12.1 The Arithmetic of Equations 12 - Mr. Leiker's Classroom

12.1 The Arithmetic of Equations 12.1

Connecting to Your World Silk is one of the most beautiful Guide for Reading 1 FOCUS

and luxurious of all fabrics. It is spun from the cocoons of silkworms. Silk Key Concepts Objectives
manufacturers know from experience that to pro-
duce enough silk to make just one elegant Japa- • How is a balanced equation like 12.1.1 Explain how balanced equa-
nese kimono they will need over 3000 a recipe? tions apply to both chemistry
cocoons. In a similar fashion, chemists need and everyday situations.
to know how much reactant is needed to • How do chemists use balanced
make a certain amount of product. The chemical equations? 12.1.2 Interpret balanced chemical
answer lies in chemical equations. From a equations in terms of moles,
balanced chemical equation, you can deter- • In terms of what quantities can representative particles, mass,
mine the quantities of reactants and products you interpret a balanced and gas volume at STP.
in a reaction. chemical equation?
12.1.3 Identify the quantities that are
Using Everyday Equations • What quantities are conserved always conserved in chemical
in every chemical reaction? reactions.
When you bake cookies, you probably use a recipe. A cookie recipe tells you
the precise amounts of ingredients to mix to make a certain number of Vocabulary Guide for Reading
cookies, as shown in Figure 12.1. If you need a larger number of cookies
than the recipe provides, you can double or triple the amounts of ingredi- stoichiometry Build Vocabulary L2
ents. A balanced chemical equation provides the same kind of quantita-
tive information that a recipe does. In a cookie recipe, you can think of the Reading Strategy Paraphrase Introduce the term sto-
ingredients as the reactants, and the cookies as the products. ichiometry in your own words. Stress
Using Prior Knowledge Before that stoichiometry allows students to
Here is another example. Imagine you are in charge of manufacturing for you read, jot down three things calculate the amounts of chemical sub-
the Tiny Tyke Tricycle Company. The business plan for Tiny Tyke requires the you know about balanced chemi- stances involved in chemical reactions
production of 640 custom-made tricycles each week. One of your responsibi- cal equations. When you have using information obtained from bal-
lities is to be sure that there are enough parts available at the start of each read the section, explain how anced chemical equations.
workweek to make these tricycles. How can you determine the number of what you already knew helped
parts you need per week? you learn something new.

Figure 12.1 A cookie recipe Reading Strategy L2
tells you the number of cookies
that you can expect to make Relate Text and Visuals Have stu-
from the listed amounts of dents construct tables similar to Figure
ingredients. Using Models 12.3 to interpret balanced chemical
How can you express a equations for stoichiometric prob-
cookie recipe as a balanced lems. Students need to use only those
equation? levels of interpretation—particles,
molecules, moles, mass, or gas vol-
umes—appropriate to the problem.

2 INSTRUCT

Section 12.1 The Arithmetic of Equations 353

Section Resources Technology Have students study the photograph
• Interactive Textbook with ChemASAP, and read the text that opens the sec-
Print Problem Solving 12.1, 12.4, Assessment tion. Ask, How many cocoons would
• Guided Reading and Study Workbook, 12.1 be required to produce enough silk
Section 12.1 • Go Online, Section 12.1 for two Japanese kimonos? (twice as
• Core Teaching Resources, Section 12.1 many, or 6000 cocoons) How did you
Review calculate the number of cocoons? (by
• Transparencies, T122–T125 multiplying the number needed for one
• Laboratory Manual, Lab 19 kimono by two)

Answers to...
Figure 12.1 Acceptable answers
will list ingredients as reactants and
cookies as the product.

Stoichiometry 353

Section 12.1 (continued) To simplify this discussion, assume that the major components of the
tricycle are the frame (F), the seat (S), the wheels (W), the handlebars (H),
Using Everyday Equations and the pedals (P), in other words, your reactants. The figure below illus-
trates how an equation can represent the manufacturing of a single tricycle.

