# Moles and Percents

## Why do we need Moles?

A chemical mole, or mol, is a unit of measure, just
like a gram or an ounce. It is used internationally so that all chemists speak the
same measurement language. The mole was invented because, well, it made sense. Scientists
were having a hard time converting between atoms of an element and grams of an element
(grams were the previous standard of measure), so scientists came up with a "mole"
of substance, which is defined as anything that has 6.02x10^{23} particles
in it.

This is based off of a mole of carbon, which has 6.02x10^{23} carbon atoms
in it. Carbon is often used to define chemical principles and constants, because
it is one of the oldest and most easily observed elements, and it occurs naturally.
You might recognize 6.02x10^{23} as Avogadro’s number; this number is used
as a constant throughout chemistry, and here we’re going to use it to define the
mole. Usually, moles refer to particles that make up a certain amount of an element,
and we use moles to measure how much of a substance is reacting in a chemical equation.
However, you can also measure other things in moles—for example, a mole of hippos
would be 6.02x10^{23} hippos . . . which is actually quite a lot. When you
think about a mole as 602,000,000,000,000,000,000,000 hippos, it seems like way
too big of a number to be describing something that fits in a beaker in the chem
lab! However, because atoms are so small (remember, we can’t even see an atom with
just the human eye) there are bunches of atoms in everything we’re measuring. Therefore,
a mole is actually a very appropriate way to measure chemical substances.

Another benefit of using moles to measure substances is that it directly correlates
to the number of atoms and molecules and grams. A mole tells you what fractional
part of Avogadro’s number you’re working with; for example, if you have .25 mol,
you would have .25 (or 25%) of Avogadro’s number, which is 1.505x10^{23}
(.25 * 6.02x10^{23}=1.505x10^{23}).

## Avogadro's Number

Every chemist has dreamed that atoms were large enough to see and manipulate one at a time. The same chemist realizes after considering it, that if individual molecules were available for manipulation, it would take far too long to get anything done. The view from the atom is very different from the view of trillions and trillions of atoms. The mass action of the atoms that we see on our "macro" view of the world is the result of the action of an incredibly large number of atoms averaged in their actions. The most usual way we count the atoms is by weighing them. The mass of material as weighed on a balance and the atomic weight of the material being weighed is the way we have of knowing how many atoms or molecules we are working with. Instead of counting eggs, we can count cartons of eggs, each carton of which has a given number, a dozen. Instead of counting B-B's, we can count liters of B-B's and find out how many B-B's are in a liter. Instead of counting rice grains, we buy kilograms or pounds of rice and have an idea of how many rice grains are in the container.

There are less than one hundred naturally occurring elements. Each element has a characteristic atomic weight. Most Periodic Charts include the atomic weight of an element in the box with the element. The atomic weight is usually not an integer because it is close to being the number of protons plus the average number of neutrons of an element. Let's use the atomic weight as a number of grams. This will give us the same number of any atom we choose. If we weigh out 1.008 grams of hydrogen and 35.45 grams of chlorine and 24.3 grams of magnesium, we will have the same number of atoms of each one of these elements. The neat trick with this system is that we can weigh the atoms on a grand scale of number of atoms and get a count of them.

This number of atoms that is the atomic weight expressed in grams is Avogadro's number, 6.022 E 23.

The name for Avogadro's number of ANYTHING is a mole. A mol of aluminum is 27.0 grams of aluminum atoms. Aluminum is a metal element, so the particles of aluminum are atoms. There are Avogadro's number of aluminum atoms in 27.0 grams of it. But 1.008 grams of hydrogen is NOT a mol of hydrogen! Why not? Remember that hydrogen is one of the diatomic gases. There is really no such thing as loose hydrogen atoms. The total mass of a single hydrogen diatomic molecule (H2) is 2. 016 AMU. A mol of hydrogen gas has a mass of 2.016 grams. In that 2.016 gram mass is Avogadro's number of H2 molecules because that is the way hydrogen comes. A mol of water is 18.016 grams because each water molecule has two hydrogen atoms and one oxygen atom. A mol of water has in it Avogadro's number of water molecules. Another way to view the same thing is that a formula weight is the total mass of a formula in AMU expressed with units of grams per mol.

So Avogadro's number is just a number, like dozen or score or gross or million or billion, but it is a **very** large number. You could consider a mol of sand grains or a mol of stars. We are more likely to speak of a mol of some chemical, for which we can find the mass of a mol of the material by adding the atomic weights of all the atoms in a formula of the chemical. The unit of atomic weight or formula weight is grams/mol.

The chemical formula of a material should tell you; (a) which elements are in the material, (b) how many atoms of each element are in the formula, (c) the total formula weight, and (d) how the elements are attached to each other. The symbols of the elements tell you which elements are in the material. The numbers to the right of each symbol tells how many atoms of that element are in the formula. The type of atoms and their arrangement in the formula will tell how the elements are attached to each other. A metal and a nonmetal or negative polyatomic ion shows an ionic compound. A pair of non-metals are bonded by covalent bonds. Some crystals have water of hydration loosely attached in the crystal. This is indicated by the dot such as in blue vitriol, Cu(SO4) · 5H2O, showing five molecules of water of hydration to one formula of cupric sulfate.

