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## Chemistry library

### Course: Chemistry library > Unit 1

Lesson 1: Introduction to the atom- Introduction to chemistry
- Preparing to study chemistry
- Elements and atoms
- Average atomic mass
- Worked example: Atomic weight calculation
- The mole and Avogadro's number
- Atomic number, mass number, and isotopes
- Worked example: Identifying isotopes and ions
- Isotope composition: Counting protons, electrons, and neutrons

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# Worked example: Atomic weight calculation

How to calculate atomic weight from atomic mass and percent abundance of carbon isotopes. Created by Sal Khan.

## Want to join the conversation?

- If Carbon-12 has an atomic mass of 12 amu, why does Carbon-13 have 13.0034 amu? Why isn't 13 amu?(74 votes)
- It's because of something called
**binding energy**.

Carbon-12 has a mass of 12 amu by definition. Theoretically, this would mean that each proton and each neutron has a mass of one amu, but this turns out not to be so. The actual mass of a proton is about 1.007 amu, and the mass of a neutron is about 1.008 amu. If you add the masses of six protons and six neutrons, you get 12.09. So why does C-12 have less mass than the sum of its components? The extra mass is converted to energy (E = mc2), which is released. This energy is called*binding energy*. Because of this energy being released, a carbon-12 nucleus is more stable than six protons and six neutrons are by themselves. If you put the energy back in, then the nucleus would fly apart.

Anyway, back to carbon-13. Now, it seems like carbon-13 should have a mass of 13, because it seems like all of the extra mass from the neutrons and protons turns into binding energy, right? Nope! In this case, we don't need to lose the whole 0.008 amu worth of energy to keep that extra neutron bound in place. In fact, we only need to convert 0.0046 amu into energy. The rest of it stays, and that is why carbon-13 has a mass of 13.0034 amu.

Hope this helps!(276 votes)

- Just wondering: can an atom be an isotope and an ion at the same time?(40 votes)
- All atoms are isotopes. All ions are atoms.(117 votes)

- why is only carbon-12 and carbon-13 used to find the atomic weight, aren't you supposed add the total weight of all carbon to find the atomic weight? i'm really confused about this?(42 votes)
- Do you mean why don't we also include other isotopes in our calculations, such as carbon-14? Only carbon-12 and carbon-13 are present in significant amounts, so it's okay to include just these two in our calculations. Of course, very precise calculations would need to include all isotopes, even those that are very rare.(61 votes)

- How do they determine the amount of each elements' different isotopes there are on the planet? Even if we could accurately measure this, wouldn't it fluctuate and change the average constantly?(30 votes)
- I would guess that somebody went around and took enough samples to have statistically significance. They then checked the samples to find the ratios. The interesting thing is that if the samples are taken on say Venus it would be different. Thus the periodic table on Venus would have different atomic weight values.(29 votes)

- Carbon-12 is exactly 12 amu as definition and it has 6 protons and 6 neutron (neglecting electrons) then 1 proton or neutron should also equal 1 amu exactly?? I asked it before and carried to binding energy but it is confusing... plz explain briefly.(10 votes)
- The mass of a neutral Carbon-12 atom is exactly 12 u, which means it includes the bound mass of protons and neutrons, as well as the mass of the electrons.

The binding energy that holds the protons and neutrons together comes from some amount of mass such that E=mc^2. If you change the number of bound neutrons or protons, you also change the energy required to bind them together, thus the total mass changes. This is also why individual, unbound protons or neutrons have a mass more than 1 u.(28 votes)

- This question is for both 12C and 13C. Where is the 98.89% and the 1.110% derived from?

I was under the impression 12C and 13C are both different isotopes for Carbon.

Secondly, I wanted to ask what is the reasoning for adjusting the decimal to the left?

Thank you!(7 votes)- If you hypothetically take a bag of 1000 carbon atoms on earth, you find that on average ~989 of them are carbon-12 and ~11 are carbon-13. If you repeat that a billion times you'll get the odd atom of carbon-14 here and there too, but still basically the same amount of carbon-12 to carbon-13, about 98.9% to 1.1%. The percentages of these isotope can be measured by using a special mass spectrometer.

