Mass Relationships in Chemical Reactions

CHAPTER 3 MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.5 (34.968 amu)(0.7553) + (36.956 amu)(0.2447) = 35.45 amu 3.6 Strategy: Each isotope contributes to the average atomic mass based on its relative abundance. Multiplying the mass of an isotope by its fractional abundance (not percent) will give the contribution to the average atomic mass of that particular isotope. 6 It would seem that there are two unknowns in this problem, the fractional abundance of Li and the fractional 7 abundance of Li. However, these two quantities are not independent of each other; they are related by the 6 fact that they must sum to 1. Start by letting x be the fractional abundance of Li. Since the sum of the two abundance’s must be 1, we can write 7 Abundance Li = (1 − x) Solution: Average atomic mass of Li = 6.941 amu = x(6.0151 amu) + (1 − x)(7.0160 amu) 6.941 = −1.0009x + 7.0160 1.0009x = 0.075 x = 0.075 6 7 x = 0.075 corresponds to a natural abundance of Li of 7.5 percent. The natural abundance of Li is (1 − x) = 0.925 or 92.5 percent. 3.7 ⎛ 6.022 × 1023 amu ⎞ The unit factor required is ⎜ ⎟ ⎜ ⎟ 1g ⎝ ⎠ ? g = 13.2 amu × 3.8 6.022 × 1023 amu = 2.19 × 10−23 g ⎛ 6.022 × 1023 amu ⎞ The unit factor required is ⎜ ⎟ ⎜ ⎟ 1g ⎝ ⎠ ? amu = 8.4 g × 3.11 1g 6.022 × 1023 amu = 5.1 × 1024 amu 1g In one year: (6.5 × 109 people) × Total time = 365 days 24 hr 3600 s 2 particles × × × = 4.1 × 1017 particles/yr 1 yr 1 day 1 hr 1 person 6.022 × 1023 particles 17 4.1 × 10 particles/yr = 1.5 × 106 yr 36 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.12 The thickness of the book in miles would be: 0.0036 in 1 ft 1 mi × × × (6.022 × 1023 pages) = 3.42 × 1016 mi 1 page 12 in 5280 ft The distance, in miles, traveled by light in one year is: 1.00 yr × 365 day 24 h 3600 s 3.00 × 108 m 1 mi × × × × = 5.88 × 1012 mi 1 yr 1 day 1h 1s 1609 m The thickness of the book in light-years is: 1 light-yr (3.42 × 1016 mi) × 12 5.88 × 10 mi = 5.8 × 103 light - yr 3 It will take light 5.8 × 10 years to travel from the first page to the last one! 6.022 × 1023 S atoms = 3.07 × 1024 S atoms 1 mol S 3.13 5.10 mol S × 3.14 (6.00 × 109 Co atoms) × 3.15 77.4 g of Ca × 3.16 Strategy: We are given moles of gold and asked to solve for grams of gold. What conversion factor do we need to convert between moles and grams? Arrange the appropriate conversion factor so moles cancel, and the unit grams is obtained for the answer. 1 mol Co 6.022 × 10 23 Co atoms = 9.96 × 10−15 mol Co 1 mol Ca = 1.93 mol Ca 40.08 g Ca Solution: The conversion factor needed to covert between moles and grams is the molar mass. In the periodic table (see inside front cover of the text), we see that the molar mass of Au is 197.0 g. This can be expressed as 1 mol Au = 197.0 g Au From this equality, we can write two conversion factors. 1 mol Au 197.0 g Au and 197.0 g Au 1 mol Au The conversion factor on the right is the correct one. Moles will cancel, leaving the unit grams for the answer. We write ? g Au = 15.3 mol Au × 197.0 g Au = 3.01 × 103 g Au 1 mol Au Check: Does a mass of 3010 g for 15.3 moles of Au seem reasonable? What is the mass of 1 mole of Au? CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.17 3.18 (a) 200.6 g Hg 1 mol Hg × = 3.331 × 10−22 g/Hg atom 1 mol Hg 6.022 × 1023 Hg atoms (b) 20.18 g Ne 1 mol Ne × = 3.351 × 10−23 g/Ne atom 1 mol Ne 6.022 × 1023 Ne atoms 37 (a) Strategy: We can look up the molar mass of arsenic (As) on the periodic table (74.92 g/mol). We want to find the mass of a single atom of arsenic (unit of g/atom). Therefore, we need to convert from the unit mole in the denominator to the unit atom in the denominator. What conversion factor is needed to convert between moles and atoms? Arrange the appropriate conversion factor so mole in the denominator cancels, and the unit atom is obtained in the denominator. Solution: The conversion factor needed is Avogadro's number. We have 1 mol = 6.022 × 10 23 particles (atoms) From this equality, we can write two conversion factors. 1 mol As and 6.022 × 1023 As atoms 6.022 × 1023 As atoms 1 mol As The conversion factor on the left is the correct one. Moles will cancel, leaving the unit atoms in the denominator of the answer. We write ? g/As atom = (b) 74.92 g As 1 mol As × = 1.244 × 10−22 g/As atom 23 1 mol As 6.022 × 10 As atoms Follow same method as part (a). ? g/Ni atom = 58.69 g Ni 1 mol Ni × = 9.746 × 10−23 g/Ni atom 1 mol Ni 6.022 × 1023 Ni atoms Check: Should the mass of a single atom of As or Ni be a very small mass? 1 mol Pb 207.2 g Pb = 3.44 × 10−10 g Pb 1 mol Pb 3.19 1.00 × 1012 Pb atoms × 3.20 Strategy: The question asks for atoms of Cu. We cannot convert directly from grams to atoms of copper. What unit do we need to convert grams of Cu to in order to convert to atoms? What does Avogadro's number represent? 6.022 × 10 23 Pb atoms × Solution: To calculate the number of Cu atoms, we first must convert grams of Cu to moles of Cu. We use the molar mass of copper as a conversion factor. Once moles of Cu are obtained, we can use Avogadro's number to convert from moles of copper to atoms of copper. 1 mol Cu = 63.55 g Cu The conversion factor needed is 1 mol Cu 63.55 g Cu 38 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS Avogadro's number is the key to the second conversion. We have 1 mol = 6.022 × 10 23 particles (atoms) From this equality, we can write two conversion factors. 1 mol Cu and 6.022 × 1023 Cu atoms 6.022 × 1023 Cu atoms 1 mol Cu The conversion factor on the right is the one we need because it has number of Cu atoms in the numerator, which is the unit we want for the answer. Let's complete the two conversions in one step. grams of Cu → moles of Cu → number of Cu atoms ? atoms of Cu = 3.14 g Cu × 1 mol Cu 6.022 × 1023 Cu atoms × = 2.98 × 1022 Cu atoms 63.55 g Cu 1 mol Cu Check: Should 3.14 g of Cu contain fewer than Avogadro's number of atoms? What mass of Cu would contain Avogadro's number of atoms? 3.21 1 mol H 6.022 × 1023 H atoms × = 6.57 × 1023 H atoms 1.008 g H 1 mol H For hydrogen: 1.10 g H × For chromium: 14.7 g Cr × 1 mol Cr 6.022 × 1023 Cr atoms × = 1.70 × 1023 Cr atoms 52.00 g Cr 1 mol Cr There are more hydrogen atoms than chromium atoms. 3.22 2 Pb atoms × 1 mol Pb 6.022 × 10 (5.1 × 10−23 mol He) × 23 Pb atoms × 207.2 g Pb = 6.881 × 10−22 g Pb 1 mol Pb 4.003 g He = 2.0 × 10−22 g He 1 mol He 2 atoms of lead have a greater mass than 5.1 × 10 3.23 −23 mol of helium. Using the appropriate atomic masses, (a) (b) (c) (d) (e) (f) (g) CH4 NO2 SO3 C6H6 NaI K2SO4 Ca3(PO4)2 12.01 amu + 4(1.008 amu) = 16.04 amu 14.01 amu + 2(16.00 amu) = 46.01 amu 32.07 amu + 3(16.00 amu) = 80.07 amu 6(12.01 amu) + 6(1.008 amu) = 78.11 amu 22.99 amu + 126.9 amu = 149.9 amu 2(39.10 amu) + 32.07 amu + 4(16.00 amu) = 174.27 amu 3(40.08 amu) + 2(30.97 amu) + 8(16.00 amu) = 310.18 amu CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.24 39 Strategy: How do molar masses of different elements combine to give the molar mass of a compound? Solution: To calculate the molar mass of a compound, we need to sum all the molar masses of the elements in the molecule. For each element, we multiply its molar mass by the number of moles of that element in one mole of the compound. We find molar masses for the elements in the periodic table (inside front cover of the text). 3.25 (a) molar mass Li2CO3 = 2(6.941 g) + 12.01 g + 3(16.00 g) = 73.89 g (b) molar mass CS2 = 12.01 g + 2(32.07 g) = 76.15 g (c) molar mass CHCl3 = 12.01 g + 1.008 g + 3(35.45 g) = 119.37 g (d) molar mass C6H8O6 = 6(12.01 g) + 8(1.008 g) + 6(16.00 g) = 176.12 g (e) molar mass KNO3 = 39.10 g + 14.01 g + 3(16.00 g) = 101.11 g (f) molar mass Mg3N2 = 3(24.31 g) + 2(14.01 g) = 100.95 g To find the molar mass (g/mol), we simply divide the mass (in g) by the number of moles. 152 g = 409 g/mol 0.372 mol 3.26 Strategy: We are given grams of ethane and asked to solve for molecules of ethane. We cannot convert directly from grams ethane to molecules of ethane. What unit do we need to obtain first before we can convert to molecules? How should Avogadro's number be used here? Solution: To calculate number of ethane molecules, we first must convert grams of ethane to moles of ethane. We use the molar mass of ethane as a conversion factor. Once moles of ethane are obtained, we can use Avogadro's number to convert from moles of ethane to molecules of ethane. molar mass of C2H6 = 2(12.01 g) + 6(1.008 g) = 30.068 g The conversion factor needed is 1 mol C2 H 6 30.068 g C2 H 6 Avogadro's number is the key to the second conversion. We have 1 mol = 6.022 × 10 23 particles (molecules) From this equality, we can write the conversion factor: 6.022 × 10 23 ethane molecules 1 mol ethane Let's complete the two conversions in one step. grams of ethane → moles of ethane → number of ethane molecules ? molecules of C2 H 6 = 0.334 g C2 H 6 × = 6.69 × 10 21 1 mol C2 H 6 6.022 × 1023 C 2 H 6 molecules × 30.068 g C2 H 6 1 mol C 2 H 6 C2H6 molecules 40 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS Check: Should 0.334 g of ethane contain fewer than Avogadro's number of molecules? What mass of ethane would contain Avogadro's number of molecules? 3.27 1.50 g glucose × = 3.01 × 10 1 mol glucose 6.022 × 1023 molecules glucose 6 C atoms × × 180.2 g glucose 1 mol glucose 1 molecule glucose 22 C atoms The ratio of O atoms to C atoms in glucose is 1:1. Therefore, there are the same number of O atoms in 22 glucose as C atoms, so the number of O atoms = 3.01 × 10 O atoms. The ratio of H atoms to C atoms in glucose is 2:1. Therefore, there are twice as many H atoms in glucose as 22 22 C atoms, so the number of H atoms = 2(3.01 × 10 atoms) = 6.02 × 10 H atoms. 