Converting Between Moles, Atoms, and Molecules

How many atoms in 5.5 moles? How many moles is 4.6 x 10^24 sulfur atoms? We'll solve problems like these, where we convert back and forth between moles and the number of atoms or molecules that we have. We'll be using both a common sense approach, and also a standard conversion factor method. We'll learn how to use a scientific calculation to do this math, and we'll see how to round our answers with scientific notation and significant figures.

In this video we're gonna look at how to
convert back and forth between moles and

the number of atoms or
molecules that we have.

Now, when we do conversions
like this atoms and

molecules are sometimes both
referred to as particles.

A particle is just a word for
any individual thing.

So, a particle could be a jelly bean or
a coin or a paperclip or

an atom or a molecule.

So, we'll work through problems like
this where we have to go from moles to,

say atoms, where we have to go from
atoms and convert back to moles.

Okay, so here's our first question.

For each one of these problems and
do it two ways.

First, I’m gonna show you
how to think through it,

in kinda simple straightforward way, so
you can understand what you’re doing.

Then, I’m gonna show you how
to use conversion factors.

I think conversion factors don’t
always make a lot of sense and

I know a lot of students
are confused by them.

But, teachers and textbooks tend
to really like conversion factors.

So, it's important to know how to solve
questions like this using conversion

factors, too.

Okay so,
how many atoms are in 5.5 moles of atoms?

We're talking about moles and atoms here,

so lets just refresh our
memory about moles, okay?

A mole is like a dozen, but

there are 12 things in a dozen,and there
are 602 hexillion things in a mol.

We often abbreviate this super
long number with all these 0s,

602 hexillion, as 6.02 x 10 to the 23rd.

Mols can be a little bit tricky at first,
and so

I wanna keep talking about
the similarity to dozen.

As we work through this first problem,
okay?

We wanna know how many atoms
are in 5.5 moles of atoms.

But, to get a handle on
how to think through this,

let's first think about how we
would do this kind of problem

if we were talking about
dozens instead of moles.

So, what if instead of 5.5 moles
we're talking about 5.5 dozen?

Well, this math is probably
pretty straightforward.

There are 12 things in a dozen, so

if you figure out how many
atoms are in 5.5 dozen.

We take 5.5 and then we multiply it by 12,
the number of things in one dozen, and

that would tell us how many atoms, or
how many things, are in 5.5 dozen, okay?

But, we're not talking about dozen here,
we're talking about moles instead.

So, instead of multiplying this by 12,
the number of things in a dozen.

We're gonna take 5.5 and
we're gonna multiply it

by 602,000,000,000,000,000,000,000,
the number of things in one mole.

Now, this big number here is
a real pain with all the zeroes.

And if you were actually
gonna do this math,

chances are you wouldn't wanna
use this long version here.

You'd wanna use the shorter
version in scientific notation.

So, let's take this big number,
602 hexillion,

and write it in a more manageable
form of 6.02 x10 to the 23rd.

This is the same number,

6.02, as 602 hexillion,
it's just an abbreviated version here.

So, you've written this out.

Chances are you're going to use
a calculator, scientific calculator or

graphic calculator to solve this problem.

So, here's how you're gonna
type it 5.5*(6.02E23).

Yes, E23 is usually how we do
exponents in a scientific calculator.

The E is 10 to the exponent.

And then 23 here is the exponent.

Plug this into the calculator, and
we're gonna get this as our final answer.

There are two things that I
need to do to this answer.

The first thing I need to do is sorta take
it out of calculator scientific notation,

and put it into normal
person scientific notation.

So, I'm gonna write 3.311E24

is 10 to the 24th, okay?

So now, it's in regular person
scientific notation but

the next thing that we gotta do is take
into account significant figures, okay?

We'll look at the numbers
that went into this,

to figure out how to round it correctly,
okay?

There are two significant figures in
5.5 and there three significant figures

in 6.02, so we're gonna round this number
to lower number of significant figures.

We are gonna round it to two.

Okay, so we are gonna take 3 and
we gonna take this 3 and

then we are gonna love the one to figure
whether to round up or keep the same.

It's a 1 its lower than 5 so
we keep it the same.

We'll do 3.3 x 10 to the 24th and

what we're solving for here is atoms.

This is our final answer.

Now, so
many people see a number like this,

3.3 x 10 to the 24th, and
they don't think of it as a real number.

So, please keep in mind that this
number is just a shorthand for

this super,
super long number with all these zeros.

This is 3 heptillion,
3 hundred hexillion atoms.

Okay, so 3.3 x 10 is 24.

It's not some weird martian number.

Keep in mind, that it is just a shorthand
version of this very long number here.

And if for some reason, your teacher
doesn't let you use a calculator and

you have to do this out by hand.

I have another video on
doing more calculations

by hand instead of a calculator.

So, you can check that out.

So anyway,

this is how we do this problem using this
sort of simple straight forward method.

We multiply 5.5 by the number of things
in one mole, plug in the calculator and

this is what we get.

Now, let's look at how we
could solve the same problem

using conversion factors instead.

In this case, we're gonna be starting with
this number here, 5.5 moles 5.5 moles.

Now, we are going to want to multiply this
by a conversion factor that's going to get

rid of moles and
is going to give us atoms.

To write this conversion factor,
we're going to think about moles.

Let's look at this definition up here.

I wanna rewrite this just as
an equation with an equals sign, okay?

So, here we have 1 mole equals this much.

I really haven't changed anything.

But, I put the equals sign in here,
because we use relationships like

this with one thing on either
side of the equals sign.

We use relationships like this to
write conversion factors, okay?

So, here's how we'll take
this relationship and

write a conversion factor.