He's adjusting the decimal because he is trying to calculate the average mass of one random carbon atom on earth. It's easier to use decimals in a calculator than percentages. To convert a percentage to a decimal you divide by 100, which is the same as moving the decimal point two places to the left.(13 votes)

- I'm confused, if this is an average, why 12.01 was not divided by 2? is it not suppose that the average of something is the sum of its parts and then divided by that same number?(4 votes)
- If each isotope was in equal proportions (eg. each made up 50%) that would work, but that isn't the case here. One isotope makes up ~99% of all carbon, the other makes up ~1%. Clearly the isotope that makes up 99% needs to be given more importance.

There is more than one way to take an average. What you generally think of when you hear average is called the arithmetic mean, this average is called the weighted mean.(16 votes)

- does that mean that the atomic weights are only related to Earth not as a whole universe??(3 votes)
- Correct, we only measure the average atomic weights based on the abundances of isotopes on Earth. We can expect the average atomic weights for elements to be slightly different between Earth and say Mars.

Hope that helps.(6 votes)

- I know that different isotopes of a same element have same chemical properties. So, does the difference in number of neutrons have any effect on isotopes? I mean, are there any cases when different isotopes show different properties?(4 votes)
- An isotope can affect the chemistry. However, this is really only noticeable for hydrogen and its isotopes. This is because deuterium has twice the mass of hydrogen and tritium has three times the mass of hydrogen - these big differences in mass can affect chemical (and biochemical) reactions.

Physical properties can also be affected. For example, D2O melts at 3.8 C whereas H2O melts a 0 C. Likewise, D2O boils at 101.4 C, whereas H2O boils at 100 C.

For isotopes of heavier elements (eg, carbon-12 and carbon-13) the difference in mass is generally too small to noticeably affect the chemistry.(2 votes)

- how did humans find out the accurate percentage of isotopes?(3 votes)

## Video transcript

- [Instructor] We have, listed here... We know that carbon 12 is the most common isotope of carbon on Earth. 98.89% of the carbon
on Earth in carbon 12. And we know that, by definition, its mass is exactly 12 atomic mass units. Now that's not the only
isotope of carbon on Earth. There are other isotopes. The next most frequent one is carbon 13. 1.11% of the carbon on Earth is carbon 13. And we can experimentally find that its mass is 13.0034 atomic mass units. So, these numbers that we have here, just as a review, these are atomic mass. These are atomic mass. And so, what we're gonna
think about, in this video, is how do they come up with
the atomic weight number that they'll give you on a
periodic table like that? So, atomic weight. Where does that come from? Well, in the video on atomic
weight and on atomic mass, we see that the atomic weight
is the weighted average of the atomic masses of the various isotopes of that element. So, to find this roughly 12.01, we take the weighted
average of these two things. And what do we weight it by? We weight it by how common
that isotope actually is. So, what we wanna do is, we could take 98.89%
and multiply it by 12. And I'll rewrite this
percentage as a decimal. So it'll be 0.9889 times 12. And, to that, we are going to add... We are going to add 1.11% times 13.0034. So, as a decimal, that's
going to be 0.011. That's 1.11% is 0.011, oh, 111. And I'm gonna multiply that
times 13.0034 atomic mass units. So, what does that give us? Let's get our calculator out here. So, we are going to have 0.9889 times 12 is equal to 11.8668. And, to that, we are going to add... We are going to add 0.0111 times 13.0034. And I know it's going to do
this multiplication first because it's a calculator knows
about order of operations. And so, that's all going to be,
as you can see, 12.01113774, which, if you were to round
to the hundredths place, is how this atomic weight was gotten. So that's that. There you go. That's how we calculate atomic weight. So, I can write this
as approximately 12.01. It's the weighted average
of the atomic masses. Now, another thing that
you might want to note is, what's the difference between
carbon 12 and carbon 13? Carbon 12, this right
over here, is six protons. The six protons are what make it carbon, so both of these will have six protons. And the difference is in the neutrons. This right over here has
six neutrons, six neutrons. And this, right over here, is gonna have one more
neutron, seven neutrons. So, when you look at the
difference in atomic mass, notice the change is... Looks like it's plus
1.0034 atomic mass units. So, from this, you can say, "Hey, look, if I add a neutron... Plus one neutron. Plus one neutron. It's roughly equal to
an atomic mass unit." It's not exactly an atomic mass unit, but, roughly speaking,
in a lot of very broad, high-level terms, you can kind of view it as being very close to
one atomic mass unit. And the same thing is true of protons. Anyway, hopefully you now
have an appreciation for the difference between atomic
mass, which is the mass, and atomic weight, which
is the weighted average of the various isotopes
of that element on Earth, how to calculate it, and roughly what the mass of a neutron is.