3.28 4 Strategy: We are asked to solve for the number of N, C, O, and H atoms in 1.68 × 10 g of urea. We cannot convert directly from grams urea to atoms. What unit do we need to obtain first before we can convert to atoms? How should Avogadro's number be used here? How many atoms of N, C, O, or H are in 1 molecule of urea? 4 Solution: Let's first calculate the number of N atoms in 1.68 × 10 g of urea. First, we must convert grams of urea to number of molecules of urea. This calculation is similar to Problem 3.26. The molecular formula of urea shows there are two N atoms in one urea molecule, which will allow us to convert to atoms of N. We need to perform three conversions: grams of urea → moles of urea → molecules of urea → atoms of N The conversion factors needed for each step are: 1) the molar mass of urea, 2) Avogadro's number, and 3) the number of N atoms in 1 molecule of urea. We complete the three conversions in one calculation. ? atoms of N = (1.68 × 104 g urea) × = 3.37 × 10 26 1 mol urea 6.022 × 1023 urea molecules 2 N atoms × × 60.062 g urea 1 mol urea 1 molecule urea N atoms The above method utilizes the ratio of molecules (urea) to atoms (nitrogen). We can also solve the problem by reading the formula as the ratio of moles of urea to moles of nitrogen by using the following conversions: grams of urea → moles of urea → moles of N → atoms of N Try it. 4 Check: Does the answer seem reasonable? We have 1.68 × 10 g urea. How many atoms of N would 60.06 g of urea contain? We could calculate the number of atoms of the remaining elements in the same manner, or we can use the atom ratios from the molecular formula. The carbon atom to nitrogen atom ratio in a urea molecule is 1:2, the oxygen atom to nitrogen atom ratio is 1:2, and the hydrogen atom to nitrogen atom ration is 4:2. ? atoms of C = (3.37 × 1026 N atoms) × 1 C atom = 1.69 × 1026 C atoms 2 N atoms CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.29 ? atoms of O = (3.37 × 1026 N atoms) × 1 O atom = 1.69 × 1026 O atoms 2 N atoms ? atoms of H = (3.37 × 1026 N atoms) × 4 H atoms = 6.74 × 1026 H atoms 2 N atoms The molar mass of C19H38O is 282.5 g. 1.0 × 10−12 g × 1 mol 6.022 × 1023 molecules × = 2.1 × 109 molecules 282.49 g 1 mol Notice that even though 1.0 × 10 pheromone molecules! 3.30 41 Mass of water = 2.56 mL × −12 g is an extremely small mass, it still is comprised of over a billion 1.00 g = 2.56 g 1.00 mL Molar mass of H2O = (16.00 g) + 2(1.008 g) = 18.016 g/mol ? H 2O molecules = 2.56 g H 2 O × = 8.56 × 10 3.33 22 1 mol H 2 O 6.022 × 10 23 molecules H 2 O × 18.016 g H 2 O 1 mol H 2 O molecules + Since there are only two isotopes of carbon, there are only two possibilities for CF4 . 12 19 + 6 C 9 F4 (molecular mass 88 amu) and 13 19 + 6 C 9 F4 (molecular mass 89 amu) There would be two peaks in the mass spectrum. 3.34 1 1 1 2 2 2 Since there are two hydrogen isotopes, they can be paired in three ways: H- H, H- H, and H- H. There will then be three choices for each sulfur isotope. We can make a table showing all the possibilities (masses in amu): 32 33 34 36 S S S S 1 H2 1 2 2 H H H2 34 35 36 35 36 37 36 37 38 38 39 40 There will be seven peaks of the following mass numbers: 34, 35, 36, 37, 38, 39, and 40. 1 Very accurate (and expensive!) mass spectrometers can detect the mass difference between two H and one 2 H. How many peaks would be detected in such a “high resolution” mass spectrum? 3.39 Molar mass of SnO2 = (118.7 g) + 2(16.00 g) = 150.7 g %Sn = 118.7 g/mol × 100% = 78.77% 150.7 g/mol %O = (2)(16.00 g/mol) × 100% = 21.23% 150.7 g/mol 42 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.40 Strategy: Recall the procedure for calculating a percentage. Assume that we have 1 mole of CHCl3. The percent by mass of each element (C, H, and Cl) is given by the mass of that element in 1 mole of CHCl3 divided by the molar mass of CHCl3, then multiplied by 100 to convert from a fractional number to a percentage. Solution: The molar mass of CHCl3 = 12.01 g/mol + 1.008 g/mol + 3(35.45 g/mol) = 119.4 g/mol. The percent by mass of each of the elements in CHCl3 is calculated as follows: %C = 12.01 g/mol × 100% = 10.06% 119.4 g/mol %H = 1.008 g/mol × 100% = 0.8442% 119.4 g/mol %Cl = 3(35.45) g/mol × 100% = 89.07% 119.4 g/mol Check: Do the percentages add to 100%? The sum of the percentages is (10.06% + 0.8442% + 89.07%) = 99.97%. The small discrepancy from 100% is due to the way we rounded off. 3.41 The molar mass of cinnamic alcohol is 134.17 g/mol. (a) (b) %C = (9)(12.01 g/mol) × 100% = 80.56% 134.17 g/mol %H = (10)(1.008 g/mol) × 100% = 7.51% 134.17 g/mol %O = 16.00 g/mol × 100% = 11.93% 134.17 g/mol 0.469 g C9 H10 O × = 2.11 × 10 3.42 21 1 mol C9 H10 O 6.022 × 1023 molecules C9 H10 O × 134.17 g C9 H10 O 1 mol C9 H10 O molecules C9H10O Compound Molar mass (g) N% by mass 2(14.01 g) × 100% = 46.65% 60.06 g (a) (NH2)2CO 60.06 (b) NH4NO3 80.05 2(14.01 g) × 100% = 35.00% 80.05 g (c) HNC(NH2)2 59.08 3(14.01 g) × 100% = 71.14% 59.08 g (d) NH3 17.03 14.01 g × 100% = 82.27% 17.03 g Ammonia, NH3, is the richest source of nitrogen on a mass percentage basis. CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.43 43 Assume you have exactly 100 g of substance. nC = 44.4 g C × 1 mol C = 3.697 mol C 12.01 g C nH = 6.21 g H × 1 mol H = 6.161 mol H 1.008 g H nS = 39.5 g S × 1 mol S = 1.232 mol S 32.07 g S nO = 9.86 g O × 1 mol O = 0.6163 mol O 16.00 g O Thus, we arrive at the formula C3.697H6.161S1.232O0.6163. Dividing by the smallest number of moles (0.6163 mole) gives the empirical formula, C6H10S2O. To determine the molecular formula, divide the molar mass by the empirical mass. molar mass 162 g = ≈ 1 empirical molar mass 162.28 g Hence, the molecular formula and the empirical formula are the same, C6H10S2O. 3.44 METHOD 1: Step 1: Assume you have exactly 100 g of substance. 100 g is a convenient amount, because all the percentages sum to 100%. The percentage of oxygen is found by difference: 100% − (19.8% + 2.50% + 11.6%) = 66.1% In 100 g of PAN there will be 19.8 g C, 2.50 g H, 11.6 g N, and 66.1 g O. Step 2: Calculate the number of moles of each element in the compound. Remember, an empirical formula tells us which elements are present and the simplest whole-number ratio of their atoms. This ratio is also a mole ratio. Use the molar masses of these elements as conversion factors to convert to moles. nC = 19.8 g C × 1 mol C = 1.649 mol C 12.01 g C nH = 2.50 g H × 1 mol H = 2.480 mol H 1.008 g H nN = 11.6 g N × 1 mol N = 0.8280 mol N 14.01 g N nO = 66.1 g O × 1 mol O = 4.131 mol O 16.00 g O Step 3: Try to convert to whole numbers by dividing all the subscripts by the smallest subscript. The formula is C1.649H2.480N0.8280O4.131. Dividing the subscripts by 0.8280 gives the empirical formula, C2H3NO5. To determine the molecular formula, remember that the molar mass/empirical mass will be an integer greater than or equal to one. molar mass ≥ 1 (integer values) empirical molar mass 44 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS In this case, molar mass 120 g = ≈ 1 empirical molar mass 121.05 g Hence, the molecular formula and the empirical formula are the same, C2H3NO5. METHOD 2: Step 1: Multiply the mass % (converted to a decimal) of each element by the molar mass to convert to grams of each element. Then, use the molar mass to convert to moles of each element. nC = (0.198) × (120 g) × 1 mol C = 1.98 mol C ≈ 2 mol C 12.01 g C nH = (0.0250) × (120 g) × 1 mol H = 2.98 mol H ≈ 3 mol H 1.008 g H nN = (0.116) × (120 g) × 1 mol N = 0.994 mol N ≈ 1 mol N 14.01 g N nO = (0.661) × (120 g) × 1 mol O = 4.96 mol O ≈ 5 mol O 16.00 g O Step 2: Since we used the molar mass to calculate the moles of each element present in the compound, this method directly gives the molecular formula. The formula is C2H3NO5. Step 3: Try to reduce the molecular formula to a simpler whole number ratio to determine the empirical formula. The formula is already in its simplest whole number ratio. The molecular and empirical formulas are the same. The empirical formula is C2H3NO5. 1 mol Fe 2 O3 2 mol Fe × = 0.308 mol Fe 159.7 g Fe2 O3 1 mol Fe 2 O3 3.45 24.6 g Fe2 O3 × 3.46 Using unit factors we convert: g of Hg → mol Hg → mol S → g S ? g S = 246 g Hg × 3.47 1 mol Hg 1 mol S 32.07 g S × × = 39.3 g S 200.6 g Hg 1 mol Hg 1 mol S → 2AlI3(s) The balanced equation is: 2Al(s) + 3I2(s) ⎯⎯ Using unit factors, we convert: g of Al → mol of Al → mol of I2 → g of I2 20.4 g Al × 3.48 3 mol I 2 253.8 g I 2 1 mol Al × × = 288 g I 2 26.98 g Al 2 mol Al 1 mol I2 Strategy: Tin(II) fluoride is composed of Sn and F. The mass due to F is based on its percentage by mass in the compound. How do we calculate mass percent of an element? Solution: First, we must find the mass % of fluorine in SnF2. Then, we convert this percentage to a fraction and multiply by the mass of the compound (24.6 g), to find the mass of fluorine in 24.6 g of SnF2. CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 45 The percent by mass of fluorine in tin(II) fluoride, is calculated as follows: mass of F in 1 mol SnF2 × 100% molar mass of SnF2 mass % F = = 2(19.00 g) × 100% = 24.25% F 156.7 g Converting this percentage to a fraction, we obtain 24.25/100 = 0.2425. Next, multiply the fraction by the total mass of the compound. ? g F in 24.6 g SnF2 = (0.2425)(24.6 g) = 5.97 g F Check: As a ball-park estimate, note that the mass percent of F is roughly 25 percent, so that a quarter of the mass should be F. One quarter of approximately 24 g is 6 g, which is close to the answer. Note: This problem could have been worked in a manner similar to Problem 3.46. You could complete the following conversions: g of SnF2 → mol of SnF2 → mol of F → g of F 3.49 In each case, assume 100 g of compound. (a) 2.1 g H × 1 mol H = 2.08 mol H 1.008 g H 65.3 g O × 1 mol O = 4.081 mol O 16.00 g O 32.6 g S × 1 mol S = 1.017 mol S 32.07 g S This gives the formula H2.08S1.017O4.081. Dividing by 1.017 gives the empirical formula, H2SO4. (b) 20.2 g Al × 1 mol Al = 0.7487 mol Al 26.98 g Al 79.8 g Cl × 1 mol Cl = 2.251 mol Cl 35.45 g Cl This gives the formula, Al0.7487Cl2.251. Dividing by 0.7487 gives the empirical formula, AlCl3. 