percent composition problem solving guide

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CHEMFILE MINI-GUIDE TO PROBLEM SOLVING

C H A P T E R 6

Percentage Composition

Suppose you are working in an industrial laboratory. Your supervisorgives you a bottle containing a white crystalline compound and asks youto determine its identity. Several unlabeled drums of this substance havebeen discovered in a warehouse, and no one knows what it is. You take itinto the laboratory and carry out an analysis, which shows that the com-pound is composed of the elements sodium, carbon, and oxygen. Imme-diately, you think of the compound sodium carbonate, Na2CO3, a verycommon substance found in most laboratories and used in many indus-trial processes.

Before you report your conclusion to your boss, you decide to check areference book to see if there are any other compounds that contain onlythe elements sodium, carbon, and oxygen. You discover that there is an-other compound, sodium oxalate, which has the formula Na2C2O4. Whenyou read about this compound, you find that it is highly poisonous andcan cause serious illness and even death. Mistaking sodium carbonate forsodium oxalate could have very serious consequences. What can you doto determine the identity of your sample? Is it the common industrialsubstance or the dangerous poison?

Fortunately, you can determine not only whichelements are in thecompound, but also how muchof each element is present. As you havelearned, every compound has a definite composition. Every water mol-ecule is made up of two hydrogen atoms and one oxygen atom, no matterwhere the water came from. A formula unit of sodium chloride is com-posed of one sodium atom and one chlorine atom, no matter whether thesalt came from a mine or was obtained by evaporating sea water.

Likewise, sodium carbonate always has two sodium atoms, one carbonatom, and three oxygen atoms per formula unit, giving it the formulaNa2CO3; and a formula unit of sodium oxalate always contains twosodium atoms, two carbon atoms, and four oxygen atoms, giving it theformula Na2C2O4. Because each atom has a definite mass, each com-pound will have a distinct composition by mass. This composition is usually expressed as the percentage composition of the compound—the percentage by mass of each element in a compound. To identify acompound, you can compare the percentage composition obtained bylaboratory analysis with a calculated percentage composition of eachpossible compound.

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SAMPLE PROBLEM 1

Determine the percentage composition of sodium carbonate,Na2CO3.

SOLUTION

1. ANALYZE• What is given in the the formula of sodium carbonate

problem?

• What are you asked to find?the percentage of each element insodium carbonate (the percentagecomposition)

General Plan for Determining PercentageComposition of a Compound

Convert using theformula of the compound.

Convert by expressingpercentage as a fractionand then multiplying bythe mass of the sample.

Molar massof

element

Convert by multiplyingby the inverse of the

molar mass of thecompound. Then convert

to a percentage bymultiplying by 100.

1

Percentageelement in the

compound

3Mass ofelement in a

sample ofcompound

5

Repeat 1, 2,and 3 for eachremainingelement in thecompound.

Percentagecomposition ofthe compound

4

Mass ofelement per

mole ofcompound

2

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Items Data

Formula of sodium carbonate Na2CO3

Molar mass of each element* Na � 22.99 g/molC � 12.01 g/molO � 16.00 g/mol

Molar mass of sodium carbonate 105.99 g/mol

Percentage composition of sodium ?%carbonate

* determined from the periodic table

2. PLAN• What step is needed to Multiply the molar mass of each ele-

determine the mass of each ment by the ratio of the number of element per mole of moles of that element in a mole of compound? the compound (the subscript of that

element in the compound’s formula).

• What steps are needed to Multiply the mass of each element determine the portion of by the inverse of the molar mass of each element as a percent- the compound, and then multiply by age of the mass of the 100 to convert to a percentage.compound?

Step 1

Step 2

multiply by the subscript of Nain Na2CO3

Molar mass of Na1

Mass Na per moleNa2CO3

2

ratio of mol Na per molNa2CO3 from formula

2 mol Na��

1 mol Na2CO3

molar mass Na22.99 g Na1 mol Na

g Na1 mol Na2CO3

multiply by the inverse of themolar mass of Na2CO3 and

multiply by 100

Mass Na permole Na2CO3

2Percentage Na in

Na2CO3

3

�105.99 g Na2CO31 mol Na2CO3

from Step 1g Na

1molar mass Na2CO3

1 mol Na2CO3 � 100 � percentage Na in Na2CO3

Now you can combine Step 1 and Step 2 into one calculation.

