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Lecture Notes on Physical Chemistry

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R + R A B v Δt Chemistry 223: Introductory Physical Chemistry I David Ronis McGill UniversityTable of Contents -2- Chemistry223 9.2. Reversible, Adiabatic Expansion or Compression of an Ideal Gas ... 61 9.3. Reversible, Isothermal Expansion or Compression of an Ideal Gas ... 61 9.4. EntropyChanges in the Ideal Gas Carnot Cycle ......... 62 10. Ideal Gas Carnot Engines and Efficiency. . . . . . . . . . . 64 10.1. EfficiencyofReal Carnot Engines ............ 67 10.2. The Clausius Inequality and the Second Law. . . . . . . . . 71 10.3. EntropyCalculations . ................ 73 11. The Third LawofThermodynamics . ........... 75 12. The Chemical Potential ................ 77 13. State Functions, Exact Differentials, and Maxwell Relations ..... 79 13.1. Applications to Thermodynamics: Maxwell Relations ...... 80 13.2. Maxwell Relations: AComplicated Example ......... 80 13.3. Appendix: Proof of Green’sTheorem in the Plane ....... 82 14. ThermodynamicStability: Free Energy and Chemical Equilibrium .. 85 14.1. Spontaneity and Stability Under Various Conditions ....... 85 14.2. Examples of free energy calculations ............ 89 14.2.1. Coupled Reactions ................. 90 14.2.2. General Trends . ................. 90 14.3. Chemical Equilibrium ................ 92 14.3.1. Thermodynamics of Chemically Reacting Systems ....... 92 14.4.1. Chemical Potentials in Pure Materials ........... 94 14.4.2. Chemical Potentials in Ideal Gas Mixtures ......... 95 14.5.1. Determination of Free Energies of Formation . ....... 96 14.5.2. Determination of the Extent of a Reaction ......... 97 14.5.3. Temperature Dependence of K ............. 97 P 14.5.4. Free Energy and EntropyofMixing . .......... 98 15. ThermodynamicStability . ...............100 16. Entropy&Randomness ................104 17. ElectrochemicalCells . ................107 17.1. General Considerations ................107 17.2. Concentration Cells .................109 17.3. Connection to Equilibrium Constants............110 17.4. Temperature effects .................110 18. Problem Sets . ...................111 18.1. Problem Set 1 ...................111 18.2. Problem Set 2...................112 18.3. Problem Set 3...................114 18.4. Problem Set 4...................116 18.5. Problem Set 5...................117 18.6. Problem Set 6...................119 19. Past Midterm Exams .................122 19.1. 2012 Midterm Exam .................122 19.2. 2013 Midterm Exam. ................125 19.3. 2014 Midterm Exam.................128 19.4. 2015 Midterm Exam.................131 20. Past Final Exams ..................134 20.1. 2012 Final Exam ..................134 Fall Term, 2015Chemistry 223 -3- GeneralInformation 1. General Information CHEMISTRY223: Introductory Physical Chemistry I. Kinetics 1: Gas laws, kinetic theory of collisions. Thermodynamics: Zeroth lawofthermodynamics. First lawofthermodynamics, heat capacity, enthalpy, thermochemistry,bond energies. Second lawofthermodynamics; the entropyand free energy functions. Third lawofthermodynamics, absolute entropies, free energies, Maxwell relations and chemical and thermodynamic equilibrium states. Prerequisites:CHEM 110, CHEM 120 or equivalent, PHYS 142, or permission of instructor. Corequisite:MATH222 or equivalent. Restrictions:Not open to students who have taken or are taking CHEM 203 or CHEM 204. 1.1. ContactInformation Professor: David Ronis Office: Otto Maass 426 E-mail: David.RonisMcGill.CA (Help my e-mail client direct your email; Please put CHEM 223 somewhere in the subject.) Tutor/Grader: SamuelPalato E-mail: Samuel.PalatoMail.McGill.CA Office: Otto Maass 25 Lectures: Tuesday and Thursday 11:35 - 12:25 Makeups, Tutorials, Friday 11:35 - 12:25 or ReviewSessions: Location: Otto Maass 217 Course Web Site: Note: username and password are needed for full access. 2015, Fall TermGeneral Information -4- Chemistry223 Iwill be awayonthe following dates and will makeupthe missed class in the Friday slot of the same week: Canceled Classes Makeup (Fall, 2015) (OM 217, 11:35-12:25) Tuesday,September 15 Friday,September 18 Tuesday,September 29 Friday,October 2 Tuesday,October 6 Friday,October 9 1.2. Texts Thomas Engel and Philip Reid, Thermodynamics, Statistical Thermodynamics, and Kinetics, 3rd edition (Pearson Education, Inc., 2013). J.R. Barrante, Applied Mathematics for Physical Chemistry,3rd edition (Pearson Education, Inc., 2004). 