3.50 (a) Strategy: In a chemical formula, the subscripts represent the ratio of the number of moles of each element that combine to form the compound. Therefore, we need to convert from mass percent to moles in order to determine the empirical formula. If we assume an exactly 100 g sample of the compound, do we know the mass of each element in the compound? How do we then convert from grams to moles? Solution: If we have 100 g of the compound, then each percentage can be converted directly to grams. In this sample, there will be 40.1 g of C, 6.6 g of H, and 53.3 g of O. Because the subscripts in the formula represent a mole ratio, we need to convert the grams of each element to moles. The conversion factor needed is the molar mass of each element. Let n represent the number of moles of each element so that 46 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS nC = 40.1 g C × 1 mol C = 3.339 mol C 12.01 g C nH = 6.6 g H × 1 mol H = 6.55 mol H 1.008 g H nO = 53.3 g O × 1 mol O = 3.331 mol O 16.00 g O Thus, we arrive at the formula C3.339H6.55O3.331, which gives the identity and the mole ratios of atoms present. However, chemical formulas are written with whole numbers. Try to convert to whole numbers by dividing all the subscripts by the smallest subscript (3.331). C: 3.339 ≈ 1 3.331 6.55 ≈ 2 3.331 H: O: 3.331 = 1 3.331 This gives the empirical formula, CH2O. Check: Are the subscripts in CH2O reduced to the smallest whole numbers? (b) Following the same procedure as part (a), we find: nC = 18.4 g C × 1 mol C = 1.532 mol C 12.01 g C nN = 21.5 g N × 1 mol N = 1.535 mol N 14.01 g N nK = 60.1 g K × 1 mol K = 1.537 mol K 39.10 g K Dividing by the smallest number of moles (1.532 mol) gives the empirical formula, KCN. 3.51 The molar mass of CaSiO3 is 116.17 g/mol. %Ca = 40.08 g = 34.50% 116.17 g %Si = 28.09 g = 24.18% 116.17 g %O = (3)(16.00 g) = 41.32% 116.17 g Check to see that the percentages sum to 100%. (34.50% + 24.18% + 41.32%) = 100.00% 3.52 The empirical molar mass of CH is approximately 13.018 g. Let's compare this to the molar mass to determine the molecular formula. Recall that the molar mass divided by the empirical mass will be an integer greater than or equal to one. CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 47 molar mass ≥ 1 (integer values) empirical molar mass In this case, molar mass 78 g = ≈ 6 empirical molar mass 13.018 g Thus, there are six CH units in each molecule of the compound, so the molecular formula is (CH)6, or C6H6. 3.53 Find the molar mass corresponding to each formula. For C4H5N2O: 4(12.01 g) + 5(1.008 g) + 2(14.01 g) + (16.00 g) = 97.10 g For C8H10N4O2: 8(12.01 g) + 10(1.008 g) + 4(14.01 g) + 2(16.00 g) = 194.20 g The molecular formula is C8H10N4O2. 3.54 METHOD 1: Step 1: Assume you have exactly 100 g of substance. 100 g is a convenient amount, because all the percentages sum to 100%. In 100 g of MSG there will be 35.51 g C, 4.77 g H, 37.85 g O, 8.29 g N, and 13.60 g Na. Step 2: Calculate the number of moles of each element in the compound. Remember, an empirical formula tells us which elements are present and the simplest whole-number ratio of their atoms. This ratio is also a mole ratio. Let nC, nH, nO, nN, and nNa be the number of moles of elements present. Use the molar masses of these elements as conversion factors to convert to moles. nC = 35.51 g C × 1 mol C = 2.9567 mol C 12.01 g C nH = 4.77 g H × 1 mol H = 4.732 mol H 1.008 g H nO = 37.85 g O × nN = 8.29 g N × 1 mol O = 2.3656 mol O 16.00 g O 1 mol N = 0.5917 mol N 14.01 g N nNa = 13.60 g Na × 1 mol Na = 0.59156 mol Na 22.99 g Na Thus, we arrive at the formula C2.9567H4.732O2.3656N0.5917Na0.59156, which gives the identity and the ratios of atoms present. However, chemical formulas are written with whole numbers. Step 3: Try to convert to whole numbers by dividing all the subscripts by the smallest subscript. 2.9567 = 4.9981 ≈ 5 0.59156 0.5917 N: = 1.000 0.59156 C: 4.732 = 7.999 ≈ 8 0.59156 0.59156 Na : = 1 0.59156 H: This gives us the empirical formula for MSG, C5H8O4NNa. O: 2.3656 = 3.9989 ≈ 4 0.59156 48 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS To determine the molecular formula, remember that the molar mass/empirical mass will be an integer greater than or equal to one. molar mass ≥ 1 (integer values) empirical molar mass In this case, molar mass 169 g = ≈ 1 empirical molar mass 169.11 g Hence, the molecular formula and the empirical formula are the same, C5H8O4NNa. It should come as no surprise that the empirical and molecular formulas are the same since MSG stands for monosodiumglutamate. METHOD 2: Step 1: Multiply the mass % (converted to a decimal) of each element by the molar mass to convert to grams of each element. Then, use the molar mass to convert to moles of each element. nC = (0.3551) × (169 g) × 1 mol C = 5.00 mol C 12.01 g C nH = (0.0477) × (169 g) × 1 mol H = 8.00 mol H 1.008 g H nO = (0.3785) × (169 g) × 1 mol O = 4.00 mol O 16.00 g O nN = (0.0829) × (169 g) × 1 mol N = 1.00 mol N 14.01 g N nNa = (0.1360) × (169 g) × 1 mol Na = 1.00 mol Na 22.99 g Na Step 2: Since we used the molar mass to calculate the moles of each element present in the compound, this method directly gives the molecular formula. The formula is C5H8O4NNa. 3.59 The balanced equations are as follows: (a) 2C + O2 → 2CO (h) N2 + 3H2 → 2NH3 (b) 2CO + O2 → 2CO2 (i) Zn + 2AgCl → ZnCl2 + 2Ag (c) H2 + Br2 → 2HBr (j) S8 + 8O2 → 8SO2 (d) 2K + 2H2O → 2KOH + H2 (k) 2NaOH + H2SO4 → Na2SO4 + 2H2O (e) 2Mg + O2 → 2MgO (l) Cl2 + 2NaI → 2NaCl + I2 (f) 2O3 → 3O2 (m) 3KOH + H3PO4 → K3PO4 + 3H2O (g) 2H2O2 → 2H2O + O2 (n) CH4 + 4Br2 → CBr4 + 4HBr CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.60 3.63 49 The balanced equations are as follows: (a) 2N2O5 → 2N2O4 + O2 (h) 2Al + 3H2SO4 → Al2(SO4)3 + 3H2 (b) 2KNO3 → 2KNO2 + O2 (i) CO2 + 2KOH → K2CO3 + H2O (c) NH4NO3 → N2O + 2H2O (j) CH4 + 2O2 → CO2 + 2H2O (d) NH4NO2 → N2 + 2H2O (k) Be2C + 4H2O → 2Be(OH)2 + CH4 (e) 2NaHCO3 → Na2CO3 + H2O + CO2 (l) 3Cu + 8HNO3 → 3Cu(NO3)2 + 2NO + 4H2O (f) P4O10 + 6H2O → 4H3PO4 (m) S + 6HNO3 → H2SO4 + 6NO2 + 2H2O (g) 2HCl + CaCO3 → CaCl2 + H2O + CO2 (n) 2NH3 + 3CuO → 3Cu + N2 + 3H2O On the reactants side there are 8 A atoms and 4 B atoms. On the products side, there are 4 C atoms and 4 D atoms. Writing an equation, 8A + 4B → 4C + 4D Chemical equations are typically written with the smallest set of whole number coefficients. Dividing the equation by four gives, 2A + B → C + D The correct answer is choice (c). 3.64 On the reactants side there are 6 A atoms and 4 B atoms. On the products side, there are 4 C atoms and 2 D atoms. Writing an equation, 6A + 4B → 4C + 2D Chemical equations are typically written with the smallest set of whole number coefficients. Dividing the equation by two gives, 3A + 2B → 2C + D The correct answer is choice (d). 3.65 The mole ratio from the balanced equation is 2 moles CO2 : 2 moles CO. 3.60 mol CO × 3.66 2 mol CO2 = 3.60 mol CO 2 2 mol CO ⎯→ SiCl4(l) Si(s) + 2Cl2(g) ⎯ Strategy: Looking at the balanced equation, how do we compare the amounts of Cl2 and SiCl4? We can compare them based on the mole ratio from the balanced equation. Solution: Because the balanced equation is given in the problem, the mole ratio between Cl2 and SiCl4 is known: 2 moles Cl2 Q 1 mole SiCl4. From this relationship, we have two conversion factors. 2 mol Cl2 1 mol SiCl4 and 1 mol SiCl4 2 mol Cl2 50 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS Which conversion factor is needed to convert from moles of SiCl4 to moles of Cl2? The conversion factor on the left is the correct one. Moles of SiCl4 will cancel, leaving units of "mol Cl2" for the answer. We calculate moles of Cl2 reacted as follows: ? mol Cl 2 reacted = 0.507 mol SiCl4 × 2 mol Cl2 = 1.01 mol Cl 2 1 mol SiCl4 Check: Does the answer seem reasonable? Should the moles of Cl2 reacted be double the moles of SiCl4 produced? 3.67 Starting with the amount of ammonia produced (6.0 moles), we can use the mole ratio from the balanced equation to calculate the moles of H2 and N2 that reacted to produce 6.0 moles of NH3. 3H2(g) + N2(g) → 2NH3(g) 3.68 ? mol H 2 = 6.0 mol NH3 × 3 mol H 2 = 9.0 mol H 2 2 mol NH3 ? mol N 2 = 6.0 mol NH3 × 1 mol N 2 = 3.0 mol N 2 2 mol NH3 Starting with the 5.0 moles of C4H10, we can use the mole ratio from the balanced equation to calculate the moles of CO2 formed. 