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INTERNATIONAL ATOMIC WEIGHTS, 1956-Journal of the American Chemical Society

1957

Element Symbol AtomicNumber AtomicWeight Element SymbolAtomicNumber AtomicWeight"

Actinium Ac 89 Mercury Kg 80 200.61Aluminum Al 13 26.98 Molybdenum Mo 42 95.95Americium Am 95 Neodymium Nd 60 144 . 27Antimony Sb 5). 121.76 Neon Ne 10 20.183ArgonArsenic

Ar 18 39,944 Neptunium Np 93As 38 74.9) Nickel Ni 28 58.71.

Astatine At 85 Niobium Nb 41 92.91Barium Ba 56 3 37 . 36 Nitrogen N 7 14.008Berkelium Bk 97 Nobelium No 102Beryllium Be 4 9.013 Osmium Os 76 190,2Bismuth Bj 83 209 . 00 Oxygen O 8 16Boron B 5 10.82 Palladium Pd 40 106 .Bromine Br 35 79.916 Phosphorus P ' 15 " 30.975Cadmium Cd 48 112.41 Platinum n 78 195.09Calcium Ca 20 40.08 Plutonium Pu 94Californium Cf 98 Polonium Po 84Carbon C 6 12.011 Potassium K 19 39 . 100Cerium Ce 58 140.13 Praseodymium Pr 59 140.92Cesium Cs 55 132.91 Promethium Pm 61Chlorine CI 17 35.457 Protactinium Pa 91Chromium Or 24 52.01 Radium Ra 88Cobalt Co 27 58.94 Radon Rn 86Copper Cu 29 63.54 Rhenium Re 75

186.22Curium Cm 96 Rhodium Rh 45 102.91Dysprosium Dy 66 162.51 Rubidium Rb 37 85.48Einsteinium Es 99 Ruthenium Ru 44 101.1Erbium Er 68 167.27 Samarium Sm 62 150.35Europium Eu 63 152.0 Scandium Sc 21 44.96Fermium Fm .100 Selenium Se 34 78.96Fluorine Y 9 19.00 Silicon Si 14 28.09Francium Fr 87 Silver AS 47 107.880Gadolinium Gd 64 157.26 Sodium Na 11 22.991Gallium Ga 31 69.72 Strontium Sr 38 87.63Germanium Ge 32 72.60 Sulfur s 16 32.060*Gold Au 79 197.0 Tantalum Ta 73 180.95Hafnium Hf 72 178 50 Technetium Tc 43Helium He 2 4.003 Tellurium Te 52 127.61Holmium Ho 67 164.94 Terbium Tb 65 158.93Hydrogen H 1 1.0080 Thallium Tl 81 204.39Indium In 49 114.82 Thorium Th 90 232.05Iodine I 53 126.91 Thulium Tm 69 168.94Iridium Ir 77 192.2 Tin Sn 50 118.70Tron Fe 26 55.85 Titanium Ti 22 47.90Krypton Kr 36 83.80 Tungsten W 74 183.86Lanthanum La 57 138.92 Uranium U 92 238.07Lean Pb 82 207.21 Vanadium V 23 50.95Lithium Li 3 6 940 Xenon Xe 54 131.30Lutetium I 71 174 . 99 Ytterbium Yb 70 173.04Magnesium Mg 12 24.32 Yttrium Y 39 88.92Manganese Mn 25 54 . 94 Zinc Zn 30 65.38Mendelevium Mo 101 Zirconium Zr 40 91.22

" Mass numbers for radioactive elements are omitted except for naturally occurring uraniumand thorium and for certain other elements that are only very slightly radioactive.1 Because of natural variations in relative abundance of the sulfur isotopes, its atomic weighthas a range of +0.003.

9 .. K a k ahj a -j y t> uPRESTON POLYTECHNICLIBRARY & LEARNING RESOURCES SERVICE

This book must be returned on or before the date last stampedTELEPnONE:PREST0N 262?13 !TT


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Experimental Physical ChemistryFARRINGTON DANIELSProfessor Emeritus of ChemistryJ. W. WILLIAMSProfessor of ChemistryPAUL BENDERProfessor of ChemistryROBERT A. ALBERTYProfessor of Chemistry 1962C. D. CORNWELLProfessor of Chemistry

New YorkSan FranciscoToronto

University of Wisconsin Sixth Edition London

M cG HAW -HILL BOOK COMPANY, INC


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#-#**

EXPERIMENTAL PHYSICAL CHEMISTRYCopyright 1956,/t9j2/by the McGraw-Hill Book Company, Inc.Copyright, 1929, 1934, 1941, 1949, by the McGraw-Hill Book Company, Inc.All Rights Reserved. Printed in the United States of America. Thisbook, or parts thereof, may not be reproduced in any form without permis-sion of the publishers. Library of Congress Catalog Card Number 62-1403815336

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Preface

To illustrate the principles of physical chemistry, to train in carefulexperimentation, to develop familiarity with apparatus, to encourageability in researchthese are the purposes of this book, as stated in thefirst edition a third of a century ago. In each of the five revised editionsan attempt has been made to keep pace with the new developments inphysical chemistry and to have the book representative of the teachingof the laboratory course in physical chemistry at the University ofWisconsin.There are many more experiments in this book than can be performedby any one student. Selection will be made on the basis of the time

and apparatus available and on the capacity and ultimate aims of thestudent. If an experiment is too short, the student will find interestingprojects under Suggestions for Further Work; if it is too long, the instruc-tor may designate parts of the Procedure to be omitted.The imperative is not used. Procedures are described, but orders arenot given. The student studies the experiment first and then plans his

worka method which develops both his power and his interest.The high cost of laboratory apparatus restricts the choice of experi-

ments, particularly where classes are small. Nevertheless, there has beenno hesitation in introducing advanced apparatus and concepts. If stu-dents are not given an opportunity to become familiar with a varietyof modern developments and new techniques, they will be handicappedin their later practice of chemistry. Space for additional material hasbeen obtained by abbreviating parts of the last edition and omittingsome of the older classical experiments which have found their way into


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VI PREFACEfrom liquid-vapor equilibria and from freezing-point equilibria, thermo-dynamic analysis of the mechanical properties of rubber, and spectro-photometric studies of chemical equilibrium.A new practice is introduced in this edition. Students should haveexperience in making detailed calculations from exact data obtained withexpensive apparatus such as high-resolution infrared spectrometers,nuclear-magnetic-resonance spectrometers, and microwave spectrom-eters. Such elaborate equipment is not generally available in the teach-ing laboratories of physical chemistry. Accordingly, an experiment hasbeen introduced in which the student does no experimental laboratorywork but is provided instead with raw data and charts obtained with theseinstruments, from which he calculates various molecular parameters.The second part of the book describes apparatus and techniques, par-

ticularly for more advanced work. It is designed not only to encouragestudents to undertake special work, but to aid them in later years in thesolution of their laboratory problems. No claim whatsoever is madefor completeness. In their selection of material the authors have beenguided by their own experience. The difficulty of selection increases witheach new edition because the literature on new apparatus and techniquesis expanding at an explosive rate, and because commercial instrumentmanufacturers have developed so many new and improved products.Many comprehensive descriptions of experimental methods may be foundin Weissberger's "Technique of Organic Chemistry."

This edition, like its predecessors, owes much to many peoplecol-leagues at the University of Wisconsin, students, laboratory assistants,and teachers in other universities and collegeswho, over the years,have offered thoughtful criticisms and provided many worthwhile sug-gestions for improvements. The authors greatly appreciate this helpand welcome further suggestions for future editions. They are indebtedto Professors John D. Ferry, Lawrence F. Dahl, Edward O. Stejskal,Monroe V. Evans, John L. Margrave and other staff members for criticalreading of parts of the manuscript. They wish to acknowledge the helpof Lawrence Barlow, Lee Thompson, and others connected with the teach-ing of physical chemistry in this laboratory. They are indebted to HarryA. Schopler for many of the drawings retained from earlier editions, andto Mrs. Irene Frey for careful typing and preparation of the manuscript.


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Symbols

A reciprocal moment of inertia, absorbancyA angstromB rotational constant, reciprocal moment of inertiaC heat capacity, capacitanceD diffusion coefficientE potential difference, voltage, electric fieldEa Arrhenius activation energyF faradayG Gibbs free energyH enthalpy, magnetic field strengthI ionic strength, moment of inertia, intensity of light, current,

angular momentum (nuclear spin)J angular momentum (generalized), spin-spin coupling constant,

rotational quantum numberK equilibrium constantK Kelvin scaleKb boiling-point elevation constantKf freezing-point depression constantL angular momentum (orbital)M molecular weightM molar scaleiV Avogadro's numberP pressure, polarization, angular momentum (molecular rotation)R gas constantR Rydberg constantS entropyT absolute temperature, Kelvin scaleT+, T- transference numberU internal energy, potential energy function for diatonic molecule

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CHAPTER 2

Thermochemistry3. HEAT OP COMBUSTION: BOMB CALORIMETERThe enthalpy of combustion is determined in this experiment by means

Of a bomb calorimeter.Theory. 3 ' 6 . 8 The standard enthalpy of combustion for a substance isdefined as the enthalpy change, A#, which accompanies a process inwhich the given substance undergoes reaction with oxygen gas to formspecified combustion products [such as C0 2 (ff), H 20(Z), N 2 (ff), S0 2 (ff)],all reactants and products being in their respective standard states atthe given temperature T. Thus the standard enthalpy of combustionof benzoic acid at 298.15K is AH298.i5 for the process

