How is average atomic mass calculated?

Multiply the mass of each isotope by its frequency (as a decimal) and add the results together. If the frequency is given as a percentage, divide it by 100 first. The formula is: AM = f1*m1+f2*m2 + ... + fn*mn. In this formula each "f" represents the frequency of an isotope (as a decimal), and each "m" represents the mass of that isotope, in units of amu (atomic mass unit).

Why aren't atomic weights always whole numbers?

Many elements are composed of a mixture of different isotopes with differing masses. The numbers on the periodic table are weighted averages of these isotopes and therefore do not occur at specific whole numbers, but rather between their masses. Chlorine has an average value of 35.45 amu while its isotopes have values near 35 and 37.

What is the difference between atomic mass and average atomic mass?

The atomic mass usually refers to the mass of a particular atom or isotope and is close to the sum of its protons and neutrons. The average atomic mass is the weighted average of all the naturally occurring isotopes of an element, taking into account the abundance of each isotope. This is also the value given in the periodic table.

Does the sum of frequencies have to be 100%?

Yes, this must be true for the average to be calculated correctly. Small rounding errors are acceptable. If the sum of frequencies is not exactly 100%, then this calculator will scale the frequencies so that they add up to the total and display a normalized average, so that rounding errors do not give incorrect results.

How do you find out the percent frequency from average atomic mass?

Elements with two isotopes use a different formula. The abundance of the first isotope is (AM - m2)/(m1 - m2), while the second isotope makes up the rest. The result will only be a valid percentage if the average atomic mass falls between the masses of the two isotopes.

Atomic Mass Calculator

Atomic mass

What would you like to calculate?

Average atomic mass (from isotopes)

Find a missing abundance

Atomic mass from protons and neutrons

Mass of isotope 1 (amu)

Abundance of isotope 1

Mass of isotope 2 (amu)

Abundance of isotope 2

Average atomic mass (amu)

Each isotope adds its mass times its share of the sample. Together they average to 35.45 amu.

Breakdown

Each isotope contributes its mass multiplied by its fractional abundance; the contributions sum to the average atomic mass.
Isotope	Mass (amu)	Abundance (%)	Contribution (amu)
Isotope 1	34.969	75.78	26.499
Isotope 2	36.966	24.22	8.953

Where the average comes from

Each slice is how much an isotope adds to the average atomic mass (its mass times its abundance).

Average atomic mass adds each isotope's mass times its fractional abundance. The missing-abundance mode reverses that for two isotopes, and the proton-plus-neutron mode gives the mass number of a single atom.

It is rare for the atomic mass on the periodic table to be a whole number. This is because many elements are made up of a mixture of different isotopes. There are three problems associated with "atomic mass" in this calculator: calculating the weighted average for each isotope of an element, working backwards from knowing the average to find out what proportion of isotopes there are, and calculating the mass of a single atom based on the number of protons and neutrons.

Select a mode from the top and enter the corresponding values into the fields provided. The results will be shown on the right (or directly below the input fields on smartphones).

What is average atomic mass?

Atoms of the same element always have the same number of protons, but they can vary in the number of neutrons. These different atoms are called isotopes and each isotope has a different mass. For example there is chlorine-35 and chlorine-37.

The average atomic mass is the weighted average of all isotopes that make up an element, with the weighting depending on how common each isotope is in nature. Since it represents the overall contribution of a mixture with one number, this value is listed on the periodic table.

Formula for calculating average atomic mass:

Multiply each isotope's mass by its relative abundance, and then add up all of your results.

\text{AM} = f_1 m_1 + f_2 m_2 + \dots + f_n m_n

where AM is the average atomic mass, each m represents the mass of an isotope expressed in amu (atomic mass units), and each f is the fractional abundance of that isotope. Since percent abundance is given as 75.78%, which corresponds to a value of 0.7578, this calculator will automatically divide by 100 if you enter the abundance as a percentage.

Example: chlorine

Chlorine has two stable isotopes. The isotope Chlorine-35 has a mass of 34.96885 amu and an abundance of 75.78%. The isotope Chlorine-37 has a mass of 36.96590 amu and an abundance of 24.22%. Plug these values into the formula.

\text{AM} = 34.96885 \times 0.7578 + 36.96590 \times 0.2422

The contribution of the first isotope is about 26.50 amu and that of the second is about 8.95 amu so the average atomic mass is 35.45 amu. This agrees with the value listed for chlorine in the periodic table (35.45). Since this calculation tool contains these values for chlorine by default, you can check your result immediately.

Estimating unknown relative abundance.

