Weighted Average Atomic Mass Calculator

This weighted average atomic mass calculator estimates an element’s atomic mass from the masses of its isotopes and their fractional abundances. Enter each isotope mass in atomic mass units (u) and its abundance as a percentage; the calculator returns the weighted mean in u.

Formula

The weighted average atomic mass is the sum of each isotope’s mass multiplied by its fractional abundance:

Average atomic mass = Σ(m_i × f_i)

• m_i is the mass of isotope i, measured in atomic mass units (u).
• f_i is the isotope’s fractional abundance, a unitless proportion between 0 and 1.

When abundances are entered as percentages p_i, the same calculation is Σ(m_i × p_i) / Σp_i. Dividing by the total percentage makes the calculator work even if rounded abundance values do not sum to exactly 100%. The calculator applies each mass to the abundance in the matching position in the two lists.

Worked Example

The defaults use two carbon isotopes: carbon-12 at 12 u and carbon-13 at 13 u, with abundances of 98.93% and 1.07%.

Weighted mean = (12 × 98.93 + 13 × 1.07) / (98.93 + 1.07)

= (1187.16 + 13.91) / 100

= 1201.07 / 100 = 12.0107 u

The total abundance is 100.00%. The result rounds to four decimal places, as displayed by the calculator. The example is a convenient illustration of the arithmetic; actual isotope masses are not always whole numbers, and measured abundances depend on the sample and reference data used.

Assumptions

• Each isotope mass is paired with its abundance at the same position in the other list.
• Abundances are percentages, not fractions such as 0.9893.
• The entered values describe the isotopic composition being averaged; the calculation does not identify isotopes or look up their masses.
• For lab work, follow your lab method, the safety data sheet (SDS) and your supervisor.

What the Result Means

An isotope is a version of an element with the same number of protons but a different number of neutrons. For example, carbon-12 and carbon-13 are both carbon, but their nuclei have different masses. A sample containing more of the heavier isotope has a higher weighted mean than a sample containing more of the lighter one.

The result is not usually a whole number because it represents an average over atoms in a naturally occurring or specified mixture. It is also not the mass of one particular atom. The mass number in an isotope’s name, such as 13 in carbon-13, counts protons and neutrons; isotope mass in u is a measured physical mass and is not necessarily that integer.

Atomic mass and molar mass have closely related numerical values but different units and meanings. An atomic mass in u describes the mass scale for an atom or isotope. The corresponding molar mass is expressed in grams per mole (g/mol). For ordinary chemistry calculations, periodic-table atomic weights are commonly used as molar masses in g/mol, but a calculator result in u should not be relabeled as g/mol.

How to Use the Calculator

1. Enter isotope masses as numbers in u, using the same order as the abundance list.
2. Enter each isotope’s percent abundance in the corresponding position. Use values such as 98.93, not 0.9893, for 98.93%.
3. Read the weighted average in u and check the total abundance shown below it.

For more than two isotopes, add every isotope mass and its matching abundance to the lists. Keep the lists the same length. The calculator’s input limits may depend on the page implementation; scientifically, isotope masses must be nonnegative and abundance percentages cannot be negative. A percentage for a single isotope cannot exceed 100% when the values represent a normalized mixture.

Common Errors and Limits

• Using mass numbers instead of isotope masses: Whole-number mass numbers can be suitable for a rough classroom estimate, but accurate results require isotope masses in u.
• Entering fractions as percentages: If you enter 0.9893 instead of 98.93, the ratio may still return the same mean if all abundances use that scale. But mixing fractions and percentages produces an incorrect result.
• Mismatching list order: If an isotope’s mass is paired with another isotope’s abundance, the calculated mean is wrong even if the total percentage looks plausible.
• Rounding too early: Retain the precision of the source masses and abundances through the calculation, then round the final answer to a sensible number of significant figures.
• Assuming every sample has the standard composition: Natural isotope abundances can vary among samples and locations. A measured sample’s composition may differ from a periodic-table reference value.

The calculator computes a mean from the values supplied; it does not establish whether those values are accurate, representative, or appropriate for a particular purpose. For reference atomic weights, consult a current authoritative table, and note whether it reports a standard atomic weight, an interval, or values for a specific material. Some elements have no stable isotopes, so a conventional atomic weight may be based on a selected isotope rather than a natural mixture. In that case, a weighted-average calculation is meaningful only if you have a defined set of isotopes and abundances to use.

Frequently Asked Questions

How do you calculate weighted average atomic mass?
Multiply each isotope’s mass by its fractional abundance, then add the products. If using percentage abundances, divide the sum of mass × percentage by the sum of the percentages.

Why is atomic mass on the periodic table usually a decimal?
For elements with naturally occurring isotopes, the listed value commonly reflects a weighted average of isotope masses rather than the mass of a single isotope.

Do isotope abundances have to add up to 100%?
For a complete, normalized isotope mixture, yes. Rounded values may total slightly above or below 100%; dividing by the total abundance accounts for that rounding, but a large discrepancy suggests missing or incorrect data.

Can I use mass numbers instead of isotope masses?
You can use them for an approximate estimate, but they are integer counts of protons and neutrons, not precise isotope masses. Use measured isotope masses for a more accurate result.

What units should the answer have?
The weighted atomic mass is reported in u. The corresponding molar mass has the same numerical value to a useful approximation but is expressed in g/mol and refers to one mole of atoms.

Related

• Weighted Average Atomic Mass Calculator With Abundance
• Isotope Atomic Mass Calculator: Mass of One Atom From amu
• What Does the Rydberg Equation Calculate? Hydrogen Line Wavelength Calculator

list masses | Isotope masses | 12, 13 | 0
list abundances | Abundances (%) | 98.93, 1.07 | 0
out mass | Weighted average atomic mass | u | sum(masses * abundances) / sum(abundances) | 4
out total | Total abundance | % | sum(abundances) | 2