2C4H10(g) + 13O2(g) → 8CO2(g) + 10H2O(l) ? mol CO2 = 5.0 mol C4 H10 × 3.69 It is convenient to use the unit ton-mol in this problem. We normally use a g-mol. 1 g-mol SO2 has a mass of 64.07 g. In a similar manner, 1 ton-mol of SO2 has a mass of 64.07 tons. We need to complete the following conversions: tons SO2 → ton-mol SO2 → ton-mol S → ton S. (2.6 × 107 tons SO 2 ) × 3.70 8 mol CO 2 = 20 mol CO2 = 2.0 × 101 mol CO 2 2 mol C4 H10 1 ton-mol SO 2 1 ton-mol S 32.07 ton S × × = 1.3 × 107 tons S 64.07 ton SO 2 1 ton-mol SO 2 1 ton-mol S (a) ⎯→ Na2CO3 + H2O + CO2 2NaHCO3 ⎯ (b) Molar mass NaHCO3 = 22.99 g + 1.008 g + 12.01 g + 3(16.00 g) = 84.008 g Molar mass CO2 = 12.01 g + 2(16.00 g) = 44.01 g The balanced equation shows one mole of CO2 formed from two moles of NaHCO3. mass NaHCO 3 = 20.5 g CO 2 × 2 mol NaHCO3 84.008 g NaHCO3 1 mol CO 2 × × 44.01 g CO 2 1 mol CO 2 1 mol NaHCO3 = 78.3 g NaHCO3 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.71 The balanced equation shows a mole ratio of 1 mole HCN : 1 mole KCN. 0.140 g KCN × 3.72 51 1 mol KCN 1 mol HCN 27.03 g HCN × × = 0.0581 g HCN 65.12 g KCN 1 mol KCN 1 mol HCN ⎯→ 2C2H5OH + 2CO2 C6H12O6 ⎯ glucose ethanol Strategy: We compare glucose and ethanol based on the mole ratio in the balanced equation. Before we can determine moles of ethanol produced, we need to convert to moles of glucose. What conversion factor is needed to convert from grams of glucose to moles of glucose? Once moles of ethanol are obtained, another conversion factor is needed to convert from moles of ethanol to grams of ethanol. Solution: The molar mass of glucose will allow us to convert from grams of glucose to moles of glucose. The molar mass of glucose = 6(12.01 g) + 12(1.008 g) + 6(16.00 g) = 180.16 g. The balanced equation is given, so the mole ratio between glucose and ethanol is known; that is 1 mole glucose Q 2 moles ethanol. Finally, the molar mass of ethanol will convert moles of ethanol to grams of ethanol. This sequence of three conversions is summarized as follows: grams of glucose → moles of glucose → moles of ethanol → grams of ethanol ? g C2 H 5OH = 500.4 g C6 H12 O6 × 1 mol C6 H12 O6 2 mol C2 H5 OH 46.068 g C2 H5OH × × 180.16 g C6 H12 O6 1 mol C6 H12 O6 1 mol C2 H5OH = 255.9 g C2H5OH Check: Does the answer seem reasonable? Should the mass of ethanol produced be approximately half the mass of glucose reacted? Twice as many moles of ethanol are produced compared to the moles of glucose reacted, but the molar mass of ethanol is about one-fourth that of glucose. The liters of ethanol can be calculated from the density and the mass of ethanol. volume = mass density Volume of ethanol obtained = 3.73 255.9 g = 324 mL = 0.324 L 0.789 g/mL The mass of water lost is just the difference between the initial and final masses. Mass H2O lost = 15.01 g − 9.60 g = 5.41 g moles of H 2 O = 5.41 g H 2 O × 3.74 1 mol H 2 O = 0.300 mol H 2 O 18.016 g H 2 O The balanced equation shows that eight moles of KCN are needed to combine with four moles of Au. ? mol KCN = 29.0 g Au × 1 mol Au 8 mol KCN × = 0.294 mol KCN 197.0 g Au 4 mol Au 52 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.75 The balanced equation is: 1.0 kg CaCO3 × 3.76 CaCO3(s) ⎯⎯ → CaO(s) + CO2(g) 1 mol CaCO3 1000 g 1 mol CaO 56.08 g CaO × × × = 5.6 × 102 g CaO 1 kg 100.09 g CaCO3 1 mol CaCO3 1 mol CaO (a) ⎯→ N2O(g) + 2H2O(g) NH4NO3(s) ⎯ (b) Starting with moles of NH4NO3, we can use the mole ratio from the balanced equation to find moles of N2O. Once we have moles of N2O, we can use the molar mass of N2O to convert to grams of N2O. Combining the two conversions into one calculation, we have: mol NH4NO3 → mol N2O → g N2O ? g N 2 O = 0.46 mol NH 4 NO3 × 3.77 1 mol N 2 O 44.02 g N 2 O × = 2.0 × 101 g N 2O 1 mol NH 4 NO3 1 mol N 2 O The quantity of ammonia needed is: 2 mol NH3 17.034 g NH3 1 mol (NH 4 )2 SO4 1 kg × × × 132.15 g (NH 4 )2 SO 4 1 mol (NH 4 )2 SO 4 1 mol NH3 1000 g 4 = 2.58 × 10 kg NH3 1.00 × 108 g (NH 4 )2 SO4 × 3.78 The balanced equation for the decomposition is : 2KClO3(s) ⎯ ⎯→ 2KCl(s) + 3O2(g) ? g O 2 = 46.0 g KClO3 × 3.81 1 mol KClO3 3 mol O2 32.00 g O2 × × = 18.0 g O 2 122.55 g KClO3 2 mol KClO3 1 mol O2 2A + B → C (a) The number of B atoms shown in the diagram is 5. The balanced equation shows 2 moles A Q 1 mole B. Therefore, we need 10 atoms of A to react completely with 5 atoms of B. There are only 8 atoms of A present in the diagram. There are not enough atoms of A to react completely with B. A is the limiting reagent. (b) There are 8 atoms of A. Since the mole ratio between A and B is 2:1, 4 atoms of B will react with 8 atoms of A, leaving 1 atom of B in excess. The mole ratio between A and C is also 2:1. When 8 atoms of A react, 4 molecules of C will be produced. B C CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.82 53 N2 + 3H2 → 2NH3 (a) The number of N2 molecules shown in the diagram is 3. The balanced equation shows 3 moles H2 Q 1 mole N2. Therefore, we need 9 molecules of H2 to react completely with 3 molecules of N2. There are 10 molecules of H2 present in the diagram. H2 is in excess. N2 is the limiting reagent. (b) 9 molecules of H2 will react with 3 molecules of N2, leaving 1 molecule of H2 in excess. The mole ratio between N2 and NH3 is 1:2. When 3 molecules of N2 react, 6 molecules of NH3 will be produced. H2 NH3 3.83 This is a limiting reagent problem. Let's calculate the moles of NO2 produced assuming complete reaction for each reactant. 2NO(g) + O2(g) → 2NO2(g) 0.886 mol NO × 0.503 mol O2 × 2 mol NO2 = 0.886 mol NO 2 2 mol NO 2 mol NO2 = 1.01 mol NO 2 1 mol O2 NO is the limiting reagent; it limits the amount of product produced. The amount of product produced is 0.886 mole NO2. 3.84 Strategy: Note that this reaction gives the amounts of both reactants, so it is likely to be a limiting reagent problem. The reactant that produces fewer moles of product is the limiting reagent because it limits the amount of product that can be produced. How do we convert from the amount of reactant to amount of product? Perform this calculation for each reactant, then compare the moles of product, NO2, formed by the given amounts of O3 and NO to determine which reactant is the limiting reagent. Solution: We carry out two separate calculations. First, starting with 0.740 g O3, we calculate the number of moles of NO2 that could be produced if all the O3 reacted. We complete the following conversions. grams of O3 → moles of O3 → moles of NO2 Combining these two conversions into one calculation, we write ? mol NO 2 = 0.740 g O3 × 1 mol O3 1 mol NO2 × = 0.01542 mol NO 2 48.00 g O3 1 mol O3 54 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS Second, starting with 0.670 g of NO, we complete similar conversions. grams of NO → moles of NO → moles of NO2 Combining these two conversions into one calculation, we write ? mol NO 2 = 0.670 g NO × 1 mol NO2 1 mol NO × = 0.02233 mol NO 2 30.01 g NO 1 mol NO The initial amount of O3 limits the amount of product that can be formed; therefore, it is the limiting reagent. The problem asks for grams of NO2 produced. We already know the moles of NO2 produced, 0.01542 mole. Use the molar mass of NO2 as a conversion factor to convert to grams (Molar mass NO2 = 46.01 g). ? g NO 2 = 0.01542 mol NO 2 × 46.01 g NO2 = 0.709 g NO 2 1 mol NO 2 Check: Does your answer seem reasonable? 0.01542 mole of product is formed. What is the mass of 1 mole of NO2? Strategy: Working backwards, we can determine the amount of NO that reacted to produce 0.01542 mole of NO2. The amount of NO left over is the difference between the initial amount and the amount reacted. Solution: Starting with 0.01542 mole of NO2, we can determine the moles of NO that reacted using the mole ratio from the balanced equation. We can calculate the initial moles of NO starting with 0.670 g and using molar mass of NO as a conversion factor. mol NO reacted = 0.01542 mol NO2 × mol NO initial = 0.670 g NO × 1 mol NO = 0.01542 mol NO 1 mol NO2 1 mol NO = 0.02233 mol NO 30.01 g NO mol NO remaining = mol NO initial − mol NO reacted. mol NO remaining = 0.02233 mol NO − 0.01542 mol NO = 0.0069 mol NO 3.85 3.86 C3H8(g) + 5O2(g) ⎯⎯ → 3CO2(g) + 4H2O(l) (a) The balanced equation is: (b) The balanced equation shows a mole ratio of 3 moles CO2 : 1 mole C3H8. The mass of CO2 produced is: 3 mol CO 2 44.01 g CO 2 3.65 mol C3H8 × × = 482 g CO 2 1 mol C3 H8 1 mol CO 2 This is a limiting reagent problem. Let's calculate the moles of Cl2 produced assuming complete reaction for each reactant. 0.86 mol MnO2 × 48.2 g HCl × 1 mol Cl2 = 0.86 mol Cl2 1 mol MnO2 1 mol Cl2 1 mol HCl = 0.3305 mol Cl2 × 36.458 g HCl 4 mol HCl CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 55 HCl is the limiting reagent; it limits the amount of product produced. It will be used up first. The amount of product produced is 0.3305 mole Cl2. Let's convert this to grams. ? g Cl 2 = 0.3305 mol Cl2 × 3.89 The balanced equation is given: 70.90 g Cl2 = 23.4 g Cl 2 1 mol Cl2 CaF2 + H2SO4 ⎯⎯ → CaSO4 + 2HF The balanced equation shows a mole ratio of 2 moles HF : 1 mole CaF2. The theoretical yield of HF is: (6.00 × 103 g CaF2 ) × 1 mol CaF2 2 mol HF 20.008 g HF 1 kg × × × = 3.075 kg HF 78.08 g CaF2 1 mol CaF2 1 mol HF 1000 g The actual yield is given in the problem (2.86 kg HF). 3.90 (a) % yield = actual yield × 100% theoretical yield % yield = 2.86 kg × 100% = 93.0% 3.075 kg Start with a balanced chemical equation. It’s given in the problem. We use NG as an abbreviation for nitroglycerin. The molar mass of NG = 227.1 g/mol. 4C3H5N3O9 ⎯ ⎯→ 6N2 + 12CO2 + 10H2O + O2 Map out the following strategy to solve this problem. g NG → mol NG → mol O2 → g O2 Calculate the grams of O2 using the strategy above. ? g O 2 = 2.00 × 102 g NG × (b) 3.91 1 mol O2 32.00 g O 2 1 mol NG × × = 7.05 g O 2 227.1 g NG 4 mol NG 1 mol O 2 The theoretical yield was calculated in part (a), and the actual yield is given in the problem (6.55 g). The percent yield is: % yield = actual yield × 100% theoretical yield % yield = 6.55 g O 2 × 100% = 92.9% 7.05 g O 2 The balanced equation shows a mole ratio of 1 mole TiO2 : 1 mole FeTiO3. The molar mass of FeTiO3 is 151.73 g/mol, and the molar mass of TiO2 is 79.88 g/mol. The theoretical yield of TiO2 is: 8.00 × 106 g FeTiO3 × 3 1 mol FeTiO3 1 mol TiO2 79.88 g TiO2 1 kg × × × 151.73 g FeTiO3 1 mol FeTiO3 1 mol TiO2 1000 g = 4.21 × 10 kg TiO2 56 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3 The actual yield is given in the problem (3.67 × 10 kg TiO2). % yield = 3.92 actual yield 3.67 × 103 kg × 100% = × 100% = 87.2% theoretical yield 4.21 × 103 kg The actual yield of ethylene is 481 g. Let’s calculate the yield of ethylene if the reaction is 100 percent efficient. We can calculate this from the definition of percent yield. We can then calculate the mass of hexane that must be reacted. % yield = actual yield × 100% theoretical yield 42.5% yield = 481 g C2 H 4 × 100% theoretical yield 3 theoretical yield C2H4 = 1.132 × 10 g C2H4 The mass of hexane that must be reacted is: (1.132 × 103 g C2 H 4 ) × 3.93 1 mol C6 H14 86.172 g C6 H14 1 mol C2 H 4 × × = 3.48 × 103 g C6 H14 28.052 g C2 H 4 1 mol C2 H 4 1 mol C6 H14 This is a limiting reagent problem. Let's calculate the moles of Li3N produced assuming complete reaction for each reactant. 