Quantitative Relationships Based on Chemical Equations

5.3 Quantitative Relationships Based on Chemical Equations

Learning Objective

Calculate the amount of one substance that will react with or be produced from a given amount of another substance.

A balanced chemical equation not only describes some of the chemical properties of substances—by showing us what substances react with what other substances to make what products—but also shows numerical relationships between the reactants and the products. The study of these numerical relationships is called stoichiometryThe study of the numerical relationships between the reactants and the products in a balanced chemical equation.. The stoichiometry of chemical equations revolves around the coefficients in the balanced chemical equation because these coefficients determine the molecular ratio in which reactants react and products are made.

Note

The word stoichiometry is pronounced “stow-eh-key-OM-et-tree.” It is of mixed Greek and English origins, meaning roughly “measure of an element.”

Looking Closer: Stoichiometry in Cooking

Let us consider a stoichiometry analogy from the kitchen. A recipe that makes 1 dozen biscuits needs 2 cups of flour, 1 egg, 4 tablespoons of shortening, 1 teaspoon of salt, 1 teaspoon of baking soda, and 1 cup of milk. If we were to write this as a chemical equation, we would write

2 c flour + 1 egg + 4 tbsp shortening + 1 tsp salt + 1 tsp baking soda + 1 c milk → 12 biscuits

(Unlike true chemical reactions, this one has all 1 coefficients written explicitly—partly because of the many different units here.) This equation gives us ratios of how much of what reactants are needed to make how much of what product. Two cups of flour, when combined with the proper amounts of the other ingredients, will yield 12 biscuits. One teaspoon of baking soda (when also combined with the right amounts of the other ingredients) will make 12 biscuits. One egg must be combined with 1 cup of milk to yield the product food. Other relationships can also be expressed.

We can use the ratios we derive from the equation for predictive purposes. For instance, if we have 4 cups of flour, how many biscuits can we make if we have enough of the other ingredients? It should be apparent that we can make a double recipe of 24 biscuits.

But how would we find this answer formally, that is, mathematically? We would set up a conversion factor, much like we did in Chapter 1 "Chemistry, Matter, and Measurement". Because 2 cups of flour make 12 biscuits, we can set up an equivalency ratio:

\frac{12 biscuits}{2 c flour}

We then can use this ratio in a formal conversion of flour to biscuits:

4 c flour \times \frac{12 biscuits}{2 c flour} = 24 biscuits

Similarly, by constructing similar ratios, we can determine how many biscuits we can make from any amount of ingredient.

When you are doubling or halving a recipe, you are doing a type of stoichiometry. Applying these ideas to chemical reactions should not be difficult if you use recipes when you cook.

Consider the following balanced chemical equation:

2C₂H₂ + 5O₂ → 4CO₂ + 2H₂O

The coefficients on the chemical formulas give the ratios in which the reactants combine and the products form. Thus, we can make the following statements and construct the following ratios:

Statement from the Balanced Chemical Reaction	Ratio	Inverse Ratio
two C₂H₂ molecules react with five O₂ molecules	$\frac{2 C_{2} H_{2}}{5 O_{2}}$	$\frac{5 O_{2}}{2 C_{2} H_{2}}$
two C₂H₂ molecules react to make four CO₂ molecules	$\frac{2 C_{2} H_{2}}{4 {CO}_{2}}$	$\frac{4 {CO}_{2}}{2 C_{2} H_{2}}$
five O₂ molecules react to make two H₂O molecules	$\frac{5 O_{2}}{2 H_{2} O}$	$\frac{2 H_{2} O}{5 O_{2}}$
four CO₂ molecules are made at the same time as two H₂O molecules	$\frac{2 H_{2} O}{4 {CO}_{2}}$	$\frac{4 {CO}_{2}}{2 H_{2} O}$

Other relationships are possible; in fact, 12 different conversion factors can be constructed from this balanced chemical equation. In each ratio, the unit is assumed to be molecules because that is how we are interpreting the chemical equation.

Any of these fractions can be used as a conversion factor to relate an amount of one substance to an amount of another substance. For example, suppose we want to know how many CO₂ molecules are formed when 26 molecules of C₂H₂ are reacted. As usual with a conversion problem, we start with the amount we are given—26C₂H₂—and multiply it by a conversion factor that cancels out our original unit and introduces the unit we are converting to—in this case, CO₂. That conversion factor is $\frac{4 {CO}_{2}}{2 C_{2} H_{2}},$ which is composed of terms that come directly from the balanced chemical equation. Thus, we have

{26C}_{2} H_{2} \times \frac{4 {CO}_{2}}{2 C_{2} H_{2}}

The molecules of C₂H₂ cancel, and we are left with molecules of CO₂. Multiplying through, we get

26 C_{2} H_{2} \times \frac{4 {CO}_{2}}{2 C_{2} H_{2}} = {52CO}_{2}

Thus, 52 molecules of CO₂ are formed.

This application of stoichiometry is extremely powerful in its predictive ability, as long as we begin with a balanced chemical equation. Without a balanced chemical equation, the predictions made by simple stoichiometric calculations will be incorrect.

Example 2

Start with this balanced chemical equation.

KMnO₄ + 8HCl + 5FeCl₂ → 5 FeCl₃ + MnCl₂ + 4H₂O + KCl

Verify that the equation is indeed balanced.
Give 2 ratios that give the relationship between HCl and FeCl₃.