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combining Steps 1 and 2

Finally, determine the percentage of carbon and oxygen in Na2CO3

by repeating the calculation above with each of those elements.

� percentage Na in Na2CO3

22.99 g Na

1 mol Na�

2 mol Na

1 mol Na2CO3�

1 mol Na2CO3

105.99 g Na2CO3� 100

repeat Steps 1 and 2 foreach remaining element

Percentage of eachelement in Na2CO3

3Percentage composition

4

3. COMPUTEpercentage sodium

percentage carbon

percentage oxygen

� 45.29% O

16.00 g O

1 mol O�

3 mol O

1 mol Na2CO3�

1 mol Na2CO3

105.99 g Na2CO3� 100

� 11.33% C

12.01 g C

1 mol C�

1 mol C

1 mol Na2CO3�

1 mol Na2CO3

105.99 g Na2CO3� 100

� 43.38% Na

22.99 g Na

1 mol Na�

2 mol Na

1 mol Na2CO3�

1 mol Na2CO3

105.99 g Na2CO3� 100

4. EVALUATE• Are the units correct? Yes; the composition is given in

percentages.

• Is the number of significant Yes; four significant figures is cor-figures correct? rect because the molar masses have

four significant figures.

• Is the answer reasonable? Yes; the percentages add up to 100percent.

Element Percentage

sodium 43.38% Na

carbon 11.33% C

oxygen 45.29% O

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1. Determine the percentage composition of each of the followingcompounds:

a. sodium oxalate, Na2C2O4 ans: 34.31% Na, 17.93%C, 47.76% O

b. ethanol, C2H5OH ans: 52.13% C, 13.15% H,34.72% O

c. aluminum oxide, Al2O3 ans: 52.92% Al, 47.08% O

d. potassium sulfate, K2SO4 ans: 44.87% K, 18.40% S,36.72% O

2. Suppose that your laboratory analysis of the white powder discussed at the beginning of this chapter showed 42.59% Na, 12.02% C, and 44.99% oxygen. Would you report that the compound is sodium oxalate or sodium carbonate (use the results of Practice Problem 1 and SampleProblem 1)? ans: sodium carbonate

SAMPLE PROBLEM 2

Calculate the mass of zinc in a 30.00 g sample of zinc nitrate,Zn(NO3).

SOLUTION

1. ANALYZE• What is given in the the mass in grams of zinc nitrate

problem?

• What are you asked to find?the mass in grams of zinc in thesample

PRACTICE

Items Data

Mass of zinc nitrate 30.00 g

Formula of zinc nitrate Zn(NO3)2

Molar mass of zinc nitrate 189.41 g /mol

Mass of zinc in the sample ? g

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3. COMPUTE

Note that mass percentage is the same as grams per 100 g, so 34.52% Zn in Zn(NO3)2 is the same as 34.52 g Zn in 100 g Zn(NO3)2.

4. EVALUATE• Are the units correct? Yes; units cancel to give the correct

units, grams of zinc.

• Is the number of significant Yes; four significant figures is cor-figures correct? rect because the data given have

four significant figures.

34.52 g Zn

100 g Zn(NO3)2� 30.00 g Zn(NO3)2 � 10.36 g Zn

� 100� 34.52% Zn

65.39 g Zn

1 mol Zn�

1 mol Zn

1 mol Zn(NO3)2�

1 mol Zn(NO3)2

189.41 g Zn(NO3)2

express percentageas a fraction andmultiply by themass of the sample

multiplyby the mole

ratio of Zn toZn(NO3)2

Molar mass of Zn1

Mass Zn in g in sample5

Mass Zn permole Zn(NO3)2

2Percentage Zn

in Zn(NO3)2

3

multiply by the inverse ofthe molar mass of

Zn(NO3)2, then multiplyby 100

2. PLAN• What steps are needed to The percentage of Zn in Zn(NO3)2

determine the mass of Zn in can be calculated and used to find a given mass of Zn(NO3)2? the mass of Zn in the sample.

