AP Chemistry - Stoichiometry
The Mole: The Start of Chemical Calculations
The Carbon-12 Based Atomic Mass Scale
In the years following Dalton's presentation of his atomic theory, chemists worked very hard at determining a complete set of relative masses for all the elements that were known.  By knowing these relative masses the chemists were able to select amounts of elements in grams needed for any desired atom ratio.    In establishing a table of atomic amsses, it is necessary to have a reference point against which to compare the relative masses. Currently, the agreed-upon reference is the most abundant isotope of carbon, which is carbon-12. By definition, an atom of this isotope is defined as having the mass of exactly 12.000 amu. (atomic mass units)  In other words, an amu is defined as 1/12th of the mass of one atom of carbon-12.

The definition of the size of the atomic mass unit was quite arbitrary.  It could just as easily have been selected to be 1/24th of the mass of one atom of a carbon atom, or 1/10th the mass of a calcium atom, or any other value. Why 1/12th the mass of carbon-12?  Carbon is a very common element, available to any scientist  and by choosing the amu to be of this size, the atomic masses of nearly all the other elements are almost whole numbers, with the lightest atom having a mass of approximately 1. (hydrogen-1 has a mass of 1.007825 amu when carbon-12 is assigned a mass of exactly 12 amu.)

The Mole
Suppose we want to make molecules of carbon dioxide, CO2, in such as way that there would be no extra carbon and oxygen atoms left. If we took ten atoms of carbon and twenty atoms of oxygen we would make 10 molecules of carbon dioxide.  Suppose we wanted to make more, lets say 40,000 molecules of carbon dioxide.  Once again, we could count out 40,000 atoms of carbon and 80,000 atoms of oxygen, let them react and we'd have 40,000 molecules of carbon dioxide.  This sounds all very nice and neat unless you realize the trap.  Have you ever seen an atom?  Atoms are too tiny to count individually.  Saying they are tiny is even wrong because tiny can be seen.   Even with the best scanning tunneling electron microscope ever invented the largest atoms known, look just like fuzzy cloud tops.  We have never seen individual atoms.  There are no lenses with the resolving power or balances fine enough to measure an individual atom.

We get around this problem because each element has its own characteristic atomic mass and each formula has its own unique molecular mass.    We know, that oxygen weighs in at 1.33 times that of carbon, because of this we get a ratio of their masses:

                   16.0 amu (for one atom of O) = 1.33
                   12.0 amu (for one atom of C)       1

If we take a sample of oxygen and carbon in a ratio of 1.33 to 1, we must obtain equal numbers of their atoms.  That is, if we actually had a balance that could measure amu directly we could mass out  32 amu of oxygen and 12 amu's of carbon - a mass ratio of 2.66 to 1 - we would have exactly 2 atoms of oxygen for every atom of carbon and a 2 to 1 ratio by atoms.

When we mass out a sample of an element such that its mass in grams is numerically equal to the element's atomic weight, we always obtain the same number of atoms no matter what element we choose.  Thus 12.0 g of carbon has the same number of atoms as 16.0 grams of oxygen, or 32.1 g of sulphur, or 55.8 g or iron.

This relationship also extends to compounds. The formula mass of water, H2O, is 18.0 amu. If we take 18.0 grams of water then it should have the same number of molecules as there were atoms in 12.0 grams of carbon.  The carbon-12 isotope, which makes up 98.89% of all naturally occuring carbon, is the reference used by SI Metric in its definition of the base unit for a chemical substance, the mole, abbreviated mol. 

This mole concept is the most important in all of chemistry. Once this concept is grasped all the rest of chemistry will appear easy.

Avogadro's Number
The mole is defined as 6.02 x 1023 units.  It is called Avogadro's number in honour of Italian scientist, Amadeo Avogadro (1776-1856). It is a pure number with a special name, just like so many others.  For example:
                                                 2 = pair
                                               12 = dozen
                                             144 = gross
                                             500 = ream
                                6.02 X 1023 = Avogadro's number
This number is not an odd number at all. It became inevitable once the amu was defined.   The relationships needed are:

                                          1 mole = 6.02 X 1023 particles     (General expression)
                                         12 amu = 1 atom of  C      (Specific example)
                                    1 mol of C = 6.02 X 1023 atoms of C
                                    1 mol of C = 12 g of  C

Another useful relationship is that 1 amu = 1.66 X 10-24 g

SO                            1  amu               =  6.0 X 1023 particles
                      1.66 X 10-24  grams

The mass in grams of a substance, that equals one mole is often called its molar mass, and the units are grams/mole or g/mol.   For example, aspirin has a molecular mass of 180 grams. Therefore if we massed out exactly 180 grams of aspirin we would have Avogadro's number of aspirin molecules.
Avogadeo's Number and Moles Exercise Sheet

Using the Mole Concept
One mole of any substance can be calculated from its formula mass. Since this is true it is absolutely essential that when you are using the mole concept that the correct formula be used. It is not enough to say "use 1 mole of nitrogen".  Do we mean atomic elemental nitrogen or nitrogen gas?  There is a difference!  One mole of  N consists of Avogadro's number of nitrogen atoms (and has a mass of 14.01 g), whereas 1 mole of N2 consists of Avogadro's number of molecules, each molecule having two nitrogen atoms. One mole of N2 molecules would have a mass of 2 X 14.01 g = 28.02 g.