Discuss L2

Write this statement on the board:“A F ؉ S ؉ 3W ؉ H ؉ 2P FSW3HP2
frame, a seat, wheels, a handlebar, and
pedals are needed to assemble a com- The finished tricycle, your product, has a “formula” of FSW3HP2. The
plete tricycle.” Have students look at balanced equation for making a single tricycle is
the visual representation of how a tri-
cycle is assembled and ask, Does this F ϩ S ϩ 3W ϩ H ϩ 2P ¡ FSW3HP2
statement correctly describe the
process? (no) What is the correct This balanced equation is a “recipe” to make a single tricycle: Making a tri-
statement? (A frame, a seat, three cycle requires assembling one frame, one seat, three wheels, one handle-
wheels, a handlebar, and two pedals are bar, and two pedals. Now look at Sample Problem 12.1. It shows you how to
needed to assemble a complete tricycle.) use the balanced equation to calculate the number of parts needed to man-
ufacture a given number of tricycles.
Using Balanced Chemical

Equations

Discuss L2 Using Balanced Chemical Equations

Remind students that during a chemi- Word Origins Nearly everything you use is manufactured from chemicals—soaps, sham-
cal reaction, atoms are rearranged into poos and conditioners, CDs, cosmetics, medicines, and clothes. In manu-
new combinations and groupings. It is Stoichiometry comes from facturing such items, the cost of making them cannot be greater than the
somewhat similar to changing silk the combination of the price they are sold at. Otherwise, the manufacturer will not make a profit.
cocoons into a kimono. Just as a Greek words stoikheioin, Therefore, the chemical processes used in manufacturing must be carried
chemical reaction has reactants and meaning “element,” and out economically. This is where balanced equations help.
products, the cocoon is the reactant metron, meaning “to mea-
and the kimono is the product. Much sure.” Stoichiometry is the A balanced chemical equation tells you what amounts of reactants to
like having the proper supply of silk calculation of amounts of mix and what amounts of product to expect. Chemists use balanced
cocoons, chemists must have an substances involved in chemical equations as a basis to calculate how much reactant is needed or
adequate supply of reactants for a chemical reactions. What product is formed in a reaction. When you know the quantity of one sub-
chemical reaction. do think the term spec- stance in a reaction, you can calculate the quantity of any other substance
trometry means? consumed or created in the reaction. Quantity usually means the amount
Word Origins L2 of a substance expressed in grams or moles. However, quantity could just
as well be in liters, tons, or molecules.
The term spectrometry means “the
study of light or spectrum.” The calculation of quantities in chemical reactions is a subject of
chemistry called stoichiometry. Calculations using balanced equations are
called stoichiometric calculations. For chemists, stoichiometry is a form of
bookkeeping. For example, accountants can track income, expenditures,
and profits for a small business by tallying each in dollars and cents. Chem-
ists can track reactants and products in a reaction by stoichiometry. It
allows chemists to tally the amounts of reactants and products using ratios
of moles or representative particles.

Checkpoint How is stoichiometry similar to bookkeeping?

354 Chapter 12

354 Chapter 12

Discuss L2

Write the equation that represents the
production of a tricycle.

SAMPLE PROBLEM 12.1 F + S + 3W + H + 2P → FSW3HP2
Point out that the balanced equation
Using a Balanced Equation as a Recipe contains information that not only
relates reactant to products but relates
In a five-day workweek, Tiny Tyke is scheduled to make 640 tricycles. one reactants to another.
How many wheels should be in the plant on Monday morning to make
these tricycles? Ask, How many handlebars are
needed to complete tricycles if 24
Analyze List the knowns and the unknown. pedals are available? (12) Show stu-
Knowns dents how to set up the proportions to
• number of tricycles ϭ 640 tricycles ϭ 640 FSW3HP2 solve this problem.
• F ϩ S ϩ 3W ϩ H ϩ 2P ¡ FSW3HP2
? H = 24 P ϫ (1 H/2P) = 12 H

Unknown Ask, How many handlebars are
• number of wheels ϭ ? wheels needed to complete tricycles if 24
seats are available? (24) 24 wheels?
The desired conversion is tricycles (FSW3HP2) ¡ wheels (W). The (8)
balanced equation tells you that each tricycle has three wheels, or
1 FSW3HP2 ϭ 3 W. The problem can be solved by using the proper Sample Problem 12.1
conversion factor derived from this expression.

