Solutions Manual For Mathematics For Physical Chemistry
Solutions Manual for Mathematicsfor Physical Chemistry
Solutions Manualfor Mathematics forPhysical ChemistryFourth EditionRobert G. MortimerProfessor EmeritusRhodes CollegeMemphis, TennesseeAMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARISSAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYOAcademic Press is an Imprint of Elsevier
ContentsPreface vii1 Problem Solving and NumericalMathematics e110 Mathematical Series 2 Mathematical Functions 11 Functional Series and IntegralTransforms e89e7e793 Problem Solving and SymbolicMathematics: Algebra e134 Vectors and Vector Algebra e1913 Operators, Matrices, andGroup Theory 5 Problem Solving and the Solutionof Algebraic Equations e236 Differential Calculus e3514 The Solution of SimultaneousAlgebraic Equations with MoreThan Two Unknowns e1257 Integral Calculus e5115 Probability, Statistics, andExperimental Errors e1358 Differential Calculus with SeveralIndependent Variables e5916 Data Reduction and thePropagation of Errors e1459 Integral Calculus with SeveralIndependent Variables e6912 Differential Equations e97e113v
PrefaceThis book provides solutions to nearly of the exercises andproblems in Mathematics for Physical Chemistry, fourthedition, by Robert G. Mortimer. This edition is a revisionof a third edition published by Elsevier/Academic Press in2005. Some of exercises and problems are carried over fromearlier editions, but some have been modified, and somenew ones have been added. I am pleased to acknowledgethe cooperation and help of Linda Versteeg-Buschman, BethCampbell, Jill Cetel, and their collaborators at Elsevier. It isalso a pleasure to acknowledge the assistance of all thosewho helped with all editions of the book for which this isthe solutions manual, and especially to thank my wife, Ann,for her patience, love, and forbearance.There are certain errors in the solutions in this manual, andI would appreciate learning of them through the publisher.Robert G. Mortimervii
Chapter 1Problem Solving and NumericalMathematicsEXERCISESExercise 1.1. Take a few fractions, such as 23 , 49 or 37 andrepresent them as decimal numbers, finding either all of thenonzero digits or the repeating pattern of digits.2 0.66666666 · · ·34 0.4444444 · · ·93 0.428571428571 · · ·7Exercise 1.2. Express the following in terms of SI baseunits. The electron volt (eV), a unit of energy, equals1.6022 10 18 J. 1.6022 10 19 Ja. (13.6 eV) 2.17896 10 19 J1 eV 2.18 10 18 J 5280 ft12 in0.0254mb. (24.17 mi)1 mi1 ft1 in 3.890 104 m 12 in0.0254 m5280 ftc. (55 mi h 1 )1 mi1 ft1 in 1h 1 1 24.59 m s 25 m s3600 s 12 1m10 ps 1d. (7.53 nm ps )109 nm1s 7.53 103 m s 1Exercise 1.3. Convert the following numbers to scientificnotation:a. 0.00000234 2,34 10 6b. 32.150 3.2150 101Mathematics for Physical Chemistry. -2 2013 Elsevier Inc. All rights reserved.Exercise 1.4. Round the following numbers to threesignificant digitsa. 123456789123 123,000,000,000b. 46.45 46.4Exercise 1.5. Find the pressure P of a gas obeying theideal gas equationP V n RTif the volume V is 0.200 m3 , the temperature T is 298.15 Kand the amount of gas n is 1.000 mol. Take the smallestand largest value of each variable and verify your numberof significant digits. Note that since you are dividing byV the smallest value of the quotient will correspond to thelargest value of V.P Pmax Pmin n RTV(1.000 mol)(8.3145 J K 1 mol 1 )(298.15 K)0.200 m3 312395 J m 12395 N m 2 1.24 104 Pan RTV(1.0005 mol)(8.3145 J K 1 mol 1 )(298.155 K)0.1995 m341.243 10 Pan RTV(0.9995 mol)(8.3145 J K 1 mol 1 )(298.145 K)0.2005 m341.236 10 Pae1