1.3. Supplementary Texts 1. G. W. Castellan, Physical Chemistry 3rd edition (Benjamin Cummings Pub.Co., 1983) (Out of print but excellent. This would be the text for the course if I could get copies). Note that Castellan doesn’tuse SI units and uses a older sign convention for an key thermodynamic quantity,namely work. 2. R.J. Silbey, R.A. Alberty and M.G. Bawendi, Physical Chemistry,4th edition (John Wiley& Sons, Inc., 2005). This was used as the text in the past. It’sOKbut Engel and Reid or Castellan are better. 3. R. Kubo, Thermodynamics (Physics orientation, excellent, but somewhat advanced with fewer chemical examples). 1.4. Grades There will be approximately one problem set every 2-3 lectures, one midterm and a final exam. The midterm will be givenbetween 6 and 9 P.M. on Tuesday,October 27, 2015 in Otto Maass 112 and 217 (a seating plan will be posted). Completion and submission of the homework is mandatory.Wehav e atutor/grader for the course, Samuel Palato, and the problems will be graded. Solutions to the problem sets will be posted on the course web page. In addition, there will be a tutorial roughly every second Friday where the tutor will go overproblems or reviewother topics. Youare strongly encouraged to do the homework by yourself. The problems will cover manydetails not done in class and will prepare you for the exams. The exams will involve extensive problem solving and may contain problems from the homework The course grad- ing scheme is: 2015, Fall TermChemistry 223 -5- GeneralInformation Grade Distribution Problems 10% Midterm 40% Final 50% 1.5. Random, McGill Specific, Notes McGill University values academic integrity.Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offenses under the Code of Student Conduct and Disciplinary Procedures (see dents/srr/honest/ for moreinformation).(approvedbySenate on 29 January 2003) In accord with McGill University’sCharter of Students’ Rights, students in this course have the right to submit in English or in French anywritten work that is to be graded. (approvedbySenate on 21 January 2009) In the event of extraordinary circumstances beyond the University’scontrol, the content and/or evaluation scheme in this course is subject to change. 2015, Fall TermGeneral Information -6- Chemistry223 1.6. Tentative Course Outline Text Chapter Lecture Topic Silbey Reid Castellan Lecture1.Introduction: Kinetics&Thermodynamics, an overview 1 12 Lecture2.Empirical properties of gases 11,7 2 Lecture3.Empirical properties of liquids and solids 11 5 Lecture4.Molecular basis: Kinetic theory of gases 17 12,16 4 Lecture5.Surface reactions & Effusion 17 16 30 Lecture6.Gas phase collision rates 17 16 30 Lecture7. Kinetics I: Collision theory of elementary gas phase reactions: Collision rates and activation energies 19 17 33 Lecture8.Mean free path & Diffusion Lecture9.Kinetics I: Review of n’th order reaction kinetics. 18 18 32 Lecture10. Intro. to mechanims & steady state approximation. Lecture11. Temperature: the zeroth law of thermodynamics 11 6 Lecture12. Mechanics,Work, and Heat 22 7 Lecture13. Reversible and irrev ersible changes 22 7 Lecture14. The First Law of Thermodynamics: Energy 2 2, 3 7 Lecture15. Enthalpy,Hess’sLaw 2 3, 4 7 Lecture16. Heat Capacities, Kirchoff ’sLaw 247 Lecture17. Estimating Enthalpy Changes: Bond Enthalpies 24 7 Lecture18. The Carnot Engine/Refrigerator 35 8 Lecture19. The Second Law of Thermodynamics: Entropy 358 Lecture20. Entropy Calculations 35 8 Lecture21. The Third Law of Thermodynamics: Absolute Entropies 3 59 Lecture22. ConditionsforStable Equilibrium: Free Energies 4610 Lecture23. Equilibrium Conditions (continued) 46 10 Lecture24. Maxwell Relations and applications 46 9.4 Lecture25. Chemicalequilibrium 5611 Lecture26. Chemical equilibrium calculations 56 11Chemistry 223 -7- Divertissements 2. Divertissements From: Ryogo Kubo, Thermodynamics (North Holland, 1976) 2.1. Divertissement 1: Founders of the first law of thermodynamics If a tomb of the Unknown Scientists had been built in the 1850’s, the most appropriate inscription would have been "In memory of the grief and sacrifice of those who fought to realize aperpetuum mobile". But the lawofconservation of energy,orthe first lawofthermodynamics, is associated primarily with three great names, Mayer,Helmholtz and Joule. Julius Robert Mayer (1814-1878) was really a genius who was born in this world only with the errand to makethis great declaration. Hermann Ludwig Ferdinand von Helmholtz (1821-1894) gav e this lawthe name "Erhaltung der Kraft" or "the conservation of energy". Like Mayer,hestarted his career as a medical doctor but livedaglorious life as the greatest