Solution

Each side has 1 K atom and 1 Mn atom. The 8 molecules of HCl yield 8 H atoms, and the 4 molecules of H₂O also yield 8 H atoms, so the H atoms are balanced. The Fe atoms are balanced, as we count 5 Fe atoms from 5 FeCl₂ reactants and 5 FeCl₃ products. As for Cl, on the reactant side, there are 8 Cl atoms from HCl and 10 Cl atoms from the 5 FeCl₂ formula units, for a total of 18 Cl atoms. On the product side, there are 15 Cl atoms from the 5 FeCl₃ formula units, 2 from the MnCl₂ formula unit, and 1 from the KCl formula unit. This is a total of 18 Cl atoms in the products, so the Cl atoms are balanced. All the elements are balanced, so the entire chemical equation is balanced.
Because the balanced chemical equation tells us that 8 HCl molecules react to make 5 FeCl₃ formula units, we have the following 2 ratios: $\frac{8 HCl}{{5FeCl}_{3}} and \frac{5 {FeCl}_{3}}{8 HCl} .$ There are a total of 42 possible ratios. Can you find the other 40 relationships?

Skill-Building Exercise

Start with this balanced chemical equation.

{2KMnO}_{4} + {3CH}_{2} {=CH}_{2} + {4H}_{2} O \to {2MnO}_{2} + {3HOCH}_{2} {CH}_{2} OH + 2KOH

Verify that the equation is balanced.
Give 2 ratios that give the relationship between KMnO₄ and CH₂=CH₂. (A total of 30 relationships can be constructed from this chemical equation. Can you find the other 28?)

Concept Review Exercises

Explain how stoichiometric ratios are constructed from a chemical equation.
Why is it necessary for a chemical equation to be balanced before it can be used to construct conversion factors?

Answers

Stoichiometric ratios are made using the coefficients of the substances in the balanced chemical equation.
A balanced chemical equation is necessary so one can construct the proper stoichiometric ratios.

Key Takeaway

A balanced chemical equation gives the ratios in which molecules of substances react and are produced in a chemical reaction.

Exercises

Balance this equation and write every stoichiometric ratio you can from it.
NH₄NO₃ → N₂O + H₂O
Balance this equation and write every stoichiometric ratio you can from it.
N₂ + H₂ → NH₃
Balance this equation and write every stoichiometric ratio you can from it.
Fe₂O₃ + C → Fe + CO₂
Balance this equation and write every stoichiometric ratio you can from it.
Fe₂O₃ + CO → Fe + CO₂
Balance this equation and determine how many molecules of CO₂ are formed if 15 molecules of C₆H₆ are reacted.
C₆H₆ + O₂ → CO₂ + H₂O
Balance this equation and determine how many molecules of Ag₂CO₃(s) are produced if 20 molecules of Na₂CO₃ are reacted.
Na₂CO₃(aq) + AgNO₃(aq) → NaNO₃(aq) + Ag₂CO₃(s)
Copper metal reacts with nitric acid according to this equation:
3Cu(s) + 8HNO₃(aq) → 3Cu(NO₃)₂(aq) + 2NO(g) + 4H₂O(ℓ)
1. Verify that this equation is balanced.
2. How many Cu atoms will react if 488 molecules of aqueous HNO₃ are reacted?
Gold metal reacts with a combination of nitric acid and hydrochloric acid according to this equation:
Au(s) + 3HNO₃(aq) + 4HCl(aq) → HAuCl₄(aq) + 3NO₂(g) + 3H₂O(ℓ)
1. Verify that this equation is balanced.
2. How many Au atoms react with 639 molecules of aqueous HNO₃?
Sulfur can be formed by reacting sulfur dioxide with hydrogen sulfide at high temperatures according to this equation:
SO₂(g) + 2H₂S(g) → 3S(g) + 2H₂O(g)
1. Verify that this equation is balanced.
2. How many S atoms will be formed from by reacting 1,078 molecules of H₂S?
Nitric acid is made by reacting nitrogen dioxide with water:
3NO₂(g) + H₂O(ℓ) → 2HNO₃(aq) + NO(g)
1. Verify that this equation is balanced.
2. How many molecules of NO will be formed by reacting 2,268 molecules of NO₂?

Answers

NH₄NO₃ → N₂O + 2H₂O; the stoichiometric ratios are $\frac{{1NH}_{4} {NO}_{3}}{{1N}_{2} O},$ $\frac{{1NH}_{4} {NO}_{3}}{{2H}_{2} O},$ $\frac{{1N}_{2} O}{{2H}_{2} O},$ and their reciprocals.
2Fe₂O₃ + 3C → 4Fe + 3CO₂; the stoichiometric ratios are $\frac{{2Fe}_{2} O_{3}}{3C},$ $\frac{{2Fe}_{2} O_{3}}{4Fe},$ $\frac{{2Fe}_{2} O_{3}}{{3CO}_{2}},$ $\frac{3C}{4Fe},$ $\frac{3C}{{3CO}_{2}},$ $\frac{4Fe}{{3CO}_{2}},$ and their reciprocals.
2C₆H₆ + 15O₂ → 12CO₂ + 6H₂O; 90 molecules
1. It is balanced.
2. 183 atoms
1. It is balanced.
2. 1,617 atoms