Science and Joe

What is relative atomic mass and how is it calculated? Let’s find out.

Definition of Relative Atomic Mass (Ar)

Okay let’s first go over the technical definition, the one you’ll see in textbooks and the one that examiners will probably want us to use. We’ll then break it down.

“Relative atomic mass is the weighted average, or weighted mean, of the mass of an atom of an element compared to 1/12 of the mass of one atom of Carbon-12.”

I think the most common initial reaction when encountering this definition for the first time is what on earth does that mean? The language is probably a bit stronger than that. Now you can simply memorise this definition and just quote it verbatim if you’re asked to define relative atomic mass in an exam. But that approach does have its limitations, particularly if you need to prove a deeper understanding but calculating it. So, let’s break the definition down.

‘Weighted’, Isotopes and Relative Abundances

Let’s start with ‘weighted average’ or ‘weighted mean’. Average and mean, we know what these terms mean from mathematics, it’s the value you get when you add up all the values in a dataset and then divide by the number of values in the dataset. But hang on, it’s not that straightforward when it comes to relative atomic mass because of the term ‘weighted’.

Weighted signifies that you have accounted for two variables, both of which are to do with isotopes. Quick reminder, isotopes are atoms of the same element that have the same number of protons and electrons, but they have different numbers of neutrons and this is represented by variation in mass number which is the unitless value that represents how many protons and neutrons (collectively known as nucleons) are present in the nucleus of the isotope. Now the two variables involving isotopes that are considered, or weighted, are the relative abundance and the relative isotopic mass, or, mass number, of each isotope of an element. Let’s look at relative abundance first.

Relative abundance, also called natural abundance, is a percentage that represents how much of a given sample of an element is constituted by each isotope. Let me explain with an example, it’s much easier.

Let’s say we have a sample of magnesium. In that sample there will be three naturally occurring isotopes of magnesium: Magnesium-24, Magnesium-25 and Magnesium-26. (This superscript on the left side of the chemical symbol represents the mass number of the isotope.) How much of the sample does each of these three isotopes make up? That’s what the relative abundances will tell us. So, approximately 79% of the sample is Magnesium-24, approximately 10% is Magnesium-25 and approximately 11% is Magnesium-26. That means that if we had 1000 magnesium atoms in the sample, 790 of them would be Magnesium-24, 100 would be Magnesium-25 and 110 would be Magnesium-26. However, relative abundances will typically be given as percentages, particularly when calculating relative atomic mass which we’ll do in a bit. Let’s first quickly go over the other variable that is considered or weighted.

Isotopic Mass, Relative Isotopic Mass and Carbon-12

The other variable can be either the mass number or the relative isotopic mass of each isotope of the element. Now we know what the mass number is, but what about relative isotopic mass?

Relative isotopic mass also has a confusing technical definition, but don’t panic we’ll break it down:

“Relative isotopic mass is the mass of one atom of an isotope of an element compared to 1/12 of the mass of one atom of Carbon-12.”

Like the relative atomic mass definition, this is the one that examiners will probably want you to quote if your asked to define relative isotopic mass. But here’s another way at looking at it:

Relative isotopic mass describes how many times heavier an atom of an isotope is than 1/12 of the mass of Carbon-12.

There’s that 1/12 of the mass of Carbon-12 again, which we saw in the relative atomic mass definition. What does it mean?

1/12 of the mass of Carbon-12 is equal to 1 unified atomic mass unit (amu), or 1 Dalton (Da). For simplicity, let’s stick to unified atomic mass units. Now unified atomic mass units essentially perform the same job that grams and kilograms do in that they measure mass. So, 1/12 of the mass of Carbon-12 is 1 amu (unified atomic mass unit), which means that the mass of an entire atom of Carbon-12 is 12 amu, 12 x 1 amu = 12amu. This is the isotopic mass of Carbon-12, so isotopic mass is the actual mass of an isotope, and it has the units of unified atomic mass units. However, relative isotopic mass doesn’t have units because all this represents is how many times heavier a given isotope is than 1/12 of the mass of Carbon-12. This value is usually given as a whole number. Let’s look at an example.

