The Mathematics of Chemical Reactions

Interconverting Masses, Moles and Numbers of Particles - Chemistry Tutorial
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Previously as we have discussed reactions, we have discussed the changes in terms of either molecules or atoms:

2H₂ + O₂ →2 H₂O

We would say 2 atoms of hydrogen react with 1 atom of oxygen to form 2 water molecules. And we introduced balancing reactions in terms of keeping the number of atoms the same on each side of the reaction equation. While this is not completely incorrect it is not the way a chemist would state the equation. A chemist would say that 2 moles of hydrogen react with 1 mole of oxygen to form 2 moles of water. So the question you are probably asking is "what is a mole?" and "why do we need to use it?"

Here are the answers to those questions:

The MOLE (mol) is a unit of measurement that is the amount of a pure substance containing the same number of chemical units (atoms, molecules etc.) as there are atoms in exactly 12 grams of carbon-12 (i.e., 6.023 X 10²³).

So the mole is the title used for the amount 6.023 x 10²³ much the same way the word "dozen" is used for the amount 12.

So if you had a mole of donuts you would have 6.023 x 10²³ donuts and a serious stomach ache.

We use the mole (mol) to represent the amount of substances in chemistry because the numbers of atoms and molecules in each substance is so large. The value given 6.023 x 10²³ is called Avagadro's number for the scientist that found the number of atoms in 12 grams of carbon 12. Why use 12 grams? This is the theoretical atomic mass of the Carbon-12 isotope (6 protons and 6 neutrons). This means that the atomic mass or atomic weight (12 grams) of carbon is equal to exactly 1 mole of carbon.

Using carbon as a reference, the atomic weights you see in the periodic table are also equal to one mole of those substances:

Lithium for instance has an atomic mass of 6.941 grams and this is equal to one mole of lithium. This is why we state the atomic and molecular masses in units of grams per mole or g/mol.

What can we do with moles? We use the unit to make calculations based on balanced chemical equations. We use the stoichiometry (fancy way of saying mole ratios in an equation) to make predictions about how much product will be made or reactant needed if we know one mole amount in a reaction.

Example

H₂SO₄ + 2NaOH → Na₂SO₄ + 2H₂O

In this reaction the stoichiometry (mole ratio) is 1 mole of sulfuric acid reacts with 2 moles of sodium hydroxide (1:2 ratio) . One mole of sulfuric acid produces one mole of sodium sulfate (1:1 ratio) and two moles of water (1:2 ratio). Two moles of sodium hydroxide on the other hand produces one mole of sodium sulfate (2:1 ratio) and 2 moles of water(1:1 ratio).

We can use this information to make predictions about the amount of products and reactants in the reaction.

If you have four moles of sulfuric acid and 2 moles of sodium hydroxide, how many moles of water can you make?

$$ 4 \text{ mol sulfuric acid} \times {2 \text{ mol water} \over 1 \text{ mol sulfuric acid}} = 8 \text{ mol of water} $$

$$ 2 \text{ mol sodium hydroxide} \times {2 \text{ mol water} \over 2 \text{ mol sodium hydroxide}} = 2 \text{ mol of water} $$

With the amounts of reactants given, the product amount of water is very different. But since sulfuric acid can only make as much water as the amount of sodium hydroxide will allow, the actual amount of product made will be only two moles of water. This is because once you run out of sodium hydroxide the reaction will stop. Sodium hydroxide is what we call the LIMITING REAGENT in this reaction scenario.

We can also use the ratio between the two reactants to determine just how much of the sulfuric acid is used up and how much will be left over:

$$ 2 \text{ mol sodium hydroxide} \times {1 \text{ mol sulfuric acid} \over 2 \text{ mol sodium hydroxide}} = 2 \text{ mol of sulfuric acid used} $$

According to the calculation, the reaction will only use 2 moles of sulfuric acid before running out of sodium hydroxide. Since we had 4 moles of sulfuric acid to start with, there will be two moles of sulfuric acid left over after the reaction is complete. Sulfuric acid is called the EXCESS REAGENT in the reaction.

Below shows some to the ways the mole can be used to calculate materials in a reaction. Note that we have completed examples of the first type of calculation. We now need to complete some examples of the other two.

In the lab, we don't actually measure amounts of chemicals in moles directly. That is to say, we measure chemicals either by using a balance, which generates values of grams or by use of a graduated cylinder, which generates units of milliliters or liters. In order to complete the calculations shown above we need to convert these values into moles first.

There are three steps to converting grams of a substance to moles.

