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# Calculating Moles

#### How Many Moles is That?

This is a critical question that we ask over and over in a chemistry course such as this one. How many moles is that? Or to be specific, how many moles of carbon are there in 48 grams of carbon? The answer is 4 moles... I divided 48 by 12 and got 4. Let's back up a bit though...

#### Atomic Weights are Scaled for Mole Counting

Take a look at the periodic table and look at those decimal numbers - the atomic weights. All of those numbers are scaled relative to each other by mass. This simply means that if you chose ANY counting number of atoms (call it a batch) of the elements from the table - the masses of each batch will all scale to each other the same exact way all the atomic weights scale to each other. We could've picked ANY standard counting number of atoms - a dozen (12, too small), a billion (109, still too small), a billion billion (also known as quintillion or 1018, still a bit too small). Let me elaborate on that last one there... a batch that is equal to a quintillion atoms of iron would weigh 93 µg. Those are micrograms there! - you wouldn't even be able to see that amount on the scale. We are going to have to scale up to a much bigger number.

#### The Origin of 6.02214076 × 1023

How 'bout septillion? That is 1024! That is more like it... Put a septillion atoms of iron on the scale and we'd have 93 grams of iron! Bingo! We have an amount that is easily seen and easily weighed. So why isn't a septillion our standard amount of substance? Well it could have been, but since the amount we pick is somewhat arbitrary, why not pick the number that would make the mass on the scale MATCH the atomic weight of the element? Totally doable and that is exactly how the mole was born and to be specific - how that 6.022 × 1023 number (Avogodro's number) was decided on back in the day. Go get yourself Avogadro's number of atoms of any element and weight it. You'll get exactly the atomic weight value (the number) in grams. This works for molecules too - you'll get the molecular weight in grams when you have a mole's worth on the scale.

So we chemists think in moles of substance. The SI unit for "amount of substance" is the mole. We count in moles. What is cool about choosing the number we did for moles - all our 1 mole amounts match the molecular weight of the molecule in grams. It is a unit factor that we use for converting from grams to moles and moles to grams. That equality or equivalency is:

molar mass of X in grams = one mole of X

We do the algebra here and rewrite as a unit factor we get that the units for molar mass (molecular weight) are g/mol (grams-per-mole).

#### An Example

Let's do an example with water. Water has the chemical formula of H2O. It's molecular weight is 18 g/mol. So how many moles of water are in a glass of water? Well we need to be specific about the "glass" of water. Well, the general consensus on a glass of water is 8 fl oz (fluid ounces). 1 fl oz is equal to 29.57 mL, and the density of water is about 1.0 g/mL. Let's do the conversion now:

$\require{cancel} \newcommand\ccancel[2][black]{\color{#1}{\bcancel{\color{black}{#2}}}} \left({8\,\ccancel[red]{\rm fl\;oz}\over 1}\right) \left({29.57\,\ccancel[green]{\rm mL}\over 1\,\ccancel[red]{\rm fl\;oz}}\right) \left({1\,\ccancel[blue]{\rm g}\over 1\,\ccancel[green]{\rm mL}}\right) \left({1\,{\rm mol}\over 18.0\,\ccancel[blue]{\rm g}}\right) = 13.1\,{\rm mol\;of\;water}$

In general, if you know the mass of a substance and the number of mole, you can calculate the molecular weight (MWt):

$\rm{mass\over moles} = MWt$

Rearrange that equation algebraically to get moles from MWt and mass, or mass from MWt and moles.