How To Calculate Average Mass

One of the common tasks you will have to perform as a budding scientist who is able to work with data is understanding the concept of an average. Often, you will encounter a sample of similar objects that differ according to a single characteristic you are studying, such as mass.

You may even have to calculate the average mass of a group of objects you cannot weigh directly, such as atoms.

Most of the 92 atoms that occur in nature come in two or more slightly different forms, called isotopes. Isotopes of the same element differ from each other only in the number of neutrons contained in their nuclei.

It can be useful to apply all of these principles together to come up with the average mass of a selection of atoms drawn from a known pool of different isotopes.

Atoms are the smallest individual unit of an element consisting of all of the properties of that element. Atoms consist of a nucleus containing protons and neutrons that is orbited by nearly massless electrons.

Protons and neutrons weigh about the same as each other. Each proton contains a positive electrical charge equal in magnitude and opposite in sign to that of an electron (negative), while neutrons carry no net charge.

Atoms are characterized primarily by their atomic number, which is just the number of protons in the atom. Adding or subtracting electrons creates a charged atom called an ion, while changing the number of neutrons creates an isotope of the atom, and thus element, in question.

The mass number of an atom is the number of protons plus neutrons it has. Chromium (Cr), for example, has 24 protons (thus defining the element as chromium) and in its most stable form − that is, the isotope that appears most often in nature − it has 28 neutrons. Its mass number is thus 52.

Isotopes of an element are specified by their mass number when written out. Thus the isotope of carbon with 6 protons and 6 neutrons is carbon-12, whereas the heavier isotope with one additional neutron is carbon-13.

Most elements occur as a mixture of isotopes with one significantly predominating over the others in terms of "popularity." For example, 99.76 percent of naturally occurring oxygen is oxygen-16. Some elements, however, such as chlorine and copper, show a wider distribution of isotopes.

A mathematical average is simply the sum of all of the individual results in a sample divided by the total number of items in a sample. For example, in a class with five students who achieved quiz scores of 3, 4, 5, 2 and 5, the class average on the quiz would be

\(\frac{3+4+5+2+5}{5}=3.8\)

The average mass equation can be written in numerous ways, and in some cases you will need to know features related to the average, such as standard deviation. For now, just focus on the basic definition.

Knowing the relative fraction of each isotope of a particular element that occurs in nature allows you to calculate the ​atomic mass​ of that element, which, because it is an average, is not the mass of any one atom but a number that is between the heaviest and lightest isotopes present.

If all of the isotopes were present in the same amount, you could just add up the mass of each kind of isotope and divide by the number of different kinds of isotopes present (usually two or three).

Average atomic mass, given in atomic mass units (amu), is always similar to mass number, but it is not a whole number.

Chlorine-35 has an atomic mass of 34.969 amu and accounts for 75.77% of chlorine on Earth.

Chlorine-37 has an atomic mass of 36.966 amu and a percent abundance of 24.23%.

To calculate the average atomic mass of chlorine, use the information in a periodic table of the element (see Resources) to find the (weighted) average but changing the percents to decimals:

\((34.969 \times 0.7577) + (36.966 \times 0.2423) = 35.45\text{ amu}\)