physiolo- gist and physicist of the day.James Prescott Joule (1818-1889) worked overforty years to estab- lish the experimental verification of the equivalence of work and heat. Among the three, Mayer was the first who arrivedatthis lawand the last whose work was recognized. His life was most dramatic. A lightening strokeofgenius overtook him, a German doctor of the age of twenty six, one day on the sea near Java when he noticed that venous blood of a patient under surgical operation appeared an unusually fresh red. He considered that this might be connected with Lavoisier’stheory of oxidation in animals, which process becomes slower in tropical zones because the rate of heat loss by animals will be slower there. A great generalization of this observation lead him to the idea of the equivalence of heat and mechanical work. For three years after his voyage, while he was working as a medical doctor at home, he devoted himself to complete the first work on the conservation of energy "Bemerkungen uber die Krafte der unbelebten Natur" which was sent to the Poggendorf Annalen and was neverpub- lished by it. In 1842 Liebig published this paper in his journal (Annalen der Chemie und Pharma- cie) but it was ignored for manyyears. Mayer wrote four papers before 1851. During these years of unusual activity he cared for nothing other than his theory.In1852 he became mentally deranged and was hospitalized. He recovered after twoyears but neverreturned to science. 2.2. Divertissement 2: Whydowehav e winter heating? Whydowehav e winter heating? The layman will answer: "Tomakethe room warmer." The student of thermodynamics will perhaps so express it: "Toimport the lacking (inner,ther- mal) energy." If so, then the layman’sanswer is right, the scientist’siswrong. We suppose, to correspond to the actual state of affairs, that the pressure of the air in the room always equals that of the external air.Inthe usual notation, the (inner,thermal) energy is, per unit mass, u = c T . v The author has assumed that the specific heat of the gas is independent of temperature; a reasonable approximation for the oxygen and nitrogen around room temperature. 2015, Fall TermDivertissements -8- Chemistry223 (An additive constant may be neglected.) Then the energy content is, per unit of volume, u = c ρT , v or,taking into account the equation of state, we have P = RT , ρ we have u = c P/R. v Forair at atmospheric pressure, 3 u = 0. 0604cal/cm . The energy content of the room is thus independent of the temperature, solely determined by the state of the barometer.The whole of the energy imported by the heating escapes through the pores of the walls of the room to the outside air. Ifetch a bottle of claret from the cold cellar and put it to be tempered in the warm room. It becomes warmer,but the increased energy content is not borrowed from the air of the room but is brought in from outside. Then whydowehav e heating? For the same reason that life on the earth needs the radiation of the sun. But this does not exist on the incident energy,for the latter apart from a negligible amount is re-radiated, just as a man, in spite of continual absorption of nourishment, maintains a constant body-weight. Our conditions of existence require a determi- nate degree of temperature, and for the maintenance of this there is needed not addition of energy butaddition of entropy. As a student, I read with advantage a small book by F.Wald entitled "The Mistress of the World and her Shadow". These meant energy and entropy. Inthe course of advancing knowledge the twoseem to me to have exchanged places. In the huge manufactory of natural processes, the principle of entropyoccupies the position of manager,for it dictates the manner and method of the whole business, whilst the principle of energy merely does the bookkeeping, balancing cred- its and debits. R. EMDEN Kempterstrasse 5, Zurich. The above isanote published in Nature 141 (l938) 908. A. Sommerfeld found it so inter- esting that he cited it in his book Thermodynamic und Statistik (Vorlesungen uber theoretische Physik, Bd. 5, Dietrich’sche Verlag, Wiesbaden; English translation by F.Kestin, Academic Press Tic., NewYork, 1956). R. Emden is known by his work in astrophysics and meteorology as represented by an article in der Enzyklopadie der mathematischen Wissenschafte Thermody- namik der Himmelskorper (Teubuer,Leipzig-Berlin, 1926). 