6Li(s) + N2(g) → 2Li3N(s) 12.3 g Li × 2 mol Li3 N 1 mol Li × = 0.5907 mol Li3 N 6.941 g Li 6 mol Li 33.6 g N 2 × 2 mol Li3 N 1 mol N 2 × = 2.398 mol Li3 N 28.02 g N 2 1 mol N 2 Li is the limiting reagent; it limits the amount of product produced. The amount of product produced is 0.5907 mole Li3N. Let's convert this to grams. ? g Li3 N = 0.5907 mol Li3 N × 34.833 g Li3 N = 20.6 g Li 3 N 1 mol Li3 N This is the theoretical yield of Li3N. The actual yield is given in the problem (5.89 g). The percent yield is: % yield = 3.94 actual yield 5.89 g × 100% = × 100% = 28.6% theoretical yield 20.6 g This is a limiting reagent problem. Let's calculate the moles of S2Cl2 produced assuming complete reaction for each reactant. S8(l) + 4Cl2(g) → 4S2Cl2(l) 4.06 g S8 × 1 mol S8 4 mol S2 Cl2 × = 0.0633 mol S2 Cl2 256.56 g S8 1 mol S8 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 6.24 g Cl2 × 57 1 mol Cl2 4 mol S2 Cl2 × = 0.0880 mol S2 Cl2 70.90 g Cl2 4 mol Cl2 S8 is the limiting reagent; it limits the amount of product produced. The amount of product produced is 0.0633 mole S2Cl2. Let's convert this to grams. ? g S2 Cl2 = 0.0633 mol S2 Cl2 × 135.04 g S2 Cl2 = 8.55 g S 2Cl 2 1 mol S2 Cl2 This is the theoretical yield of S2Cl2. The actual yield is given in the problem (6.55 g). The percent yield is: % yield = 3.95 actual yield 6.55 g × 100% = × 100% = 76.6% theoretical yield 8.55 g All the carbon from the hydrocarbon reactant ends up in CO2, and all the hydrogen from the hydrocarbon reactant ends up in water. In the diagram, we find 4 CO2 molecules and 6 H2O molecules. This gives a ratio between carbon and hydrogen of 4:12. We write the formula C4H12, which reduces to the empirical formula CH3. The empirical molar mass equals approximately 15 g, which is half the molar mass of the hydrocarbon. Thus, the molecular formula is double the empirical formula or C2H6. Since this is a combustion reaction, the other reactant is O2. We write: C2H6 + O2 → CO2 + H2O Balancing the equation, 2C2H6 + 7O2 → 4CO2 + 6H2O 3.96 2H2(g) + O2(g) → 2H2O(g) We start with 8 molecules of H2 and 3 molecules of O2. The balanced equation shows 2 moles H2 Q 1 mole O2. If 3 molecules of O2 react, 6 molecules of H2 will react, leaving 2 molecules of H2 in excess. The balanced equation also shows 1 mole O2 Q 2 moles H2O. If 3 molecules of O2 react, 6 molecules of H2O will be produced. After complete reaction, there will be 2 molecules of H2 and 6 molecules of H2O. The correct diagram is choice (b). 3.97 First, let's convert to moles of HNO3 produced. 1.00 ton HNO3 × 1 mol HNO3 2000 lb 453.6 g × × = 1.44 × 104 mol HNO3 1 ton 1 1b 63.018 g HNO3 Now, we will work in the reverse direction to calculate the amount of reactant needed to produce 1.44 × 10 mol of HNO3. Realize that since the problem says to assume an 80% yield for each step, the amount of 100% , compared to a standard stoichiometry reactant needed in each step will be larger by a factor of 80% calculation where a 100% yield is assumed. 3 58 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS Referring to the balanced equation in the last step, we calculate the moles of NO2. (1.44 × 104 mol HNO3 ) × 2 mol NO2 100% × = 3.60 × 104 mol NO 2 1 mol HNO3 80% 4 Now, let's calculate the amount of NO needed to produce 3.60 × 10 mol NO2. Following the same procedure as above, and referring to the balanced equation in the middle step, we calculate the moles of NO. (3.60 × 104 mol NO 2 ) × 1 mol NO 100% × = 4.50 × 104 mol NO 1 mol NO 2 80% 4 Now, let's calculate the amount of NH3 needed to produce 4.5 × 10 mol NO. Referring to the balanced equation in the first step, the moles of NH3 is: (4.50 × 104 mol NO) × 4 mol NH3 100% × = 5.625 × 104 mol NH3 4 mol NO 80% Finally, converting to grams of NH3: 5.625 × 104 mol NH3 × 3.98 17.034 g NH3 = 9.58 × 105 g NH 3 1 mol NH3 We assume that all the Cl in the compound ends up as HCl and all the O ends up as H2O. Therefore, we need to find the number of moles of Cl in HCl and the number of moles of O in H2O. mol Cl = 0.233 g HCl × 1 mol HCl 1 mol Cl × = 0.006391 mol Cl 36.458 g HCl 1 mol HCl mol O = 0.403 g H 2 O × 1 mol H 2 O 1 mol O × = 0.02237 mol O 18.016 g H 2 O 1 mol H 2 O Dividing by the smallest number of moles (0.006391 mole) gives the formula, ClO3.5. Multiplying both subscripts by two gives the empirical formula, Cl2O7. 3.99 The number of moles of Y in 84.10 g of Y is: 27.22 g X × 1 mol X 1 mol Y × = 0.81448 mol Y 33.42 g X 1 mol X The molar mass of Y is: molar mass Y = 84.10 g Y = 103.3 g/mol 0.81448 mol Y The atomic mass of Y is 103.3 amu. 3.100 The symbol “O” refers to moles of oxygen atoms, not oxygen molecule (O2). Look at the molecular formulas given in parts (a) and (b). What do they tell you about the relative amounts of carbon and oxygen? (a) 0.212 mol C × (b) 0.212 mol C × 1 mol O = 0.212 mol O 1 mol C 2 mol O = 0.424 mol O 1 mol C CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 59 3.101 The observations mean either that the amount of the more abundant isotope was increasing or the amount of the less abundant isotope was decreasing. One possible explanation is that the less abundant isotope was undergoing radioactive decay, and thus its mass would decrease with time. 3.102 This is a calculation involving percent composition. Remember, percent by mass of each element = mass of element in 1 mol of compound × 100% molar mass of compound The molar masses are: Al, 26.98 g/mol; Al2(SO4)3, 342.17 g/mol; H2O, 18.016 g/mol. Thus, using x as the number of H2O molecules, ⎛ ⎞ 2(molar mass of Al) mass % Al = ⎜ ⎟ × 100% ⎝ molar mass of Al2 (SO4 )3 + x (molar mass of H 2 O) ⎠ ⎛ ⎞ 2(26.98 g) 8.20% = ⎜ ⎟ × 100% 342.17 g (18.016 g) + x ⎝ ⎠ x = 17.53 Rounding off to a whole number of water molecules, x = 18. Therefore, the formula is Al2(SO4)3⋅18 H2O. 3.103 Molar mass of C4H8Cl2S = 4(12.01 g) + 8(1.008 g) + 2(35.45 g) + 32.07 g = 159.07 g %C = 4(12.01 g/mol) × 100% = 30.20% 159.07 g/mol %H = 8(1.008 g/mol) × 100% = 5.069% 159.07 g/mol %Cl = 2(35.45 g/mol) × 100% = 44.57% 159.07 g/mol %S = 3.104 The number of carbon atoms in a 24-carat diamond is: 24 carat × 3.105 32.07 g/mol × 100% = 20.16% 159.07 g/mol 200 mg C 0.001 g C 1 mol C 6.022 × 1023 atoms C = 2.4 × 1023 atoms C × × × 1 carat 1 mg C 12.01 g C 1 mol C The amount of Fe that reacted is: 1 × 664 g = 83.0 g reacted 8 The amount of Fe remaining is: 664 g − 83.0 g = 581 g remaining Thus, 83.0 g of Fe reacts to form the compound Fe2O3, which has two moles of Fe atoms per 1 mole of compound. The mass of Fe2O3 produced is: 83.0 g Fe × 1 mol Fe2 O3 159.7 g Fe2 O3 1 mol Fe × × = 119 g Fe2O 3 55.85 g Fe 2 mol Fe 1 mol Fe2 O3 The final mass of the iron bar and rust is: 581 g Fe + 119 g Fe2O3 = 700 g 60 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.106 The mass of oxygen in MO is 39.46 g − 31.70 g = 7.76 g O. Therefore, for every 31.70 g of M, there is 7.76 g of O in the compound MO. The molecular formula shows a mole ratio of 1 mole M : 1 mole O. First, calculate moles of M that react with 7.76 g O. mol M = 7.76 g O × molar mass M = 1 mol O 1 mol M × = 0.485 mol M 16.00 g O 1 mol O 31.70 g M = 65.4 g/mol 0.485 mol M Thus, the atomic mass of M is 65.4 amu. The metal is most likely Zn. 3.107 (a) Zn(s) + H2SO4(aq) ⎯⎯ → ZnSO4(aq) + H2(g) (b) We assume that a pure sample would produce the theoretical yield of H2. The balanced equation shows a mole ratio of 1 mole H2 : 1 mole Zn. The theoretical yield of H2 is: 3.86 g Zn × 1 mol H 2 2.016 g H 2 1 mol Zn × × = 0.119 g H 2 65.39 g Zn 1 mol Zn 1 mol H 2 percent purity = (c) 3.108 0.0764 g H 2 × 100% = 64.2% 0.119 g H 2 We assume that the impurities are inert and do not react with the sulfuric acid to produce hydrogen. The wording of the problem suggests that the actual yield is less than the theoretical yield. The percent yield will be equal to the percent purity of the iron(III) oxide. We find the theoretical yield : (2.62 × 103 kg Fe 2 O3 ) × 3 1000 g Fe2 O3 1 mol Fe2 O3 2 mol Fe 55.85 g Fe 1 kg Fe × × × × 1 kg Fe2 O3 159.7 g Fe2 O3 1 mol Fe 2 O3 1 mol Fe 1000 g Fe = 1.833 × 10 kg Fe percent yield = percent yield = 3.109 actual yield × 100% theoretical yield 1.64 × 103 kg Fe 1.833 × 103 kg Fe × 100% = 89.5% = purity of Fe 2O 3 The balanced equation is: C6H12O6 + 6O2 ⎯⎯ → 6CO2 + 6H2O 6 mol CO 2 44.01 g CO2 365 days 5.0 × 102 g glucose 1 mol glucose × × × × × (6.5 × 109 people) 1 day 180.16 g glucose 1 mol glucose 1 mol CO2 1 yr = 1.7 × 10 15 g CO2/yr CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.110 The carbohydrate contains 40 percent carbon; therefore, the remaining 60 percent is hydrogen and oxygen. The problem states that the hydrogen to oxygen ratio is 2:1. We can write this 2:1 ratio as H2O. Assume 100 g of compound. 40.0 g C × 1 mol C = 3.331 mol C 12.01 g C 60.0 g H 2 O × 1 mol H 2 O = 3.330 mol H 2 O 18.016 g H 2 O Dividing by 3.330 gives CH2O for the empirical formula. To find the molecular formula, divide the molar mass by the empirical mass. molar mass 178 g = ≈ 6 empirical mass 30.026 g Thus, there are six CH2O units in each molecule of the compound, so the molecular formula is (CH2O)6, or C6H12O6. 