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CHEMFILE MINI-GUIDE TO PROBLEM SOLVING CHAPTER 6

Percentage Composition
Suppose you are working in an industrial laboratory. Your supervisor gives you a bottle containing a white crystalline compound and asks you to determine its identity. Several unlabeled drums of this substance have been discovered in a warehouse, and no one knows what it is. You take it into the laboratory and carry out an analysis, which shows that the compound is composed of the elements sodium, carbon, and oxygen. Immediately, you think of the compound sodium carbonate, Na2CO3 , a very common substance found in most laboratories and used in many industrial processes. Before you report your conclusion to your boss, you decide to check a reference book to see if there are any other compounds that contain only the elements sodium, carbon, and oxygen. You discover that there is another compound, sodium oxalate, which has the formula Na2C2O4 . When you read about this compound, you nd that it is highly poisonous and can cause serious illness and even death. Mistaking sodium carbonate for sodium oxalate could have very serious consequences. What can you do to determine the identity of your sample? Is it the common industrial substance or the dangerous poison? Fortunately, you can determine not only which elements are in the compound, but also how much of each element is present. As you have learned, every compound has a denite composition. Every water molecule is made up of two hydrogen atoms and one oxygen atom, no matter where the water came from. A formula unit of sodium chloride is composed of one sodium atom and one chlorine atom, no matter whether the salt came from a mine or was obtained by evaporating sea water. Likewise, sodium carbonate always has two sodium atoms, one carbon atom, and three oxygen atoms per formula unit, giving it the formula Na2CO3 ; and a formula unit of sodium oxalate always contains two sodium atoms, two carbon atoms, and four oxygen atoms, giving it the formula Na2C2O4 . Because each atom has a denite mass, each compound 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 a compound, you can compare the percentage composition obtained by laboratory analysis with a calculated percentage composition of each possible compound.

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General Plan for Determining Percentage Composition of a Compound

Molar mass of element

2
Convert using the formula of the compound.

Mass of element per mole of compound

Convert by multiplying by the inverse of the molar mass of the compound. Then convert to a percentage by multiplying by 100.

Mass of element in a sample of compound

3
Convert by expressing percentage as a fraction and then multiplying by the mass of the sample.

Percentage element in the compound Repeat 1, 2, and 3 for each remaining element in the compound.

Percentage composition of the compound

SAMPLE PROBLEM 1 Determine the percentage composition of sodium carbonate, Na2CO3 . SOLUTION 1. ANALYZE What is given in the problem? What are you asked to nd? the formula of sodium carbonate the percentage of each element in sodium carbonate (the percentage composition)

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Items Formula of sodium carbonate Molar mass of each element* Data Na2CO3 Na 22.99 g/mol C 12.01 g/mol O 16.00 g/mol 105.99 g/mol ?%

Molar mass of sodium carbonate Percentage composition of sodium carbonate

* determined from the periodic table

2. PLAN What step is needed to determine the mass of each element per mole of compound?

What steps are needed to determine the portion of each element as a percentage of the mass of the compound? Step 1
1
Molar mass of Na

Multiply the molar mass of each element by the ratio of the number of moles of that element in a mole of the compound (the subscript of that element in the compounds formula). Multiply the mass of each element by the inverse of the molar mass of the compound, and then multiply by 100 to convert to a percentage.

2
multiply by the subscript of Na in Na2CO3 ratio of mol Na per mol Na2CO3 from formula

Mass Na per mole Na2CO3

molar mass Na

22.99 g Na 1 mol Na

2 mol Na 1 mol Na2CO3

g Na 1 mol Na2CO3
3

Step 2
2
Mass Na per mole Na2CO3
multiply by the inverse of the molar mass of Na2CO3 and multiply by 100

Percentage Na in Na2CO3

from Step 1

g Na 1 mol Na2CO3

1 mol Na2CO3 105.99 g Na2CO3

1 molar mass 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 22.99 g Na 1 mol Na 2 mol Na 1 mol Na2CO3 1 mol Na2CO3 100 105.99 g Na2CO3 percentage Na in Na2CO3

Finally, determine the percentage of carbon and oxygen in Na2CO3 by repeating the calculation above with each of those elements.
3
Percentage of each repeat Steps 1 and 2 for element in Na2CO3 each remaining element