2015, Fall TermChemistry 223 -9- Divertissements 2.3. Divertissement 3: Nicolas Leonard Sadi Carnot In the first half of the last century,the steam engine, completed by introduction of the con- denser (the low-temperature heat reservoir), due to James Watt (1765) had come to produce more and more revolutionary effects on developments in industry and transportation. Manyeminent physicists likeLaplace and Poisson set about to study the Motive Power of Fire. Sadi Carnot (1796-1832) was a son of Lazare Carnot, Organizer of Victory in the French Revolution, and was born and died in Paris. He probably learned the caloric theory of heat, in which heat was assumed to be a substance capable either of flowing from body to body (heat conduction) or of making chemical compound with atoms (latent heat). He wrote a short but very important book, Reflexions sur la puissance motrice du feu et sur les machines propres a developper cette puis- sance (Paris, 1824), which was reprinted by his brother (1878) together with some of Carnot’s posthumous manuscripts. Carnot directed his attention to the point that, in the heat engine, work was done not at the expense of heat but in connection with the transfer of heat from a hot body to a cold body,and thus heat could not be used without a cold body,inanalogy of water falling from a high reservoir to a lowreservoir.Inhis book he assumed the lawofconversation of heat, namely that the quan- tity of heat was a state function, although he later abandoned this lawand arrivedatthe lawof equivalence of heat and work: he actually proposed manymethods to estimate the mechanical equivalent of heat. He introduced what came to be known as Carnot’scycle, and established Carnot’sprinciple. Carnot’sbook had been overlooked until B. P.E.Clapeyron (1834) gav e Carnot’stheory an analytical and graphical expression by making use of the indicator diagram devised by Watt. The lawofconservation of heat assumed by Carnot was corrected by R. Clausius (1850), based on the work of J. R. von Mayer (1841) and J. P.Joule (1843-49), into the form that not only a change in the distribution of heat but also a consumption of heat proportional to the work done is necessary to do work, and vice versa. Clausius named this modification the First LawofThermo- dynamics. H. L. F.van Helmholtz (1847) and Clausius generalized this lawtothe principle of the conservation of energy.W.Thomson (Lord Kelvin), who introduced Kelvin’sscale of tempera- ture (1848) based on Carnot’swork, also recognized the lawofequivalence of heat and work. The Second LawofThermodynamics was formulated by Thomson (1851) and Clausius (1867). Asketch of the history of early thermodynamics is givenbyE.Mendoza, Physics Today 14 (1961) No. 2, p. 32. See also E. Mach: Principien der Warmelehre (vierte Aufl. 1923, Verlag vonJohann Ambrosius Barth, Leipzig). 2.4. Divertissement 4: Absolute Temperature The absolute temperature scale means that temperature scale which is determined by a thermodynamic method so that it does not depend on the choice of thermometric substance, the zero of the scale being defined as the lowest temperature which is possible thermodynamically. Absolute temperature, which is nowused in thermal physics, was introduced by Lord Kelvin (William Thomson) in 1848 and is also called the Kelvin temperature. Forthe complete definition of the scale, we have two choices; one is to use twofixed points above zero and assign their temperature difference and the other is to use one fixed point 2015, Fall TermDivertissements -10- Chemistry223 and assign its numerical value. Until recently the calibration of the Kelvin temperature scale was o o performed using twofixedpoints: the ice point T K and the boiling point T + 100 K of pure 0 0 water under 1 standard atm (= 101325 Pa). Wecan measure T by a gas thermometric method. 0 At lowpressures, the equation of state of a real gas can be written in the form pV = α + κ p. We measure the values of pV, α and κ at the above two fixedpoints.Considering that α is equal to nRT,wehav e 100α 0 T = 0 α -α 100 0 o If we put T = 0, we get the thermodynamic Celsius temperature scale. Hence, - T C means 0 0 absolute zero as measured by this scale. The precise gas thermometric investigations of the Frenchman P.Chappuis from 1887 to 1917 gav e the value of T between 273.048 and 273.123. Inspired by this work, more than one 0 hundred determinations of T were performed until 1942. Among them, the results of W.Heuse 0 and J. Otto of Germany, W.H.Keesom et al. of the Netherlands, J. A. Beattie et al. of the U.S.A. and M. Kinoshita and J. Oishi of Japan are noted for their high precision. Their values are found to lie between 273.149 and 273.174. Considering these results and the fact that the triple point of pure water is very near to o 0.0100 C,the 10th General Conference on Weights and Measures in 1954 decided to use the triple point of the water as the fixed point and to assign the value 273.16 as its temperature. It o also redefined the thermodynamic Celsius temperature t C as t = T-273.15, where T is the value of the absolute temperature determined from the above decision. The zero of the newthermody- o namic Celsius temperature differs by about 0.0001 from the ice point. Forordinary purposes, the differences in these newand old scales are quite negligible. - 4 However, for cases where a precision of 1O degree in the absolute value is required, we must takethe differences into consideration. 