C 6H5C02H(s) + \5-0 2 (g) = 7C02 (9) + 3H20(Z) (1)with reactants and products in their standard states for this temperature.As will be shown below, the enthalpy of combustion can be calculatedfrom the temperature rise which results when the combustion reaction

occurs under adiabatic conditions in a calorimeter. It is important thatthe reaction in the calorimeter take place rapidly and completely. Tothis end, the material is burned in a steel bomb with oxygen under apressure of about 25 atm. A special acid-resistant alloy is used for theconstruction of the bomb because water and acids are produced in thereaction.In the adiabatic-jacket bomb calorimeter (Fig. 5) the bomb is immersed

in a can of water, fitted with a precise thermometer. This assembly is


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20 LABORATORY EXPERIMENTS [EXP. 3(With the Parr bomb, one ignition wire is connected to the terminal onthe cover, the other to the can, which makes electrical contact with thebomb.)The water in the can must cover the bomb. If gas bubbles escape, the

assembly ring may require tightening, or the gaskets may need to bereplaced.The cover of the adiabatic jacket is set in place, and the thermometers

lowered into position. The stirrer is started, and the jacket temperatureis then adjusted to within 0.03 of that of the water in the can. Thethermometer in the can is read for a few minutes to be sure that equi-librium has been attained. This temperature is recorded as the initialtemperature T\. The ignition switch is then closed until fusion of thewire is indicated by glowing of the lamp (Fig. 7). However, the switchshould not be held closed for more than about 5 sec, because damage tothe ignition unit or undue heating by passage of current through the watermay result.

If combustion has occurred, the temperature of the water in the canwill be seen to rise within a few seconds. Otherwise the leads should beexamined, the voltage output of the ignition circuit checked, or the bombopened and examined for possible sources of trouble.After a successful ignition, the temperature of the calorimeter risesquickly. The jacket temperature should be kept as close as possible tothat of the can. After several minutes the rate of change of the tempera-ture becomes sufficiently small that the difference between the can andjacket temperatures can be reduced to a few hundredths of a degree. Thefinal steady temperature of the can is then recorded as T2 .Next, the bomb is removed, the pressure relieved by opening the valve,and the cover removed. If the sample contained nitrogen, the acidresidue in the bomb is washed quantitatively with water into a flask andtitrated with 0.1 N NaOH. However, the nitric acid titration may beomitted in work of moderate accuracy if the only source of nitrogen isthe air initially present in the bomb. In any case, the length of theresidue of unoxidized fuse wire is measured. The bomb and calorimeterare carefully cleaned and dried after each experiment.Two runs are made with benzoic acid for determination of the heat


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EXP- 4] THERMOCHEMISTRY 23The differentiation indicated on the right side of Eq. (1) may be carriedout explicitly to obtain

/dAH,\ /3AffM.\Vdnrjr^ = ** + m \rmr) (2)Note that all the quantities on the right side of Eq. (2) are dependent onmolality, so that the differential heat of solution also is a function ofmolality.The magnitudes of these heats of solution depend specifically on thesolute and solvent involved. The value of the heat of solution at highdilutions is determined by the properties of the pure solute and by theinteractions of the solute with the solvent. As the concentration of thesolution increases, the corresponding changes in the differential andintegral heats of solution reflect the changing solute-solvent and solute-solute interaction effects.

For the interpretation of heat-of-solution data for some systems, it isinstructive to compare the results with the behavior predicted for idealsolutions. An ideal solution may be defined as one which obeys Raoult'slaw over the entire range of concentrations being considered. It can beshown3 that the mixing, at constant T and P, of pure liquid solvent withsuch a solution produces no change in enthalpy. From this it followsthat for the case of a liquid solute, dissolving to form an ideal solution,Aifi.s. = 0. For the case of a solid solute which dissolves to form anideal solution, ATi. a . equals the molar heat of fusion to give the (super-cooled) liquid at the temperature of the solution. Such behavior isapproximated in some actual cases, which involve nonelectrolyte solutesand nonpolar solvents (e.g., naphthalene in benzene). For electrolytesolutes the actual behavior is very different from the ideal case, becauseof marked solute-solvent and solute-solute interactions.The integral heat of dilution AHD.m^m, between two molalities mi andm2 is defined as the heat effect, at constant temperature and pressure,accompanying the addition of enough solvent to a quantity of solutionof molality mi containing one mole of solute to reduce the molality to thelower value m2 . The process to which the integral heat of solution atmolality m2 refers is equivalent to the initial formation of the more


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EXP- 4] THEBMOCHEMISTBY 27through the sample tube. The emptied weighing bottle is set aside to bereweighed later. The heating-circuit switch is then closed. The heatingcurrent is recorded, and any salt adhering to the surface of the filling tubeis pushed down with a blunt glass rod or a camel's-hair brush.The galvanometer deflection is checked at frequent intervals; thebridge dial settings are not changed. When the unbalance has beenreduced far enough so that the galvanometer light spot remains on thescale, the galvanometer circuit switch is closed and the heating currentand timer are turned off. The number n of scale divisions traversedthereafter by the spot due to the lag in the heater and thermometer isnoted. The heating current and timer are switched on again and turnedoff when the light spot has reached the point n scale divisions short of theinitial balance reference point. The spot will then come to rest veryclose to the latter, and in this way the final temperature of thesolution is matched to the initial temperature. The total heating timeis recorded.The thermometer bridge balance is checked, and the second deter-

mination is made as above with the addition of the 4-g sample. Sinceheat exchange with the surroundings is influenced by the magnitude ofthe temperature differential between the calorimeter and the room, thesolute may profitably be added gradually during the heating periodrather than all at once. The remaining samples are used in turn toextend the concentration range studied to near 2 molal. The emptyweighing bottles are then reweighed.To determine the work of stirring per second, the solution is stirred fora time of the order of 15 min and the resulting change in the thermistorresistance, ARt , is noted. Next, a measured current is passed through theheating coil for about one minute, and the change ARt again noted.From these data, the work of stirring per second may be calculated if it isassumed that the resistance of the thermistor Rt varies linearly withtemperature. This assumption is entirely satisfactory over the verysmall temperature intervals involved.It is wise to make one or two test experiments to learn the propertechnique.Calculations. For this experiment the first law of thermodynamics


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EXP. 5] THERMOCHEMISTRY 29

References1. Benson and Benson, Rev. Sci. Instr., 26, 477 (1955).2. Campbell and Campbell, J. Am. Chem. Soc, 62, 291 (1940).3. Lewis and Randall, "Thermodynamics," 2d ed., rev. by Pitzer and Brewer,

McGraw-Hill Book Company, New York (1961).4. Selected Values of Chemical Thermodynamic Properties, Natl. Bur. Standards

(U.S.), Circ, 500 (1952).5. Sturtevant in Weissberger (ed.): "Technique of Organic Chemistry," 3d ed.,Vol. I, Pt. I, Chap. 10, Interscience Publishers, Inc., New York (1959).6. Williamson, Trans. Faraday Soc, 40, 421 (1944).7. Zaslow, J. Chem. Educ, 37, 577 (1960).

5. HEAT OF REACTION IN SOLUTION:CONSTANT-PRESSURE CALORIMETERThe enthalpy change for a reaction in solution at constant pressure and

temperature is found from the measured temperature change whichoccurs when the reaction takes place in a thermally insulated vessel.Theory. The present experiment is concerned with the determination

of the enthalpy change for a chemical reaction in solution. As anexample, consider the reaction

Hg++(aq) + 2Cl-(aq) = HgCl 2(aq) (1)where the symbol Hg++(aq) means one mole of Hg++ in an infinitelydilute solution in water solvent, and HgCU(aq) refers to a mole ofundissociated HgCU in solution at infinite dilution. The value of AHfor process (1) at a given temperature is practically equal to AC/, thedifference in energy between reactants and products. Because of thespecification attached to Eq. (1) that reactants and products be atinfinite dilution, the enthalpy change equals AH, the standard-stateenthalpy change for the given temperature.As the measurement cannot actually be performed on infinitely dilute

solutions, recourse is had to a scheme such as that depicted in Fig. 10.Hg+^aq) + 2N03-(aq) + 2Na+(aq) + 2Cl"(aq) -ML, HgCl2 (aq) + 2Na+(aq) + 2N03>C|)


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34 LABORATORY EXPERIMENTS [EXP. 5


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the case, then 0% for the solvent is just its molar heat capacity. Forpure water, this is

CP = (0.9989) (18.016) = 17.996 cal deg- 1 mole" 1 at 25CApparatus. Calorimeter (Fig. 13) consisting of 500-ml vacuum bottle, sensitive

(0.01) thermometer, stirrer, special inner flask, and 15-watt heater coil; source ofwater-saturated, pressurized air; 6 or 12 volts d-c supply for heater; heater timer;double-pole double-throw switch controlling both heater and heater timer ; additionaltimer or clock with sweep second hand ; calibrated ammeter ; potentiometer ; precisionpotential divider ; reactant solutions, 500 ml of [NaCl, 100H2O], 100 ml of [Hg(NOs )2,50HiO].Procedure. The calorimeter is shown in Fig. 13. A vacuum bottle

provides a simple yet effective means of keeping the rate of heat exchangewith the environment to a very lowvalue. The cover is fitted with athermometerf sensitive to 0.01 anda polystyrene stirrer. The innerPyrex vessel serves to hold one react-ant separate from the other, whilepermitting the two solutions to cometo the same temperature before thereaction takes place. At the propertime, air pressure is used to force thereactant out of the inner flask.Water-saturated air should be usedto minimize the cooling effect ofevaporation.A simpler method which yieldssatisfactory though somewhat lessaccurate results is to place the re-actants initially in separate Dewarflasks, in one of which a heating coilis immersed, and to mix the two solu-tions at the proper time by pouringone reactant into the other. Theproduct mixture should then be in

Dewarflask -E^U-1 Reactant 2

Reactant 1

Fig. 13. Vacuum-flask calorimeter forsolution reactions.


