we introduced ourselves to the idea of average atomic mass,

which we began to realize could be a very useful way

of thinking about a mass at an atomic level,

or at a molecular level.

But, what we're gonna do in this video

is connect it to the masses that we might actually see

in a chemistry lab.

You're very unlikely to just be dealing with one atom,

or just a few atoms, or just a few molecules.

You're more likely to deal with several grams

of an actual substance.

So, how do we go from the masses at an atomic scale

to the masses, masses of samples

that you see in an actual chemistry lab,

or in, I guess you could say, r-scale of the world.

Well, the chemistry community has come up with

a useful tool.

They said, all right, let's think about a given element.

So, say, lithium.

We know its average atomic mass is 6.94,

6.94 unified atomic mass units per atom, atom of lithium.

What if there were a certain number of atoms of lithium

such that if I have that number,

so times certain, certain number of atoms,

then I will actually end up with 6.94 grams,

grams of lithium.

And, this number of atoms is 6.02214076

times 10 to the 23rd power.

So, if you have a sample with this number of lithium atoms,

that sample is going to have a mass of 6.94 grams.

Whatever its average atomic mass is

in terms of unified atomic mass units,

if you have that number of the atom,

you will have a mass of that same number in terms of grams.

Now, you might be saying, is there a name for this number,

and there is indeed a name,

and it is called Avogadro's number,

named in honor of the early 19th century Italian chemist,

Amedeo Avogadro.

And, in most contexts, because you're not normally

dealing with data with this many significant digits,

we will usually approximate it as

6.022 times 10 to the 23rd power.

Now, there's another word that it's very useful

to familiarize yourself with in chemistry,

and that's the idea of a mole.

Now, what is a mole?

It is not a little mark on your cheek.

It is not a burrowing animal.

Actually, it is both of those things,

but, in a chemistry context,

a mole is just saying you have this much of something.

The word mole was first used by the German chemist

Wilhelm Ostwald at the end of the 19th century,

and he came up with the word

because of its relation to molecule.

Now, what does that mean?

Well, think about the word dozen.

If I say I've got a dozen of eggs,

how many eggs do I have?

Well, if I have a dozen of eggs,

that means I have 12 eggs.

So, if I say I have a mole of lithium atoms,

how many lithium atoms do I have?

That means that I have 6.02214076

times 10 the to 23rd lithium atoms.

Exact same idea, it's just that Avogadro's number

is much hairier of a number than a dozen.

So, let's use our new found powers of the mole

and Avogadro's number to start doing some useful things.

Let's say that someone were to walk up to you and say,

hey, you, I have a 15.4 milligram sample of germanium.

How many atoms of germanium am I dealing with?

Pause this video and try to think about that.

So, let me clear out some space

the periodic table of elements was taking up.

All right, so we started off

with 15.4 milligrams of germanium.

The first step might be hey, let's convert this

to grams of germanium.

And so, we can do a little bit of dimensional analysis.

We can just multiply this,

for every one gram of germanium

that is equivalent to 1,000 milligrams,

milligrams of germanium.

And so, if you essentially multiply by one thousandth

or divide by 1,000, we're gonna get the grams of germanium.

And, you can see that in the dimensional analysis

by seeing that that is going to cancel out with that

leaving us with just the grams of germanium.

And, now that we have an expression for grams of germanium,

we can think about moles of germanium.

So, how do we do that?

Well, we're going to multiply by some quantity,

and in the denominator we're going to want

grams of germanium for the dimensional analysis to work out,

grams of germanium, and in the numerator we want

the new expression to be in terms of moles of germanium.

So, one mole of germanium is equal to

how many grams of germanium?

Well, we see it right over here.

Germanium's molar mass is 72.63 grams per mole.

So, for every mole, we have 72.63 grams of germanium.

And, you can see that the units work out.

These grams of germanium are going to cancel

with the grams of germanium

just leaving us with moles of germanium.

In an actual chemistry practice,

finding out the moles of a substance

might actually be the most useful thing,

but if you wanted to find out the actual atoms of germanium

that we're dealing with, we will just multiply by

the number of atoms you have per mole.

And, this is going to be true for any element.

For every mole, you have Avogadro's number of atoms.

And, we're going to approximate that as

6.022 times 10 to the 23rd atoms, atoms of germanium,

for every one mole, mole of germanium.

And so, just to review what we just did,

we had milligrams of germanium.

You multiply these two together,

you'll have grams of germanium, which makes sense,

you're essentially just dividing by 1,000.

If you were to multiply your grams of germanium

times the moles per gram,

which is really just the reciprocal

of this molar mass we got here,

and just to make sure where it makes sense,

the units work out nice with the dimensional analysis,

this right over here tells you your moles,

moles of germanium.

And then, if you take your moles

and then you multiply it by Avogadro's number,

it tells you how many atoms of germanium we have,

and that makes sense.

If I told you I had a certain number of dozen of eggs,

if I wanted to know how many eggs that is

I would multiply by 12.

So, this whole expression is the number of atoms,

atoms of germanium.

So, we have 15.4 milligrams.

If we wanna figure out how many grams we have,

we then divide by 1,000,

that's what our dimensional analysis tells us,

and it also makes logical sense,

divided by 1,000.

So, this is how many grams we have.

And then, if we wanna figure out how many moles,

and it's going to be a small fraction of a mole

because a mole is 72.63 grams per mole,

we have a small fraction of a gram,

much less 72.63 grams.

And so, we saw from our analysis

to figure out the number of moles,

we're now going to essentially divide by 72.63,

so divided by 72.63 is equal to,

this is the number of moles of germanium we have.

And, if we wanna figure out the number of atoms

of germanium, we'll then multiply that

times Avogadro's number.

So, times 6.022 times 10 to the 23rd,

and this EE button means times-10-to-the,

EE 23rd power, so that's how you do it on a calculator.

And then, that gives us this many atoms.

And, let's see, just to get our significant digits here,

our significant figures, out of all of the things

we multiplied, see we had four significant digits here,

four significant digits here,

but we only had three over here,

so I'm going to round to three significant digits.

So, I'll go to 1.28 times 10 to the 20th atoms.

So, we have approximately

1.28 times 10 to the 20th atoms of germanium,

which is a lot.