Relative atomic mass from isotopic abundances

Find chlorine on the periodic table you are given. The number printed with it is thirty-five point five. But no chlorine atom has that mass. Every chlorine atom is either chlorine-thirty-five or chlorine-thirty-seven, with nothing in between. Yet thirty-five point five is the number the exam tells you to use.

This is video five of seven in Atomic structure, and it follows on from Atomic number, mass number and isotopes.

Not every course asks for this calculation. On some it is Higher tier only, and one leaves it out altogether, so check the list for your own course. Everything else in this chapter is on every course.

Now, the number on the table is the relative atomic mass. It is an average mass for the element's atoms, and it takes account of how common each isotope is. Isotopes are atoms of one element with different numbers of neutrons. How common an isotope is, as a percentage, is its abundance. In chlorine, about seventy-five per cent of the atoms are chlorine-thirty-five. The other twenty-five per cent are chlorine-thirty-seven. Picture a hundred chlorine atoms in a bag. Seventy-five of them have a mass of thirty-five, and twenty-five have a mass of thirty-seven. Add thirty-five and thirty-seven, halve it, and you get thirty-six. But the table says something smaller. Why is the true value lower than thirty-six? Because most of the atoms are the lighter isotope. There are three light atoms for every heavy one, and they pull the average down. So weight each mass by how many atoms have it. Seventy-five atoms times thirty-five is two thousand six hundred and twenty-five. Twenty-five atoms times thirty-seven is nine hundred and twenty-five. Add them, and the hundred atoms have a total mass of three thousand five hundred and fifty. Divide by one hundred for one atom's share: thirty-five point five. And that is why relative atomic masses are usually not whole numbers. Each one is a weighted average over isotopes with different masses. And thirty-five point five landed between thirty-five and thirty-seven, nearer thirty-five, the common one. That check is your handle for this video: it lands between the two, nearer the common one.

Now, the full method, on gallium. Gallium-sixty-nine makes up sixty point one per cent of gallium. Gallium-seventy-one makes up thirty-nine point nine per cent. Before any arithmetic, two things go at the top of your working. The answer is wanted to one decimal place. And it must land between sixty-nine and seventy-one, nearer sixty-nine. Next comes the expression, with the hundred in it from the start. Sixty-nine times sixty point one, plus seventy-one times thirty-nine point nine, all over one hundred. Sixty-nine times sixty point one is four thousand one hundred and forty-six point nine. Seventy-one times thirty-nine point nine is two thousand eight hundred and thirty-two point nine. Add the two, and you get six thousand nine hundred and seventy-nine point eight. Divide by one hundred: sixty-nine point seven nine eight. Now round once, at the end, to one decimal place: sixty-nine point eight. It lands between sixty-nine and seventy-one, nearer sixty-nine, so it passes the check. The periodic table gives gallium as about seventy, so the answer agrees with it as well. A student writes the same expression, then gives the answer as six thousand nine hundred and seventy-nine point eight. The expression was right - so the slip came after it. What went wrong, and how could the student have caught it? The total was never divided by a hundred. A relative atomic mass near seven thousand cannot sit between sixty-nine and seventy-one, so the check rejects it on sight. One examiner's report on this calculation notes that "some students did not divide one thousand and eighty point three by one hundred". It adds that "a small number of students who evaluated the expressions as ten point eight nought three did not round their answer to one decimal place." So the cure is where you write things. The hundred lives inside the expression, and the rounding instruction sits at the top, so neither is left to memory. One more element, this time with three isotopes. Neon-twenty is ninety point five per cent, neon-twenty-one is nought point three per cent, and neon-twenty-two is nine point two per cent. The check says between twenty and twenty-two, close to twenty. Twenty times ninety point five is one thousand eight hundred and ten. Twenty-two times nine point two is two hundred and two point four. That leaves one line to fill in, the rare isotope, and then the finish. What does the neon-twenty-one line add, and what is the final answer? Twenty-one times nought point three is six point three. The total is two thousand and eighteen point seven. Divided by one hundred, that is twenty point one eight seven, which rounds to twenty point two. That rare neon-twenty-one barely moved the answer, because so few atoms have that mass. The weighting is doing its job, and the table's value of about twenty agrees.

A last use for the answer: naming the element. Two isotopes have mass numbers sixty-three and sixty-five. A student works out sixty-three point five, then names the element as zinc, because zinc sits at sixty-five on the table. The student's arithmetic was fine; the slip is in the final step. Is zinc the right answer, and what should the student have matched? No. Match the calculated value, sixty-three point five, against the relative atomic masses. That is copper. Sixty-five is the mass number of one isotope, not the element's average. A different report, on a question like this, records it directly: "those who looked at the isotope of mass number sixty-five and gave the answer zinc did not gain credit."

Over to you now. The second question is a calculation you have not seen. Why are most relative atomic masses not whole numbers? Because each is a weighted average over isotopes with different masses. Next: boron-ten is twenty per cent, boron-eleven eighty. Relative atomic mass? Ten point eight. Ten times twenty plus eleven times eighty is one thousand and eighty, and over a hundred, that is ten point eight, between ten and eleven and nearer eleven. Now back to chlorine: why does no atom have a mass of thirty-five point five? Because thirty-five point five is an average. Each chlorine atom is thirty-five or thirty-seven, and the table prints their weighted average, nearer thirty-five because that isotope is commoner.

If you can set up the expression with the hundred already in it and say where the answer lands before you start, you've got this down. Tap the thumb. And if your course leaves this calculation out, you lose nothing by moving on; nothing later in the chapter depends on it.

Next in the chapter: Electronic structure of the first twenty elements.

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