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Practical Applications. The study of heats of reaction in solution has con-tributed to an understanding of the behavior of some systems, particularly electro-lytes. Enthalpy data for reactions are used extensively for calculating the rate ofchange of equilibrium constant with temperature, through the relation d In K/dT =AH/RT2 . Enthalpy data may also be combined with entropy data for the calcula-tion of K at a single temperature by use of the relation BT In K = AG = AH" TAS".Suggestions for Further Work. It is interesting to study heats of neutralization

of a series of weak acids by NaOH. Miller, Lowell, and Lucasse2 suggest the acidssulfamic, acetic, monochloroacetic, oxalic, and tartaric. The results may be com-bined with the accurately known enthalpy change for the process

H+(aq) + OH-(aq) = H 2to obtain the enthalpy changes for dissociation of the acids at infinite dilution.While the association of ions to form the weak acids could be studied directly by theuse of reactions of the sodium or potassium salts with a strong acid, the neutralizationreactions offer the advantage of not involving ternary (three-component) solutions,for which heat-capacity and heat-of-dilution data are usually not available.

Pattison, Miller, and Lucasse3 describe procedures for determining heats of severaltypes of reactions. A few examples are:

1. Gas evolution: decomposition of H2O22. Oxidation-reduction: reaction of KBrOa with HBr3. Precipitation: reaction of MgSCh with NaOH4. Dilution: H2S04 with water; ethyl alcohol with water5. Organic reaction: reaction of hydroxylamine with acetoneAn instructive exercise is the following. Given4 the enthalpy of formation of

Pbls(c), devise a suitable procedure and then use it to determine the enthalpy offormation of the dissolved, ionized salt at infinite dilution in water, Pb++(aq) +2I~(aq). The symbol (c) indicates the crystalline state.

Approximate values can be obtained in favorable cases for the heat of formation ofa complex ion from its constituents even though the stability of the complex ion isnot high. An illustration is the determination of AH for the reaction

Ni++ + 6NH S -* Ni(NH 3)J +by Yatsimirskii and Grafova. 8 A large excess of NH 3 is used, and the heat of reactionis measured as a function of the concentration of NH3 in the final solution. Caremust be taken in interpreting such data if there is any possibility that several speciesof complex ions are present in significant amounts.

References


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where Te is the critical temperature. The compressibility factor is cal-culated for a particular point on the vapor-pressure curve, and the slopeof the plot of In P versus 1/f at that point is used. The ideal-gas law isused to calculate V in the factor (1 Vi/Vv).

Practical Applications. Vapor-pressure measurements are important in alldistillation problems and in the calculation of certain other physical properties.They are used in the correction of boiling points and in the recovery of solvents.The concentration of vapor in a gas space may be regulated nicely by controlling thetemperature of the evaporating liquid. Humidity conditions, which are so importantin many manufacturing processes, depend largely on the vapor pressure of water.

Suggestions for Further Work. The vapor pressures of other liquids may bedetermined, using, if possible, liquids whose vapor pressures have not yet been recordedin tables. The sublimation temperature of a solid may be obtained by covering thethermometer bulb with a thin layer of the solid.The vapor pressure may be determined by an entirely different method, evaluating

the amount of liquid evaporated by a measured volume of air, as described in Chap. 21.References

1. Brown, J. Chem. Educ., 28, 428 (1951).2. Ramsay and Young, /. Chem. Soc., 47, 42 (1885).3. Thomson in Weissberger (ed.): "Physical Methods of Organic Chemistry,"Vol. I, Pt. I, Chap. 9, Interscience Publishers, Inc., New York (1959).4. Tobey, /. Chem. Educ, 35, 352 (1958).5. Willingham, Taylor, Pignocco, and Rossini, J. Research, Natl. Bur. Standards

(U.S.), 35,219(1945).

7. KNUDSEN SUBLIMATION-PRESSURE MEASUREMENTThis experiment illustrates a method for the determination of the

sublimation pressure of a solid and provides experience in the use of high-vacuum equipment.Theory. The rate of escape of molecules of a gas through a small

hole into a vacuum is directly proportional to the pressure under certainconditions, and the proportionality constant may be calculated fromkinetic theory.4,68 In order to obtain a simple relation it is necessarythat the pressure outside of the hole be sufficiently low so that the mean


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EXP - 7J VAPOR PRESSURES OF PURE SUBSTANCES 471. The water to the condenser of the mercury diffusion pump is


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turnedon.2. Stopcock G is opened, the water aspirator is turned on full force,and then stopcock G is closed.3. A slush of finely powdered dry ice in trichloroethylene is prepared inthe Dewar flask. The dry ice, as obtained from the mechanical crusher,

is too coarse for this purpose and should be reduced to fine powder byplacing a cupful at a time in a towel and beating it with a hammer. Thefinal mixture should contain enough liquid so that it can be placed aroundthe cold trap without danger of breakage. The Dewar flask should bealmost half filled. When the bath has been prepared, it is placed verycarefully around the cold trap.4. Initially stopcocks A, D, E, F, H, J, K, and L should be closed.The forepump is started. E and then D are opened to bypass thediffusion pump. Pumping is continued for about one minute, until thepump is relatively quiet. It is well to listen to the pump while openingstopcock H. If the right-hand portion of the system was full of air

initially, the characteristic sound of large quantities of air passingthrough the forepump will be heard. This sound should be remembered.5. A test is carried out to see what the pressure is after the forepumphas been running about 10 to 15 min:The stopcock A to the McLeod gauge is opened cautiously.a. If the mercury begins rising in the McLeod gauge, this stopcock isclosed and an instructor is consulted.b. If the mercury does not begin rising into the gauge, stopcock B isslowly opened to the atmosphere. The mercury will then begin to rise inthe McLeod. The stopcock is left open until the mercury is about 2 cmabove the midpoint of the large glass tube marked C on the diagram.The stopcock B is closed. The mercury will continue to rise until itreaches the capillaries. The pressure can now be read.To pull the mercury down, stopcock B is opened to the aspirator. (Theaspirator is left on throughout the experiment.) Then stopcock A is

closed. When the mercury is down, stopcock B is closed.6. When the pressure is below 100 n, stopcock E is turned to open thediffusion pump to the forepump. By listening to the forepump the pres-

48 LABORATORY EXPERIMENTS [EXP. 7result from a leak, from faulty operation of the pumps, or from outgassing.


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If the pressure is below 1 n, the experiment is continued. The mercurylevel is lowered and the McLeod gauge is closed off by shutting stopcocksA and B. These are not opened again during the experiment.B. Preparing the Knudsen Cells. Two Knudsen cells are used. A

sketch showing the design of a Knudsen cell appears in Fig. 16a. Thetop plate is a thin disk of brass (about 0.002 in. thick) with an orifice inthe center. The orifice can best be fabricated by means of a drill press,with the top plate sandwiched between two Ts-in. plates of brass theupper of which has a pilot hole to guide the drill.

Because of a tendency for the orifice to collect obstructing particles ofdust, it should be washed with acetone and carefully blown out withfiltered compressed air before the plate is inserted into the effusion cell.Furthermore, just before introducing each cell into the vacuum system,its orifice should be cleaned by inserting a fine wire and very lightlyrubbing it against the edges.The Knudsen cells are carefully cleaned and dried and are half filled

with finely powdered naphthalene. Top plates having the desired orificesizes (see below) are placed on top of the Knudsen cells, and the caps arescrewed on. Fingerprints are wiped off the cells, and the filled cells areweighed to 0.1 mg.

C. Placing the Knudsen Cell in the Vacuum System. Observations areto be made at four different temperatures, for which four different topplates (three orifice sizes) are provided. The largest orifice size is usedat 0C, the medium size at 15, and the two smallest sizes at 30 and 45.The two higher temperatures may be attained in the regulated water

baths, the two lower in Dewar flasks containing water to which ice maybe added from time to time as required. Two runs may be made on eachday.The following procedure is suitable for introduction of the Knudsen

cell:1. Stopcock H is closed.2. Stopcocks K and L are opened. Then J is opened slowly to admit

dry air.3. The sample tubes are removed.

EXP. 7] VAPOR PRESSURES OF PURE SUBSTANCES 497. Stopcock J is closed.


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8. Stopcocks K and L are opened.9. Stopcock H is opened. The time is recorded, as the vaporizationbegins at this time.D. Closing Down the Vacuum System. After 2 or 3 hr, the cells areremoved and the vacuum system closed down according to the following

procedure.1. Stopcock H is closed.2. Stopcock J is opened to the atmosphere. The time at which this isdone is recorded.3. Stopcocks K and L to the sample tubes are closed; then the samples

are removed. (K and L are closed to prevent moist laboratory air fromentering the manifold.)

4. Stopcocks D and E are closed to prevent the vacuum from beinglost in the diffusion pump.5. Stopcock F is opened to the atmosphere. Then the forepump ispromptly turned off.6. Stopcock G on the water trap leading to the aspirator is opened.Then the water aspirator is turned off.7. The diffusion pump heater is turned off.8. The cooling water to the diffusion pump is turned off.9. Stopcocks F and J are closed.E. Weighing the Knudsen Cells. If the cell has been at a temperature

appreciably different from room temperature, it must be allowed to cometo the temperature of the balance case before weighing.