Determine how many grams are given in the problem.
Calculate the molar mass of the substance.
Divide step one by step two.

The three steps above can be expressed in the following proportion:

$$ {\text{grams of the substance} \over \text{moles of the substance}} = {\text{molar mass of the substance in grams} \over \text{one mole}} $$

Example

How many moles are in 17.0 grams of H₂O₂?

17.0 grams are given in the text of the problem.

First we need to calculate the molar mass (molecular weight) of H₂O₂ (hydrogen peroxide) so we know how many grams equal one mole of the molecule. The molar mass is calculated using the formula and the atomic weights on a periodic table.

Hydrogen has an atomic mass of 1.008 g/mol and oxygen has an atomic mass of 15.9993 g/mol. According to the formula there are 2 moles of hydrogen and 2 moles of oxygen in each hydrogen peroxide molecule.

$$ 2(1.008 \text{g}/\text{mol}) + 2(15.9993\text{g}/\text{mol}) = 34.0146 \text{grams}/\text{mole} $$

So hydrogen peroxide has a molecular weight of 34.0146 grams per mole.

We can use this amount to convert the grams given of hydrogen peroxide into moles:

$$ 17.0 \text{g} \text{ H}_2\text{O}_2 \times { 1 \text{ mol} \text{ H}_2\text{O}_2 \over 34.0146 \text{g} \text{ H}_2\text{O}_2 } = 0.500 \text{ mol} \text{ H}_2\text{O}_2 $$

This answer has been rounded to three significant figures because of the 17.0.

Now that you know how to convert grams to moles, it is simple enough to reverse the process and convert moles to grams:

There are three steps to converting moles of a substance to grams:

Determine how many moles are given in the problem.
Calculate the molar mass of the substance.
Multiply step one by step two.

The three steps above can be expressed in the following proportion:

$$ {\text{grams of the substance} \over \text{moles of the substance}} = {\text{molar mass of the substance in grams} \over \text{one mole}} $$

Try to complete the following example on your own:

How many grams are in 0.700 moles of H₂O₂? The correct answer is 28.3 g.

We also stated above that we measure compounds in graduated cylinders which have units of liters or milliliters. How do we convert from units of volume to a unit of moles?

Molarity

Concentrations of Solutions Molarity (M)
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Solutions of chemicals are described in terms of their concentration and the most common concentration unit used in the laboratory is MOLARITY (M). The molarity of a solution is the number of moles of solute in that solution per liters of the solution.

$$ \text{Molarity (M)} = { \text{Moles of Solute} \over \text{Liters of Solution} } $$

Now it should be more obvious how we would convert a solution volume measured in the lab in milliliters or liters into moles.

There are two steps to converting the volume of a solution to moles:

Convert the given amount of volume into liters (L).
Multiply the liters of the solution by its given molarity.

Example

How many moles of Sodium Chloride (NaCl) are in a 24.5 mL volume of a 1.3M solution of NaCl?

First convert 24.5 mL into liters. Remember that there are a thousand milliliters in one liter

$$ 25.4 \text{ mL} \times {1 \text{ L} \over 1000 \text{ mL}} = 0.0254 \text{ L} $$

Now take the liters of solution and calculate the number of moles NaCl in the solution:

$$ 0.0254 \text{ L} \times {1.3 \text{ Moles of NaCl} \over \text{Liters of Solution}} = 0.033 \text{ mol NaCl} $$

Another way volumes can be converted to moles is by the density of the substance. In cases where the amount of a pure liquid or solid is given in units of volume, the density, which is given in terms of grams per unit volume, is used to convert the volume into grams. Once in grams, you already know how to convert the value further into moles.

There are two steps:

Multiply the volume by the density to get the mass.
Divide the mass by the molar mass to get the number of moles.

Example

A lead block was found to have a volume of 2.66 cm³. How many moles of lead are in the block?

The density of lead is 9.78 g/cm³. Convert the volume to grams first:

$$ 2.66 \text{ cm}^3 \times 9.78 \text{ g}/\text{cm}^3 = 26.0 \text{ g lead} $$

Now convert to moles using the molar mass of lead, 207.2 g/mol.

$$ 26.0 \text{ g Pb} \times 1 \text{ mol}/207.2 \text{ g Pb} = 0.126 \text{ mol Pb} $$

You now have been shown how to write and balance chemical reactions, how to convert their amounts into moles and use those values to calculate products and/or reactant amounts, and how to convert units of grams and volume into moles. So why was all this information needed?

We need to know this information to both create and use the forensic solutions we will need to complete our investigations in the lab.