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Popularity Contests and Atomic Mass
We should all be familiar with the basic information that is displayed with the symbol for each element on
the periodic table. You should remember that the Atomic Number (AN) refers to the number of protons in
the atomic nucleus and is unique for each element. You should also remember that the Atomic Mass (AM)
is just the number of protons in the nucleus plus the number of neutrons, which we’ve decided are about
the same mass and are given a value of 1 atomic mass unit (amu). This works out quite well for the Atomic
Number, which is always a whole number, and the periodic table is set up according to increasing atomic
number, starting at 1 and going to over 100. What’s strange is that the Atomic Mass is usually not a whole
number. Even if it is shown with a decimal and zeros, there is always a suggestion that the AM is a decimal
number. This usually strikes people as odd, because we clearly state that the AM is always equal to “the
number of protons” + “the number of neutrons.” Which should give us a simple whole number.
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We’ve asked the question, “does this mean that the neutron can be sliced into small pieces?” The proton is
OK because we have a whole number of those represented by the Atomic Number. The quick answer is, of
course, no - all particles are wholes, there are no pieces or parts in the particle world. So the number you
see for the AM is just a calculated value based on the fact that:
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Individual atoms of the same element can have different numbers of neutrons.
Which brings us to that great descriptive word for these different varieties of an element - isotopes - which
just means that despite having different numbers of neutrons, the atoms are still the same element (iso meaning the same, and topos - meaning the place on the periodic table). As we said before, having
different numbers of neutron doesn’t affect the chemistry of an atom too much, since for our purpose the
neutron is just dead weight - i.e. sandbags in the back of a pickup. Really dedicated chemically minded, or
physically oriented people might see this statement as a bit of a stretch, but like with our previous
discussions of atomic structure - we can go about the world, oblivious to this excess detail that has little
affect on our daily lives.
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We just need to realise that different atoms of the same element can come in different varieties, with these
variations just having to do with how many neutrons have joined the protons in the nucleus when these
atoms were formed. That formation is an interesting topic in itself, but let’s just pretend that we are living in
the present and only need to deal with what we have, not how we got there. Anyone interested in how
elements were formed, feel free to ask, I can bluff my way through the basics of nuclear fusion...
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But let’s get back to that decimal issue of Atomic Mass.
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In a nutshell, the Atomic Mass is an average number that is trying to represent all the various isotopes that
are possible in any given sample of an element. However, it’s not just any old average, it’s a weighted
average, which is used to demonstrate which isotope is the most popular.
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Before we can appreciate a weighted average, we first need to make sure we know what we mean by a
regular arithmetic average (that’s a math joke - funny?...OK maybe not). Of course we all do averages
frequently, and as students you probably are calculating averages very frequently. But, do we know what
the average really represents?
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Let’s do the student thing and calculate the average grade for a class of 5 people:
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Student
Grade
Phil
85
Jane
75
Mary
92
Dan
65
Joe
72
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We should all know that we calculate the average by adding up all the grades and dividing by the number
of students:
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=(85+75+92+65+72) / 5
=77.8
=78 (sig figs)
What we’re really saying is that each of these individual grades has an equal importance or weight and
contributes the same to the average.
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We could rewrite the equation to show that each individual grade contributes 1/5 of the value of the
average
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= 85 + 75 + 92 + 65 + 72
5 5 5 5 5
= 17 + 15 + 18.4 + 13 + 14.4
=78
We could also write the 1/5 as a percent value of 0.2 and get the average by multiplication (easier for me,
since I don’t have an equation editor...)
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= (0.2 x 85) + (0.2 x 75) + (0.2 x 92) + (0.2 x 65) + (0.2 x 72)
= 17 + 15 + 18.4 + 13 + 14.4
=78
The point is that each of the individual grades has the same weight in the
average. No one person’s grade is more important or has higher weight
towards that average mark for the class.
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Now lets consider the example of the Silicon atomic mass, which is shown at
the bottom of the symbol on the periodic table as 28.09 - recreated for you on
the right. We’ve already mentioned that this atomic mass is an average of all
the naturally occurring isotopes (versions of silicon with different numbers of
neutrons). However, unlike the normal averaging that we would do for a class grade, in which every
person’s grade is equally important, for our isotope averaging we have to take into account that each of the
various isotopes may have a different contribution (weight) to the value of atomic mass shown on the
periodic table. In other words, some of the isotopes are more popular than the others - they are not all
equally important.
For interest sake, let’s start by treating each of the isotope masses as if they were equally important performing a simple average calculation in which each atomic mass contributes 1/3 of the average:
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= 28 + 29 + 30
3 3 3
= 9.33 + 9.67 + 10.
= 29
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Again, we could do it as multiplication using the percent value of 1/3 or 0.33
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= (0.33 x 28) + (0.33 x 29) + (0.33 x 30)
= 9.33 + 9.67 + 10.
= 29
We know that 29 is not the real atomic mass, because this normal averaging technique doesn’t consider
that each of the isotopes have a different contribution to the average - i.e. they are not all equally popular.
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So instead of treating them as equals we use their relative popularity (their real percent abundance) and we
get the average that is weighted towards the more abundant or popular isotopes.
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= (0.922 x 28) + (0.047 x 29) + (0.0310 x 30)
= 25.82 + 1.363 + 0.930
= 28.11 (sig figs)
The answer is not exactly equal to the value on the periodic table for a couple of reasons, rounding being
one, but it is close enough and does demonstrate the point. With the use of the weighted average, the AM
is close to the more popular value of Si-28, which it should be since that isotope represents over 92% of all
Silicon atoms on earth. Compare this with the value calculated using the simple average technique, which
suggests that most Si is Si-29. Just like the most popular people in high school, the most abundant isotope
is not going to want to be misrepresented and confused with regular folks. Fortunately, the weighted
average highlights which isotopes are the most abundant and popular.