The unit of the formula weight or molecular weight or atomic weight is "grams per mol," so it provides a relationship between mass in grams and mols of material: nFw = m

Now we’ll show you how to do common conversions from grams to moles to atoms, and back again.

## Gram to Mole Conversion

Convert 15.5 grams C to mol.

First, locate your periodic table. Then, locate C (carbon) on the periodic table. It is in row 2, column IVA. Be sure to locate the atomic weight of this element, which will most likely be listed under the symbol for the element, but could also be near the upper right hand corner of the element’s box on the periodic table. Please note that the atomic weight is also referred to as relative atomic mass. For carbon, it should read 12.01, or 12.011 depending on how many significant figures your periodic tables uses. We are going to use 12.01 in this calculation.

Now that we’ve found all of our information, we’re going to set up dimensional analysis. You’ll start with a table, like this:

And then you can begin filling in the information. You’ll always start in the upper left hand corner with the information given to you in the problem. In this case, we were given 15.5 g C to convert, so we’re going to put in 15.5 g C in the upper left hand corner. Notice that we can abbreviate grams as g, but also note that we always include the symbol for the element we’re working with. Although it may not seem important now, it will be in future conversions. Here’s what the first step would look like:

Now, we have to fill in the conversion part, which is the two boxes on the right hand side. We want the items with the same units (grams, moles, etc) to be diagonal to each other, because we want similar units to cancel. In other words, if g C are across from each other, we can cross out g C and use the units we're converting to. Technically, we are performing division, which is what cancels the units. We’re going to use 12.01 g/mol as our conversion factor. Any of the atomic weights you get from the periodic table will have the units of g/mol, which stands for grams/mole. Now, one thing that we should explain before we get started is that by giving 12.01 g/mol that unit label, we are technically saying that there are 12.01 grams for every 1 mole. This is important to remember as we fill out our dimensional analysis box. Now, remembering that we are going to place amounts with the same units diagonally from each other, we’re going to put 12.01 g C in the bottom right hand corner of our dimensional analysis, like this:

Now, we have to put in the "1 mol." We're going to put it above the 12.01 g C, because we know that those two are equal. 12.01 g C = 1 mol C. That’s why we can write them as a grams per mole fraction, because they express the same value. Here’s what our dimensional analysis looks like now:

Now, notice that the units we don’t want, grams, are across from each other. We’re going to circle them in red so that you can see them. We have the units we do want, mol, in the upper right hand corner. No matter how many columns of conversion factors you may have, the units you want to end up with should always be furthest to the right, in the last column. The units you want can be on the top or the bottom, but they should never cancel with other units. Notice that in this problem, we have grams canceling (because that's not what we want) whereas moles do not cancel, because we want our answer in moles. We’re going to circle moles in green, so that you can see it more easily.

At last, we’re ready to do our calculation. In our dimensional analysis box, we are going to multiply by any numbers on the "top" row, and divide by any numbers on the "bottom" row. It would look like this:

Now, the calculation would go as follows:

15.5 g C * 1 mol C / 12.01 g C = 1.29 mol C

Notice that we multiplied the top numbers, and divided by the bottom number in order to get our final answer of 1.29 mol C. Notice also that we rounded the answer to three significant figures (sig figs) because the least accurate number contains 3 sig figs and the answer needs to follow the sig fig rules.

## Mole to Gram Conversion

Convert .798 mol O into grams of O.

This example should go much more quickly than the first one, now that we know how to set up the dimensional analysis. First, find O (oxygen) on the periodic table. Its atomic weight is 16.00 g/mol. Now that we know the atomic weight, we can set up dimensional analysis. This time, we are given a measurement in moles, and want to convert it to grams, so we place the mol number given to us in the top left corner of the dimensional analysis box. Then, we have to set up our conversion factor. Our conversion factor this time is 16.00 g/mol O, which means 16.00 g O = 1 mol O. Since we want the same units placed diagonally from each other, so they can cancel, we’re going to place 1 mol O in the bottom right hand corner, which means 16.00 g O is placed in the upper right hand corner. This is because we want to find the amount of grams in the specified amount. Our dimensional analysis looks like this:

So, we multiply across the top, and divide by the bottom number. In this case, the bottom number is 1, so we do not need to divide (because anything divided by 1 is that same number). After performing the multiplication, we get 12.8 g O. (Notice that we rounded to three sig figs, because our least accurate number had 3 digits.) Therefore, our final answer is 12.8 g O.

Now, here are some for you to try! We’ll give you two to start with.

**Practice Problem 1**

The dimensional analysis looks like this:

To get the answer, we multiplied across the top and divided by the bottom number. After our multiplication and division, we rounded the number to two digits because the least accurate number we were given had two sig figs as well.

Final answer: 50. g N

**Practice Problem 2 **

The dimensional analysis looks like this:

To get the answer, we multiplied across the top and divided by the bottom number. After the multiplication and division, we rounded the number to three digits because the least accurate number we were given had two sig figs as well.

Final answer: 3.06 mol F