2.5. Divertissement 8: On the names of thermodynamic functions The word "energy εν ε ργ εια"can be seen in the works of Aristotle but "internal energy" is due to W.Thomson (1852) and R. J. E. Clausius (1876). The portion "en" means inhalt=capac- ity and "orgy", likethe unit "erg", derivesfrom εργ oν=work. "Entropy" is also attributed to Clausius (1865) who took it from εν τ ρε π ειν =verwandeln and means verwandlungsin- halt=change quantity."Enthalpy" was introduced by H. Kamerlingh Onnes (1909) from εν θ αλ πε ιν=sich erwarmen which means warmeinhalt.J.W.Gibbs called it the heat function (for constant pressure). "Free energy" is due to H. van Helmholtz (1882), and means that part of the internal energy that can be converted into work, as seen in the equation dF=d’A for an isothermal quasi-static process. It was customary to call the remaining part, TS, of the internal energy,U=F+TS, the gebundene energie (bound energy), but this is not so common now. The Gibbs free energy (for constant pressure) was introduced by Gibbs, but German scientists used to call it die freie enthalpie.Thus the thermodynamic functions often have different names in 2015, Fall TermChemistry 223 -11- Divertissements German and in English. Further,onthe equation of state: Kamerlingh Onnes gav e the names, thermische zustands- gleichung to p = p(T, V)and the name kalorische zustandsgleichung to E = E(S,V). M. Planck (1908) called the latter kanonische zustandsgleichung. 2015, Fall TermIdeal & Non-Ideal Materials -12- Chemistry223 3. Some Properties of Ideal and Non-Ideal Materials 3.1. IdealGases Very dilute gases obeythe so-called ideal gas laworequation of state, initially deduced † from Boyle’sLaw and Charles’sLaw ,which when combined showthat PV = NRT,(3.1) where SI Units for some common quantities arising in the study of gases. SI Symbol Name Abbreviation Unit 2 PPressure (Pascale) Pa kg/(msec) 3 VVolume m NNumber of moles. mol moles TAbsolute Temperature K Kelvin RGas Constant 8.314442 J/(Kmol) 23 N Av ogadro’sNumber 6.0225×10 molecules/mol A 5 Note that 1 standard atmosphere is 1.01325 × 10 Pa = 101. 325kPa.Under Standard Temperature and Pressure (STP) conditions T ≡ 273. 15K (0C)and P ≡ 101. 325kPa (1ˆatm); hence, by rearranging Eq. (3.1) we see that 3 V ≡ V /N = RT /P = 0. 0224 m /mol,ormore commonly as 22. 4 liters/mol. 3.2. Dalton’sLaw ‡ In mixtures of dilute gases, Dalton showed that the ideal gas equation, cf., Eq. (3.1), needed to be modified by replacing N by the total number of moles in the gas, i.e., N → N ≡ N,i.e., total i Σi RT P = N = P,(3.2) i i Σ Σ V i i where P ≡ N RT /V = x P,(3.3) i i i is known as the partial pressure and is the pressure a pure gas of component i would have for a givenmolar volume and temperature. We hav e also introduced the mole fraction, x ≡ N /N , i i total in writing the last equality.Since x = 1, it can be summed to give Eq. (3.2). Σi i Robert Boyle, 1627-1691, showed that P∝1/V . † Jacques Alexandre Ce ´sar Charles, 1746-1823, showed that V ∝T . ‡ John Dalton, FRS, 1766-1844. 2015, Fall TermChemistry 223 -13- Ideal&Non-Ideal Materials Youmight think that the partial pressure concept is some sort of mathematical game and that the partial pressures are not physically relevant. After all, only N matters in the equation total of state. As Dalton showed, this is not correct. To see why, consider the following experiment. N , N , N , ... 1 2 3 P 1 T, V constant Fig. 3.1. Dalton’sExperiment: a rigid cylinder containing a gas mix- ture at temperature T and N moles of gas "i". Two pressure measuring i devices (e.g., manometers) are attached to the cylinder.The one on the left is directly connected to the gas mixture via a hole (or valve) in the top of the cylinder.The meter on the right is connected to the mixture through a porous plug that only allows component 1 to pass. As expected, the left meter reads P in accord with Dalton’sLaw.The meter on the total right reads P ,the partial pressure of the permeable component. Thus, in establishing its equilib- 1 rium with the meter,the permeable component acts as if the other components weren’tthere As we shall see later,this plays a central role in chemical equilibrium.. 