e2Mathematics for Physical ChemistryExercise 1.6. Calculate the following to the propernumbers of significant digits.a. 17.13 14.6751 3.123 7.654 8.123 34.359 34.36b. ln (0.000123)ln (0.0001235) 8.99927ln (0.0001225) 9.00740The answer should have three significant digits:6. The distance by road from Memphis, Tennesseeto Nashville, Tennessee is 206 miles. Express thisdistance in meters and in kilometers. (206 mi)5380 ft1 mi 12 in1 ft 0.0254 m1 in 3.32 105 m 332 km7. A U. S. gallon is defined as 231.00 cubic inches.ln (0.000123) 9.00a. Find the number of liters in 1.000 gallon.PROBLEMS1. Find the number of inches in 1.000 meter. 1 in(1.000 m) 39.37 in0.0254 m2. Find the number of meters in 1.000 mile and thenumber of miles in 1.000 km, using the definition ofthe inch. 12 in0.0254 m5280 ft(1.000 mi)1 mi1 ft1 in 1609 m 1000 m1 in1 ft(1.000 km)1 km0.0254 m12 in 1 mi 0.6214 5280 ft3. Find the speed of light in miles per second. 1 in1 ft 1(299792458 m s )0.0254 m12 in 1 mi 186282.397 mi s 15280 ft4. Find the speed of light in miles per hour. 1 in1 ft 1(299792458 m s )0.0254 m12 in 1 mi3600 s 670616629 mi h 15280 ft1h5. A furlong is exactly one-eighth of a mile and afortnight is exactly 2 weeks. Find the speed of lightin furlongs per fortnight, using the correct number ofsignificant digits. 1 in1 ft 1(299792458 m s )0.0254 m12 in 8 furlongs1 mi 5280 ft1 mi 3600 s24 h14 d 1h1d1 fortnight 1.80261750 1012 furlongs fortnight 1 231.00 in3(1 gal)1 gal 3.785 l 0.0254 m1 in 3 1000 l1 m3 b. The volume of 1.0000 mol of an ideal gasat 0.00 C (273.15 K) and 1.000 atm is22.414 liters. Express this volume in gallons andin cubic feet. 3 1 in1 m3(22.414 l)1000 l0.0254 m3 1 gal 5.9212 gal231.00 in3 3 1 in1 m3(22.414 l)1000 l0.0254 m3 31 ft 0.79154 ft312 in 8. In the USA, footraces were once measured in yards andat one time, a time of 10.00 seconds for this distancewas thought to be unattainable. The best runners nowrun 100 m in 10 seconds or less. Express 100.0 m inyards. If a runner runs 100.0 m in 10.00 s, find histime for 100 yards, assuming a constant speed. 1 in1 yd(100.0 m) 109.4 m0.0254 m36 in 100.0 yd 9.144 s(10.00 s)109.4 m9. Find the average length of a century in seconds and inminutes. Use the rule that a year ending in 00 is not aleap year unless the year is divisible by 400, in whichcase it is a leap year. Therefore, in four centuries therewill be 97 leap years. Find the number of minutes in amicrocentury.
CHAPTER 1 Problem Solving and Numerical Mathematicse3b. Find the Rankine temperature at 0.00 F.Number of days in 400 years (365 d)(400 y) 97 d 146097 d273.15 K 18.00 K 255.15 K 9 F(255.15 K) 459.27 R5KAverage number of days in a century146097 d 36524.25 d 4 24 h60 min1 century (36524.25 d)1d1h12. The volume of a sphere is given by7 5.259492 10 min 1 century(5.259492 107 min)1 106 microcenturies 52.59492 min 60 s(52.59492 min) 3155.695 s1 min10. A light year is the distance traveled by light in oneyear.a. Express this distance in meters and in kilometers.Use the average length of a year as described inthe previous problem. How many significant digitscan be given?(299792458 m s 1 ) 60 s1 min (9.46055060 1015 Vmin V m) 1 kmm)1000 m44 3πr V π(0.005250 m)3336.061 10 7 m34π(0.005245 m)3 6.044 10 7 m334π(0.005255 m)3 6.079 10 7 m336.06 10 7 m3The rule of thumb gives four significant digits, but thecalculation shows that only three significant digits canbe specified and that the last digit can be wrong byone.13. The volume of a right circular cylinder is given by 9.460551 1012 kmSince the number of significant digits in thenumber of days in an average century is seven,we round to seven significant digits.b. Express a light year in miles. 