The Chemical Equation

Chemical transformations take place according to strict 'rules' because of the nature of molecular structure. Chemical reactions are really the rearrangement of the connectivity between the atoms in the reagenmts to produce products. The Ancients saw that chemical reactions obeyed the law of multiple proportions very precisely and correctly concluded that this was significant. Because the creation or destruction of nuclei is very unlikely without a nuclear reactor, the same number of atoms of each type (element) must exist both before and after any chemical transformation. This means that molecules must react in simple, whole-number ratios (i.e. the law of multiple proportions), as can be seen by the following example:

Even though we could try to write down the chemical reaction as
`X1 * CH4  +  X2 * O2  =  X3 * CO2  +  X4 * H2O`
but this equation is not useful because it is not 'balanced'. Not balanced means it does not have the correct (or determined) stoichiometric coefficients. The process of 'balancing' a chemical equation is simply determining a set of coefficients {Xi}, which represent the proportions of whole molecules in the reaction that balance the number of atoms of each element on both sides of the equal sign. Note: an equal sign, as well as single and double headed arrows, is sometimes used to separate reactants and products in a chemical equation. Nomatter what is used as a separator, a chemical reaction must be written as an equation, which means that the same number of each type of atom exist each side. (Sometimes unscrupulous chemistry intructors provide unbalanced chemical equations (which are really not equations at all) in problems given to students, but they argue that these are the cards that life deals us. Nonetheless, the equations always must be checked before proceeding further with the use of said equation.) How do you balance a chemical equation?
1. Identify all the different elements in the chemical equation.
2. Count the number of atoms on both sides of the equal sign of the element that appears in the fewest different molecules in the equation. Make sure to use the correct molecular formula (structure) in this arithmetic
3. Adjust the Stoichiometric Coefficents of the species that contain this element to balance the count of this element in the equation.
4. Repeat the last two steps for each element identified in step 1. You must maintain the ratio of coefficients of every molecule that has been balanced for a previous element.
5. When all the elements have been balanced, multiply the coefficients in the entire equation by any number you wish. This is usually done to obtain the smallest whole number (integer) coefficients, {Xi}
For the above equation:
`Carbon:   X1 = X3Hydrogen: X4 = (2)*X1Oxygen:   X2 = X3 + (1/2)*X4`
If one chooses X1 =1, then X3 = 1, then X4 = 2, then X2 = 2, and the equation is balanced. Practice some more equation balancing on your own.
Counting Atoms by Weight

Hardware store owners solved a problem long ago with a method that can be applied to almost everything, including chemistry. People need nails. Store owners have nails, but want money for them. People need lots of nails, but counting lots of nails so that you can charge for them is a drag, because they are small and pointy. Shopkeepers decided on an easier method than counting each individual nail, they simply sold nails by weight. But how many nails do you get when you buy a pound of nails? It obviously depends on the weight of an individual nail. Even now, the weight of an individual nail is still found on in Hardware Stores; It is listed as Penny Weight. A 10 penny nail weighs 1/2 oz (1 penny = 1/20 oz). So, how many 10 penny nails are in 1 lb of nails? answer.

Remember that molecules are so tiny, that instead of counting individual particles, we may want to count a "chemist's dozen", or mole, of molecules instead. So, three possible ways of specifying the 'amount' of a chemical are possible, by mass (by weight) / by mole / by molecular count. These are obviously interconvertible measures:

Isotopes and the 'Average' Mass of an Atom

In order to count molecules, we need to weigh them, because they are too small to count individually. Molecules also come in different sizes, just like nails, but there are actually an infinite number of possible molecular weights! We are saved by the fact that all molecules are made up of atoms, and there are only about 100 of these, so all we need to know is the atomic weights of the elements.

We can measure the charge to mass ration of ions in a mass spectrometer, shown below.

When we do this, we find out that the atoms of a given element do not all weigh exactly the same, but come is varieties with the same number of protons but different numbers of neutrons called isotopes. (Note: the mass of the electron lost upon ionization in this measurement is negligible, and known, and since the absolute charge of the ions is also known, the mass spectrometer may yield the exact mass of each elemental atom) Since we usually need to count billions and billions of atoms, and we cannot normally tell the difference between the isotopes since they are identical chemically, we use the average atomic weights of the elements in mole calculations and in the determination of average molecular weights. For example, for every 100,000 Hydrogen (Z=1) atoms(ions) we count in our mass spectrometer, 99,985 atoms weigh 1.007825 grams/mole and 15 weigh 2.0140 grams/mole. The minor isotope of hydrogen is so important it is called deuterium, but it is not a different element. The presence of naturally occuring deuterium makes the average atomic mass of the element with Z=1 as follows:

Atomic Mass of Hydrogen = (0.99985)*(1.007825) + (0.000015)*(2.0140) = 1.00797 g/mol

The natural abundance of the isotopes of Neon are as follows:

`Abundance(20Ne) = 90.92  %Abundance(21Ne) =  0.257 %Abundance(22Ne) =  8.82  %`
and the atomic masses of these isotopes are:
`Atomic Mass(20Ne) = 19.99244 g/molAtomic Mass(21Ne) = 20.99395 g/molAtomic Mass(22Ne) = 21.99138 g/mol`
What is the average atomic weight of Neon from these data? answer
Calculating Molecular Weight
Once we know the average weight (mass) of all the elements (these are usually listed in the Periodic Table), we can calculate the molar mass of all the molecules just by knowing their molecular formula. For example, the molecular weight of Methane, CH4, is simply

`MW{Methane} =    (1 carbon atom per molecule)*(12.011 grams per mole for carbon atoms)  + (4 atoms of hydrogen per molecule)*(1.00797 grams per mole for hydrogen atoms)  = 16.0429 grams per mole methane`
To calculate the isotopically averaged molecular mass of a substance (the weight in grams of one mole of molecules of the compound) simply use the average atomic weight (in g/mol) of each of the elements in the molecular formula multiplied by the number of times each element appears in each molecule.

Limiting Reagents
Consider the reacton of Hydrogen gas and Oxygen gas to for water. Lets assume for a moment we can 'see' each of the reacting molecules. The reaction might look like this:

Why weren't all of the reactants consumed in the reaction? Because the initial mixture was not in the proper stoichiometric proportions! 'Proper stoichiometric proportions' means in thre proportions that appear in the balanced chemical reaction. (What is the balanced chemical equation for the transformation? answer.) Whenever the reaction mixture does not contain stoichiometric proportions, one of the reagents is said to be limiting; When it runs out, the reaction must stop. In the case above, the hydrogen gas was the limiting reagent. How do you know which one of the reagents is limiting? You calculate the yield of the products from the amount of each one of the reactants, and the one that produces the least products is limiting.
Here is a little simulation that you might enjoy showing the effect of non-stoichiometric reactant concentration ratios on reaction yield for a couple of simple transformations.