Calculate Solve for the unknown. Answers
You can write two conversion factors relating wheels to tricycles.
1. 288 seats, 864 wheels, 576 pedals
3W and 1 FSW3HP2 2. Answers will vary but should
1 FSW3HP2 3W
include the correct number of
The desired unit is W, so use the conversion factor on the left—the one Math Handbook “parts” to make the product.
For help with dimensional
that has W in the numerator. Multiply the number of tricycles by the analysis, go to page R66. Practice Problems Plus L2

conversion factor. Problem Solving 12.1 Solve
Problem 1 with the help of an
640 FSW3HP2 ϫ 1 3W ϭ 1920 W interactive guided tutorial. Chapter 21 Assessment problem 57 is
FSW3HP2 related to Sample Problem 12.1.
with ChemASAP
Evaluate Does the result make sense? Math Handbook

If three wheels are required for each tricycle, and a total of more than For a math refresher and preview,
600 tricycles are being made, then a number of wheels in excess of direct students to dimensional
1800 is a logical answer. The unit of the known cancels with the unit in analysis, page R66.
the denominator of the conversion factor, and the answer is in the unit
of the unknown.

Practice Problems

1. Tiny Tike has decided to make 2. Write an equation that gives
288 tricycles each day. How your own “recipe” for making
many tricycle seats, wheels, a skateboard.
and pedals are needed?

Section 12.1 The Arithmetic of Equations 355

Answers to...

Checkpoint In stoichiome-
try, the amount of product formed
by a chemical reaction can be calcu-
lated based on the amounts of the
reactants. In bookkeeping, the prof-
its of a business can be calculated
based on income and expenditures.

Stoichiometry 355

Section 12.1 (continued)

Interpreting Chemical

Equations L2

Discuss

Review balancing chemical reactions Interpreting Chemical Equations
by writing several unbalanced equa-
tions on the board. For example: In gardens such as the one shown in Figure 12.2, fertilizers are often used to
improve the growth of flowers. As you may recall from Chapter 10, ammo-
CuO(s) +NH3(aq) → Figure 12.2 Gardeners use nia is widely used as a fertilizer. Ammonia is produced industrially by the
Cu(s) + H2O(l) + N2(g) ammonium salts as fertilizer. The reaction of nitrogen with hydrogen.
nitrogen in these salts is essential
NH3(g) + O2(g) → NO(g) + H2O(g) to plant growth. N2(g) ϩ 3H2(g) ¡ 2 NH3(g)
KClO3(s) → KCl(s) + O2(g).
356 Chapter 12 The balanced chemical equation tells you the relative amounts of reac-
Have students balance the equations as tants and product in the reaction. However, your interpretation of the
equation depends on how you quantify the reactants and products. A
shown. balanced chemical equation can be interpreted in terms of different quanti-
ties, including numbers of atoms, molecules, or moles; mass; and volume. As
3CuO(s) + 2NH3(aq) → you study stoichiometry, you will learn how to read a chemical equation in
3Cu(s) + 3H2O(l) + N2(g) terms of any of these quantities.

4NH3(g) + 5O2(g) → 4NO(g) +6H2O(g) Number of Atoms At the atomic level, a balanced equation indicates
2KClO3(s) → 2KCl(s) + 3O2(g)
Ask, Why is it not correct to balance that the number and type of each atom that makes up each reactant also
an equation by changing the sub- makes up each product. Thus, both the number and types of atoms are not
scripts in one or more of the formu- changed in a reaction. In the synthesis of ammonia, the reactants are com-
las? (Changing the subscripts in a posed of two atoms of nitrogen and six atoms of hydrogen. These eight
formula changes the chemical identity of atoms are recombined in the product.

the substance.) Number of Molecules The balanced equation indicates that one mol-