4
Percentage composition

3. COMPUTE percentage sodium 22.99 g Na 1 mol Na 2 mol Na 1 mol Na2CO3 1 mol Na2CO3 105.99 g Na2CO3 100 43.38% Na percentage carbon 12.01 g C 1 mol C 1 mol C 1 mol Na2CO3 1 mol Na2CO3 105.99 g Na2CO3 100 11.33% C percentage oxygen 16.00 g O 1 mol O 3 mol O 1 mol Na2CO3
Element sodium carbon oxygen

1 mol Na2CO3 105.99 g Na2CO3

Percentage 43.38% Na 11.33% C 45.29% O

100 45.29% O

4. EVALUATE Are the units correct? Is the number of signicant gures correct? Is the answer reasonable?

Yes; the composition is given in percentages. Yes; four signicant gures is correct because the molar masses have four signicant gures. Yes; the percentages add up to 100 percent.

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PRACTICE 1. Determine the percentage composition of each of the following compounds: a. sodium oxalate, Na2C2O4 ans: 34.31% Na, 17.93% C, 47.76% O ans: 52.13% C, 13.15% H, b. ethanol, C2H5OH 34.72% O c. aluminum oxide, Al2O3 ans: 52.92% Al, 47.08% O ans: 44.87% K, 18.40% S, d. potassium sulfate, K2SO4 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 Sample Problem 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 problem? What are you asked to nd? the mass in grams of zinc nitrate the mass in grams of zinc in the sample

Items Mass of zinc nitrate Formula of zinc nitrate Molar mass of zinc nitrate Mass of zinc in the sample

Data 30.00 g Zn(NO3)2 189.41 g /mol ?g

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2. PLAN What steps are needed to determine the mass of Zn in a given mass of Zn(NO3)2?
1
Molar mass of Zn
multiply by the mole ratio of Zn to Zn(NO3)2

The percentage of Zn in Zn(NO3)2 can be calculated and used to nd the mass of Zn in the sample.
5
Mass Zn in g in sample
express percentage as a fraction and multiply by the mass of the sample

2
Mass Zn per multiply by the inverse of mole Zn(NO3 )2 the molar mass of
Zn(NO3)2 , then multiply by 100

3
Percentage Zn in Zn(NO3 )2

molar mass Zn

65.39 g Zn 1 mol Zn

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

1 mol Zn 1 mol Zn(NO3)2

1 mol 189.41 g Zn(NO3)2

1 molar mass Zn(NO3)2

100

percentage Zn

percentage Zn expressed as a fraction

g Zn 100 g Zn(NO3)2

given

g Zn(NO3)2

g Zn in sample

3. COMPUTE 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)2 100 34.52% Zn

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 . 34.52 g Zn 100 g Zn(NO3)2 4. EVALUATE Are the units correct? Is the number of signicant gures correct? 30.00 g Zn(NO3)2 10.36 g Zn

Yes; units cancel to give the correct units, grams of zinc. Yes; four signicant gures is correct because the data given have four signicant gures.

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

PRACTICE 1. Calculate the mass of the given element in each of the following compounds: a. bromine in 50.0 g potassium bromide, ans: 33.6 g Br KBr b. chromium in 1.00 kg sodium dichromate, ans: 397 g Cr Na2Cr2O7 c. nitrogen in 85.0 mg of the amino ans: 16.3 mg N acid lysine, C6H14N2O2 d. cobalt in 2.84 g cobalt(II) acetate, ans: 0.945 g Co Co(C2H3O2)2

HYDRATES
Many compounds, especially ionic compounds, are produced and puried by crystallizing them from water solutions. When this happens, some compounds incorporate water molecules into their crystal structure. These crystalline compounds are called hydrates because they include water molecules. The number of water molecules per formula unit is specic for each type of crystal. When you have to measure a certain quantity of the compound, it is important to know how much the water molecules 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 include ve water molecules for each formula unit of CuSO4 . The true name of the substance is copper(II) sulfate pentahydrate, and its formula is written correctly as CuSO4 5H2O. Notice that the ve water molecules are written separately. They are preceded by a dot, which means they are attached to the copper sulfate molecule. On a molar basis, a mole of CuSO4 5H2O contains 5 mol of water per mole of CuSO4 5H2O. The water molecules contribute to the total mass of CuSO4 5H2O. When you determine the percentage water in a hydrate, the water molecules are treated separately, as if they were another element. SAMPLE PROBLEM 3 Determine the percentage water in copper(II) sulfate pentahydrate, CuSO4 5H2O.