1 ft1 in(9.460551 1015 m)0.0254 m12 in 1 mi 5.878514 1012 mi 5280 ft11. The Rankine temperature scale is defined so that theRankine degree is the same size as the Fahrenheitdegree, and absolute zero is 0 R, the same as 0 K.a. Find the Rankine temperature at 0.00 V 15 9.4605506 1012 km9 F0.00 C (273.15 K)5K 4 3πr3where V is the volume and r is the radius. If a certainsphere has a radius given as 0.005250 m, find itsvolume, specifying it with the correct number of digits.Calculate the smallest and largest volumes that thesphere might have with the given information andcheck your first answer for the volume.Vmax (5.259492 105 min) (9.46055060 10V V πr 2 h,where r is the radius and h is the height. If a rightcircular cylinder has a radius given as 0.134 m and aheight given as 0.318 m, find its volume, specifyingit with the correct number of digits. Calculate thesmallest and largest volumes that the cylinder mighthave with the given information and check your firstanswer for the volume.V π(0.134 m)2 (0.318 m) 0.0179 m3Vmin π(0.1335 m)2 (0.3175 m) 0.01778 m3Vmax π(0.1345 m)2 (0.3185 m) 0.0181 m3 C. 491.67 R14. The value of an angle is given as 31 . Find the measureof the angle in radians. Find the smallest and largestvalues that its sine and cosine might have and specify
e4Mathematics for Physical ChemistryV 10.00 l, and T 298.15 K. Convert your answerto atmospheres and torr.the sine and cosine to the appropriate number of digits.(31 ) 2π rad 0.54 rad360 sin (30.5 ) 0.5075sin (31.5 ) 0.5225sin (31 ) 0.51cos (30.5 ) 0.86163cos (31.5 ) 0.85264cos (31 ) 0.8615. Some elementary chemistry textbooks givethe value of R, the ideal gas constant, as0.0821 l atm K 1 mol 1 .a. Using the SI value, 8.3145 J K 1 mol 1 , obtainthe value in l atm K 1 mol 1 to five significantdigits. 1 atm1 Pa m3(8.3145 J K 1 mol 1 )1J101325 Pa 1000 l 0.082058 l atm K 1 mol 1 1 m3b. Calculate the pressure in atmospheres and inN m 2 (Pa) of a sample of an ideal gas with n 0.13678 mol, V 10.000 l and T 298.15 K.P (8.3145 J K 1 mol 1 )(298.15 K)7.3110 10 2 m3 mol 1 4.267 10 5 m3 mol 10.3640 Pa m6 mol 2 (7.3110 10 2 m3 mol 1 )2aRT 2P Vm bVm (8.3145 J K 1 mol 1 )(298.15 K)7.3110 10 2 m3 mol 1 4.267 10 5 m3 mol 10.3640 Pa m6 mol 2 (7.3110 10 2 m3 mol 1 )2 3.3927 104 J m 3 68.1 Pa 3.3927 104 Pa 68.1 Pa 3.386 Pa 1 atm 0.33416 atm(3.3859 Pa)101325 PaThe prediction of the ideal gas equation is(8.3145 J K 1 mol 1 )(298.15 K)7.3110 10 2 m3 mol 1 3.3907 104 J m 3 3.3907 104 PaP 17.n RTP V(0.13678 mol)(0.082058 l atm K 1 mol 1 )(298.15 K) 1.000 l 0.33464 atmn RTP V(0.13678 mol)(8.3145 J K 1 mol 1 )(298.15 K) 10.000 10 3 m3 3.3907 104 J m 3 3.3907 104 N m 218. 3.3907 104 Pa16. The van der Waals equation of state gives betteraccuracy than the ideal gas equation of state. It is P aVm2 (Vm b) RTwhere a and b are parameters that have differentvalues for different gases and where Vm V /n, the molar volume. For carbon dioxide,a 0.3640 Pa m6 mol 2 , b 4.267 10 5 m3 mol 1 . Calculate the pressure of carbondioxide in pascals, assuming that n 0.13678 mol,aRT 2Vm bVmThe specific heat capacity (specific heat) of a substanceis crudely defined as the amount of heat required toraise the temperature of unit mass of the substance by1 degree Celsius (1 C). The specific heat capacity ofwater is 4.18 J C 1 g 1 . Find the rise in temperatureif 100.0 J of heat is transferred to 1.000 kg of water. 1 kg100.0 J T (4.18 J C 1 g 1 )(1.000 kg) 1000 g 0.0239 CThe volume of a cone is given byV 1 2πr h3where h is the height of the cone and r is the radiusof its base. Find the volume of a cone if its radius isgiven as 0.443 m and its height is given as 0.542 m.V 1 31πr h π(0.443 m)2 (0.542 m) 0.111 m33319. The volume of a sphere is equal to 43 πr 3 where r is theradius of the sphere. Assume that the earth is sphericalwith a radius of 3958.89 miles. (This is the radius ofa sphere with the same volume as the earth, which