189.41 g Zn(NO3)2

� � � 100 � percentage Zn

1molar mass Zn(NO3)2

1 mol1 mol Zn

molar mass Zn65.39 g Zn

1 mol Zn(NO3)2

ratio of mol Zn per molZn(NO3)2 from formula

1 mol Zn

g Zn� g Zn(NO3)2 � g Zn in sample

100 g Zn(NO3)2

given

percentage Znexpressed as a fraction

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• Is the answer reasonable? Yes; the molar mass of zinc is aboutone third of the molar mass ofZn(NO3)2, and 10.36 g Zn is aboutone third of 30.00 g of Zn(NO3)2.

1. Calculate the mass of the given element in each of the followingcompounds:

a. bromine in 50.0 g potassium bromide, ans: 33.6 g BrKBr

b. chromium in 1.00 kg sodium dichromate, ans: 397 g CrNa2Cr2O7

c. nitrogen in 85.0 mg of the amino ans: 16.3 mg Nacid lysine, C6H14N2O2

d. cobalt in 2.84 g cobalt(II) acetate, ans: 0.945 g CoCo(C2H3O2)2

HYDRATESMany compounds, especially ionic compounds, are produced and puri-fied by crystallizing them from water solutions. When this happens, somecompounds incorporate water molecules into their crystal structure.These crystalline compounds are called hydratesbecause they includewater molecules. The number of water molecules per formula unit is specific for each type of crystal. When you have to measure a certainquantity of the compound, it is important to know how much the watermolecules contribute to the mass.

You may have seen blue crystals of copper(II) sulfate in the laboratory.When this compound is crystallized from water solution, the crystals in-clude five water molecules for each formula unit of CuSO4. The truename of the substance is copper(II) sulfate pentahydrate, and its formulais written correctly as CuSO4�5H2O. Notice that the five water moleculesare written separately. They are preceded by a dot, which means they areattached to the copper sulfate molecule. On a molar basis, a mole ofCuSO4�5H2O contains 5 mol of water per mole of CuSO4�5H2O. Thewater molecules contribute to the total mass of CuSO4�5H2O. When youdetermine the percentage water in a hydrate, the water molecules aretreated separately, as if they were another element.

SAMPLE PROBLEM 3

Determine the percentage water in copper(II) sulfate penta-hydrate, CuSO4�5H2O.

PRACTICE

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2. PLAN• What steps are needed to Find the mass of water per mole of

determine the percentage of hydrate, multiply by the inverse water in CuSO4�5H2O? molar mass of the hydrate, and

multiply that by 100 to convert to apercentage.

249.72 g CuSO4�5H2O� �

� 100 � percentage H2O

1molar mass CuSO4�5H2O

1 mol CuSO4�5H2O1 mol H2O

molar mass H2O18.01 g H2O

1 mol CuSO4�5H2O

ratio of moles H2O per moleCuSO4�5H2O from formula

5 mol H2O

Items Data

Formula of copper(II) sulfate pentahydrate CuSO4�5H2O

Molar mass of H2O 18.02 g/mol

Molar mass of copper(II) sulfate 249.72 g/molpentahydrate*

Percentage H2O in CuSO4�5H2O ?%

* molar mass of CuSO4 � mass of 5 mol H2O

SOLUTION

1. ANALYZE• What is given in the the formula of copper(II) sulfate

problem? pentahydrate

• What are you asked to find?the percentage water in the hydrate

multiply by the inverse ofthe molar mass ofCuSO4�5H2O; thenmultiply by 100

multiply by themole ratio of

H2O toCuSO4�5H2O

Molar mass of H2O1 3

Mass H2O per moleCuSO4�H2O

Percentage H2O in CuSO4�5H2O

2

3. COMPUTE

� 100� 36.08% H2O

�1 mol CuSO4�5H2O

249.72 g CuSO4�5H2O

18.01 g H2O

1 mol H2O�

5 mol H2O

1 mol CuSO4�5H2O

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4. EVALUATE• Are the units correct? Yes; the percentage of water in

copper(II) sulfate pentahydrate wasneeded.

• Is the number of significant Yes; four significant figures is cor-figures correct? rect because molar masses were

given to at least four significant figures.

• Is the answer reasonable? Yes; five water molecules have amass of about 90 g, and 90 g is a little more than 1/3 of 250 g; the calculated percentage is a little morethan 1/3.