A conversion factor has both a top and
a bottom.

And we take something on one side
of the equation, one mole, and

we can put it on, say,
the top of the conversion factor.

And then, the thing that's on
the other side of the equal sign,

we put that on the bottom.

So, I'll do 6.02 x 10 to the 23rd.

Things here, but
we're talking about atoms.

And this conversion factor is just telling
me that in one mole there are 6.02

times 10 to the 23rd atoms.

But, for every equation

like this with an equal sign there are two
conversion factors that we can write.

We can write it like this or
we can flip it, that's cool too.

So, I could also write 6.02
x 10 to the 23rd atoms

on top, with 1 mole on the bottom.

Now, both of these conversion
factors are totally valid.

Which one do we wanna use for
this problem?

Well, we wanna multiply this by a
conversion factor that's gonna get rid of

moles, is gonna leave me with atoms.

So moles is on top here.

I'm gonna wanna choose the version
of this conversion factor

that's going to give me moles on
the bottom so they'll cancel out.

So I'm gonna use this one.

And then,
I've got moles on the top here cancel out,

moles on the bottom cancel out here.

And that's gonna leave me with atoms,
okay.

What's the math I'm gonna do?

I'm gonna do 5.5 x 6.02 x10
to the 23rd divided by 1.

Now, you might realize that
dividing this number by one

doesn't really change anything.

So, all the math we're really
doing is just 5.5 x 6.02 x 10^23.

Which is exactly what we did up here.

So, you could just type this into your
calculator and get this as an answer, or

which would be totally fine.

You could decide that you wanna put
this whole conversion factor in and

you could type it in like this 5.5 x and
then parentheses 6.02E23 divided by 1.

Whichever one you type in,
you are going to get the same number here,

which in regular person scientific
notation is gonna look like this, and

then we round it using sig
figs to get this number here.

Now, once again,
don't forget that 3.3 x10 24,

is just an abbreviated version of this
very long number of atoms here, okay.

So, that's how we go
from moles to say atoms.

Now, let's look at how to do problems in
the other direction from say atoms or

molecules to number of moles.

How many moles is 4.6 x 10
to the 24th Sulphur atoms?

Okay, check out this number.

I just wanna remind you that this
isn't some weird martian number,

that this is just
a shorthand abbreviation for

this very long number with
a whole bunch of zeros.

As we did before,

instead of jumping right to moles, Let's
redo this common sense approach, where we

think about what we would do if instead
of moles we were talking about dozen.

If we wanna know how many
dozen this big number were.

Well, we'd recognize that there
are 12 things in dozen, and so

we would divide this number by 12.

12 things in a dozen,we wanna know how
many times will 12 go into this one, okay?

So, we are gonna be dividing by
the number of things in a dozen.

But, as before we are not talking about
dozens, we are talking about moles.

So, instead of dividing by
the number of things in a dozen,

we wanna find out how many moles is it,
so we are going to divide.

By the number of things in one mole.

So, we're going to
divide by 602 hexillion.

As before, you're probably not
gonna wanna use these giant

versions of each number
with all these zeroes.

So, this is where the scientific
notation comes in handy.

Let's rewrite this in scientific notation,
okay.

We're 4.6 x 10 to the 24

divided 6.02 x 10 to

the 23, put this into the calculator.

And you'll wanna type it in like
this where we replace the 10 to

the 24 with E24 or 10 x 23 with E23.

Hit return, and
we're gonna get a number like this.

Now, it's not in scientific notation,
so we don't have to worry about that.

But, we are going to want to round
this with significant figures.

There are two significant figures here,
three significant figures here.

So, we're gonna round this
to two significant figures.

We're gonna take the 7 and the 6,

look next door to figure out whether
we round up or keep the same.

It's a 4, so we keep it the same.

And we're solving here for moles.

It's gonna be 7.6 moles of sulfur atoms

are in this super huge
number of Sulfur atoms.

Okay, I am going to just stop.

Slip it in right here, and

now let's look at how we'd use conversion
factors to solve this same problem.

Okay, here we're gonna be starting
with 4.6 x 10 to the 24th atoms.

And we wanna multiple this by a conversion
factor that's gonna get rid of atoms.

And is going to move me to moles,
so let’s look at the two conversion

factors that we could write using
this relationship here, okay.

The first one is gonna
put one mole on top.

We are talking about atoms here,
so there are 6.02 x 10 to

the 23rd atoms, in 1 mole.

Or we could write this other conversion
factor where we put 6.02 x 10 to

the 23rd atoms on top and
1 mole on the bottom.

Which of these do we wanna use?

We want to use the 1 that
is gonna get rid of atoms.

Atoms is on the top up here,
it's on the bottom here.

So, they're gonna cancel out.

Okay, get rid of this.

And now what's the math we're going to do?

The math is going to be 4.6 x 10 to the
24th x 1 divided by 6.02 x 10 to the 23rd.

Now, multiplying this number by 1
isn't really gonna change anything.

So, all we're really doing is
we're taking this number and

dividing it by this number,
the exact same math that we did up here.

But, just as we did previously, if you
prefer to put this as a big fraction into

the calculator, that's totally cool too.

It's gonnalook like this
4.6E24 x this whole fraction,

1/6.02 x10 to the 23, but all you're
doing is just dividing this by this,

because this 1 doesn't really matter.

We're gonna get the same number here,
which rounds to 7.6 moles.

So, that's how we go from a number
of things, like atoms, molecules,

jellybeans or coins,
to figure out how many moles are in it.

We divide it by the number
of things in 1 mole.

Okay, so if you want some more practice
with these kind of problems, check out

the next video, converting between moles,
atoms, and molecules part two.