CHAPTER 1 Problem Solving and Numerical Mathematicsis flattened at the poles by about 30 miles.) Find thevolume of the earth in cubic miles and in cubic meters.Use a value of π with at least six digits and give thecorrect number of significant digits in your answer.44 3πr π(3958.89 mi)333 2.59508 1011 mi3 5280 ft 3 12 in 3113(2.59508 10 mi )1 mi1 ft 3 0.0254 m 1.08168 1021 m31 inV 20. Using the radius of the earth in the previous problemand the fact that the surface of the earth is about 70%covered by water, estimate the area of all of the bodiesof water on the earth. The area of a sphere is equal tofour times the area of a great circle, or 4πr 2 , where ris the radius of the sphere.A (0.7)4πr 2 (0.7)4π(3958.89 mi)2 1.4 108 mi2e5We give two significant digits since the use of 1 asa single digit would specify a possible error of about50%. It is a fairly common practice to give an extradigit when the last significant digit is 1.21. The hectare is a unit of land area defined to equalexactly 10,000 square meters, and the acre is a unitof land area defined so that 640 acres equals exactlyone square mile. Find the number of square meters in1.000 acre, and find the number of acres equivalent to1.000 hectare. (5280 ft)212 in 21.000 acre 6401 ft 20.0254 m 4047 m21 in 10000 m21.000 hectare (1.000 hectare)1 hectare 1 acre 2.471 acre4047 m2
Chapter 2Mathematical Functions0.1 1/10EXERCISESExercise 2.1. Enter a formula into cell D2 that willcompute the mean of the numbers in cells A2,B2, and C2.log (0.1) log (10) 10.01 1/100 (A2 B2 C2)/3log (0.01) log (100) 2Exercise 2.2. Construct a graph representing the functiony(x) x 3 2x 2 3x 40.001 1/1000(2.1)log (0.001) log (1000) 3Use Excel or Mathematica or some other software toconstruct your graph.Here is the graph, constructed with Excel:0.0001 1/10000log (0.001) log (10000) 4Exercise 2.4. Using a calculator or a spreadsheet, evaluatethe quantity (1 n1 )n for several integral values of n rangingfrom 1 to 1,000,000. Notice how the value approaches thevalue of e as n increases and determine the value of n neededto provide four significant digits.Here is a table of values'Exercise 2.3. Generate the negative logarithms in the shorttable of common logarithms.' x(1 1/n)n1222.2552.48832102.59374246 1002.704813829xy log10 (x)xy log10 (x)10002.716923932100.1 691010.01 210020.001 3100030.0001 4&&%%Mathematics for Physical Chemistry. -4 2013 Elsevier Inc. All rights reserved.e7
e8Mathematics for Physical ChemistryTo twelve significant digits, the value of e is2.71828182846. The value for n 1000000 is accurateto six significant digits. Four significant digits are obtainedwith n 10000.Exercise 2.5. Without using a calculator or a table oflogarithms, find the following:a. ln (100.000) ln (10) log10 (100.000) (2.30258509 · · ·)(2.0000) 4.60517b. ln (0.0010000) ln (10) log10 (0.0010000) (2.30258509 · · ·)( 3.0000) 6.907761ln (e) 0.43429 · · ·c. log10 (e) ln (10)2.30258509 · · ·Exercise 2.6. For a positive value of b find an expressionin terms of b for the change in x required for the functionebx to double in size.There is no round-off error to 11 digits in the calculatorthat was used.Exercise 2.9. Using a calculator and displaying as manydigits as possible, find the values of the sine and cosine of49.500 . Square the two values and add the results. See ifthere is any round-off error in your calculator.sin (49.500 ) 0.7604059656cos (49.500 ) 0.64944804833(0.7604059656)2 (0.64944804833)2 1.00000000000Exercise 2.10. Construct an accurate graph of sin (x) andtan (x) on the same graph for values of x from 0 to 0.4 radand find the maximum value of x for which the two functionsdiffer by less than 1%.eb(x x)f (x x) 2 eb xf (x)ebx0.69315 · · ·ln (2) x bbExercise 2.7. A reactant in a first-order chemical reactionwithout back reaction has a concentration governed by thesame formula as radioactive decay,[A]t [A]0 e kt ,where [A]0 is the concentration at time t 0, [A]t is theconcentration at time t, and k is a function of temperaturecalled the rate constant. If k 0.123 s 1 find the timerequired for the concentration to drop to 21.0% of its initialvalue. 