I was led to thinking about relative atomic mass the other day, and asked myself a really simple question that I couldn’t answer. I was slightly embarrassed that (a) I didn’t know the answer, and more importantly that (b) I’d never considered the question before. The question is “when/where do these numbers apply?” Let me explain. But to do so, I’ll introduce the idea of relative atomic mass, leading up to it step by step.
Periodic tables
Let’s start with this section of a simplified periodic table.
It’s the sort of table that you might use as, say, a 14 year old studying chemistry (in the UK, anyway – I don’t know about elsewhere…). The smaller number for each element ‘orders the elements’, and is called the atomic number. It can also be called the ‘proton number’, because it is the number of protons in an atom of an element, and that is what determines which element it is.
The larger number is called the mass number. It is the number of (protons + neutrons) in an atom of the element. Since protons and neutrons overwhelmingly dominate the mass of an atom, the mass number is also a (crude) measure of the mass of an atom (hence the term ‘mass number’…). You may remember tables like the following showing the relative masses of protons, neutrons and electrons…
[table id=6 /]
And everything is fine until we notice chlorine, which appears to have 17 protons and 18.5 neutrons in each atom. But you can’t have half a neutron. So then we learn that there is more than one form, or isotope, of chlorine. One has 18 neutrons and has a mass number of 35 – it is called Cl-35. The other has 20 neutrons and a mass number of 37, so is called Cl-37. The Cl-35 isotope is three times more abundant, and so the ‘weighted average’ of the mass numbers is closer to 35 than to 37. That’s why the table displays 35.5.
Why don’t any of the other elements act like this? Well they do in reality (see the next periodic table) but usually there is one isotope that predominates, and the weighted average is close to a whole number. In this simple periodic table, a choice has been taken to use the mass number of the most common isotope and ignore the others. Chlorine is treated differently because its weighted average is so far from a whole number that it can’t be sensibly rounded, and maybe as a tantalising glimpse that there is more going on and another level of understanding is there to be gained.
Shortly after that stage in our chemical education we might learn about moles. The mole is the SI unit of amount of substance – it is just a large number (6.02 x 1023 to 3 significant figures) that allows you to parcel huge numbers of objects together to make the numbers easier. So rather than talking about 6.02 x 1023 atoms of helium, say, we can talk about 1 mole of helium. It’s like buying 1 dozen eggs, rather than 12 individual eggs, but rather more extreme. The magic number 6.02 x 1023 is not plucked from thin air. It is the number of atoms of carbon in 12 g of carbon-12 (until the SI redefinition in May 2019, anyway). And that choice of 12 g is not accidental either – it comes from the mass number of carbon. So 4 g of helium, 12 g of carbon, and 19 g of fluorine all contain the same number of atoms – 1 mole.
And that’s very nearly true…
When you are a 16-year old (or so) budding chemist, someone explains that the previous periodic table was a simplified one, and was based on atomic structure (numbers of protons, neutrons and electrons), and gives you a more practically useful one, like the one below.
This periodic table is built in a fundamentally different way. It is based on the empirically determined mass of one mole, rather than the number of subatomic particles in an atom. 4.0026 g is the mass of helium you would actually have to weigh out to get 1 mole of helium atoms. We learn that 4.0026 is called the relative atomic mass (RAM) of helium. It doesn’t have a unit, because it is measured relative to the mass of carbon-12, which is given a RAM of 12 exactly (hence the ‘R’ in RAM).
Why aren’t any of the relative atomic masses whole numbers? There are at least three reasons.
Isotopic abundances
The first is that most elements exist as a mixture of isotopes. (Our 14-year old chemist knew that for chlorine, but managed to ignore it for the other elements!). Well, OK, but bearing in mind the definition of the mole, why isn’t the RAM of carbon exactly 12? Well, the definition of the mole was based on carbon-12. But in a sample of carbon, you don’t just have carbon-12. There is a mixture of isotopes – 99 % carbon-12 and 1 % carbon-13. (These percentages are called the natural abundances of the isotopes – they are the cause of my unanswered question that we are heading towards!).
The average of the mass numbers of the carbon isotopes, weighted according to their abundances, is then
[latex]\frac{(99\times12)+(1\times13)}{100}=12.011[/latex]
which is the number given in the periodic table for the RAM of carbon1.