3.111 The molar mass of chlorophyll is 893.48 g/mol. Finding the mass of a 0.0011-mol sample: 0.0011 mol chlorophyll × 893.48 g chlorophyll = 0.98 g chlorophyll 1 mol chlorophyll The chlorophyll sample has the greater mass. 3.112 If we assume 100 g of compound, the masses of Cl and X are 67.2 g and 32.8 g, respectively. We can calculate the moles of Cl. 1 mol Cl 67.2 g Cl × = 1.896 mol Cl 35.45 g Cl Then, using the mole ratio from the chemical formula (XCl3), we can calculate the moles of X contained in 32.8 g. 1 mol X 1.896 mol Cl × = 0.6320 mol X 3 mol Cl 0.6320 mole of X has a mass of 32.8 g. Calculating the molar mass of X: 32.8 g X = 51.9 g/mol 0.6320 mol X The element is most likely chromium (molar mass = 52.00 g/mol). 3.113 (a) The molar mass of hemoglobin is: 2952(12.01 g) + 4664(1.008 g) + 812(14.01 g) + 832(16.00 g) + 8(32.07 g) + 4(55.85 g) 4 = 6.532 × 10 g (b) To solve this problem, the following conversions need to be completed: L → mL → red blood cells → hemoglobin molecules → mol hemoglobin → mass hemoglobin 61 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 62 We will use the following abbreviations: RBC = red blood cells, HG = hemoglobin 5.00 L × 1 mL 5.0 × 109 RBC 2.8 × 108 HG molecules 1 mol HG 6.532 × 104 g HG × × × × 0.001 L 1 mL 1 RBC 1 mol HG 6.022 × 1023 molecules HG 2 = 7.6 × 10 g HG 3.114 A 100 g sample of myoglobin contains 0.34 g of iron (0.34% Fe). The number of moles of Fe is: 0.34 g Fe × 1 mol Fe = 6.09 × 10−3 mol Fe 55.85 g Fe Since there is one Fe atom in a molecule of myoglobin, the moles of myoglobin also equal 6.09 × 10 The molar mass of myoglobin can be calculated. molar mass myoglobin = 3.115 (a) 8.38 g KBr × 100 g myoglobin 6.09 × 10 −3 mol myoglobin 1 mol KBr 6.022 × 1023 KBr 1 K + ion × × = 4.24 × 1022 K + ions 119.0 g KBr 1 mol KBr 1 KBr − + − 5.40 g Na 2SO 4 × + 7.45 g Ca 3 (PO 4 )2 × = 4.34 × 10 − Br ions 2− 2− ions = 2.29 × 10 22 SO4 2− ions 1 mol Ca 3 (PO 4 )2 6.022 × 1023 Ca 3 (PO4 ) 2 3 Ca 2+ ions × × 310.18 g Ca 3 (PO4 ) 2 1 mol Ca 3 (PO4 )2 1 Ca 3 (PO4 )2 22 Ca Since there are three Ca 2+ 4.34 × 1022 Ca 2+ ions × 3.116 22 1 mol Na 2SO 4 6.022 × 1023 Na 2SO4 2 Na + ions × × = 4.58 × 1022 Na + ions 142.05 g Na 2SO 4 1 mol Na 2SO 4 1 Na 2SO4 Since there are two Na for every one SO4 , the number of SO4 (c) mole. = 1.6 × 104 g/mol Since there is one Br for every one K , the number of Br ions = 4.24 × 10 (b) −3 2+ ions 3− for every two PO4 , the number of PO4 2 PO34− ions 3 Ca 2+ ions 3− ions is: = 2.89 × 1022 PO43− ions If we assume 100 g of the mixture, then there are 29.96 g of Na in the mixture (29.96% Na by mass). This amount of Na is equal to the mass of Na in NaBr plus the mass of Na in Na2SO4. 29.96 g Na = mass of Na in NaBr + mass of Na in Na2SO4 To calculate the mass of Na in each compound, grams of compound need to be converted to grams of Na using the mass percentage of Na in the compound. If x equals the mass of NaBr, then the mass of Na2SO4 is 100 −x. Recall that we assumed 100 g of the mixture. We set up the following expression and solve for x. 29.96 g Na = mass of Na in NaBr + mass of Na in Na2SO4 ⎡ 22.99 g Na ⎤ ⎡ (2)(22.99 g Na) ⎤ 29.96 g Na = ⎢ x g NaBr × ⎥ ⎥ + ⎢(100 − x) g Na 2SO 4 × 102.89 g NaBr ⎦ ⎣ 142.05 g Na 2SO 4 ⎦ ⎣ CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 63 29.96 = 0.22344x + 32.369 − 0.32369x 0.10025x = 2.409 x = 24.03 g, which equals the mass of NaBr. The mass of Na2SO4 is 100 − x which equals 75.97 g. Because we assumed 100 g of compound, the mass % of NaBr in the mixture is 24.03% and the mass % of Na2SO4 is 75.97%. 3.117 1 mol aspirin 1 mol salicylic acid 138.12 g salicylic acid × × = 0.307 g salicylic acid 180.15 g aspirin 1 mol aspirin 1 mol salicylic acid (a) 0.400 g aspirin × (b) 0.307 g salicylic acid × 1 = 0.410 g salicylic acid 0.749 If you have trouble deciding whether to multiply or divide by 0.749 in the calculation, remember that if only 74.9% of salicylic acid is converted to aspirin, a larger amount of salicylic acid will need to be reacted to yield the same amount of aspirin. (c) 9.26 g salicylic acid × 1 mol salicylic acid 1 mol aspirin × = 0.06704 mol aspirin 138.12 g salicylic acid 1 mol salicylic acid 8.54 g acetic anhydride × 1 mol acetic anhydride 1 mol aspirin × = 0.08365 mol aspirin 102.09 g acetic anhydride 1 mol acetic anhydride The limiting reagent is salicylic acid. The theoretical yield of aspirin is: 0.06704 mol aspirin × 180.15 g aspirin = 12.1 g aspirin 1 mol aspirin The percent yield is: % yield = 3.118 10.9 g × 100% = 90.1% 12.1 g The mass percent of an element in a compound can be calculated as follows: percent by mass of each element = mass of element in 1 mol of compound × 100% molar mass of compound The molar mass of Ca3(PO4)2 = 310.18 g/mol % Ca = (3)(40.08 g) × 100% = 38.76% Ca 310.18 g %P = (2)(30.97 g) × 100% = 19.97% P 310.18 g %O = (8)(16.00 g) × 100% = 41.27% O 310.18 g 64 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.119 (a) First, calculate the mass of C in CO2, the mass of H in H2O, and the mass of N in NH3. For now, we will carry more than 3 significant figures and then round to the correct number at the end of the problem. ? g C = 3.94 g CO 2 × 1 mol CO 2 1 mol C 12.01 g C × × = 1.075 g C 44.01 g CO 2 1 mol CO2 1 mol C ? g H = 1.89 g H 2 O × 1 mol H 2 O 2 mol H 1.008 g H × × = 0.2114 g H 18.02 g H 2 O 1 mol H 2 O 1 mol H ? g N = 0.436 g NH3 × 1 mol NH3 1 mol N 14.01 g N × × = 0.3587 g N 17.03 g NH3 1 mol NH3 1 mol N Next, we can calculate the %C, %H, and the %N in each sample, then we can calculate the %O by difference. 1.075 g C %C = × 100% = 49.43% C 2.175 g sample %H = 0.2114 g H × 100% = 9.720% H 2.175 g sample %N = 0.3587 g N × 100% = 19.15% N 1.873 g sample The % O = 100% − (49.43% + 9.720% + 19.15%) = 21.70% O Assuming 100 g of compound, calculate the moles of each element. ? mol C = 49.43 g C × 1 mol C = 4.116 mol C 12.01 g C ? mol H = 9.720 g H × 1 mol H = 9.643 mol H 1.008 g H ? mol N = 19.15 g N × 1 mol N = 1.367 mol N 14.01 g N ? mol O = 21.70 g O × 1 mol O = 1.356 mol O 16.00 g O Thus, we arrive at the formula C4.116H9.643N1.367O1.356. Dividing by 1.356 gives the empirical formula, C3H7NO. (b) The empirical molar mass is 73.10 g. Since the approximate molar mass of lysine is 150 g, we have: 150 g ≈ 2 73.10 g Therefore, the molecular formula is (C3H7NO)2 or C6H14N2O2. 3.120 Yes. The number of hydrogen atoms in one gram of hydrogen molecules is the same as the number in one gram of hydrogen atoms. There is no difference in mass, only in the way that the particles are arranged. Would the mass of 100 dimes be the same if they were stuck together in pairs instead of separated? CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.121 65 The mass of one fluorine atom is 19.00 amu. The mass of one mole of fluorine atoms is 19.00 g. Multiplying the mass of one atom by Avogadro’s number gives the mass of one mole of atoms. We can write: 19.00 amu × (6.022 × 1023 F atoms) = 19.00 g F 1 F atom or, 6.022 × 10 23 amu = 1 g This is why Avogadro’s numbers has sometimes been described as a conversion factor between amu and grams. 3.122 Since we assume that water exists as either H2O or D2O, the natural abundances are 99.985 percent and 0.015 percent, respectively. If we convert to molecules of water (both H2O or D2O), we can calculate the molecules that are D2O from the natural abundance (0.015%). The necessary conversions are: mL water → g water → mol water → molecules water → molecules D2O 400 mL water × 1 g water 1 mol water 6.022 × 1023 molecules 0.015% molecules D 2 O × × × 1 mL water 18.02 g water 1 mol water 100% molecules water = 2.01 × 10 3.123 21 molecules D2O There can only be one chlorine per molecule, since two chlorines have a combined mass in excess of 70 amu. 35 Since the Cl isotope is more abundant, let’s subtract 35 amu from the mass corresponding to the more intense peak. 50 amu − 35 amu = 15 amu 15 amu equals the mass of one 12 1 C and three H. To explain the two peaks, we have: 12 1 35 molecular mass C H3 Cl = 12 amu + 3(1 amu) + 35 amu = 50 amu 12 1 37 molecular mass C H3 Cl = 12 amu + 3(1 amu) + 37 amu = 52 amu 35 Cl is three times more abundant than the 52 amu peak. 3.124 37 Cl; therefore, the 50 amu peak will be three times more intense than First, we can calculate the moles of oxygen. 2.445 g C × 1 mol C 1 mol O × = 0.2036 mol O 12.01 g C 1 mol C Next, we can calculate the molar mass of oxygen. molar mass O = 3.257 g O = 16.00 g/mol 0.2036 mol O If 1 mole of oxygen atoms has a mass of 16.00 g, then 1 atom of oxygen has an atomic mass of 16.00 amu. 66 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.125 The molecular formula for Cl2O7 means that there are 2 Cl atoms for every 7 O atoms or 2 moles of Cl atoms for every 7 moles of O atoms. We can write: mole ratio = 3.126 1 mol Cl 2 2 mol Cl = 7 mol O 3.5 mol O 2 (a) The mass of chlorine is 5.0 g. (b) From the percent by mass of Cl, we can calculate the mass of chlorine in 60.0 g of NaClO3. mass % Cl = 35.45 g Cl × 100% = 33.31% Cl 106.44 g compound mass Cl = 60.0 g × 0.3331 = 20.0 g Cl (c) 0.10 mol of KCl contains 0.10 mol of Cl. 0.10 mol Cl × (d) 35.45 g Cl = 3.5 g Cl 1 mol Cl From the percent by mass of Cl, we can calculate the mass of chlorine in 30.0 g of MgCl2. mass % Cl = (2)(35.45 g Cl) × 100% = 74.47% Cl l 95.21 g compound mass Cl = 30.0 g × 0.7447 = 22.3 g Cl (e) The mass of Cl can be calculated from the molar mass of Cl2. 0.50 mol Cl2 × 70.90 g Cl = 35.45 g Cl 1 mol Cl2 Thus, (e) 0.50 mol Cl2 contains the greatest mass of chlorine. 3.127 The mass percent of Cl is given. From the mass of the compound and the number of hydrogen atoms given, we can calculate the mass percent of H. The mass percent of carbon is then obtained by difference. Once the mass percentages of each element are known, the empirical formula can be determined. 