Calculations. The vapor pressure of naphthalene is calculated fromthe rate of weight loss, using Eq. (4). The reliability of the calculatedvapor pressure is estimated.The experimental vapor pressures are compared with literature valuesby means of a plot of log P versus 1/T. The heat of sublimation permole AH is calculated from the slope of this plot, assuming AH is inde-pendent of temperature.Suggestions for Further Work. Other compounds which may be studied arebenzoic acid,


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52 LABORATORY EXPERIMENTS [EXP. 8solution of a nonvolatile solute such as sodium chloride in aqueous solu-


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tion. It is thus convenient to introduce the thermodynamic activity o< for aconstituent of a given phase by the relationw = M + RT In at = M + RT Infi (2)

where/? is the fugacity for a selected standard state for which the chemicalpotential is n. The activity a* is a ratio of two fugacities, which mayreadily be determined even when the individual fugacities involved can-not be. The standard state employed may be selected arbitrarily on abasis of practical convenience, but will normally be so chosen as to pro-vide the simplest possible relation between the activity and the con-centration of the constituent in the phase concerned. It thus becomescommon to select a different standard state for a component for eachphase in which it is present, so that the activity, unlike the fugacity,usually does not have a common Value for different equilibrium phases.

For nonelectrolytic solutions the standard state for each component isnormally taken to be the pure liquid at the temperature and pressure ofthe solution, and the activity is correlated with the concentration on themole-fraction scale by means of the activity coefficient y.

f.di = 75T = 7iXi,l (3)J i,l

For what is called an ideal solution, y( as defined above is identically equalto unity for any component at any concentration. For real solutions theactivity coefficients must be determined by experiment.The fugacity / ? for this standard state is calculated as follows. Thevapor pressure Pf{T) of pure liquid i at the given temperature is multi-plied by the fugacity coefficient &f(Pf,T) of the vapor as calculated fromthe equation of state of the vapor to obtain the fugacity of the saturatedvapor at the temperature T. This then gives the fugacity of pure liquidi for temperature T and pressure Pf. The fugacity/?* = fi,i(P,T) maythen be calculated by taking into account the difference between Pf andP, using the thermodynamic relation

EXP. 8] SOLUTIONS 53Assuming the effect of pressure on the fugacity of the pure liquid to be


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negligible,/?, - *fP? andXi,i = -pr p* Xi, T {p)

If the gas phases involved are considered to behave ideally,

Xi,i p^ Xi, v (7)For ideal liquid solutions, this desired relation between the liquid andvapor compositions further simplifies to

Xi,i = -p* Xj, (8)For real solutions the activity coefficients are functions of concentra-

tion, temperature, and pressure. For a binary nonelectrolytic solutionsystem the concentration dependence may often be represented to a gooddegree of approximation by the van Laar equations, which have beenwritten as follows by Carlson and Colburn: 2

log7i = 7 / ^\ 2 . log 72 - y- / v \ 2 (9)(>+m ^twThe van Laar coefficients A i, A 2 are functions of temperature and pres-sure. Even substantial changes in pressure have only a small effect. Thedependence on temperature is more important, but over a range of ten ortwenty degrees the resultant change in a typical activity coefficient willusually be only a few per cent.

Real solution systems vary widely in their degree of departure from theideal-solution rule, for which the boiling points of the solutions are alwaysintermediate between those of the pure liquids. In many cases, however,the deviation from ideality becomes so great that a minimum or maximum

or vapor composition.

54 LABORATORY EXPERIMENTS [EXP. 8Apparatus. Distilling apparatus as illustrated in Fig. 17; pipette of about 1 ml;

resistance wire for electric heater; step-down


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transformer (110 to 6 volts); thermom-eter graduated to 0.1; refractometer with thermostated prism; weighing bottle;benzene; ethanol.Procedure. The apparatus which is shown in Fig. 17 may be readily-

constructed from a 50-ml distilling flask. Superheating is avoided byinternal electrical heating with a resist-ance coil. An alternative apparatusis described by Rogers, Knight, andChoppin. 11The heating coil of No. 26 nichrome

wire about 14 cm long is wound in theform of a helix about 3 mm in diam-eter. It is brazed to No. 14 copper wireleads set into the cork. The coil shouldtouch the bottom. A small step-downtransformer capable of at least 25 wattsoutput is used.

Other types and sizes of resistance wiremay be used, but the current should besuch that the wire is heated to a dull redheat when out in the open air. A heaterof 2 ohms operating at 6 volts issatisfactory.A thermometer graduated to 0.1 andreading from 50 to 100 serves very well,but any accurate thermometer with large1 divisions will do. A short length ofglass tubing surrounds the bulb of the

thermometer; this enables the boiling liquid to circulate over the entirethermometer bulb. The bulb must not touch the heating coil.The arm of the distilling flask is bent upward to act as a reflux con-denser; at the bottom of the bend is a bulb of about 1 ml capacity to actas a pocket for retaining condensed distillate as it flows down from theshort condenser.

Fig. 17. Apparatus for determiningliquid and vapor compositions ofbinary systems as a function oftemperature.

EXP. 8] SOLUTIONS 55or pipette from the distillate in the pocket and then from the residue in theflask through the sidearm. The sample


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of distillate is removed by-inserting the end of the pipette through the open end of the reflux con-denser directly into the pocket below. A dry pipette should be used fortaking the samples. The refractive indices of the samples are determinedwith a refractometer. Samples for this determination may be preserved

Liquid

Refractive index scale

LCritical rays

Fig. 18. Optical path in Abbe refractometer.for a short time in small


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EXP. 8] SOLUTIONS 57is determined. Boiling points and refractive indices of the residue and


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distillate are then determined after successive additions of 0.2, 0.5, 1, 5, 5,and 5 ml of ethanol. The refractive indices are used to obtain the molefractions of ethanol in these solutions.

In order to construct a plot of refractive index versus mole per centethanol, the refractive indices are determined for the pure benzene andethanol and for a series of solutions containing accurately known weightsof benzene and ethanol. Mixtures about 5 ml in volume containingapproximately 1 volume of ethanol to 1, 3, and 6 volumes of benzene areconvenient.The boiling flask is drained and dried, and about 25 ml of ethanol is

introduced for a boiling-point determination. Boiling points and com^positions of the residue and distillate are then determined after successiveadditions of benzene as, for example, 2, 4, 5, 7, and 10 ml.The barometer should be read occasionally. In case the atmosphericpressure changes considerably, it is necessary to estimate a correction forthe boiling point, taking an average correction for the two liquids as anapproximation. Such a correction may usually be avoided by perform-ing all the distillation experiments within a few hours.Calculations. The refractive indices of the weighed samples and the

pure liquids are plotted against the compositions of the solutions expressedin mole fractions of ethanol. The composition of each sample of distillateand residue may then be determined by interpolation on this graph. Ina second graph three sets of curves are plotted

I. The boiling-point diagram for the system as determined experi-mentally. Two curves are plotted, one in which boiling temperature isplotted against the mole fraction of ethanol in the residue; in the other,the same boiling temperatures are plotted against the mole fraction ofethanol in the distillate. The composition as mole-fraction ethanol isplotted along the horizontal axis. Different symbols should be used forthe two sets of points. The significance of this graph is discussed withrespect to the feasibility of separating benzene and ethanol by fractionaldistillation.

II. The boiling-point diagram for the system as predicted by the ideal-solution rule. Points for the two curves involved may be calculated as

58 LABORATORY EXPERIMENTS [EXP. 82 in the solution having vapor pressure P at temperature T. Then the


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mole fraction X2 ,r for the equilibrium vapor phase is given byv - P- _ ^2.'-P*(^T) c>-.\A 2, -p p (,ll;

III. The boiling-point diagram for the system as predicted by thevan Laar equations for values of A h A 2 consistent with the experimentallydetermined azeotrope temperature and composition. In this calculationit will be necessary to assume that the activity coefficients are functions ofcomposition only (that is, A h A 2 constants), an approximationjustified by the small temperature range involved.The activity coefficient *n is given by the relation (7) (for the ideal-gas

approximation).XilV P

For the azeotropic solution, the mole fraction of each component has thesame value for the liquid and vapor phases; hence the activity coefficients7i,u and 72, as for the azeotropic solution are given by

p pFrom the pair of activity coefficients so calculated and the compositionof the azeotrope, the van Laar coefficients may be calculated. It isconvenient first to calculate the ratio Ai/A\.

A 2 _ X?,, log 71A x X?>2 log72 (i6)Then ^-(l+^fgy.log* ^-^ (14)Now select some concentration Xz,i,Xi,i = 1 X it i, and calculate fromthe van Laar equations 72 and 71.