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SOLUTION 1. ANALYZE What is given in the problem? What are you asked to nd?
Items Formula of copper(II) sulfate pentahydrate Molar mass of H2O Molar mass of copper(II) sulfate pentahydrate* Percentage H2O in CuSO4 5H2O
* molar mass of CuSO4 mass of 5 mol H2O

the formula of copper(II) sulfate pentahydrate the percentage water in the hydrate
Data CuSO4 5H2O 18.02 g/mol 249.72 g/mol ?%

2. PLAN What steps are needed to determine the percentage of water in CuSO4 5H2O?

Find the mass of water per mole of hydrate, multiply by the inverse molar mass of the hydrate, and multiply that by 100 to convert to a percentage.
3
Percentage H2O in CuSO4 5H2O
multiply by the inverse of the molar mass of CuSO4 5H2O; then multiply by 100

1
Molar mass of H2O
multiply by the mole ratio of H2O to CuSO4 5H2O

2
Mass H2O per mole CuSO4 H2O
ratio of moles H2O per mole CuSO4 5H2O from formula 1 molar mass CuSO4 5H2O

18.01 g H2O 1 mol H2O

molar mass H2O

5 mol H2O 1 mol CuSO4 5H2O

1 mol CuSO4 5H2O 249.72 g CuSO4 5H2O 100 percentage H2O

3. COMPUTE 18.01 g H2O 1 mol H2O

5 mol H2O 1 mol CuSO4 5H2O

1 mol CuSO4 5H2O 249.72 g CuSO4 5H2O 100 36.08% H2O

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4. EVALUATE Are the units correct?

Is the number of signicant gures correct?

Is the answer reasonable?

Yes; the percentage of water in copper(II) sulfate pentahydrate was needed. Yes; four signicant gures is correct because molar masses were given to at least four signicant gures. Yes; ve water molecules have a mass of about 90 g, and 90 g is a little more than 1/3 of 250 g; the calculated percentage is a little more than 1/3.

PRACTICE 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 NiI2 6H2O NiI2 6H2O c. ammonium hexacyanoferrate(III) ans: 17.89 % H2O in trihydrate (commonly called (NH4)2Fe(CN)6 3H2O ammonium ferricyanide), (NH4)2Fe(CN)6 3H2O d. aluminum bromide hexahydrate ans: 28.85% H2O in AlBr3 6H2O ADDITIONAL PROBLEMS 1. Write formulas for the following compounds and determine the percentage composition of each: a. nitric acid b. ammonia c. mercury(II) sulfate d. antimony(V) uoride 2. Calculate the percentage composition of the following compounds: a. lithium bromide, LiBr b. anthracene, C14H10 c. ammonium nitrate, NH4NO3 d. nitrous acid, HNO2

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e. silver sulde, Ag2S f. iron(II) thiocyanate, Fe(CNS)2 g. lithium acetate h. nickel(II) formate Calculate the percentage of the given element in each of the following compounds: a. nitrogen in urea, NH2CONH2 b. sulfur in sulfuryl chloride, SO2Cl2 c. 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 Calculate the mass of the given element in each of the following quantities: 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 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 uorophosphate dihydrate, CaFPO3 2H2O d. beryllium nitrate trihydrate, Be(NO3)2 3H2O Calculate the percentage of the given element in each of the following hydrates. You must rst 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 Cinnabar is a mineral that is mined in order to produce mercury. Cinnabar is mercury(II) sulde, HgS. What mass of mercury can be obtained from 50.0 kg of cinnabar? The minerals malachite, Cu2(OH)2CO3 , and chalcopyrite, CuFeS2 , can be mined to obtain copper metal. How much copper could be obtained from 1.00 103 kg of each? Which of the two has the greater copper content?

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9. Calculate the percentage of the given element in each of the following 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 from the crystals, leaving anhydrous copper sulfate, a white powder. Anhydrous means 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 by placing a copper strip into the solution. What mass of AgNO3 would you 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 element could 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 sample loses 11.8 g in the process. What was the mass of the original sample? 14. The process of manufacturing sulfuric acid begins with the burning of sulfur. What mass of sulfur would have to be burned in order to produce 1.00 kg of H2SO4 ? Assume that all of the sulfur ends up in the sulfuric acid.

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