4.19 × 1023 H atoms × mass % H = 1 mol H 6.022 × 10 23 H atoms × 1.008 g H = 0.7013 g H 1 mol H 0.7013 g H × 100% = 7.792% H 9.00 g compound mass % C = 100% − (55.0% + 7.792%) = 37.21% C To determine the empirical formula, assume 100 g of compound and convert to moles of each element present. 1 mol C mol C = 37.21 g C × = 3.098 mol C 12.01 g C CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS mol H = 7.792 g H × 1 mol H = 7.730 mol H 1.008 g H mol Cl = 55.0 g Cl × 1 mol Cl = 1.551 mol Cl 35.45 g Cl 67 Thus, we arrive at the formula C3.098H7.730Cl1.551, which gives the identity and the mole ratios of atoms present. However, chemical formulas are written with whole numbers. Try to convert to whole numbers by dividing each of the subscripts by the smallest subscript (1.551). This gives the empirical formula C2H5Cl. 3.128 Both compounds contain only Pt and Cl. The percent by mass of Pt can be calculated by subtracting the percent Cl from 100 percent. Compound A: Assume 100 g of compound. 26.7 g Cl × 1 mol Cl = 0.753 mol Cl 35.45 g Cl 73.3 g Pt × 1 mol Pt = 0.376 mol Pt 195.1 g Pt Dividing by the smallest number of moles (0.376 mole) gives the empirical formula, PtCl2. Compound B: Assume 100 g of compound. 42.1 g Cl × 1 mol Cl = 1.19 mol Cl 35.45 g Cl 57.9 g Pt × 1 mol Pt = 0.297 mol Pt 195.1 g Pt Dividing by the smallest number of moles (0.297 mole) gives the empirical formula, PtCl4. 3.129 The mass of the metal (X) in the metal oxide is 1.68 g. The mass of oxygen in the metal oxide is 2.40 g − 1.68 g = 0.72 g oxygen. Next, find the number of moles of the metal and of the oxygen. moles X = 1.68 g × 1 mol X = 0.0301 mol X 55.9 g X moles O = 0.72 g × 1 mol O = 0.045 mol O 16.00 g O This gives the formula X0.0301O0.045. Dividing by the smallest number of moles (0.0301 moles) gives the formula X1.00O1.5. Multiplying by two gives the empirical formula, X2O3. The balanced equation is: 3.130 X2O3(s) + 3CO(g) ⎯⎯ → 2X(s) + 3CO2(g) Both compounds contain only Mn and O. When the first compound is heated, oxygen gas is evolved. Let’s calculate the empirical formulas for the two compounds, then we can write a balanced equation. (a) Compound X: Assume 100 g of compound. 63.3 g Mn × 1 mol Mn = 1.15 mol Mn 54.94 g Mn 68 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 36.7 g O × 1 mol O = 2.29 mol O 16.00 g O Dividing by the smallest number of moles (1.15 moles) gives the empirical formula, MnO2. Compound Y: Assume 100 g of compound. 72.0 g Mn × 28.0 g O × 1 mol Mn = 1.31 mol Mn 54.94 g Mn 1 mol O = 1.75 mol O 16.00 g O Dividing by the smallest number of moles gives MnO1.33. Recall that an empirical formula must have whole number coefficients. Multiplying by a factor of 3 gives the empirical formula Mn3O4. (b) The unbalanced equation is: Balancing by inspection gives: 3.131 MnO2 ⎯ ⎯→ Mn3O4 + O2 3MnO2 ⎯ ⎯→ Mn3O4 + O2 The mass of the water is the difference between 1.936 g of the hydrate and the mass of water-free (anhydrous) BaCl2. First, we need to start with a balanced equation for the reaction. Upon treatment with sulfuric acid, BaCl2 dissolves, losing its waters of hydration. → BaSO4(s) + 2HCl(aq) BaCl2(aq) + H2SO4(aq) ⎯⎯ Next, calculate the mass of anhydrous BaCl2 based on the amount of BaSO4 produced. 1.864 g BaSO 4 × 1 mol BaSO 4 1 mol BaCl2 208.2 g BaCl2 × × = 1.663 g BaCl2 233.4 g BaSO4 1 mol BaSO 4 1 mol BaCl2 The mass of water is (1.936 g − 1.663 g) = 0.273 g H2O. Next, we convert the mass of H2O and the mass of BaCl2 to moles to determine the formula of the hydrate. 0.273 g H 2 O × 1 mol H 2 O = 0.0151 mol H 2 O 18.02 g H 2 O 1.663 g BaCl2 × 1 mol BaCl2 = 0.00799 mol BaCl2 208.2 g BaCl2 The ratio of the number of moles of H2O to the number of moles of BaCl2 is 0.0151/0.00799 = 1.89. We round this number to 2, which is the value of x. The formula of the hydrate is BaCl2 ⋅ 2H2O. 3.132 SO2 is converted to H2SO4 by reaction with water. The mole ratio between SO2 and H2SO4 is 1:1. This is a unit conversion problem. You should come up with the following strategy to solve the problem. tons SO2 → ton-mol SO2 → ton-mol H2SO4 → tons H2SO4 ? tons H 2SO4 = (4.0 × 105 tons SO 2 ) × 5 = 6.1 × 10 tons H2SO4 1 ton-mol SO2 1 ton-mol H 2SO 4 98.09 tons H 2SO4 × × 64.07 tons SO 2 1 ton-mol SO 2 1 ton-mol H 2SO4 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 69 Tip: You probably won’t come across a ton-mol that often in chemistry. However, it was convenient to use in this problem. We normally use a g-mol. 1 g-mol SO2 has a mass of 64.07 g. In a similar manner, 1 ton-mol of SO2 has a mass of 64.07 tons. 3.133 The mass of water lost upon heating the mixture is (5.020 g − 2.988 g) = 2.032 g water. Next, if we let x = mass of CuSO4 ⋅ 5H2O, then the mass of MgSO4 ⋅ 7H2O is (5.020 − x)g. We can calculate the amount of water lost by each salt based on the mass % of water in each hydrate. We can write: (mass CuSO4 ⋅ 5H2O)(% H2O) + (mass MgSO4 ⋅ 7H2O)(% H2O) = total mass H2O = 2.032 g H2O Calculate the % H2O in each hydrate. % H 2 O (CuSO 4 ⋅ 5H 2 O) = (5)(18.02 g) × 100% = 36.08% H 2 O 249.7 g % H 2 O (MgSO 4 ⋅ 7H 2 O) = (7)(18.02 g) × 100% = 51.17% H 2 O 246.5 g Substituting into the equation above gives: (x)(0.3608) + (5.020 − x)(0.5117) = 2.032 g 0.1509x = 0.5367 x = 3.557 g = mass of CuSO4 ⋅ 5H2O Finally, the percent by mass of CuSO4 ⋅ 5H2O in the mixture is: 3.557 g × 100% = 70.86% 5.020 g 3.134 We assume that the increase in mass results from the element nitrogen. The mass of nitrogen is: 0.378 g − 0.273 g = 0.105 g N The empirical formula can now be calculated. Convert to moles of each element. 0.273 g Mg × 0.105 g N × 1 mol Mg = 0.0112 mol Mg 24.31 g Mg 1 mol N = 0.00749 mol N 14.01 g N Dividing by the smallest number of moles gives Mg1.5N. Recall that an empirical formula must have whole number coefficients. Multiplying by a factor of 2 gives the empirical formula Mg3N2. The name of this compound is magnesium nitride. 3.135 The balanced equations are: CH4 + 2O2 ⎯⎯ → CO2 + 2H2O 2C2H6 + 7O2 ⎯⎯ → 4CO2 + 6H2O If we let x = mass of CH4, then the mass of C2H6 is (13.43 − x) g. 70 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS Next, we need to calculate the mass of CO2 and the mass of H2O produced by both CH4 and C2H6. The sum of the masses of CO2 and H2O will add up to 64.84 g. ? g CO 2 (from CH 4 ) = x g CH 4 × 1 mol CH 4 1 mol CO2 44.01 g CO 2 × × = 2.744 x g CO2 16.04 g CH 4 1 mol CH 4 1 mol CO 2 ? g H 2 O (from CH 4 ) = x g CH 4 × 1 mol CH 4 2 mol H 2 O 18.02 g H 2 O × × = 2.247 x g H 2 O 16.04 g CH 4 1 mol CH 4 1 mol H 2 O ? g CO 2 (from C2 H 6 ) = (13.43 − x ) g C2 H 6 × 1 mol C2 H 6 4 mol CO2 44.01 g CO 2 × × 30.07 g C2 H 6 2 mol C2 H 6 1 mol CO2 = 2.927(13.43 − x) g CO2 ? g H 2 O (from C2 H6 ) = (13.43 − x) g C2 H 6 × 1 mol C2 H 6 6 mol H 2 O 18.02 g H 2 O × × 30.07 g C2 H 6 2 mol C2 H 6 1 mol H 2 O = 1.798(13.43 − x) g H2O Summing the masses of CO2 and H2O: 2.744x g + 2.247x g + 2.927(13.43 − x) g + 1.798(13.43 − x) g = 64.84 g 0.266x = 1.383 x = 5.20 g The fraction of CH4 in the mixture is 3.136 5.20 g = 0.387 13.43 g Step 1: Calculate the mass of C in 55.90 g CO2, and the mass of H in 28.61 g H2O. This is a dimensional analysis problem. To calculate the mass of each component, you need the molar masses and the correct mole ratio. You should come up with the following strategy: g CO2 → mol CO2 → mol C → g C Step 2: ? g C = 55.90 g CO2 × 1 mol CO2 1 mol C 12.01 g C × × = 15.25 g C 44.01 g CO 2 1 mol CO2 1 mol C ? g H = 28.61 g H 2 O × 1 mol H 2 O 2 mol H 1.008 g H × × = 3.201 g H 18.02 g H 2 O 1 mol H 2 O 1 mol H Similarly, Since the compound contains C, H, and Pb, we can calculate the mass of Pb by difference. 51.36 g = mass C + mass H + mass Pb 51.36 g = 15.25 g + 3.201 g + mass Pb mass Pb = 32.91 g Pb CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 71 Step 3: Calculate the number of moles of each element present in the sample. Use molar mass as a conversion factor. ? mol C = 15.25 g C × 1 mol C = 1.270 mol C 12.01 g C ? mol H = 3.201 g H × 1 mol H = 3.176 mol H 1.008 g H Similarly, ? mol Pb = 32.91 g Pb × 1 mol Pb = 0.1588 mol Pb 207.2 g Pb Thus, we arrive at the formula Pb0.1588C1.270H3.176, which gives the identity and the ratios of atoms present. However, chemical formulas are written with whole numbers. Step 4: Try to convert to whole numbers by dividing all the subscripts by the smallest subscript. Pb: 0.1588 = 1.00 0.1588 C: 1.270 ≈ 8 0.1588 H: 3.176 ≈ 20 0.1588 This gives the empirical formula, PbC8H20. 3.137 First, calculate the mass of C in CO2 and the mass of H in H2O. ? g C = 30.2 g CO 2 × 1 mol CO 2 1 mol C 12.01 g C × × = 8.24 g C 44.01 g CO2 1 mol CO2 1 mol C ? g H = 14.8 g H 2 O × 1 mol H 2 O 2 mol H 1.008 g H × × = 1.66 g H 18.02 g H 2 O 1 mol H 2 O 1 mol H Since the compound contains C, H, and O, we can calculate the mass of O by difference. 12.1 g = mass C + mass H + mass O 12.1 g = 8.24 g + 1.66 g + mass O mass O = 2.2 g O Next, calculate the moles of each element. ? mol C = 8.24 g C × 1 mol C = 0.686 mol C 12.01 g C ? mol H = 1.66 g H × 1 mol H = 1.65 mol H 1.008 g H ? mol O = 2.2 g O × 1 mol O = 0.14 mol O 16.00 g O Thus, we arrive at the formula C0.686H1.65O0.14. Dividing by 0.14 gives the empirical formula, C5H12O. 72 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.138 (a) The following strategy can be used to convert from the volume of the Mg cube to the number of Mg atoms. 