P = Pi + P2 = yiXuP*(T) + ytXi,J>t(T) (15)

EXP - 8] SOLUTIONS 59The required data on the vapor pressures of benzene and ethanol maybe calculated from the following relations:


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Benzene (Ref. 12): log P = 6.90522 - 1211215t ~r~ 220.87

Ethanol (Ref. 8): log P = 8.11576 - 1595.76t + 226.5

These equations give the vapor pressure in standard millimeters ofmercury for temperature in degrees centigrade.Practical Applications. Vapor-composition curves are necessary for the efficientseparation of liquids by distillation. Fractional distillation under controlled condi-tions is essential in the purification of liquids and in many industries, such as thepetroleum industry and solvent industries.Suggestions for Further Work. Solutions of chloroform and acetone, giving amaximum in the boiling-point curve, may be studied in exactly the same manner asdescribed for ethanol and benzene.The maximum in the boiling-point curve of hydrochloric acid and water occurs at108.5 and a composition of 20.2 per cent hydrochloric acid at a pressure of 760 mm.The distillate at the maximum boiling point is so reproducible in composition at agiven pressure and so easily obtained that it may be used to prepare solutions ofHC1 for volumetric analysis. A solution of hydrochloric acid is made up roughly toapproximate the constant-boiling composition, and after boiling off the first third,the remaining distillate is retained. The barometer is read accurately, and thecorresponding composition is obtained from the literature. 1 --7Solutions of chloroform and methanol, giving a minimum m the boiling-pointcurve, may be studied by using a Weatphal density balance for determining thecompositions instead of a refractometer. A density-mole-fraction curve is plotted,

and the compositions of the samples are determined by interpolation. Since largersamples are needed for the density measurements, more material and a larger flaskare required.The gas-saturation method for vapor-pressure measurements may be used instudying binary liquids. Using this technique, Smyth and Engel 13 have determinedvapor-pressure-composition curves for a number of ideal and nonideal types.Vapor-liquid equilibria at different total pressures provide an interesting study.

The acetonitrile-water system has an azeotrope which varies considerably in composi-tion as the pressure is reduced." Othmer and Morley 10 describe an apparatus forthe study of vapor-liquid compositions at pressures up to 500 psi. The earlierpapers of Othmer may be consulted for a number of binary vapor-liquid equilibria.


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EXP. 9] SOLUTIONS 61provide good contact between the vapor and liquid phases in the column,

it is


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but undesirable for the packing to hold a large fraction of the batchbeing distilled because of the resulting decrease in sharpness of separation.A number of types of packing are listed on page 450.The effectiveness of a distilling column is expressed in terms of thetheoretical plate. The theoretical plate is a hypothetical section of

,100

90

f 70'-* 60g .c 50uu| 40.2J 30

20

10

'.: ."

-;"" :' -~ B

''." // \A1

I

'

l

O 10 20 30 40 50 60 70 80 90 100Mole per cent CC14 in liquid

Fig. 19. Vapor-liquid graph for calculating the number of theoretical plates.column which produces a separation of components such that the vaporleaving the top of the section has the composition of the vapor which is inequilibrium with the liquid leaving the bottom of the section. A columnconsisting of a simple jUcm tube 1 m long might be equivalent to only 1theoretical plate, whereas the same tube filled with adequate packing cangive the equivalent of 20 or more theoretical plates. A column with 12theoretical plates is adequate for the practical separation of benzene(bp 80.-1) and toluene (bp 110.8). The number of theoretical plates


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EXP. 9] SOLUTIONS 65plotting the temperatures of the condensing distillate against the per-centage of the total volume of the mixture distilled. The refractive


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indices of the samples of distillate are also plotted on the same graphagainst the percentage of the total volume of liquid distilled. For acolumn with a very large number of theoretical plates operated at a highreflux ratio, a square step-shaped curve is obtained. The refractive indexof the distillate changes abruptly from that of the more volatile to thatof the less volatile component when the former has all distilled out.Likewise, the temperature of distillation rises abruptly when the morevolatile component is distilled out.

Table 1. Liquid and Vapor Composition of Mixtures of CarbonTetrachloride and Benzene at 760 mm and at Temperaturesbetween the boiling points7Mole per cent CC1 in

liquidMole per cent CC1 4 invapor

13.6415.82

21.5724.15

25.7328.80

29.4432.15

36.3439.15

40.5743.50

52.6954.80

62.0263.80

72.273.3

Suggestions for Further Work. Additional pairs oi liquids may be separated byfractionation with an efficient column. A mixture of carbon tetrachloride and toluenemay be used to determine the number.of theoretical plates. The data for this systemare given in Table 2.

Table 2. Liquid and Vapor Compositions of Mixtures of CarbonTetrachloride and Toluene7Mole per cent CCL in liquid . .Mole per cent CCL in vapor . . . 5.7512.65 16.2531.05 28.8549.35 42.6064.25 56.0575.50 64.2581.22 78.2089.95 94.5597.35Some of the various types of packing referred to in Chap. 21 may be compared by

determining the H.E.T.P. for each.The value of a fractionating column depends not only on the number of theoreticalplates, but also on the amount of liquid held up by the packing. 4 '8 Equilibrium condi-tions are attained more rapidly if the holdup of the column is small. The amount ofliquid held up may be determined at the end of an experiment by removing theheating bath, taking out the column and blowing dry air through it, and condensingthe material in a weighed TJ tube surrounded by a freezing bath of dry ice. When the

66 LABORATORY EXPERIMENTS [EXP. 104. Collins and Lantz, Ind. Eng. Chem. Anal. Ed., 18, 673 (1946).5. McCabe and Thiele, Ind. Eng. Chem., 17, 605 (1925).6. Morton, "Laboratory Techniques in Organic Chemistry," McGraw-Hill Book


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Company, Inc., New York (1938).7. Perry, "Chemical Engineers' Handbook," 3d ed., Sec. 9, McGraw-Hill Book

Company, Inc., New York (1950).8. Robinson and Gilliland, "Elements of Fractional Distillation," 4th ed., McGraw-

Hill Book Company, Inc., New York (1950).9. Williams and Glazebrook in Weissberger (ed.): "Technique of Organic Chemistry,"

3d ed., Vol. IV, Chap. 2, "Distillation," Interscience Publishers, Inc., New York(1960).

10. VARIATION OF AZEOTROPE COMPOSITION WITH PRESSUREThe variation of azeotrope composition with pressure is determined

for a binary system, and the results compared with predictions based onthermodynamic principles. Measurements of heats of mixing of the twopure liquids are made, and their relation to the effect of temperature onthe activity coefficients is considered.Theory.1-3 The Gibbs free energy G of a single homogeneous phase

containing two components is a function of temperature, pressure, andthe numbers of moles i, n2 of the two components. For a change ofstate the differential change in Gibbs free energy is given by

dG = -SdT + VdP + nidrn + i^dm (1)where S is the entropy of the phase, V its volume, and m, m the chemicalpotentials of its two components. Since at any temperature and pressurethe free energy of the phase is given by

G nim + nzixi (2)the differential dG is also given by

dG = nidm + n^dya + mdrii + mdn2 (3)Combination of Eqs. (1) and (3) yields the Gibbs-Duhem equation,

SdT - VdP + mdjii + n^2 = (4)

EXP. 10] SOLUTIONS 67Consider now the simultaneous application of this relation to a liquid

phase (subscript I) and the vapor phase (subscript v) in equilibrium with


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it:SidT - VidP + Xi.tin.1 + Xt.tint.i = (6)SvdT - VvdP +~'-Xi,vdfi 1 , T + Xz.vd/xt.v = (7)

Since for equilibrium the chemical potential for each component musthave a common value for both phases,

Ml,t> = 1*1.1 = Mi M2,i = M2,i> = M2and Eqs. (6) and (7) may be combined to give

(S, - Si)dT - (t\ - Vi)dP + (X,,i - X,.,)(dMi - dm) = (8)This equation determines the relation between the changes in tempera-ture, in pressure, and in chemical potentials in going from one vapor-liquid equilibrium condition to another. Now consider a variation incomposition at constant temperature (dT = 0) away from a state forwhich the equilibrium vapor and liquid compositions are identical (theazeotropic condition), so that (X2 ,i X2,) = 0. Since in general(Vv Vi) is not zero, it follows that dP = for the indicated differentialchange, and hence there must be a minimum or maximum in the plot ofequilibrium pressure versus composition, at constant temperature, atthe azeotrope composition. Similarly, for a shift in composition atconstant pressure, since (Sv Si) ^ 0, there must be a maximum orminimum at the azeotrope composition in the plot of equilibrium tem-perature versus composition at constant pressure (the boiling-pointdiagram).The azeotrope composition changes as the equilibrium temperature andpressure are changed. If the composition is known for one pressure, thecorresponding azeotrope composition for another pressure can be esti-mated in the following way. As shown in Exp. 8, to the ideal-gasapproximation for the vapors involved, the activity coefficient for acomponent in the liquid phase is given by

7i ~ XiwlP?(T) w

68 LABORATORY EXPERIMENTS [EXP. 10which Xi,i = Xi,v . Then for the azeotropic solution


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7l,a P*(T) 72,az " p*(rp\ \*-")and hence li- = p^ (H)

72,a* "lA-UThe azeotrope composition thus is that composition for which the ratio ofthe activity coefficients of the two components is equal to the inverseratio of the vapor pressures of the two pure liquids at the temperatureconcerned. If the azeotrope composition and pressure are known for onetemperature, the coefficients Ah A 2 can be calculated for the given T, P,for the van Laar equations:

loS 7i = 7 f\, X2 log 72 = . . f2

\.~

(12)

If Ai and A 2 as thus determined are assumed to be independent of tem-perature and pressure, i.e., if the activity coefficients are considered to befunctions of composition only, then it is possible to calculate the molefraction of component 1 for the azeotrope at a different temperature asthe value of Xi,i for which Eq. (11) is satisfied at this new temperature.The new azeotrope pressure Pa can be calculated by use of the relation

7l,a. = PtiT)The accuracy of the prediction so made necessarily depends on how wellthe van Laar equations express the concentration dependence of theactivity coefficients at a given temperature and pressure; for satisfactoryresults the azeotropes involved should be reasonably intermediate incomposition.There remains the question of the validity of the assumption that the

activity coefficients are insensitive to changes in temperature and pressureat constant composition. From Eq. (12) it is seen thatAi = lim (log yi) Ai = lim (log 72) (13)

EXP. 10] SOLUTIONS 69and thus

\ dT )PtXli 2.303R T> uo -'


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where the term fix - HI actually represents the differential heat ofsolution (cf. Exp. 4) of pure-liquid component 1 in the solution of con-centration Xu at the given temperature and pressure. The temperaturecoefficient of A i is thus determined by the limiting value of the differ-ential heat of solution of component 1 at infinite dilution in component 2as solvent.

