More About Coefficients in Chemical Equations

In any given chemical reaction, the equation that describes it is made up of numbers and symbols that represent the molecules that are combining. Any molecule is represented by the atomic symbols and subscripts telling how many of each type of atom make up a single molecule.

When the equation is balanced, coefficients are added to the molecules of the reactants and to the molecules of the products until the number of atoms of each type are equal on each side of the reaction. Thus, if a reaction begins with 4 atoms of "X" and 3 of "Y", it must end up with the same number of each atom.

To balance an equation, coefficients are added in front of the molecules. The coefficient tells how many of which molecule is needed in order to come up with the right number of atoms on each side. The coefficient is a multiplier for the numbers of each atom in each molecule of the equation.

As has been explained previously (for example), 3 C6H12O6 means 3 molecules of C6H12O6, which results in 18 C, 36 H, and 18 O atoms.

However, the coefficients also give more information. They serve additionally as a ratio of molecules that will react with each other. The coefficients can be seen as the how many of any numeric units are needed to properly react.

Using water as an example, the equation is this:

2H2 + O2 --> 2H2O

At the simplest level, this means 2 molecules of H2 reacted with 1 molecule of O2 results in 2 molecules of H2O.

HOWEVER, it ALSO means that any number of molecules can be combined, so long as the ration of 2:1 --> is preserved. Thus:

2 molecules of H2 reacted with 1 molecule of O2 results in 2 molecules of H2O

2 dozen molecules of H2 reacted with 1 dozen molecules of O2 results in 2 dozen molecules of H2O

2 score molecules of H2 reacted with 1 score molecules of O2 results in 2 score molecules of H2O

2 bazillion molecules of H2 reacted with 1 bazillion molecules of O2 results in 2 bazillion molecules of H2O

And for the most relevant example…

2 moles of H2 reacted with 1 mole of O2 results in 2 moles of H2O

It is when we relate the coefficients to moles that we tap a great deal of power! While we cannot count moles, we can use the atomic mass of the atoms to find the right amounts of elements or compounds to use in reactions.

For the reaction of sodium (Na) and Chlorine (Cl) we can use the balanced equation to determine the masses of the two elements that would be needed in a reaction. Here is the reaction:

2 Na + Cl2 → 2 NaCl

According the the formula above, we need molecules of Na and Cl2 in a ration of 2:1. Thus, if we have 2 moles of Na and 1 mole of Cl2 we will have the right ratio for a complete reaction with no left-overs.

From the atomic masses, we know that one mole of NA weighs 22.989 grams. Thus, we begin with 45.979 grams of sodium, we will have the right amount.

Because chlorine is always Cl2, one mole of it will weigh 70.90 grams. Finding the mass of a gas is not as easy as finding the mass of a solid, but it can be done, so starting with 70.90 grams of chlorine is the right amount as well.

It is the coefficients of the balanced equation that guide us to these masses. Understanding that coefficients give us the correct ratio of molecules allows us to use the relationship between atomic mass and numbers of moles to come up with the correct amounts of reactants for any reaction.

Balancing Equations and Conservation of Matter

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The action of balancing chemical equations primarily sets out to find the ratios of compounds that result in there being the same number of atoms in the product as there were in the reactants. This is the most fundamental concept that must be understood. It is so important, it bears being repeated and emphasized.

At the most basic level, balancing chemical equations is the process of finding coefficients which result in the number of atoms in the reactants being equal to the number of atoms in the product.

If there are 8 atoms total in all of the molecules of the reactants, then there MUST be 8 atoms total in all of the molecules of the products.

It does not matter how many different compounds are involved. It does not matter how many molecules of any of the compounds are involved. The driving factor in balancing equations is that the number of atoms on each side of the reaction are equal.

Read that last sentence ten times!

Conservation of Matter

The reason that the number of atoms must be the same stems from the Law of Conservation of Matter. In basic terms, the Law of Conservation states that matter cannot be created or destroyed. It really, if thought about, is logical. If you have 15 atoms of something, no matter how you combine or arrange them, you end up with 15 of something.

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This law drives chemistry and chemical reactions. This is why that the number of atoms cannot change. 

One way to look at this principle as it applies to chemical reactions is to compare the mass of the reactants to the mass of the products. Because the number of atoms CANNOT change, and because each atom has a certain mass, there should be no difference between the mass of the products and the mass of the reactants.

Consider the reaction of hydrogen and oxygen as an example:

2H2 + O2 = 2H2O

This reaction begins with 2 compounds on the left and ends up with one compound on the right.

This reaction begins with 2 molecules of hydrogen and one molecule of oxygen on the left and ends up with 2 molecules of water on the right.

Neither the number of compounds nor the number of molecules remain the same.

However, it begins with 6 atoms on the left (4 hydrogen and 2 oxygen) and ends up with 6 atoms (the same) on the right.

Look at it as a diagram:

2H2 + O2 = 2H2O

HH          OO  HH

=

HOH      HOH

Though the compounds (elements in this case) recombine and rearrange, the number of (and indeed, the ratio of) atoms stays the same. The reaction begins with 4 hydrogen and 2 oxygen and ends up with 4 hydrogen and 2 oxygen.

