Atomic Mass and Average Atomic Mass

Atoms are unimaginably tiny. A single hydrogen atom weighs about $1.6736 \times 10^{-24}$ grams, a number so small that using grams to describe atomic masses would be like measuring the distance between two houses in light-years. Scientists needed a more practical scale, one that gives atoms clean, manageable numbers. That is exactly what the atomic mass unit provides.

How Atomic Mass Is Measured: The Relative Approach

Back in the nineteenth century, scientists could not weigh individual atoms on any balance. What they could do, through careful experiments, was figure out how heavy one type of atom was compared to another. This is a relative measurement: you pick one atom as your reference, call its mass some round number, and then express every other atom’s mass as a multiple or fraction of that reference.

The earliest choice for a reference atom was hydrogen, the lightest element. It was assigned a mass of 1 (with no units), and all other atoms were compared against it. This system worked, but it had practical drawbacks. Over time, scientists moved to a better standard.

The Carbon-12 Standard

In 1961, the international scientific community agreed on a new reference: carbon-12 ($^{12}C$), one of the isotopes (atoms of the same element with different numbers of neutrons) of carbon. Under this system, one atom of $^{12}C$ is assigned a mass of exactly 12 atomic mass units (amu).

From this definition, one atomic mass unit is the mass equal to exactly one-twelfth of the mass of a single $^{12}C$ atom:

$1 \text{ amu} = \frac{1}{12} \times \text{mass of one } ^{12}C \text{ atom}$

In grams, this works out to:

$1 \text{ amu} = 1.66056 \times 10^{-24} \text{ g}$

Every other atom’s mass is then expressed relative to this unit. The beauty of this approach is that you get numbers that are easy to work with: hydrogen is roughly 1 amu, oxygen is roughly 16 amu, and so on.

Calculating Atomic Mass in amu

To find the atomic mass of any element in amu, you divide its absolute mass (in grams) by the value of 1 amu. Here is how it works for hydrogen:

Step 1: The actual mass of one hydrogen atom, measured by modern techniques like mass spectrometry, is:

$\text{Mass of H atom} = 1.6736 \times 10^{-24} \text{ g}$

Step 2: Divide by the value of 1 amu:

$\text{Atomic mass of H} = \frac{1.6736 \times 10^{-24} \text{ g}}{1.66056 \times 10^{-24} \text{ g}}$

$= 1.0078 \text{ amu} \approx 1.0080 \text{ amu}$

In the same way, the atomic mass of $^{16}O$ (oxygen-16) comes out to 15.995 amu.

Notice that hydrogen’s atomic mass is very close to 1 and oxygen’s is very close to 16, which is what you would expect since the scale is built around carbon-12 being exactly 12.

From amu to u: A Change in Notation

You will often see atomic masses written with the symbol u instead of amu. The symbol u stands for unified mass and has officially replaced amu in modern usage. Both refer to exactly the same quantity:

$1 \text{ u} = 1 \text{ amu} = 1.66056 \times 10^{-24} \text{ g}$

So when a periodic table lists oxygen as 15.995 u, it means exactly the same thing as 15.995 amu. The newer symbol simply provides a single, universally agreed-upon notation.

Why Average Atomic Mass?

When you look up the atomic mass of an element in the periodic table, you will notice that the numbers are rarely whole. Carbon shows 12.011, not 12.000. Chlorine shows 35.5, not 35 or 37. Why?

The reason is isotopes. Most elements in nature do not consist of just one type of atom. They exist as a mixture of two or more isotopes, each with a slightly different mass. Since a real-world sample of an element contains all of its isotopes mixed together, the mass you measure for that element is a weighted average of all the isotope masses, where the weight assigned to each isotope is its natural abundance (how commonly it occurs in nature).

This weighted average is called the average atomic mass.

How to Calculate Average Atomic Mass

The formula is straightforward: multiply each isotope’s mass by its fractional abundance (its percentage divided by 100), then add all the products together.

$\text{Average atomic mass} = \sum (\text{fractional abundance}_i \times \text{mass}_i)$

Let us work through the example of carbon, which has three naturally occurring isotopes:

Isotope	Relative Abundance (%)	Atomic Mass (amu)
$^{12}C$	98.892	12
$^{13}C$	1.108	13.00335
$^{14}C$	$2 \times 10^{-10}$	14.00317

Step 1: Convert each percentage to a fraction by dividing by 100:

• $^{12}C$: $0.98892$
• $^{13}C$: $0.01108$
• $^{14}C$: $2 \times 10^{-12}$

Step 2: Multiply each fraction by the corresponding isotope mass:

• $^{12}C$ contribution: $(0.98892)(12 \text{ u}) = 11.86704 \text{ u}$
• $^{13}C$ contribution: $(0.01108)(13.00335 \text{ u}) = 0.14408 \text{ u}$
• $^{14}C$ contribution: $(2 \times 10^{-12})(14.00317 \text{ u}) \approx 0.00000 \text{ u}$ (negligibly small)

Step 3: Add all contributions:

$11.86704 + 0.14408 + \approx 0 = 12.011 \text{ u}$

The average atomic mass of carbon is 12.011 u. This is the number you see in the periodic table.

Notice how the result is very close to 12 but slightly above it. That small bump comes from the 1.108% contribution of $^{13}C$, which is heavier. The trace amount of $^{14}C$ is so tiny that it barely affects the average at all.

What the Periodic Table Really Shows

An important takeaway: the atomic masses printed in the periodic table are not the mass of any single isotope. They are average atomic masses, calculated from the natural isotope mixture of each element. When you use these values in chemical calculations, you are working with the mass that reflects how the element actually appears in nature.