3.3. Beyond Ideal Gases 1 Fig. 3.2. Phase diagram of water. Shown are the coexistence lines for gas-solid (sublimation), gas-liq- Fig. 3.3. Liquid-vapor pressure-volume phase dia- 2 uid (vaporization or condensation), and liquid-solid gram near the critical point. The solid curves are (freezing or melting) lines. The point where all three known as isotherms (constant temperature) and the meet is known as the triple point. The liquid-vapor dashed lines correspond to liquid-vapor phase equi- line terminates at the so-called critical point. Finally, librium where low(gas, V )and high (liquid, V ) G L for water,note that there are several solid-solid coex- phases coexist. Two features of interest are the criti- istence lines (not shown) at evenlower temperatures cal point, labeled c, and a path (1 → 2 → 3 → 4) and/or higher pressures. whereby a liquid is vaporized without boiling. Forexample, thin Pd sheets are porous to H and not much else. Alternately,small-pore zeolites can also 2 be used to filter/selectively pass gases. 2015, Fall Term P totIdeal & Non-Ideal Materials -14- Chemistry223 Figures 3.2 and 3.3 showexamples of pressure-temperature and pressure-volume phase diagrams, respectively.Manyofthe details contained in these phase diagrams will be considered next term. Fornow,simply note that Fig. 3.3 shows that ideal gas behavior is observed only at high enough temperature and molar volume. Also note that only the liquid-gas equilibrium’s showcritical points. That’sbecause liquids and gases differ only in details (e.g., density,index of refraction, etc.) and not in symmetries, i.e., both are isotropic and homogeneous, something that solids are not by virtue of their crystal lattices. There are manyobjections that can be raised against the ideal gas and Dalton’slaws. Here are a few: a) whydon’ttheydepend on the chemical identity of the gas? b) theypredict finite pressure for all but V → 0; and c) theypredict vanishing volume as T → 0K.Some of these objections can be dismissed if we consider howfar apart the gas molecules are under typical con- ditions, i.e., around ambient temperature and pressure. According Eq. (3.3), under STP conditions, the volume per molecule is 3 23 - 10 3 V /N = 0. 0224 m /6.0225 × 10 = (33. 39 × 10 m).Thus, we see that the typical distance A o between molecules in this gas is 33. 39A,which is large compared to the size of gaseous elements and manysmall molecules. Nonetheless, the distance shrinks as the pressure increases or in con- densed liquid or solid phases. Forexample, giventhat the molar volume of water (molecular 3 3 - 23 3 weight 18 g/mol, density at STP 1 g/cm )is18cm /mol or 2.98 × 10 cm /molecule we see o that the typical distance between water molecules is around 3.1A,which is approximately the size of a water molecule; hence, in liquid water the molecules are more or less in direct contact, and we would expect that molecular details (geometry,bonding, dipole moment, etc.) to play an important role, as theydo. This discussion can be made more quantitative ifweconsider the so-called compressibility factor or ratio, Z, PV V Z = =,(3.4) RT V ideal where V = RT /P is the molar volume an ideal gas would at the same temperature and pres- ideal sure. Some examples for the van der Waals model are shown in Fig. 3.4. One general way to deal with deviations from ideal behavior in the gas phase, at least for lowdensities, is to write down the so-called virial expansion In more modern terms, the virial expansion is a Taylor polynomial approximation, i.e., 2 i- 1 Z = 1 + Bn + Cn +... = B n,(3.5) Σ i i=1 where n ≡ 1/V is the molar density,and where B and C are known as the second and third viral coefficients, respectively.The second equality is an alternate notational convention with B = 1, 1 1 G. W.Castellan, Physical Chemistry,3rd ed.,(Benjamin Pub.Co., 1983), p. 266. 2 R.J. Silbeyand R.A. Alberty, Physical Chemistry,3rd ed.,(John Wiley&Sons, Inc. 2001) p. 16. Virial n. L. vis, viris, force. Acertain function relating to a system of forces and their points of applica- tion, first used by Clausius in the investigation of problems in molecular physics/physical chemistry. 