In textbooks tasks are often asked to solve problems inversely. The masses of two isotopes of an element and the average atomic mass of this element are given, and it is necessary to indicate the relative frequencies of each isotope in percent. A simple method can be used since the sum of the relative frequencies must be 100%. When switching to another mode, this calculator uses the following formula:

f_1 = \frac{\text{AM} - m_2}{m_1 - m_2}, \qquad f_2 = 1 - f_1

Copper is a good example to check. The masses of the two copper isotopes are 62.9296 amu and 64.9278 amu, respectively, while the average atomic mass is 63.546 amu. The formula gives 69.15% for Cu-63 and 30.85% for Cu-65, which agrees with the normally observed natural abundances.

Calculating atomic mass from number of protons and neutrons:

The mass of a single atom is almost entirely determined by the number of protons and neutrons it has since the mass of an electron is about 1,836 times smaller than that of a proton. To quickly estimate the atomic mass, one can simply add up the number of protons and neutrons using the unit "atomic mass units".

A = Z + N

A is the mass number, Z is the number of protons and N is the number of neutrons. Carbon (12) has 6 protons and 6 neutrons so A = 12. The masses are also shown in kilograms in the third mode since one atomic mass unit is approximately equal to 1.66054 multiplied by 10 to the power of -27 kg. The actual mass is slightly less than Z+N because a small amount of mass is converted into binding energy that holds the nucleus together. This conversion does not take this effect into account.

Common isotopic data for some elements:

Element	Isotopes (mass amu, abundance %)	Average atomic mass (amu)
Carbon	12.00000 (98.94%), 13.00335 (1.06%)	12.011
Chlorine	34.96885 (75.78%), 36.96590 (24.22%)	35.45
Copper	62.9296 (69.15%), 64.9278 (30.85%)	63.55
Boron	10.0129 (19.9%), 11.0093 (80.1%)	10.81
Bromine	78.9183 (50.69%), 80.9163 (49.31%)	79.90

This is how this calculation tool works:

To calculate the average atomic mass, select the number of isotopes for that element and enter the mass of each isotope as well as its abundance. Make sure to add up all the abundances so they total 100%.
To determine an unknown abundance, switch to the appropriate mode and enter the masses of both isotopes as well as the average atomic mass for the element.
To calculate the mass of a single atom switch to the protons and neutrons mode and enter the number of both particles.
Check the result which is prominently shown on the right side. In mass-average mode a detailed table and pie chart are also shown.

The masses and abundances of isotopes can vary slightly depending on the source, and for some elements may be different for various natural samples. For graded assignments use values given in class or in materials.

Frequently asked questions

How is average atomic mass calculated?: Multiply the mass of each isotope by its frequency (as a decimal) and add the results together. If the frequency is given as a percentage, divide it by 100 first. The formula is: AM = f1*m1+f2*m2 + ... + fn*mn. In this formula each "f" represents the frequency of an isotope (as a decimal), and each "m" represents the mass of that isotope, in units of amu (atomic mass unit).
Why aren't atomic weights always whole numbers?: Many elements are composed of a mixture of different isotopes with differing masses. The numbers on the periodic table are weighted averages of these isotopes and therefore do not occur at specific whole numbers, but rather between their masses. Chlorine has an average value of 35.45 amu while its isotopes have values near 35 and 37.
What is the difference between atomic mass and average atomic mass?: The atomic mass usually refers to the mass of a particular atom or isotope and is close to the sum of its protons and neutrons. The average atomic mass is the weighted average of all the naturally occurring isotopes of an element, taking into account the abundance of each isotope. This is also the value given in the periodic table.
Does the sum of frequencies have to be 100%?: Yes, this must be true for the average to be calculated correctly. Small rounding errors are acceptable. If the sum of frequencies is not exactly 100%, then this calculator will scale the frequencies so that they add up to the total and display a normalized average, so that rounding errors do not give incorrect results.
How do you find out the percent frequency from average atomic mass?: Elements with two isotopes use a different formula. The abundance of the first isotope is (AM - m2)/(m1 - m2), while the second isotope makes up the rest. The result will only be a valid percentage if the average atomic mass falls between the masses of the two isotopes.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

IUPAC: Atomic weights of the elements (Technical Report)
The IUPAC Commission on Isotopic Abundances and Atomic Weights, the authority on standard atomic weights and isotopic compositions.
NIST: Atomic Weights and Isotopic Compositions
Reference isotope masses and natural abundances for every element from the National Institute of Standards and Technology.
LibreTexts Chemistry: Calculating Atomic Mass
Open educational reference on isotopes, natural abundance, and the weighted-average calculation.
NIST: Fundamental physical constants, atomic mass unit
The CODATA value of the unified atomic mass unit in kilograms (1 u = 1.66053906660e-27 kg).