~dT/p = 2 303RT2 [i(.Xi,i)- ^lta.r-x) (16)The bracketed enthalpy term can be calculated from integral-heat-of-solution data; it can be approximated by measuring the heat of solutionper mole of component 1 in ^t sufficiently large excess of component 2 togive a very dilute product solution. Its value may be as large as severalkilocalories per mole or more, particularly when hydroxylic compounds,such as alcohols, are involved. Similarly, the pressure coefficient of A 1is found as follows/dlogjyA = 1 f/gMuN _ (d\ 1 ,\ dP )T 2.30SRTl\dPjT,^ \dP Jt^,} W

(:

and dAA = Vi(Xi,i - 0) - V*dPjT 2.Z0ZRT (19)

where V\ = Vf, the molar volume of pure liquid 1 at the temperatureand pressure of the solution. The partial molar volume of component 1at infinite dilution in component 2 as solvent, Vi(Xi,i> 0), can be cal-culated from density measurements on the solutions (cf. Exp. 13). Sincefor solutions of this type Vi changes but slightly with concentration andis usually not much different from Vf, the pressure coefficient of Ai willbe very small.

70 LABORATORY EXPERIMENTS [EXP. 10flask, which is then connected to the fractionating column. The watervalve is opened slowly to start the flow of cooling water through the dis-tilling head, which is set for total reflux. The autotransformer switch


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is turned on, and the control knob advanced from zero as required toraise the temperature of the solution to the boiling point. The boil-uprate is increased until condensation takes place in the distilling head, witha steady but not excessive stream of condensate returned to the column.When the temperature shown by the thermometer immersed in the vaporat the top of the column has become constant, indicating that equilibriumhas been reached, a preliminary sample of condensate is drained into thefirst receiver (and subsequently discarded) and the sample collector isrotated to obtain a sample for analysis in the second receiver. Thebarometric pressure and condensation temperature are recorded. Analy-sis of this sample gives the azeotrope composition for the measuredtemperature and pressure, since the fractionating column separates outthe minimum-boiling azeotrope due to its high volatility. If the analysisis not made immediately, the sample container should be tightly stoppered.The head is again set for total reflux, and the autotransformer is

turned off. The sample collecting unit is removed, and the containerholding the azeotrope sample is tightly stoppered and set aside for lateranalysis. The sample collector is cleaned, dried, reassembled, and putback in place on the column, which is then connected to the pressure-control system. The pressure in the column is then gradually reduced toabout 500 mm and maintained there. The liquid boil-up rate is increased,and the column is brought to equilibrium as before. A sample for analy-sis is then obtained as described above; the pressure and condensationtemperature are recorded. The column is reset to total reflux, and theheating mantle turned off. The pressure in the system is now raisedslowly to atmospheric. The second azeotrope sample is set aside foranalysis, and the apparatus prepared for the next run. Further deter-minations are then made for pressures of approximately 350 and 200 mm.Care should be taken to record the condensation temperature and pressurefor each azeotropic solution collected.The densities of the samples are measured by means of the Westphalbalance (page 454) or by the pycnometric method (page 90), and their


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72 LABORATORY EXPERIMENTS [EXP. 10Here Cp is the sum of the heat capacities of the product solution andcalorimeter. The former is to be calculated as the sum of the heat capaci-ties4 of the pure liquids, an approximation whichis adequate for the present


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purposes. The latter will be furnished for the unit provided. The inte-gral heat of solution is then calculated by dividing the observed AH by thenumber of moles of solute added. It is now assumed that the productsolution is sufficiently dilute so that the integral heat of solution and differ-ential heat of solution differ negligibly and their common value would notchange significantly for work at lower concentrations. The temperaturecoefficient of the van Laar parameter A\ is then calculated by use ofEq. (16) . In a similar fashion the temperature coefficient for the quantityA 2 is calculated. The implications of these results (concerning theassumption earlier made that the activity coefficients are functions ofcomposition only) are considered. It should be recognized that theheat-of-solution data, here obtained at room temperature, are subject tosome change with temperature.A summary of experimental results for the acetonitrile-water systemhas been given by Timmermans. 9 Of particular pertinence are thepapers of Othmer and Josefowitz, 6 Maslan and Stoddard, 6 and Vierk. 9 - 10The heat-of-mixing data given in the latter reference were calculatedfrom temperature-pressure-composition data for the system and do notresult from direct calorimetric measurements.

Practical Applications. The prediction of the behavior of real-solution systemsfrom a minimum of experimental data is a common problem in modern technology.Suggestions for Further Work. The heat^of-solution measurements may bemade for a range of concentrations to permit a more accurate estimation of the dif-

ferential heats of solution at infinite dilution. Similar measurements may be madefor the system acetonitrile-ethanol, and the results compared with those calculatedfrom the heats of mixing given by Thacker and Rowlinson. 7 The term heat of mixing,Aflm , as regularly employed in solution thermochemistry, refers to the increase inenthalpy in the formation of one mole of solution from the pure-liquid components atthe temperature and pressure of the solution.

References1. Guggenheim, "Thermodynamics," pp. 188-190, Interscience Publishers, Inc.,New York (1949).

EXP - H] SOLUTIONS 738. Timmermans, "Physico-chemical Constants of Pure Organic Compounds,"Elsevier Press, Inc., Houston, Tex. (1950).9. Timmermans, "Physico-chemical Constants of Binary Systems," Vol. IV, Inter-science Publishers, Inc., New York (1960).


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10. Vierk, Z. anorg. Chem., 261, 283 (1950).

11. ELEVATION OF THE BOILING POINTThe boiling points of a solution and of the pure solvent are determinedand used for calculating the molecular weight of a nonvolatile solute.Theory. When a nonvolatile solute is dissolved in a solvent, the vapor

pressure of the latter is decreased; as a consequence, the boiling point ofthe solution is higher than that of the pure solvent. The extent of theelevation depends upon the concentration of the solute, and for dilute,ideal solutions it may be shown that

Th - r = 8 = Kbm (1)where Kh = RTl/1000\v (2)and To = boiling point of pure solventTb = boiling point of solution of molality m, at same pressureX = enthalpy of vaporization of pure solvent per gram at temper-

ature ToKb = molal boiling-point elevation constantNote that Kt is a constant characteristic of the solvent. Relation (1)permits calculation of the molecular weight of the solute, since it maybe transformed into the equivalent form- (3)where is the elevation of the boiling point for a solution containing ggrams of solute of molecular weight M in G grams of solvent of boiling-point elevation constant Kb.

It should be noted that even for ideal solutions the foregoing relationsare valid only if the solution is also dilute, i.e., if the mole fraction of


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76 LABORATORY EXPERIMENTS [EXP. 11If the liquid does not pump steadily over the thermometer bulb, the

rate of heating is changed. The rate should be adjusted so thatebullition takes place primarily within the funnel in order to produce the


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most efficient pumping action. The rate of heating should be steadyand should not be so great as to drive the liquid condensate film too closeto the upper end of the condenser, since this may result in loss of solventand also cause superheating. A metal chimney placed around theburner helps to reduce fluctuations in the rate of heating.An absolutely constant boiling-point reading cannot be expected, butwhen equilibrium has been reached, the observed temperature willfluctuate slightly around a mean value and in particular will not show aslow drift except when there is a corresponding drift in barometricpressure. The thermometer, which must be handled carefully, is tappedgently before a reading is taken. Since the boiling point is sensitive tochanges in pressure, the barometer should be read just after the tempera-ture reading is recorded.

After the boiling point of the pure solvent has been determined, theliquid is allowed to cool. The condenser is then removed, and a weighedquantity of benzoic acid sufficient to produce a 2 to 3 per cent solution isadded. To prevent loss, the benzoic acid is made up into a pellet in apellet machine or is placed in a short glass tube and rammed tight with acentral rod acting as a plunger.

After the boiling point of the pure solvent has been determined, theliquid is allowed to cool, the condenser is removed, and the first solutesample is added; alternatively, if it is feasible to do so, the pellets may bedropped in through the condenser. The steady boiling point of the solu-tion is then determined in the manner previously described. Additionaldeterminations are made by adding more pellets. In each case, thebarometric pressure is recorded just after the temperature reading hasbeen made.When possible, a second series of measurements should be made, start-ing again with pure solvent and covering the same general range of con-centrations as before. In this way, a valuable check is obtained onreproducibility.