Mass Conservation in ecule of nitrogen reacts with three molecules of hydrogen. Nitrogen and
hydrogen will always react to form ammonia in a 1:3:2 ratio of molecules. If
Chemical Reactions L1 you could make 10 molecules of nitrogen react with 30 molecules of hydro-
gen, you would expect to get 20 molecules of ammonia. Of course, it is not
Use Visuals practical to count such small numbers of molecules and allow them to
react. You could, however, take Avogadro’s number of nitrogen molecules
Figure 12.3 Remind students that the and make them react with three times Avogadro’s number of hydrogen mol-
term STP represents “standard temper- ecules. This would be the same 1:3 ratio of molecules of reactants. The reac-
ature and pressure.” Ask, What are the tion would form two times Avogadro’s number of ammonia molecules.
values of STP? (0°C and 101.3 kPa)
Why is the volume of a gas usually Moles A balanced chemical equation also tells you the number of moles of
measured at STP? (The volume of a gas
is usually measured at STP because its reactants and products. The coefficients of a balanced chemical equation
indicate the relative numbers of moles of reactants and products in a chemi-
volume varies with temperature and cal reaction. This is the most important information that a balanced chemi-
cal equation provides. Using this information, you can calculate the amounts
pressure.) What is the molar volume of reactants and products. In the synthesis of ammonia, one mole of nitrogen
of any gas at STP? (22.4 L/mol) How molecules reacts with three moles of hydrogen molecules to form two moles
many particles does it contain? of ammonia molecules. As you can see from this reaction, the total number
(22.4 L of any ideal gas at STP contains of moles of reactants does not equal the total number of moles of product.
6.02 × 1023 particles of that gas.)
Checkpoint What do the coefficients of a balanced chemical equation
indicate?

Facts and Figures clover plants.These bacteria change atmo-
spheric nitrogen into ammonia molecules or
Ammonia in the Nitrogen Cycle ammonium ions. Other bacteria break down
Earth’s atmosphere contains 0.01 parts per the nitrogenous material in dead plants and
million of ammonia, and small amounts of animals into ammonia molecules. Certain
ammonia occur in volcanic gases. Most soil bacteria oxidize these molecules into
ammonia cycles through the living world nitrate ions, the form readily absorbed by
without returning to the atmosphere. plant roots. When a plant dies, this cycle
Ammonia plays a role in several stages of the begins again.
nitrogen cycle. Nitrogen-fixing bacteria form
nodules, or swellings, on the roots of plants
in the legume family, such as beans and

356 Chapter 12

Mass A balanced chemical equation obeys the law of conservation of For: Links on Conservation Download a worksheet on
of Mass Conservation of Mass for students
mass. This law states that mass can be neither created nor destroyed in an to complete and find additional
ordinary chemical or physical process. As you recall, the number and type Visit: www.SciLinks.org support for NSTA SciLinks.
of atoms does not change in a chemical reaction. Therefore, the total mass Web Code: cdn-1121
of the atoms in the reaction does not change. Using the mole relationship, TEACHER Demo
you can relate mass to the number of atoms in the chemical equation. The
mass of 1 mol of N2 (28.0 g) plus the mass of 3 mol of H2 (6.0 g) equals the Interpreting a Chemical Equation
mass of 2 mol of NH3 (34 g). Although the number of moles of reactants Purpose Students interpret a bal-
does not equal the number of moles of product, the total number of grams anced equation of the reaction of mag-
of reactants does equal the total number of grams of product. nesium and hydrochloric acid.
Materials 2.5–3.5-cm strip of magne-
Volume If you assume standard temperature and pressure, the equation sium, 50 mL 1M HCl(aq) in a 100-mL
beaker, baking soda
also tells you about the volumes of gases. Recall that 1 mol of any gas at STP Safety Wear safety glasses and an
occupies a volume of 22.4 L. The equation indicates that 22.4 L of N2 reacts apron. Neutralize remaining HCl(aq)
with 67.2 L (3 ϫ 22.4 L) of H2. This reaction forms 44.8 L (2 ϫ 22.4 L) of NH3. with baking soda before flushing down
the drain.
Mass Conservation in Chemical Reactions Procedure Identify the two reactants
as magnesium and hydrochloric acid.
Figure 12.3 summarizes the information derived from the balanced chemi- Have students observe the reaction as
you carefully add the magnesium strip
cal equation for the formation of ammonia. As you can see, the mass of the to the acid. Ask students to write a bal-
anced chemical equation for the reac-
reactants equals the mass of the products. In addition, the number of tion of magnesium and hydrochloric
acid. [Mg(s) + 2HCl(aq) → MgCl2(aq) +
atoms of each type in the reactants equals the number atoms of each type H2(g)] Have students interpret the
equation in terms of particles, moles,
in the product. Mass and atoms are conserved in every chemical and molar masses.
Expected Outcome Students should
reaction. However, molecules, formula units, moles, and volumes are not express the balanced equation at the
particle level as one atom of magne-
necessarily conserved—although they may be. Consider, for example, the sium reacts with two molecules of
hydrogen chloride to produce one for-
formation of hydrogen iodide, mula unit of magnesium chloride and
one molecule of hydrogen gas. Simi-
H2(g) ϩ I2(g) ¡ 2HI(g) larly, one mole of magnesium reacts
with two moles of hydrogen chloride
In this reaction, molecules, moles, and volume are all conserved. But in the to produce one mole of magnesium
majority of chemical reactions, they are not. chloride and one mole of hydrogen
gas. Finally, 24.31 g Mg + 72.92 g HCl
. produces 95.21 g MgCl2 + 2.02 g H2.