1. Calculate the percentage of water in each of the following hydrates:

a. sodium carbonate decahydrate, ans: 62.97% H2O in Na2CO3�10H2O Na2CO3�10H2O

b. nickel(II) iodide hexahydrate, ans: 25.71% H2O in NiI 2�6H2O NiI2�6H2O

c. ammonium hexacyanoferrate(III) ans: 17.89 % H2O intrihydrate (commonly called (NH4)2Fe(CN)6�3H2Oammonium ferricyanide),(NH4)2Fe(CN)6�3H2O

d. aluminum bromide hexahydrate ans: 28.85% H2O inAlBr3�6H2O

1. Write formulas for the following compounds and determine the per-centage composition of each:

a. nitric acid

b. ammonia

c. mercury(II) sulfate

d. antimony(V) fluoride

2. Calculate the percentage composition of the following compounds:

a. lithium bromide, LiBr

b. anthracene, C14H10

c. ammonium nitrate, NH4NO3

d. nitrous acid, HNO2

ADDITIONAL PROBLEMS

PRACTICE

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e. silver sulfide, Ag2S

f. iron(II) thiocyanate, Fe(CNS)2

g. lithium acetate

h. nickel(II) formate

3. Calculate the percentage of the given element in each of the follow-ing compounds:

a. nitrogen in urea, NH2CONH2

b. sulfur in sulfuryl chloride, SO2Cl2c. thallium in thallium(III) oxide, Tl2O3

d. oxygen in potassium chlorate, KClO3

e. bromine in calcium bromide, CaBr2

f. tin in tin(IV) oxide, SnO2

4. Calculate the mass of the given element in each of the followingquantities:

a. oxygen in 4.00 g of manganese dioxide, MnO2

b. aluminum in 50.0 metric tons of aluminum oxide, Al2O3

c. silver in 325 g silver cyanide, AgCN

d. gold in 0.780 g of gold(III) selenide, Au2Se3

e. selenium in 683 g sodium selenite, Na2SeO3

f. chlorine in 5.0� 104 g of 1,1-dichloropropane, CHCl2CH2CH3

5. Calculate the percentage of water in each of the following hydrates:

a. strontium chloride hexahydrate, SrCl2�6H2O

b. zinc sulfate heptahydrate, ZnSO4�7H2O

c. calcium fluorophosphate dihydrate, CaFPO3�2H2O

d. beryllium nitrate trihydrate, Be(NO3)2 �3H2O

6. Calculate the percentage of the given element in each of the follow-ing hydrates. You must first determine the formulas of the hydrates.

a. nickel in nickel(II) acetate tetrahydrate

b. chromium in sodium chromate tetrahydrate

c. cerium in cerium(IV) sulfate tetrahydrate

7. Cinnabar is a mineral that is mined in order to produce mercury.Cinnabar is mercury(II) sulfide, HgS. What mass of mercury can beobtained from 50.0 kg of cinnabar?

8. The minerals malachite, Cu2(OH)2CO3, and chalcopyrite, CuFeS2,can be mined to obtain copper metal. How much copper could beobtained from 1.00� 103 kg of each? Which of the two has thegreater copper content?

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9. Calculate the percentage of the given element in each of the follow-ing hydrates:

a. vanadium in vanadium oxysulfate dihydrate, VOSO4�2H2O

b. tin in potassium stannate trihydrate, K2SnO3�3H2O

c. chlorine in calcium chlorate dihydrate, CaClO3�2H2O

10. Heating copper sulfate pentahydrate will evaporate the water fromthe crystals, leaving anhydrous copper sulfate, a white powder. An-hydrousmeans “without water.” What mass of anhydrous CuSO4

would be produced by heating 500.0 g of CuSO4�5H2O?

11. Silver metal may be precipitated from a solution of silver nitrate byplacing a copper strip into the solution. What mass of AgNO3 wouldyou dissolve in water in order to get 1.00 g of silver?

12. A sample of Ag2S has a mass of 62.4 g. What mass of each elementcould be obtained by decomposing this sample?

13. A quantity of epsom salts, magnesium sulfate heptahydrate,MgSO4�7H2O, is heated until all the water is driven off. The sampleloses 11.8 g in the process. What was the mass of the original sample?

14. The process of manufacturing sulfuric acid begins with the burningof sulfur. What mass of sulfur would have to be burned in order toproduce 1.00 kg of H2SO4? Assume that all of the sulfur ends up inthe sulfuric acid.

percent composition problem solving guide

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