So perhaps all the RAM numbers can be explained with just this idea of a weighted average of mass numbers.
The masses of protons, neutrons and electrons
But if this were true, any elements with only one isotope would have whole number RAMs (no averaging would be necessary). Fluorine is one such element – the only stable isotope of fluorine is F-19, and this isotope’s abundance is essentially 100 %. And yet in the table, fluorine has a RAM of 18.998403 g. Close to 19, admittedly, but not exactly 19…
Which brings us to the second and third reasons.
The second is that the table of particle masses and charges shown earlier is simplified. The mass of the proton is not exactly the same as the mass of the neutron, and the electrons mass is small, but not zero! In ‘atomic mass units’, u, the mass of a proton is 1.00728 u, the mass of a neutron is 1.00866 u, and an electron has a mass of 0.00055 u. (The atomic mass unit, u, is defined as 1/12 the mass of a carbon-12 atom – it is similar to the mass of a hydrogen atom). The result of this is that if two elements have different ratios of protons and neutrons, their RAMs will not be the same ratio as the nice, simple, whole-number ratio of their mass numbers! This makes whole number RAMs statistically improbable, to say the least!
Isotopic masses
And the third reason is the most startling of all – when you add up the masses of the protons, neutrons and electrons that make up an atom (with our new, precise masses), it is different from the mass of the atom!
You can see this for fluorine. It’s mass number is 19. The mass of each proton and neutron is slightly more than 1 (and it has some electrons too, whose mass is not zero). And yet the RAM is less than 19.
This happens because the arrangement of the particles as an atom is a lower potential energy state than them being separate, and Einstein’s mass-energy equivalence law ([latex]E=mc^2[/latex]) means that the atom also has a lower mass than the separate particles. Not by much. But by a measurable amount. The mass of a sample of a single isotope (as compared to C-12 being given the value 12) is not, then, the same as the mass number. It tends to be an awkward decimal. It is called the isotopic mass.
Moving on…
So, to recap…
- carbon has a non-integer RAM because it is composed of a mixture of isotopes.
- And fluorine has a non-integer RAM because protons and neutrons have non-identical masses, and an atom’s mass is not the sum of the masses of the protons, neutrons and electrons. The same applies to 21 other elements.
For all other elements, there is a combination of these effects at play. Still, for many elements, there is one predominant isotope, which leads to the weighted average being close to that…
This much I knew already. To calculate a relative atomic mass, you need the masses of the isotopes and their relative abundances.
So what’s the problem?
But then I thought… where are these relative abundances true? Across the whole universe? On Earth? Throughout Earth’s crust? What would that even mean? We haven’t exactly sampled all of Earth’s crust!
Won’t RAMs vary from sample to sample? They do! It turns out that you can define a relative atomic mass for a sample. And in fact, the term ‘relative atomic mass’, according to its Wikipedia page (who says these posts aren’t rigorously researched?) is only applicable to a given sample whose isotopic composition is known! Boron from Turkey has a lower relative atomic mass than boron from California, apparently, because of its different isotopic composition.
Then the ‘standard atomic weight’, acts as an average of many, many samples taken across time in different places. I don’t know how that’s done and who coordinates it, but I find it amazing that it has to be done and it goes on in the background. It is harder to determine the isotopic abundance of a sample than isotopic masses, so most uncertainty in ‘standard atomic weights’ comes from the abundances.
I’ve used a periodic table for many years and only just come to the realisation that the inconvenient decimal numbers aren’t strictly RAMs at all – they are standard atomic weights! So we should really talk about ‘standard atomic weight’ (which IUPAC really needs to rename in terms of mass, not weight. ‘Standard relative atomic mass’, maybe?). Or at least talk about ‘a relative atomic mass’, rather than ‘the relative atomic mass’.
Footnote
1This calculation has been simplified. It should actually involve the mass of C-13 as the isotope mass, 13.003355, rather than the mass number, 13. However, this will only make a tiny difference, since (a) the difference between the two masses is very small, and (b) the amount of that isotope is so small (its abundance is only 1.07 %.
2And for a more in-depth exploration of the periodic table in general, and its links to atomic structure, see our series beginning here.