3 cm → grams → moles → atoms 1.0 cm3 × (b) 1.74 g Mg 1 cm3 × 1 mol Mg 6.022 × 1023 Mg atoms × = 4.3 × 1022 Mg atoms 24.31 g Mg 1 mol Mg Since 74 percent of the available space is taken up by Mg atoms, 4.3 × 10 volume: 3 22 atoms occupy the following 3 0.74 × 1.0 cm = 0.74 cm We are trying to calculate the radius of a single Mg atom, so we need the volume occupied by a single Mg atom. 0.74 cm3 volume Mg atom = = 1.7 × 10−23 cm3 /Mg atom 22 4.3 × 10 Mg atoms 4 3 πr . Solving for the radius: 3 4 cm3 = πr 3 3 The volume of a sphere is V = 1.7 × 10−23 3 r = 4.1 × 10 r = 1.6 × 10 −24 −8 3 cm cm Converting to picometers: radius Mg atom = (1.6 × 10−8 cm) × 3.139 0.01 m 1 pm × = 1.6 × 102 pm 1 cm 1 × 10−12 m The balanced equations for the combustion of sulfur and the reaction of SO2 with CaO are: → SO2(g) S(s) + O2(g) ⎯⎯ → CaSO3(s) SO2(g) + CaO(s) ⎯⎯ First, find the amount of sulfur present in the daily coal consumption. (6.60 × 106 kg coal) × 1.6% S = 1.06 × 105 kg S = 1.06 × 108 g S 100% The daily amount of CaO needed is: (1.06 × 108 g S) × 3.140 1 mol SO2 1 mol CaO 56.08 g CaO 1 mol S 1 kg × × × × = 1.85 × 105 kg CaO 32.07 g S 1 mol S 1 mol SO 2 1 mol CaO 1000 g The molar mass of air can be calculated by multiplying the mass of each component by its abundance and adding them together. Recall that nitrogen gas and oxygen gas are diatomic. molar mass air = (0.7808)(28.02 g/mol) + (0.2095)(32.00 g/mol) + (0.0097)(39.95 g/mol) = 28.97 g/mol CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.141 (a) Assuming the die pack with no empty space between die, the volume of one mole of die is: 3 23 (1.5 cm) × (6.022 × 10 ) = 2.0 × 10 (b) 24 3 cm 8 6371 km = 6.371 × 10 cm Volume = area × height (h) h = 3.142 volume 2.0 × 1024 cm3 = = 3.9 × 105 cm = 3.9 × 103 m area 4π(6.371 × 108 cm)2 The surface area of the water can be calculated assuming that the dish is circular. 2 2 2 2 surface area of water = πr = π(10 cm) = 3.1 × 10 cm 2 The cross-sectional area of one stearic acid molecule in cm is: 2 ⎛ 1 × 10−9 m ⎞ ⎛ 1 cm ⎞2 0.21 nm × ⎜ = 2.1 × 10−15 cm 2 /molecule ⎟ × ⎜ 1 nm ⎟ ⎜⎝ 0.01 m ⎟⎠ ⎝ ⎠ 2 Assuming that there is no empty space between molecules, we can calculate the number of stearic acid 2 2 molecules that will fit in an area of 3.1 × 10 cm . (3.1 × 102 cm 2 ) × 1 molecule 2.1 × 10−15 cm 2 = 1.5 × 1017 molecules Next, we can calculate the moles of stearic acid in the 1.4 × 10 Avogadro’s number (the number of molecules per mole). 1.4 × 10−4 g stearic acid × g sample. Then, we can calculate 1 mol stearic acid = 4.9 × 10−7 mol stearic acid 284.5 g stearic acid Avogadro's number ( N A ) = 3.143 −4 1.5 × 1017 molecules 4.9 × 10 −7 mol = 3.1 × 1023 molecules/mol The balanced equations for the combustion of octane are: 2C8H18 + 25O2 ⎯⎯ → 16CO2 + 18H2O 2C8H18 + 17O2 ⎯⎯ → 16CO + 18H2O The quantity of octane burned is 2650 g (1 gallon with a density of 2.650 kg/gallon). Let x be the mass of octane converted to CO2; therefore, (2650 − x) g is the mass of octane converted to CO. The amounts of CO2 and H2O produced by x g of octane are: x g C8 H18 × 1 mol C8 H18 16 mol CO 2 44.01 g CO2 × × = 3.083 x g CO 2 114.2 g C8 H18 2 mol C8 H18 1 mol CO 2 x g C8 H18 × 1 mol C8 H18 18 mol H 2 O 18.02 g H 2 O × × = 1.420 x g H 2 O 114.2 g C8 H18 2 mol C8 H18 1 mol H 2 O 73 74 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS The amounts of CO and H2O produced by (2650 − x) g of octane are: (2650 − x) g C8 H18 × 1 mol C8 H18 16 mol CO 28.01 g CO × × = (5200 − 1.962 x) g CO 114.2 g C8 H18 2 mol C8 H18 1 mol CO (2650 − x) g C8 H18 × 1 mol C8 H18 18 mol H 2 O 18.02 g H 2 O × × = (3763 − 1.420 x) g H 2 O 114.2 g C8 H18 2 mol C8 H18 1 mol H 2 O The total mass of CO2 + CO + H2O produced is 11530 g. We can write: 11530 g = 3.083x + 1.420x + 5200 − 1.962x + 3763 − 1.420x x = 2290 g Since x is the amount of octane converted to CO2, we can now calculate the efficiency of the process. efficiency = 3.144 (a) g octane converted 2290 g × 100% = × 100% = 86.49% g octane total 2650 g The balanced chemical equation is: C3H8(g) + 3H2O(g) ⎯ ⎯→ 3CO(g) + 7H2(g) (b) You should come up with the following strategy to solve this problem. In this problem, we use kg-mol to save a couple of steps. kg C3H8 → mol C3H8 → mol H2 → kg H2 ? kg H 2 = (2.84 × 103 kg C3 H8 ) × 2 1 kg-mol C3 H8 7 kg-mol H 2 2.016 kg H 2 × × 44.09 kg C3 H8 1 kg-mol C3 H8 1 kg-mol H 2 = 9.09 × 10 kg H2 3.145 For the first step of the synthesis, the yield is 90% or 0.9. For the second step, the yield will be 90% of 0.9 or (0.9 × 0.9) = 0.81. For the third step, the yield will be 90% of 0.81 or (0.9 × 0.9 × 0.9) = 0.73. We see that the yield will be: n Yield = (0.9) where n = number of steps in the reaction. For 30 steps, Yield = (0.9) 3.146 17 amu, NH3 30 = 0.04 = 4% (a) 16 amu, CH4 18 amu, H2O 64 amu, SO2 (b) The formula C3H8 can also be written as CH3CH2CH3. A CH3 fragment could break off from this molecule giving a peak at 15 amu. No fragment of CO2 can have a mass of 15 amu. Therefore, the substance responsible for the mass spectrum is most likely C3H8. (c) First, let’s calculate the masses of CO2 and C3H8. molecular mass CO2 = 12.00000 amu + 2(15.99491 amu) = 43.98982 amu molecular mass C3H8 = 3(12.00000 amu) + 8(1.00797 amu) = 44.06376 amu CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 75 These masses differ by only 0.07394 amu. The measurements must be precise to ±0.030 amu. 43.98982 + 0.030 amu = 44.02 amu 44.06376 − 0.030 amu = 44.03 amu 3.147 (a) We need to compare the mass % of K in both KCl and K2SO4. %K in KCl = 39.10 g × 100% = 52.45% K 74.55 g %K in K 2SO4 = 2(39.10 g) × 100% = 44.87% K 174.27 g The price is dependent on the %K. Price of K 2SO 4 %K in K 2SO 4 = Price of KCl %K in KCl Price of K 2SO4 = Price of KCl × Price of K 2SO4 = (b) %K in K 2SO4 %K in KCl $0.55 44.87% × = $0.47 /kg kg 52.45% First, calculate the number of moles of K in 1.00 kg of KCl. (1.00 × 103 g KCl) × 1 mol KCl 1 mol K × = 13.4 mol K 74.55 g KCl 1 mol KCl Next, calculate the amount of K2O needed to supply 13.4 mol K. 13.4 mol K × 3.148 1 mol K 2 O 94.20 g K 2 O 1 kg × × = 0.631 kg K 2O 2 mol K 1 mol K 2 O 1000 g When magnesium burns in air, magnesium oxide (MgO) and magnesium nitride (Mg3N2) are produced. Magnesium nitride reacts with water to produce ammonia gas. Mg3N2(s) + 6H2O(l) → 3Mg(OH)2(s) + 2NH3(g) From the amount of ammonia produced, we can calculate the mass of Mg3N2 produced. The mass of Mg in that amount of Mg3N2 can be determined, and then the mass of Mg in MgO can be determined by difference. Finally, the mass of MgO can be calculated. 2.813 g NH3 × 1 mol NH3 1 mol Mg3 N 2 100.95 g Mg3 N 2 × × = 8.335 g Mg 3 N 2 17.034 g NH3 2 mol NH3 1 mol Mg3 N 2 The mass of Mg in 8.335 g Mg3N2 can be determined from the mass percentage of Mg in Mg3N2. (3)(24.31 g Mg) × 8.335 g Mg3 N 2 = 6.022 g Mg 100.95 g Mg3 N 2 The mass of Mg in the product MgO is obtained by difference: 21.496 g Mg − 6.022 g Mg = 15.474 g Mg 76 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS The mass of MgO produced can now be determined from this mass of Mg and the mass percentage of Mg in MgO. 40.31 g MgO × 15.474 g Mg = 25.66 g MgO 24.31 g Mg 3.149 Possible formulas for the metal bromide could be MBr, MBr2, MBr3, etc. Assuming 100 g of compound, the moles of Br in the compound can be determined. From the mass and moles of the metal for each possible formula, we can calculate a molar mass for the metal. The molar mass that matches a metal on the periodic table would indicate the correct formula. Assuming 100 g of compound, we have 53.79 g Br and 46.21 g of the metal (M). The moles of Br in the compound are: 53.79 g Br × 1 mol Br = 0.67322 mol Br 79.90 g Br If the formula is MBr, the moles of M are also 0.67322 mole. If the formula is MBr2, the moles of M are 0.67322/2 = 0.33661 mole, and so on. For each formula (MBr, MBr2, and MBr3), we calculate the molar mass of the metal. MBr: 3.150 46.21 g M = 68.64 g/mol (no such metal) 0.67322 mol M MBr2 : 46.21 g M = 137.3 g/mol (The metal is Ba. The formula is BaBr2 ) 0.33661 mol M MBr3 : 46.21 g M = 205.9 g/mol (no such metal) 0.22441 mol M The decomposition of KClO3 produces oxygen gas (O2) which reacts with Fe to produce Fe2O3. 4Fe + 3O2 → 2Fe2O3 When the 15.0 g of Fe is heated in the presence of O2 gas, any increase in mass is due to oxygen. The mass of oxygen reacted is: 17.9 g − 15.0 g = 2.9 g O2 From this mass of O2, we can now calculate the mass of Fe2O3 produced and the mass of KClO3 decomposed. 2.9 g O2 × 2 mol Fe2 O3 159.7 g Fe 2 O3 1 mol O 2 × × = 9.6 g Fe2 O 3 32.00 g O 2 3 mol O 2 1 mol Fe2 O3 The balanced equation for the decomposition of KClO3 is: 2KClO3 → 2KCl + 3O2. The mass of KClO3 decomposed is: 2.9 g O2 × 2 mol KClO3 122.55 g KClO3 1 mol O 2 × × = 7.4 g KClO 3 32.00 g O 2 3 mol O 2 1 mol KClO3 CHAPTER 3: MASS RELATIONSHIPS IN CHEMICAL REACTIONS 3.151 77 Assume 100 g of sample. Then, mol Na = 32.08 g Na × mol O = 36.01 g O × 1 mol Na = 1.395 mol Na 22.99 g Na 1 mol O = 2.251 mol O 16.00 g O 1 mol Cl = 0.5504 mol Cl 35.45 g Cl mol Cl = 19.51 g Cl × Since Cl is only contained in NaCl, the moles of Cl equals the moles of Na contained in NaCl. mol Na (in NaCl) = 0.5504 mol The number of moles of Na in the remaining two compounds is: 1.395 mol − 0.5504 mol = 0.8446 mol Na. To solve for moles of the remaining two compounds, let x = moles of Na2SO4 y = moles of NaNO3 Then, from the mole ratio of Na and O in each compound, we can write 2x + y = mol Na = 0.8446 mol 4x + 3y = mol O = 2.251 mol Solving two equations with two unknowns gives x = 0.1414 = mol Na2SO4 and y = 0.5618 = mol NaNO3 Finally, we convert to mass of each compound to calculate the mass percent of each compound in the sample. Remember, the sample size is 100 g. mass % NaCl = 0.5504 mol NaCl × 58.44 g NaCl 1 × × 100% = 32.17% NaCl 1 mol NaCl 100 g sample mass % Na 2 SO4 = 0.1414 mol Na 2SO4 × mass % NaNO 3 = 0.5618 mol NaNO3 × 142.1 g Na 2SO 4 1 × × 100% = 20.09% Na 2SO4 1 mol Na 2SO 4 100 g sample 85.00 g NaNO3 1 × × 100% = 47.75% NaNO 3 1 mol NaNO3 100 g sample