Serious error can result from failure to wait for equilibrium to be

EXP. 11] SOLUTIONS 77taken to minimize these effects. For work of highest accuracy, one mayuse an apparatusf in which provision has been made for withdrawing asample of the liquid just after the temperature measurement has been


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made. The molality is then found by weighing this sample, evaporatingthe solvent, and then weighing the residue.Calculations. The molecular weight is calculated by means of Eq. (3),from the values for Kh shown in Table 1, corrected by use of the pressure

coefficient given in the last column, unless the correction is negligible.Table 1. Molal Boiling-point Elevation Constants4

Solvent Boiling point,C at 760 mm at 760 mm AXi,AP (mm)Acetone 56.080.2

155.876.760.278.334.464.7100.0

1.712.536.205.033.631.222.020.830.51

0.00040.00070.00160.00130.00090.00030.00050.00020.0001

Benzene. ......BromobenzeneCarbon tetrachlorideChloroformEthanolEthyl etherMethanolWater

The necessary correction to required by a difference in the barometricpressures at the times the boiling points of the solvent and solution wererecorded may be made by use of Eq. (1) of Exp. 6. For this purpose, it isassumed that dp/dT may be set equal to Ap/AT. The value of X may bevalues taken from a handbook.The calculated apparent molecular weight values are graphed againstmolality and compared with the formula weight. Any discrepanciesamong these values should be discussed in the light of the estimated exper-imental error.Practical Applications. Many materials cannot be vaporized for direct determi-nations of the vapor density without decomposition. In such cases the material isdissolved in a suitable solvent, and the elevation of the boiling point furnishes a rapid


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84 LABORATORY EXPERIMENTS [exp; 12efficiently. If a thermistor is used, the resistance measurement may bemade by means of a d-c Wheatstone bridge and sensitive galvanometer,

as outlined in Exp. 4. The temperature depend-ence of the thermistor resistance may be assumed


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to have the formR = Be*iT (30)

where R is the resistance of the thermistor at thetemperature T. The parameters B and maybe evaluated by measurement of R at severalknown temperatures, which for convenience maybe chosen in the range to 25C.To the vacuum bottle is added distilled waterand an equal volume of clean cracked ice. Thewater used should be prechilled, to minimize theamount of impurities set free by melting of ice.The freezing mixture should cover the thermom-eter bulb or thermistor without being too close tothe cork. If the ice is not of adequate purity,the observed freezing point will slowly decreasewith time as occluded impurities are freed bymelting; in this event, it may be necessary toprepare ice by freezing distilled water.The ice and water are stirred vigorously untila steady temperature is attained. This value isthen recorded. If a mercury thermometer isused, it should be gently tapped before being read.The water is drained off and replaced by achilled solution of the specified solute in distilledwater at a concentration of about 3 molal.The solution and ice are stirred vigorously untila constant temperature is reached, whereupon thetemperature is recorded and a sample is with-drawn quickly with a 10-ml pipette; the tip of the pipette should be

held near the bottom of the flask to avoid getting pieces of ice. The

Fig. 22. Freezing-pointapparatus.


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86 LABORATORY EXPERIMENTS [exp. 12Equation (31) is obtained by expanding T as a function of R in a Taylorseries about T ; the derivatives required are most easily calculated from

dRdT Rrp2 (32)


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etc. The first neglected term inside the brackets is of the order of[(Rf - R )/RoV.The molalities and freezing-point depressions for the various samplesare then tabulated. A graph of 6/m versus m is prepared to illustrate thedetermination of v from Eq. (22). The solvent activity and the osmoticcoefficient are calculated for each molality and graphed as functions ofm. For this calculation it is convenient to express the coefficient inEq. (12) in terms of Kf . Values of Kf and of several other pertinentquantities appear in Table 1. For most nonelectrolyte solutions in theconcentration range considered here, the Lx term in Eq. (13) is quitenegligible.

Table 1. Freezing-point Depression Constants and Related Data

Solvent Freezing point,tC cal/g Kf cal/deg moleAcetic acid" 16.61

5.525.1-22.96.5

25.10.00

46.630.121.883.97.44.27

79.72

3.585.128.37

3220.037.71.860

9 4teri-Butanol cCarbon tetrachloride"Cyclohexane6 1.1Cyclohexanol1'Water" 8.911

'Selected Values of Chemical Thermodynamic Properties, Natl. Bur. Standards(.U.S.), Circ, 500 (1952).

6 Lange, "Handbook of Chemistry," 8th ed., Handbook Publishers, Inc., Sandusky,Ohio (1952).c Getman, J. Am. Chem. Soc., 62, 2179 (1940)."Wilson and Heron, J. Soc. Chem. Ind. (London), 60, 168 (1941).The calculation of solute activities from these data is outlined below.

EXP.'13] SOLUTIONS 87constants of weak acids can be found from freezing-point data, since the apparentmolecular weight M2 can be expressed in terms of the fraction dissociated.Suggestions for Further Work. If the data are sufficiently accurate, soluteactivities may be calculated from the data obtained in this experiment; Eq. (26) isapplicable provided the graph of 0/m versus m indicates v = 1 to within experimental

error. It may be desirable


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to draw a smooth curve, representing the data, on the6/m versus m graph and to use selected points from this curve (rather than raw datapoints) for the calculation of solute activities. In any case, the quantities ( 1)and ( l)/ra are tabulated and graphed against m. For selected values of m, theintegral in Eq. (26) is evaluated from the graph, and y is calculated from Eq. (26).A good example of a dissociated solute which is readily studied is KC1. In thiscase, the samples, after being withdrawn into a 5-ml pipette, should be dischargedinto previously weighed bottles and then weighed. The weight of KC1 in eachsample is subsequently found by titrating the samples with 0.1 N AgNOs, withabout 0.2 ml of 5 per cent potassium chromate solution added as indicator. TheAgNOs solution is standardized with weighed samples of KC1.

References1. Harned and Owen, "Physical Chemistry of Electrolyte Solutions," 3d ed., Rein-hold Publishing Corporation, New York (1958).2. Lewis and Randall, "Thermodynamics," 2d ed., pp. 404-409, 412-413, rev. by

Pitzer and Brewer, McGraw-Hill Book Company, Inc., New York (1961).3. Robinson and Stokes, "Electrolyte Solutions," 2d ed., Academic Press, Inc., NewYork (1959).4. Wall, "Chemical Thermodynamics," pp. 353-355, 357-358, W. H. Freeman andCompany, San Francisco (1958).13. PARTIAL MOLAL PROPERTIES OF SOLUTIONSThe accurate determination of the density of a liquid and the precise

mathematical treatment of the properties of solutions are studied.Theory.3_67 The quantitative study of solutions has been greatlyadvanced by the introduction of the concept of partial molal quantities.A property of a solution, e.g., the volume of a mixture of alcohol andwater, changes continuously as the composition is changed, and con-siderable confusion existed formerly in expressing these properties as afunction of composition. A partial molal property of a component of asolution is defined as follows. Let Y represent any extensive property ofa binary solution; at constant temperature and pressure, Y then will be a


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90 LABORATORY EXPERIMENTS [EXP. 13pycnometer of the Weld or Ostwald-Sprengel type shown in Fig. 23 maybe used. The pycnometer is dried carefully, weighed, then filled withdistilled water, and placed in the thermostat for 10 or 15 min.The Weld pycnometer is initially filled to bring the liquid level abouthalfway up the throat T of the reservoir R. The pycnometer is placed in


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the thermostat with the cap C in position to prevent evaporation fromthe exposed liquid surface. When temperature equilibrium has beenreached, the cap C is removed and the plug P is inserted. A moderate

(a) (6)Fig. 23. (a) Weld pycnometer; (b) Ostwald-Sprengel pycnometer.

pressure is sufficient to seat the plug firmly. Any excess liquid on the tipof the plug is wiped off with a piece of filter paper, care being taken toavoid removing liquid from the plug capillary in the process. Thepycnometer is then removed from the thermostat, wiped dry with a lint-less cloth, and the (dried) cap C put in place. It is allowed to stand inthe balance case for a few minutes before being weighed.With the Ostwald-Sprengel pycnometer, the quantity of liquid is

adjusted so that the liquid meniscus is at the mark on the horizontalcapillary when the other capillary arm is filled. This adjustment may


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92 LABORATORY EXPERIMENTS [EXP. 132. Gucker, /. Phyg. Chem., 38, 307 (1934).3. Klotz, "Chemical Thermodynamics," Prentice-Hall, Inc., Englewood Cliffs, N.J.

(1950).4. Lewis and Randall, "Thermodynamics," 2d ed., rev. by Pitzer and Brewer,

McGraw-Hill Book Company, Inc., New York (1961).5. MacDougall, "Thermodynamics and Chemistry," 3d ed., pp. 23fl\, John Wiley &


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Sons, Inc., New York (1939).6. Osborne, McKelvey, and Bearce, Natl. Bur. Standards (U.S.), Bull, 9, 424 (1913).7. Wall, "Chemical Thermodynamics," W. H. Freeman and Company, San Francisco

(1958).8. Wood and Brusie, /. Am. Chem. Soc., 65, 1891 (1943).


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94 LABORATORY EXPERIMENTS [EXP. 14The equilibrium is sometimes reached very slowly, and it may be

necessary to raise the temperature or to use a catalyst to hasten theapproach to equilibrium. In this experiment the reaction is catalyzedby hydrochloric acid. Its concentration is great enough to change thecharacter of the water and alter the numerical value of the equilibrium


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constant, 6 but the results are fairly constant for a given concentrationof hydrochloric acid. The hydrochloric acid is added merely as a catalystto hasten the reaction, and it takes no part in the stoichiometric reaction.Apparatus. Burette; 5-ml pipette; 2-ml pipette; 1-ml pipette; fourteen 50-ml

glass-stoppered bottles; 0.5 N sodium hydroxide; phenolphthalein, ethyl acetate,concentrated hydrochloric acid, glacial acetic acid, absolute ethanol.