N2(g) ؉ 3H2(g) → 2NH3(g) Figure 12.3 The balanced Answers to...
chemical equation for the Figure 12.3 10 molecules NH3
؉→ formation of ammonia can be
interpreted in several ways. Checkpoint the relative
2 atoms N ؉ 6 atoms H → 2 atoms N and Predicting How many mol- numbers of representative particles,
6 atoms H the relative numbers of moles, and,
ecules of NH3 could be made for gases, the relative volumes
from 5 molecules of N2 and 15
molecules of H2? Stoichiometry 357

1 molecule N2 ؉ 3 molecules H2 → 2 molecules NH3

10 molecules N2 ؉ 30 molecules H2 → 20 molecules NH3

6.02 ؋ 1023 6.02 ؋ 1023 6.02 ؋ 1023
molecules N2 molecules H2 molecules NH3
( )1 ؋ ؉ ( ) ( )3 ؋ →
2؋

1 mol N2 ؉ 3 mol H2 → 2 mol NH3

28 g N2 ؉ 3 ؋ 2 g H2 → 2 ؋ 17 g NH3

34 g reactants → 34 g products

Assume 22.4 ؉ 22.4 22.4 22.4 → 22.4 22.4
STP L LLL LL

22.4 L N2 67.2 L H2 44.8 L NH3

Section 12.1 The Arithmetic of Equations 357

Differentiated Instruction

Less Proficient Readers L1

Have students construct a table like the one the completed table to answer questions
such as,“How many moles of water are
shown in Figure 12.3 for the reaction of produced by reacting 4 moles of hydrogen
gas with excess oxygen?” Other reactions for
hydrogen gas with oxygen gas to form water. students to analyze:
CO(g) + 2H2(g) → CH3OH(l)
Students should begin by writing the bal- CH4(g) + 2O2(g) → CO2(g) + 2H2O(g)

anced equation. Encourage students to draw

pictures as shown in Figure 12.3 to represent

reactants and products. Have students use

Section 12.1 (continued)

CONCEPTUAL PROBLEM 12.1

CONCEPTUAL PROBLEM 12.1 Interpreting a Balanced Chemical Equation

Answers Hydrogen sulfide, which smells like rotten eggs, is found in volcanic
gases. The balanced equation for the burning of hydrogen sulfide is:
3. 2 molecules H2 + 1 molecule O2→
2H2S(g) ϩ 3O2(g) ¡ 2SO2(g) ϩ 2H2O(g)
2 molecules H2O Interpret this equation in terms of
2 mol H2 + 1 mol O2 → 2 mol H2O a. numbers of representative particles and moles.
44.8 L H2 + 22.4 L O2 → 44.8 L H2O b. masses of reactants and products.
4. 2 mol C2H2 + 5 mol O2 →
Analyze Identify the relevant concepts. b. Multiply the number of moles of each reactant
4 mol CO2 + 2 mol H2O and product by its molar mass:
44.8 L C2H2 + 112 L O2 → a. The coefficients in the balanced equation
give the relative number of molecules or 2 mol H2S ϩ 3 mol O2 ¡ 2 mol SO2 ϩ 2 mol H2O.
89.6 L CO2 + 44.8 L H2O moles of reactants and products.
212 g reactants → 212 g products a2 mol ϫ 34.1mgol b + a3 mol ϫ 32.0 g b ¡
b. A balanced chemical equation obeys the mol
law of conservation of mass.
Practice Problems Plus L2 g g
Solve Apply concepts to this situation. 64.1mol mol
Balance the following equation: Praa.c2timceolePcruolebslHe2mS ϩs 3 molecules O2 ¡ a2 mol ϫ b + a2 mol ϫ 18.0 b