Let’s use the isotope Calcium-40. Calcium-40 has a mass number of 40 because it has 40 nucleons in its nucleus (20 protons and 20 neutrons), it has an isotopic mass, to four significant figures of 39.96 amu, however its relative isotopic mass would be 40 because that’s what we would round to in order to express it as a whole number. So, what does this mean? It means that one Calcium-40 isotope is, approximately, 40 times heavier than the mass of 1/12 of the mass of Carbon-12. That’s all it represents. It’s a scale based  on 1/12 of the mass of 1/12 of  Carbon-12 which acts a benchmark for determining the comparative mass of all other isotopes. So now let’s return to our definition of relative atomic mass.

The Technical Definition and a Clearer Definition of Relative Atomic Mass

Here’s that exam friendly, technical definition again:

“Relative atomic mass is the weighted average of the mass of an atom of an element compared to 1/12 of the mass of one atom of Carbon-12.”

Here’s an alternative definition that includes everything we just learned:

“Relative atomic mass is a unitless value that indicates how many times heavier the average mass of an atom of an element is (considering the relative abundances and relative isotopic masses or mass numbers of all the isotopes of that element) than 1/12 of the mass of the Carbon-12 isotope.”

That latter definition is obviously a bit longer, but it reveals all the parts involved in relative atomic mass. Okay, lets now actually calculate relative atomic mass.

Calculating Relative Atomic Mass

Okay so firstly a quick explanation as to why either the relative isotopic mass or the mass number of each isotope of an element can be used: they are numerically equal. Providing that relative isotopic mass is expressed as a whole number, which it conventionally is for this calculation, it is going to be the same as the mass number for each isotope.

Okay here’s the actual formula for calculating relative atomic mass:

For consistency, I’m going to use relative isotopic mass for the calculation, but don’t forget you can use mass numbers instead. Please check which one is preferred for the specification of your course.

Okay, so what this formula essentially means is that you multiply the relative isotopic mass of each isotope in the sample by its relative abundance. Then once you’ve done that for all the isotopes of the element in the sample, you then add those results up to get a sum, a total, that you would then divide by the total abundance which is 100%. You don’t usually have to include the percentage signs in the formula for the calculation, but you can include them if you want to.

Okay, Let’s look at an example.

How about potassium (K)?

Question
A sample of potassium contains three naturally occurring isotopes: ³⁹ Potassium with a relative abundance of approximately 93.26%, ⁴⁰ Potassium with a relative abundance of approximately 0.01% and  ⁴¹Potassium with a relative abundance of approximately 6.73%. What is the relative atomic mass of potassium in the sample? (Give your answer to 3 significant figures.)

Answer

Don’t forget the answer doesn’t have any units, we just want the numerical value. Typically, the value for relative atomic mass is expressed in either three or four significant figures. In this video we’re going to stick to 3 significant figures.

Okay let’s do another one, this time involving sulfur. It seems a little bit tricker because there are 4 naturally occurring isotopes, but the method is the same.

Question
A sample of sulfur contains 4 naturally occurring isotopes: ³² Sulfur with a relative abundance of 94.99%; ³³Sulfur with a relative abundance of 0.75%; ³⁴ Sulfur with a relative abundance of 4.25% and ³⁶Sulfur with a relative abundance of 0.01%. What is the relative atomic mass of sulfur in the sample? (Give your answer to 3 significant figures.)

Answer

                                             

Okay that’s nearly the end of this tutorial. But I’m going to leave you with another practice question.

A Question for You

A sample of silver is collected for analysis and is found to contain 51.4% of ¹⁰⁷Ag and 48.6% of ¹⁰⁹Ag. What is the relative atomic mass of silver in the sample? Give your answer to 3 significant figures. Give it a try and feel free to leave your answers in the comments section down below.