Procedure. The following solutions are prepared in 50-ml glass-stoppered bottles.

a. 5 ml 3 N HC1 + 5 ml waterb. 5 ml 3 N HC1 + 5 ml ethyl acetatec. 5 ml 3 N HC1 + 4 ml ethyl acetate + 1 ml waterd. 5 ml 3 N HC1 + 2 ml ethyl acetate + 3 ml watere. 5 ml 3 N HC1 + 4 ml ethyl acetate -f- 1 ml ethanol/. 5 ml 3 N HC1 + 4 ml ethyl acetate + 1 ml acetic acidg. 5 ml 3 N HC1 + 4 ml ethanol + 1 ml acetic acidDuplicate determinations are made. Each of the bottles is stoppered

immediately and allowed to stand in the desk for at least 48 hr and prefer-ably for a week, with occasional shaking. It is necessary that the stop-pers fit tightly to prevent evaporation. A thermostat is unnecessarybecause this equilibrium is affected only slightly by temperature changes.The weight of each reactant is determined by discharging the pipette

directly into a glass-stoppered weighing bottle and weighing. In thisway the following weighings are made:

5 ml 3 N hydrochloric acid5 ml and 2 ml ethyl acetate

EXP. 14] HOMOGENEOUS EQUILIBRIA 95After standing, each solution is titrated with the 0.5 M sodium hydrox-ide, using phenolphthalein as an indicator.Calculations. The weight of water in each bottle is obtained byadding the weight of pure water to the water contained in the 3 N hydro-chloric acid. The latter is calculated by subtracting the weight of the


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hydrochloric acid, obtained by titration, from the weight of the 5 ml ofhydrochloric acid solution.The amount of acetic acid at equilibrium in each bottle is obtained bysubtracting the number of milliliters of sodium hydroxide used in solu-tion a from that used for the titration of the equilibrium solution. Insolutions / and g, acetic acid is added to the original solution, and thisamount must be used in calculating the equilibrium amounts of the otherreactants. For every mole of acetic acid produced in the reaction, 1 moleof ethanol is produced, 1 mole of water disappears, and 1 mole of ethylacetate disappears.

If the number of moles of each of the four reactants in the originalmixture and the number of moles of acetic acid produced in the reactionare known, the apparent equilibrium constant K' may be computed.It is defined by Eq. (2).As indicated in the opening paragraph, the value of K' obtained inthis way is somewhat dependent on the various concentrations.

Practical Applications. In planning any chemical synthesis, it is desirable toknow what yield of material may be expected from a given concentration of reactants.Such a calculation may be made when the value of the equilibrium constant isknown, provided that the reaction is fast enough to come to equilibrium in the timeallowed.Suggestions for Further Work. Similar experiments may be carried out withother esters.In the absence of catalyst, it is necessary to heat the mixture to about 150 insealed tubes to affect an equilibrium within a couple of days. If sufficient precautionsare taken to avoid danger from bursting tubes, the equilibrium constant may bedetermined by titrating mixtures that have been weighed out, sealed off in small glasstubes, and heated. 3The equilibrium involved in the reaction between acetaldehyde and alcohol to giveacetal and water may be studied. 2 A little hydrochloric acid is used as a catalyst,and the equilibrium concentration of the acetaldehyde is determined volumetricallyby the sulfite method, 1 with thymolphthalein as indicator.4

96 LABORATORY EXPERIMENTS [EXP. 1515. DISSOCIATION OF NITROGEN TETROXIDEThe equilibrium constant for a reaction is determined as a function of

temperature, and the corresponding heat of reaction is calculated.Theory. Nitrogen tetroxide dissociates in accordance with the


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reaction N 2 4 = 2N0 2If the equilibrium degree of dissociation is represented by a, an initialone mole of N 2 4 gives 2a moles of NO2 and 1 a mole of N 2 4 atequilibrium. The total number of moles is thus 1 + a, and the molefraction of N02 is 2a/(l + a), and that of N 2 4 is (1 a)/(l + a).When the partial pressure p of each constituent is set equal to the productof its mole fraction and the total pressure P, the equilibrium constant forthe reaction takes the form

42P1 -H

1 + aPnOs (=')Pn,o, 1 - a pKP =^ = x ; '__ ' = ^zi CD

Experimentally, a is found by measuring M, the average molecularweight of the equilibrium gas mixture. One mole of undissociated N 2 4 ,of molecular weight ilf = 92.06, dissociates to form 1 + a moles in theequilibrium mixture. Since the total weight is unchanged, the averagemolecular weight is

, Mo M - M ..M = T+^ or a = M (2)Equation (2) is used to calculate a fromM . Equations (1) and (2) applywhen the gas mixture is considered to be an ideal mixture of perfect gases.The standard Gibbs free-energy change for a reaction can be calculated

from the thermodynamic equilibrium constant.AG = -RT In K = -2.30ZRT log K (3)

The determination of the equilibrium constant at a series of temperaturespermits the evaluation of the standard heat of reaction by application of


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100 LABORATORY EXPERIMENTS [EXP. 16made of log K versus 1/T, and the equation is found for the line con-sidered to best represent the set of points. The standard heat of reaction,the standard free-energy change, and the entropy change for the reactionare calculated for 25C.

Practical Applications. The determination of equilibrium constants is of funda-mental importance in industrial work, where the yield under specified conditions


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must be known.Suggestions for Further Work. A glass-diaphragm manometer may be used

for studying this equilibrium at various pressures. 6A simple photometer may be used for determining the partial pressure of NOj inthe mixture.'The dissociation of NsCh in carbon tetrachloride solution may be studied. 1Other rapid reversible dissociations such as that of phosphorus pentachloride andammonium chloride may be studied at higher temperatures. The method of Victor

Meyer (Exp. 2) is suitable for these determinations.References

1. Atwood and Rollefson, J. Chem. Phys., 9, 506 (1941).2. Giauque and Kemp, J. Chem. Phys., 6, 40 (1938).3. Harris and Siegel, Ind. Eng. Chem., Anal. Ed., 14, 258 (1942).4. Lewis and Randall, "Thermodynamics," 2d ed., p. 561, rev. by Pitzer and Brewer,

McGraw-Hill Book Company, Inc., New York (1961).5. Verhoek and Daniels, J. Am. Chem. Soc., 53, 1250 (1931).

16. SPECTROPHOTOMETRIC DETERMINATIONOF AN EQUILIBRIUM CONSTANTThe equilibrium constant for a reaction in solution is determined from a

study of the concentration dependence of an absorption band in the spec-trum of the solution.

, Theory. 1 '2 The present experiment illustrates an important methodfor the study of chemical equilibrium in solution. The method utilizesdifferences in the light-absorbing properties of reactants and productsand is particularly appropriate for systems in which the reaction is sorapid that classical methods of chemical analysis cannot be used to findthe concentrations of the various species present.The optical absorption spectrum, i.e., the percentage transmission oflight as a function of wavelength, has been investigated for iodine in a

EXP. 16] HOMOGENEOUS EQUILIBRIA 101tative study of the intensity of absorption at the peak of this band, as afunction of concentration, it is possible to test this interpretation and toobtain a value for the equilibrium constant for the formation of thecomplex.The study is best carried out with iodine and the complexing organic

substance, mesitylene in this experiment, both present as solutes in


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dilute solution in an inert solvent such as CC14 . (The inertness of CC14,expected on chemical grounds, is verified by the absence of new absorptionbands in solutions of I 2 in CC1 4 .) The initial step involves the measure-ment of the percentage transmission, over the appropriate wavelengthrange, of three solutions: one containing I 2 solute only, one containingmesitylene solute only, the third containing both I 2 and mesitylene assolutes. An absorption band present only in the spectrum of the thirdsolution is attributed to a 1:1 complex, M I2, existing in equilibriumwith free mesitylene (M) and I2 ,

M + I* = M I2K = (C1-X)(C2~X) (1)

where C\ = total concentration of mesitylenec2 = total concentration of I2x = concentration of complex at equilibriumK = equilibrium constant

The task remaining is to verify that an equilibrium condition of the formof Eq. (1) is actually consistent with the concentration dependence of theabsorption intensity, and to evaluate K. The new absorption bandoccurs in a wavelength region in which absorption by uncomplexed iodineand mesitylene is very slight; the investigation can therefore proceedwithout serious interference from absorption due to uncomplexed solutes.

Let / and 7 be intensities of light of a specified wavelength transmitted,respectively, by solution and by pure solvent. Then the optical absorb-ancy A, defined by A=-logi- (2)/,


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EXP. 16] HOMOGENEOUS EQUILIBRIA 103carrying out this refinement is outlined under Suggestions for FurtherWork.A word of caution is in order with regard to the interpretation of theresults. While it is possible by the procedure outlined here to learnwhether or not the absorbancies are consistent with the postulated reac-

tion equilibrium, linearity of the graph does not of itself constitute proof


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that Eq. (1) represents the true state of affairs. It has in fact beenshown3 ' 4 that a functional relationship among cu c2, and A z identical withthat of Eq. (9) will exist for a system described by two stages of complexformation,

M + X=MX K^JM^LMX + X = MX2 K* = (MX,)(MX)(X)

if the ratio of the molar absorbancies of MX and MX2 happens to satisfya certain condition. (In this case A 3 , found experimentally as before,stands for the total absorbancy due to completed M.) The best proce-dure, therefore, when it can be used, is to make measurements on morethan one of the absorption bands due to the complex. The necessarycondition on the ratio of the absorbancies is likely not to be satisfied ateach of several widely different wavelengths, so that it will then becomeobvious if the system is not obeying the equilibrium equations corre-sponding to Eq. (1).

Finally, a few comments will be made about the principle and operatio

expt physical chem

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