used in place of percentages. A general weighted-average tool would happily normalise whatever you gave it and return a plausible number, and you would have no way of knowing. The tolerance is half a percentage point either side, which absorbs rounded literature values and genuinely negligible trace nuclides while still catching a missing major isotope. When the total is inside the band, the mean is divided by the total you actually entered — not by a hard-coded 100 — so a marginally incomplete list is not biased low by treating the missing fraction as though it were massless.
Why does CIAAW give chlorine an interval instead of a number?
Because chlorine's isotopic composition genuinely varies between terrestrial sources by more than the measurement precision, so no single value is defensible. CIAAW's standard atomic weight for chlorine is [35.446, 35.457], and it publishes the abundances themselves as ranges too — [0.755, 0.761] for chlorine-35 and [0.239, 0.245] for chlorine-37 — with interval notation in use since 2009. Fourteen or so elements are in this position: hydrogen, lithium, boron, carbon, nitrogen, oxygen, magnesium, silicon, sulfur, chlorine, argon, bromine, thallium and lead. NIST takes a different approach for the same element, publishing a single representative composition of 0.7576 and 0.2424, which is what textbook exercises use and what this page's chlorine example uses. Those two are different kinds of statement, not a disagreement about a value: our computed 35.4529375826 u falls inside CIAAW's interval and rounds to the abridged conventional 35.45 that periodic tables print.
Is average atomic mass the same as molar mass, or as relative atomic mass?
Numerically yes, in all three cases, but they are different quantities with different units and it is worth keeping them straight. The relative atomic mass A_r is defined by IUPAC as an atom's mass divided by the atomic mass constant, so it is dimensionless — a pure number. The average atomic mass is the same figure carrying the unit u, the unified atomic mass unit, for which dalton (Da) is an alternative name; the constant is defined as one twelfth of the mass of a carbon-12 atom, which is why ¹²C is exactly 12 u. The molar mass is the same figure again in g/mol, and since the 2019 SI revision that correspondence is exact only to within the molar mass constant, whose relative uncertainty is 3.1 × 10⁻¹⁰. So copper's 63.5460399458 u is A_r = 63.546 and M = 63.546 g/mol, and you can move between them freely at any precision you will ever need at a bench.
Can I use this for enriched or labelled material?
Yes, but only with that material's own measured composition — never with the periodic-table abundances. Published isotopic compositions describe normal terrestrial material: unfractionated samples from the crust, hydrosphere and atmosphere. Enriched ²³⁵U, ⁶Li-depleted lithium reagent, D₂O and ¹³C-labelled internal standards have all been through a process that deliberately changes the isotope ratios, and the natural abundances are simply the wrong input. The same caution applies to meteoritic, lunar and interstellar material, and to biological or hydrological samples with real isotopic fractionation. The calculation itself is unchanged — it is still a weighted mean — but the weights have to come from a measurement of your actual material.
How does this differ from a general weighted average calculator?
The arithmetic is the same weighted mean, and if you fed isotope masses and abundances into the weighted average calculator you would get the same number. What this page adds is everything around the arithmetic: it states the units, names the terrestrial reference state that the abundances assume, refuses an abundance set that does not sum to 100 % instead of quietly normalising it, catches the amount-fraction mistake by name, reports the most abundant nuclide alongside the average so the distinction between the two is visible, and gives you the published standard atomic weight to check against. If you want the general form for weights that are not abundances — grades, prices, survey responses — the weighted average calculator is the right tool and it is linked below.

References& sources.

  1. [1]NIST Standard Reference Database 144 — 'Atomic Weights and Isotopic Compositions with Relative Atomic Masses' (Coursey, J. S., Schwab, D. J., Tsai, J. J. & Dragoset, R. A.). Source of every nuclide figure used on this page: Cu-63 62.92959772(56) / 0.6915(15), Cu-65 64.92778970(71) / 0.3085(15); Cl-35 34.968852682(37) / 0.7576(10), Cl-37 36.965902602(55) / 0.2424(10); Mg-24 23.985041697(14) / 0.7899(4), Mg-25 24.985836976(50) / 0.1000(1), Mg-26 25.982592968(31) / 0.1101(3); F-19 18.99840316273(92). Relative atomic masses derive from the AME atomic mass evaluation; isotopic compositions are IUPAC representative values. Open access; retrieved and verified 2026-07-29.
  2. [2]IUPAC CIAAW — element data page for copper: 'Ar(Cu) = 63.546(3) since 1969', with abundances 0.6915(15) and 0.3085(15) and the note that the uncertainty was expanded in 1969 to cover roughly 0.15 % natural variation in isotopic abundance between samples. Consulted as the independent second authority for this page: the weighted mean computed here, 63.5460399458 u, reproduces it to five significant figures. Open access; retrieved and verified 2026-07-29.
  3. [3]IUPAC CIAAW — element data page for chlorine: standard atomic weight [35.446, 35.457] in interval notation since 2009, with abundances published as ranges [0.755, 0.761] for Cl-35 and [0.239, 0.245] for Cl-37. Recorded here as a source-presentation conflict with NIST's single representative composition (0.7576 / 0.2424): the two are different kinds of statement, and this page uses NIST's point values because a weighted mean requires a point. The result, 35.4529375826 u, lies inside CIAAW's interval. Open access; retrieved and verified 2026-07-29.
  4. [4]IUPAC CIAAW, Standard Atomic Weights, 2024 revision (the 2021 technical report as revised for Gd, Lu and Zr) — the table of interval-valued elements: H, Li, B, C, N, O, Mg, Si, S, Cl, Ar, Br, Tl and Pb are given as intervals rather than point values because their terrestrial isotopic composition varies measurably. Open access; retrieved and verified 2026-07-29.
  5. [5]IUPAC, Quantities, Units and Symbols in Physical Chemistry ('Green Book'), 3rd edition, 2nd printing 2012 — section 2.10, p. 47 defines relative atomic mass A_r = m_a / m_u with unit 1, and the atomic mass constant m_u = m_a(carbon-12)/12; section 3.7, p. 92 records the dalton (Da) as an alternative name for the unified atomic mass unit u. Open-access searchable PDF; text-verified 2026-07-29.
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