One of the advantages of the mole concept is that it lets us think about formulas on two levels at the same time.  One level is that of atoms or molecules or ions, and the other level is that of lab-sized practical quantitites, such as moles and grams.  Look at the equation below:

    H2O   consists of                  2 H                                  +  O
1 molecule of H2O                   2 atoms of H                       1 atom of O
1 dozen H2O molecules           2 dozen atoms of H            1 dozen O atoms
6.02 X 1023 H2O molecules    12.04 X 1023 H atoms        6.02 X 1023 O atoms
1 mole of H2O molecules        2 moles of H atoms            1 moles of O atoms
18.0 g of H2O                           2.0 g of H                           16.0 g O atoms

When we think about H2O at the first level, we can easily see that a dozen of its molecules are made from two dozen atoms of H and one dozen atoms of O.  However, if we switch to the more practical lab-sized level, it is just as easy to think about one mole of H2O and to view this quantity as consisting of two moles of H and one mole of O.

The numbers in all but the last row are in the same ratio regardless of the scale, whether we deal with single particles or with moles of them. After planning an experiment at the mole level, it is easy to convert numbers of moles into corresponding masses of chemicals to meet any desired needs.

Moles of Atoms

The atomic mass of an element is a relative quantity. Originally the atomic mass of hydrogen, the lightest of the elements, was taken to be one and the atomic masses of all other elements were measured in relation to the atomic mass of hydrogen. This later proved to have been a poor choice. Not only does hydrogen naturally consist of more than one isotope, but there was the additional question (particularly among early chemists) as to whether monatomic hydrogen or diatomic hydrogen should be taken as having atomic mass one.

After some effort, and one major false start with oxygen, chemists and physicists agreed on a common relative scale of atomic mass. Carbon of isotopic mass twelve was assigned an atomic mass of exactly twelve, and all other atomic masses whether of isotopes or of elements were specified relative to carbon of atomic mass twelve. This had the effect of making the relative atomic mass of hydrogen 1.0079...rather than exactly 1.0000.... The difference of less than 1% is too small to matter in many approximate chemical calculations, but it is large enough to be significant when accurate work must be done.

Physicists, and some chemists, measure the masses of individual atoms in kg, g, or atomic mass units. For most chemists, however, the mass of a single atom is inconveniently small and the molar mass of a substance is used. The molar mass of an atom is the mass of a very large number of identical atoms-- one mole of atoms. One mole of atoms is by definition that number of atoms which exist in exactly twelve grams of carbon of isotopic mass twelve (12C). This number is called the Avogadro number, NA, and the best current determination of its value is 6.02 x 10+23.  Moles of atoms and molecules are so central to chemistry that several of the following sections are devoted to introducing them, and they will be used continually throughout all courses in chemistry.  This mole is just a number.  If someone asked you how many in a dozen, the answer would be an almost automatic 12. How many in a gross?  Again an automatic answer of 144!  This number is vital to chemists.  1 mole = 6.02 x 1023 particles (either atoms or molecules).   It would be in your best interest to memorize this number!     

Molar Atomic Masses of Elements  

The molar mass of an atom is simply the mass of one mole of identical atoms. However, most of the chemical elements are found on earth not as one isotope but as a mixture of isotopes, so the atoms of one element do not all have the same mass. Chemists therefore distinguish the molar atomic mass of an isotope, which is the mass of one mole of the identical atoms which form that isotope, from the molar atomic mass of an element, which is the mass of one mole of the atoms of the various isotopes of that element having the natural abundances as they are found on earth. For many elements the variation found in the natural abundances limits the accuracy with which the molar atomic mass of that element can be known. Those elements for which this is true are indicated in the periodic table.

Chemists deal with elements as they are naturally found. In actual fact it is very difficult to separate isotopes. Chemists like to deal with the atomic mass or atomic weight of 1 mole of a substance. The weighted molar atomic mass of an element as it naturally occurs will be referred to simply as the atomic mass of the element from now on.

What is the atomic mass of Pb?   Look on the periodic table and find Pb. You'll find the mass number listed as 207.2

One atom of Pb weights 207.2 amu. (atomic mass units)

One mole of Pb atoms weights 207.2 grams. That is 1 mole or 207.2 grams of Pb contains 602,000,000,000,000,000,000,000 atoms of Pb.

The sum of individual atoms can be used to find the mass of a molecule.

The mass of hydrogen peroxide, H2O2 would be calculated like this:

H2O2 has 2 hydrogen atoms and 2 oxygen atoms in it.

Therefore the mass is 2 X H  + 2 X O = 2 X 1.01 amu + 2 X 16.00 amu = 2.02 + 32.00 = 34.02 amu. So one molecule of hydrogen peroxide weighs in at 34.02 amu. A mole of hydrogen peroxide would weigh 34.02 grams.

The periodic table provides you with individual atomic masses.  If you know the number and type of elements in a molecule you can add up the individual masses to find the molecular mass or molecular weight.