Since the number of atoms stays the same, and since atoms have a particular mass, it logically and naturally follows that the mass on both sides of the reaction stays equal.

Of course, proving this involves math and a periodic table!

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Consider the table below for this reaction:

2H2 + O2 = 2H2O

______________________________________________________________________________

Reactants	Products
hydrogen: 2 molecules, 2 atoms each	water: 2 molecules with 2 hydrogen and 1 oxygen each
atomic mass of 1 hydrogen: 1.0008	atomic mass of 1 water molecule: 2.0016 + 15.999 = 18.0006
total atomic mass of 4 hydrogen: 4.0032 oxygen: 1 molecule of 2 atoms
atomic mass of 1 oxygen: 15.999
total atomic mass of 2 oxygen: 31.998
TOTAL NUMBER OF ATOMS: 4H and 2O	TOTAL NUMBER OF ATOMS: 4H and 2O
TOTAL ATOMIC MASS OF REACTANTS: 36.0012	TOTAL ATOMIC MASS OF PRODUCTS: 36.0012

______________________________________________________________________________

In the above, the sum of the atomic masses of the reactants is equal to the sum of the masses of the products. Just in the same way, the total number of each atom is the same.

This confirms the Law of Conservation of Matter. Because matter cannot be created or destroyed, it has to be equal, both in the number of atoms as well as in the masses.

Balancing Equations and Reactions with Polyatomic Ions

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When working with chemical notation, it has been established that the subscripts (or numbers FOLLOWING the elements) represent the number of those elements present. The coefficient tells how many of the molecule is present.

Thus,

3 H20 or sometimes 3 H2O

means 3 molecules of water in which are 2 atoms of hydrogen and 1 atom of oxygen.

Understanding Polyatomic Nomenclature

There is another variation of this notation that is applied in certain cases with some compounds. Because of how the compounds form, there is sometimes a value in keeping some of the elements as a unit and subscripting the whole unit to show how many of that unit are present.

Look at the reaction below:

CaC2   +   2H2O   --->   Ca(OH)2   +   C2H2

Notice on the product side, the OH is inside parenthesis. This represents that that is a unit of molecules that are being kept together based on how the compound is formed. The subscript indicates that there are two of these units present.

Examples:

3Ca(OH)2 has in it, 3 Ca, 6 O, and 6 H. (The subscripted 2 applies to both atoms inside the parenthesis, and the coefficient of 3 applies to the whole molecule.)

Ca3(PO4)2 has in it, 3 Ca, 2 P, and 8 O. (The subscripted 2 applies to the PO4, so there are 2 P and 8 0)

2Cu(NO3)2 has—to begin with, there are 2 molecules of Cu(NO3)2 as indicated by the coefficient.

• EACH molecule has 1 Cu and 2 (NO3). Since there are 2 (NO3) (The subscript 2 applies to everything inside the parenthesis.), that means there are 2 N and 6 O in each Cu(NO3)2 molecule.
• Since there are 2 molecules of Cu(NO3)2, there are in TOTAL:
• 2 Cu
• 4 N
• 12 O

Summary:

• Coefficients, the numbers in front, apply to the whole molecule and tell how many molecules or "sets" of molecules are present.
• Subscripts (or number FOLLOWING the atom symbols) tell how many of that atom are in the molecule.
• If a group of atoms are inside parenthesis:
• They are to be kept together as a unit.
• Any subscripts to the closing parenthesis means that there are that many units of the atoms inside the parenthesis are present.

Balancing Polyatomic Reactions

Keeping in mind that the process of balancing ANY equation means finding coefficients that result in the same numbers of the same types of atoms appearing on both sides of the reaction…

…then to do so with polyatomic reactions means doing a little more work.

The first thing you need to do is get the reaction written out unbalanced…

Fe(NO3)3 + (NH4)2CO3 → Fe2(CO3)3 + NH4NO3   <<<--- Unbalanced reaction

The second step is to do a little inspection. This is vital. And tedious. The goal is to identify the polyatomic "chunks" that move across the reaction unchanged. The parenthesis, if present, will help!

The chunk has to move across UNCHANGED. If for instance a chunk of PO4 becomes PO3, then it changed. If it does move unchanged, then we can balance the equation looking at the "chunks."

In the above example, there are three polyatomic chunks (ions) that move across:

NO3

CO3

NH4

Since they move unchanged, they can be treated chunk by chunk. Optionally, if it makes it easier on the eyes and brain, you can even use an abbreviation for or color code the chunk.

Once you have identified the chunks you can treat in whole, the third step is to do the balancing:

Fe(NO3)3 + (NH4)2CO3 → Fe2(CO3)3 + NH4NO3   <<<--- Unbalanced reaction

            Reactant Side                    Product Side

Fe                 1                                           2

NO3              3                                            1

NH4              2                                            1

CO3              1                                            3

2Fe(NO3)3 + 3(NH4)2CO3 → Fe2(CO3)3 + 6NH4NO3   <<<--- Balanced reaction

            Reactant Side                    Product Side

Fe                 2                                           2

NO3              6                                           6

NH4               6                                           6

CO3              3                                           3