Relative Atomic Mass Calculator & Formula

This calculator works out the relative atomic mass (Ar) of an element from the masses and natural abundances of its isotopes. You enter the mass of each isotope and its percentage abundance, and it returns the weighted average in atomic mass units (u). The result is the number you see on the periodic table for that element.

Formula

The relative atomic mass is a weighted average of the masses of an element’s naturally occurring isotopes. The formula is:

Ar = (m� × a� + m₂ × a₂ + …) / (a� + a₂ + …)

• mâ‚�, mâ‚‚, … – the mass of each isotope in atomic mass units (u). These are close to the mass number but not exactly equal (e.g., chlorine-35 is 34.968853 u, not 35).
• aâ‚�, aâ‚‚, … – the natural abundance of each isotope as a percentage (e.g., 75.78% for chlorine-35). The sum of all abundances should be 100% for a complete set.

The division by the sum of abundances normalises the result so that the percentages do not need to add to exactly 100 – but they should for a real element.

Worked Example

Chlorine has two stable isotopes: chlorine-35 (mass 34.968853 u, abundance 75.78%) and chlorine-37 (mass 36.965903 u, abundance 24.22%).

1. Multiply each mass by its abundance:
   34.968853 × 75.78 = 2649.58 (to 2 decimal places)
   36.965903 × 24.22 = 895.31
2. Add the products: 2649.58 + 895.31 = 3544.89
3. Add the abundances: 75.78 + 24.22 = 100.00
4. Divide: 3544.89 ÷ 100.00 = 35.4489

The calculator gives 35.4489 u, which rounds to 35.45 – the relative atomic mass of chlorine on the periodic table.

When to Use This Calculator

Use it when you need to find the relative atomic mass of an element from isotope data. This is common in:

• Chemistry homework and lab reports where you are given isotope masses and abundances.
• Mass spectrometry problems that list isotopic peaks.
• Checking whether a periodic table value makes sense given known isotope data.

You can also work backwards: if you know the relative atomic mass and the isotope masses, you can solve for the abundances. But this calculator is set up for the forward calculation (masses and abundances → Ar).

How to Read the Result

The result is the weighted average mass of one atom of the element, expressed in atomic mass units (u). One atomic mass unit is defined as one-twelfth of the mass of a carbon-12 atom. The value on the periodic table is the same number – for example, carbon is 12.011 u because it is a mixture of carbon-12 (98.93%) and carbon-13 (1.07%).

The relative atomic mass is not a whole number because it is an average. It is also not the mass of any single atom – it is the average of all naturally occurring atoms of that element. For elements with only one stable isotope (like fluorine-19), the relative atomic mass is very close to that isotope’s mass.

Common Mistakes and Pitfalls

• Using mass numbers instead of exact masses. Mass numbers (e.g., 35 for chlorine-35) are integers, but the exact isotopic masses are slightly different. Using mass numbers gives an approximate Ar, not the precise one.
• Forgetting to divide by the total abundance. If you just add the products (mâ‚�×aâ‚� + m₂×aâ‚‚) without dividing, you get a number that is too large. The division is essential.
• Abundances not summing to 100%. If you omit a rare isotope, your result will be slightly off. For example, bromine has two main isotopes (50.69% and 49.31%), but if you ignore a tiny trace isotope, the result will be slightly wrong.
• Using percentages as fractions. If you enter 0.7578 instead of 75.78, the result will be 0.3545 – wrong by a factor of 100. Always check that your abundances are in the same form as the formula (percentages).

Limitations of the Formula

This formula assumes you have a complete set of naturally occurring isotopes with their exact masses and abundances. In practice, isotopic abundances vary slightly depending on the source of the element (e.g., different geological samples). The periodic table value is a standard average, but a specific sample might differ slightly. The formula also does not account for nuclear binding energy differences – those are already embedded in the isotopic masses you enter.

For elements with many isotopes (like tin, which has ten stable isotopes), you need to enter all of them. The calculator allows up to six inputs, so for elements with more isotopes, you may need to combine or round. For most common elements (up to three or four isotopes), this is not an issue.

Frequently Asked Questions

What is the difference between relative atomic mass and mass number?

Mass number is the total number of protons and neutrons in a specific isotope (e.g., carbon-14 has mass number 14). Relative atomic mass is the weighted average of all naturally occurring isotopes of an element. For a single isotope, the relative atomic mass is close to its mass number, but not exactly equal because of binding energy and the slight mass difference between protons and neutrons.

Why is the relative atomic mass of chlorine 35.45, not 35.5?

Chlorine has two stable isotopes: chlorine-35 (75.78%) and chlorine-37 (24.22%). The weighted average is 34.968853 × 0.7578 + 36.965903 × 0.2422 = 35.45. The exact value depends on the precise isotopic masses and abundances, which are measured experimentally.

Can I use this calculator for molecules?

No, this calculator is for elements only. For molecules, you would use relative molecular mass (Mr), which is the sum of the relative atomic masses of all atoms in the molecule. For example, water (H₂O) has Mr = 2×1.008 + 15.999 = 18.015.

What are atomic mass units (u)?

An atomic mass unit (u) is defined as one-twelfth of the mass of a carbon-12 atom. It is approximately 1.66053906660 × 10�²� kg. This unit is convenient because atomic masses are extremely small when expressed in kilograms.

Why does the periodic table show a range for some elements?

For elements that have no stable isotopes (like uranium) or whose isotopic composition varies naturally (like lithium), the periodic table may show a range or a footnote. This calculator uses the standard abundance values you provide, so it will give a specific number, not a range.

How do I find isotope masses and abundances?

You can find them in chemistry textbooks, databases like the NIST Physical Measurement Laboratory, or from mass spectrometry data. For common elements, the values are well established and appear in most reference tables.

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in m1 | Mass of isotope 1 | u | 34.968853 | 0
in a1 | Abundance of isotope 1 | % | 75.78 | 0
in m2 | Mass of isotope 2 | u | 36.965903 | 0
in a2 | Abundance of isotope 2 | % | 24.22 | 0
out ar | Relative atomic mass | u | (m1 * a1 + m2 * a2) / (a1 + a2) | 4