Find the molecular mass of calcium phosphate, Ca3(PO4)2

The molecule has 3 calcium atoms, 2 phosphate atoms and 8 O atoms in it. Stop and verify this for yourself. The Ca has a subscript 3 with it. The P has an assumed 1 and the O has a 4. However the PO4  group has a set of brackets around it with a subscript 2. The 2 means multiply everything inside the brackets by 2. So we end up with the 2 P and 8 O atoms.

Calculation:  3 X Ca = 3 X 40.08 amu = 120.24 amu

      2 X P   = 2 X 30.97 amu =   61.94 amu

      8 X O  = 8 X 16.00 amu = 128.00 amu

The total of the individual types of atoms is 120.24 amu + 61.94 amu + 128.00 amu = 310.18 amu.

One molecule of calcium phosphate weighs 310.18 amu and a mole of it would weigh 310.18 grams.

Stop here and do the  Atomic Weights & Masses Exercise

Click here to go to a molecular mass calculator site

Percent Composition

Percent composition is a simple little calculation that has a large impact on chemistry. When a brand new substance is discovered one of the first things that is determined is its chemical make-up. That means how much of each type of atom is in the molecule. What is the percent composition of strontium oxide? SrO

There is 1 atom of Sr and 1 atom of O.

The mass of 1 Sr is 87.62 amu and the mass of an O is 16.00 amu.

The percent composition is the fraction of the SrO that is just Sr.  This fraction is:

Percentage of Sr = mass of Sr in the molecule / molecular mass X 100%

            = 87.62 amu / 103.62 amu X 100%

            = 0.8456 X 100%

            =  84.56%

Strontium makes up 84.56% of the molecule. The oxygen makes up 100% - 84.56% = 15.44%. The amu's are units that cancel out. The % sign means out of 100.

The general equation for finding the percentage composition of an element is:

percentage of an element = the total mass of just that element / molecular mass X 100%

Stop here and do the  Percent Composition Exercise

Click here to go to a percent composition calculator

Converting Moles into Grams
This is probably the single most used equation in chemistry.  It is the one that allows the conversion of moles into grams and grams back into moles.  Moles are a "theoretical value" which looks good on paper.  Grams is the "practical value" that you take out of a stock bottle and place on the balance.  Lab balances read in grams, not moles, and so in practical work all mole values have to first be converted.

Moles to grams formula:     grams(g)  = moles(n)  X  molecular mass

Example:  Sodium bicarbonate, NaHCO3, is one ingredient of baking powder.  How many grams of sodium bicarbonate are in 0.673 moles?
Step 1.  Find the molecular mass:
          NaHCO3 = 1 Na = 1 X 22.99 g/mol =  22.99 g/mol
                             1 H   = 1 X   1.01 g/mol =    1.01 g/mol
                               1 C = 1 X 12.01 g/mol =  12.01 g/mol
                               3 O = 3 X 16.00 g/mol =  48.00 g/mol
                                                                       84.01 g/mol
Step 2. Use the equation:   g = n  X  molecular mass
                                            = 0.673 mol X 84.01   g
                                            = 56.54 g

Step 3: Provide a written answer:  There are 56.54 grams of sodium bicarbonate in 0.673 moles.
Grams to moles formula:

 moles =         grams          OR  n =            g
                molecular mass                 molecular mass

Example:  Potassium permanganate, KMnO4, at one time was used for an anti-fungal agent. You could always tell someone who had just been treated because their feet were purple. The pharmacy gives you 250 grams of the stuff in a bottle.  How many moles of it do you have?
Step 1:  Find the molecular mass of KMnO4:
              1  K  = 1 X 39.10 g/mol  =  39.10 g/mol
              1 Mn = 1 X 54.94 g/mol  =  54.94 g/mol
              4 O   =  4 X 16.00 g/mol  = 64.00 g/mol
                                                       158.04 g/mol
Step 2:  Use the equation:  n =           g             =         250 g          =  1.58 mol
                                               molecular mass      158.04 g/mol
Step 3:  Write a written answer.
A sample of 250 grams is equivalent to 1.58 moles of potassium permanganate.

Finding the molecular mass
In the above two examples you have had a molecular formula to work with.  As long as you have a molecular formula, or name, from which you can make a formula up, you will always have a molecular mass.   In order to find the molecular mass, without a name to go on, you need two pieces of information. The number of grams and the number of moles that it represents.
        molecular mass =    grams     =        g
                                     moles               n
Example:   A fellow student comes to you with a sample of an unknown chemical. They tell you that it has a mass of 34.91 grams and that it is exactly 0.20 moles.   What is the molecular mass of the substance?

Step 1:  Use the equation:     molecular mass =   34.91  g   =  174.55     g
                                                                                0.20 mol                    mol

Step 2: Write a sentence.  The unknown substance has a molecular mass of 174.55 g/mol.

Go to the Grams, Moles and Molecular Mass Worksheet

Go to the Marvin Da Mole Worksheet

Go to the Marvin Da Mole Strikes Again Worksheet

Go to the Next Stoichiometry Section on Balancing