C5H12(g) + O2(g) → CO2(g) + H2O(g) 2 molecules SO2 ϩ 2 molecules H2O 68.2 g H2S ϩ 96.0 g O2 ¡ 128.2 g SO2 ϩ 36.0 g H2O
2 mol H2S ϩ 3 mol O2 ¡ 164.2 g ϭ 164.2 g
Interpret the balanced equation in
2 mol SO2 ϩ 2 mol H2O
terms of relative number of moles,
Practice Problems
volumes of gas at STP, and masses of
3. Interpret the equation for the formation of water relative numbers of moles, volumes of gas at STP,
reactants and products. [1 mol from its elements in terms of numbers of mole- and masses of reactants and products.
C5H12(g) + 8 mol O2(g) → 5 mol CO2(g) + cules and moles, and volumes of gases at STP.
6 mol H2O(g); 22.4 L C5H12(g) + 179 L Problem-Solving 12.4 Solve
O2(g) → 112 L CO2(g) + 134 L H2O(g); 328 2H2(g) ϩ O2(g) ¡ 2H2O(g) Problem 4 with the help of an
g reactants → 328 g products] 4. Balance the following equation. interactive guided tutorial.

3 ASSESS C2H2(g) ϩ O2(g) ¡ CO2(g) ϩ H2O(g) with ChemASAP
Interpret the balanced equation in terms of

Evaluate Understanding L2 12.1 Section Assessment

Have pairs of students write a balanced 5. Key Concept How is a balanced equation simi- 10. Balance this equation: C2H5OH(l) ϩ O2(g) ¡
chemical equation for a simple reaction. lar to a recipe? CO2(g) ϩ H2O(g). Show that the balanced equation
Have pairs exchange equations and obeys the law of conservation of mass.
write quantitative relationships between 6. Key Concept How do chemists use balanced
reactants and products in terms of mass, equations? Explanatory Paragraph Explain this statement:
moles, particles, and, where appropriate, “Mass and atoms are conserved in every chemical
volumes. 7. Key Concept Chemical reactions can be reaction, but moles are not necessarily conserved.”
described in terms of what quantities?
Reteach L1 Assessment 12.1 Test yourself on
8. Key Concept What quantities are always con- the concepts in Section 12.1.
Remind students that the coefficients served in chemical reactions?
in a balanced chemical equation state with ChemASAP
the relationships among substances 9. Interpret the given equation in terms of relative
involved in the reaction. numbers of representative particles, numbers of
moles, and masses of reactants and products.

2K(s) ϩ 2H2O(l ) ¡ 2KOH(aq) ϩ H2(g)

358 Chapter 12

Moles of reactants and products are Section 12.1 Assessment
conserved for some reactions, but
this is generally not the case. 5. Both a balanced equation and a recipe 9. 2 atoms K + 2 molecules H2O →
give quantitative information about the
If your class subscribes to the starting and end materials. 2 formula units KOH + 1 molecule H2
Interactive Textbook, use it to 2 mol K + 2 mol H2O →
review key concepts in Section 12.1. 6. as a basis to calculate how much reactant
is needed or product is formed in a reac- 2 mol KOH + 1 mol H2
with ChemASAP tion 78.2 g K + 36.0 g H2O →
358 Chapter 12
7. numbers of atoms, molecules, or moles; 112.2 g KOH + 2.0 g H2
mass; and volumes 10. C2H5OH + 3O2 → 2CO2 + 3H2O

8. mass and atoms 46.0 g C2H5OH + 96.0 g O2 →

88.0 g CO2 + 54.0 g H2O
142.0 g reactants → 142.0 g products