1913 Webster 2015, Fall TermChemistry 223 -15- Ideal&Non-Ideal Materials B = B, B = C,etc.. In general, the virial coefficients are intensive functions of temperature 2 3 i- 1 and has units of volume . The theoretical tools required to calculate the virial coefficients were developed in the mid 20th century and we’ve been able to calculate the first 10 for model potentials of molecular inter- action. Less well understood is the radius of convergence of the virial expansion, an important question, if we would liketosomehowextrapolate to the liquid phase. According to the Lee- † Yang theorem, the radius of convergence is the condensation density,which means that the series cannot be used to study the liquid phase directly. Fig. 3.4. The compressibility factor for the van der Waals model (see belowfor an expla- nation of reduced variables). Note that both positive and negative deviations from Z = 1 are possible. The change-overtemperature, the so-called reduced Boyle temperature, τ , B is that where attractive and repulsive interactions balance and the second virial coefficient vanishes. For the van der Waals model T = a/Rb which leads to τ =27 / 8= 3. 375. B B † C.N. Yang and T.D. Lee, Statistical Theory of Equations of State and Phase Transitions. I. Theory of Con- densation,Phys. Rev. 87,404-409, (1952); T.D. Lee and C.N. Yang Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model Phys. Rev. 87,410-419, (1952). Note that these papers are well beyond your current mathematics and physics skills. 2015, Fall TermIdeal & Non-Ideal Materials -16- Chemistry223 One of the first attempts at writing an equation of state that had liquid and vapor phases ‡ wasdue to the Dutch physical chemist van der Waals The van der Waals model considers the repulsive and attractive interactions separately.First, it corrects for the intrinsic or steric volume per molecule by replacing the system’svolume V by V - Nb,where b,known as the van der Waals b coefficient, and can be thought of as the minimum volume occupied by a mole of mole- cules, assuming that theydon’tdeform at high pressure. Note that this won’tbethe geometric volume of the molecule, since some space is wasted due to packing considerations. The second idea was to suggest that there are weak attractive forces between molecules (due to the so-called London dispersion forces). The attractions lead to the formation of weakly bound van der Waals dimers, thereby reduce the total number of molecules in the system. Since PV = NRT anything that reduces N lowers the pressure. To quantify this last idea, consider the dimerization reaction K → 2A A,(3.6) ← 2 where K is the equilibrium constant for the reaction, and is very small for van der Waals dimers. At equilibrium, A 2 = K,(3.7) 2 A where A, etc., denote molar concentrations. Since A is conserved in the reaction, A ≡ A + 2A(3.8) total 2 is constant, and can be used to eliminate A from Eq. (3.7), which becomes 2 2 2KA - A + A =0. (3.9) total This quadratic equation has one physical (positive)root, namely, 1/2 - 1 + (1 + 8KA ) total A = .(3.10) 4K By using this in Eq. (3.8) we can easily find A , and finally, 2 1/2 A + A 4KA - 1 + (1 + 8KA ) total total total N ≡ A + A = = (3.11a) 2 2 8K 2 A - KA + ... , for K A 1.(3.11b) total total total ‡ Johannes Diderik van der Waals, 1837-1923, was the first to suggested howrepulsive and attractive forces (nowknown as van der Waals or London dispersion interactions) lead to the existence of different phases and a critical point. 2015, Fall TermChemistry 223 -17- Ideal&Non-Ideal Materials 1/2 2 Where the last result was obtained by noting that (1 + x) 1 + x/2 - x /8+... for x1.In - 2 short, the dimerization reaction leads to a reduction in the molar density proportional to V . By combining the results of our discussion of the roles of repulsions and attractions, we can write down the van der Waals equation of state RT a P = -,(3.12) 2 V - b V where a,the van der Waals "a" constant, is the proportionality constant characterizing the reduc- tion of N due to attractions. Some results are shown in Fig. 3.5. Fig. 3.5. Semi-log plots of the reduced pressure versus reduced volume for isotherms obtained using the van der Waals equation of state. The reduced pressures divergeas ϕ → 1/3 and become ideal as ϕ → ,cf. Eq. (3.12). ∞ Notice the inflection point at the critical point. Fortemperatures belowthe critical temperature three states are possible, one at small ϕ one at large ϕ,and one in the middle. It is reasonable to identify the twooutermost as a liquid and gas, respectively.The state in the middle is unphysical 2015, Fall TermIdeal & Non-Ideal Materials -18- Chemistry223 because its slope is positive;i.e., its molar volume increases with increasing pressure and leads to aneg ative compressibility. Other thermodynamic quantities are easily found for the van der Waals model. Forexam- ple the compressibility factor becomes: 1 a/RT Z = -.