1100.0[A]01 lnlnt k[A]t0.123 s 121.0 12.7 sExercise 2.8. Using a calculator, find the value of thecosine of 15.5 and the value of the cosine of 375.5 .Display as many digits as your calculator is able to display.Check to see if your calculator produces any round-off errorin the last digit. Choose another pair of angles that differ by360 and repeat the calculation. Set your calculator to useangles measured in radians. Find the value of sin (0.3000).Find the value of sin (0.3000 2π ). See if there is anyround-off error in the last digit.cos (15.5 ) 0.96363045321 cos (375.5 ) 0.96363045321sin (0.3000) 0.29552020666sin (0.3000 2π ) sin (6.58318530718) 0.29552020666The two functions differ by less than 1% at 0.14 rad.Notice that at 0.4 rad, sin (x) x tan (x) and that thethree quantities differ by less than 10%.Exercise 2.11. For an angle that is nearly as large as π/2,find an approximate equality similar to Eq. (2.36) involving(π/2) α, cos (α), and cot (α).Construct a right triangle with angle with the angle(π/2) α, where α is small. The triangle is tall, with asmall value of x (the horizontal leg) and a larger value of y(the vertical leg). Let r be the hypotenuse, which is nearlyequal to y.xcos ((π/2) α) rcot ((π/2) α) xy rx . The measure of the anglein radians is equal to the arc length subtending the angleα divided by r and is very nearly equal to x/r . Thereforecos ((π/2) α) αcot ((π/2) α) αcos ((π/2) α) cot ((π/2) α)Exercise 2.12. Sketch graphs of the arcsine function, thearccosine function, and the arctangent function. Includeonly the principal values.
CHAPTER 2 Mathematical FunctionsHere are accurate graphs:e9We calculate sin (95.45 ) and sin (95.45 ). Using acalculator that displays 8 digits, we obtainsin (95.45 ) 0.99547946sin (95.55 ) 0.99531218We report the sine of 95.5 as 0.9954, specifying foursignificant digits, although the argument of the sine wasgiven with three significant digits. We have followed thecommon policy of reporting a digit as significant if it mightbe incorrect by one unit.Exercise 2.15. Sketch rough graphs of the followingfunctions. Verify your graphs using Excel or Mathematica.a. e x/5 sin (x). Following is a graph representing eachof the factors and their product:b. sin2 (x) [sin (x)]2Following is a graph representing sin (x) and sin2 (x).Exercise 2.13. Make a graph of tanh (x) and coth (x) onthe same graph for values of x ranging from 0.1 to 3.0.PROBLEMSExercise 2.14. Determine the number of significant digitsin sin (95.5 ).1. The following is a set of data for the vapor pressureof ethanol taken by a physical chemistry student.Plot these points by hand on graph paper, with thetemperature on the horizontal axis (the abscissa) and
e10Mathematics for Physical Chemistrythe vapor pressure on the vertical axis (the ordinate).Decide if there are any bad data points. Draw a smoothcurve nearly through the points, disregarding any badpoints. Use Excel to construct another graph and noticehow much work the spreadsheet saves you.'a function of the reciprocal of the Kelvin temperature.Why might this graph be more useful than the graphin the previous problem? Temperature/ CVapor 45.00154.150.00190.755.00241.9&%Here is a graph constructed with Excel:The third data point might be suspect. Here is agraph omitting that data point:This graph might be more useful than the first graphbecause the function is nearly linear. However, thethird data point still lies off the curve. Here is a graphwith that data point omitted.Thermodynamic theory implies that it should be nearlylinear if there were no experimental error.3. A reactant in a first-order chemical reaction withoutback reaction has a concentration governed by thesame formula as radioactive decay,[A]t [A]0 e kt ,where [A]0 is the concentration at time t 0, [A]tis the concentration at time t, an
This book provides solutions to nearly of the exercises and problems in Mathematics for Physical Chemistry, fourth edition, by Robert G. Mortimer. This edition is a revision of a third edition published by Elsevier/Academic Press in 2005. Some of exercises and problems are carried over from earlier editions, but some have been modified, and some
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