(3.13) 1 - b/V V and finally,the thermal expansion coefficient is found by differentiating the van der Waals equa- tion, with respect to T keeping Pconstant.Some examples are shown in Fig. 3.4. Note that the compressibility diverges at the critical point, cf. Eq. (3.15) below. We can makecontact with the virial expansion, cf. Eq. (3.5), by recalling the geometric series 1 2 3 1 + x + x + x +..., 1 - x which when used in Eq. (3.13) shows that 2 3 a 1 b b ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Z = 1 + b - + + +... . (3.14) ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ RT V V V i- 1 Thus, B = b - a/ RT and B = b ,for i ≥3. The higher order virial coefficients are simply 2 i related to the excluded volume effects characterized by powers of the van der Waals b coeffi- cient. This is probably not correct. Only the second virial coefficient, b - a/RT,isnontrivial. First note that it can be positive (e.g. as in H )orneg ative (e.g., as in N )depending on whether repulsions or attractions are 2 2 more important, particular,inparticular,itwill be positive for large temperatures and negative for lowtemperature. The model predicts a zero initial slope when T = T ≡ a/(Rb), known as the B Boyle temperature. Physically,itisthe temperature at which attractions and repulsions balance each other and the gas behavesmore ideally than expected. Perhaps the most interesting feature of the van der Waals model is the existence of the so- called critical point; i.e., the one where the differences between the liquid and vapor phases van- ish (see, e.e., point c in Fig. 3.3). This implies that one can choose a path (such as 1→2→3→ 4) which starts with a high-density (liquid) phase and ends up as a lowdensity (gas) phase without everhav e 2phase coexistence (no bubbles form and the system doesn’tboil). This was controversial in the 19th century,but is nowwell established. The critical state is the inflection point on the critical isotherm, cf. Figs. 3.3 or 3.5; i.e., where the first and second derivativesofthe pressure-volume critical curvevanish. For the van der Waals model this implies that 2 ∂P RT 2a ∂ P 2RT 6a ⎛ ⎞ ⎛ ⎞ 0 = =- + and 0= = - ,(3.15) 3 2 4 2 3 ⎝ ∂V ⎠ (V - b) ⎝ ⎠ (V - b) V ∂V V T T This assumes that a and b don’tdepend on temperature. 2015, Fall TermChemistry 223 -19- Ideal&Non-Ideal Materials cf. Eq. (3.12). These can be solved for a and b, giving V C 2 b = and a = 3P V,(3.16) c C 3 or 8a a V = 3b, T = ,and P =,(3.17) C C c 2 27Rb 27b where we have used Eq. (3.12) to get the critical pressure P . C Something interesting happens if we introduce reduced variables, i.e., P T V π ≡ , τ ≡ ,and ϕ ≡,(3.18) P T V C C C all of which are dimensionless. By using the reduced variables and Eq. (3.17) we can rewrite the vander Waals equation, Eq. (3.12), as 8τ 3 π = -.(3.19) 3 3ϕ - 1 ϕ All material dependent parameters(e.g.,a and b)have canceled out.Hence, if the van der Waals model wereexact, equations of state plotted in terms of reduced variables would give the same curves, cf. Figs. 3.4 and 3.5. The materials are said to be in corresponding states. This phenomena is known as the lawofcorresponding states or universality.Note that this can be done for any2parameter model. In reality,the "law" is only an approximation. In summary,the van der Waals equation is qualitatively correct, predicting 2-phase coexis- tence, a critical point, and universal behavior.Onthe other hand, it is quantitatively incorrect, and in practice, other models are used. Problem 3 of problem set 1 explores this claim more care- fully. 3.4. Liquids and Solids Depending on the question asked, solids and liquids can be easier or harder to treat than gases. For example, since both are difficult to compress, linear approximations are often satis- factory,e.g., ΔV ≈- κ ΔP,(3.20) V where the isothermal compressibility, κ,isdefined as The story is a bit more complicated. It turns out that manydisparate materials exhibit universality close enough to the critical point. An interesting observation because all of the classical models, while exhibit- ing universal behavior,fail to describe manyofthe basic details of the behavior close to the critical point. This was sorted out in the 1970’sbyB.Widom (chemistry), M. E. Fisher (chemistry), L. Kadanoff (physics), and K.G. Wilson (physics), and led to Wilson winning the 1982 Nobel prize in physics. 2015, Fall TermIdeal & Non-Ideal Materials -20- Chemistry223 1 ∂V ⎛ ⎞ κ≡-.(3.21) ⎝ ⎠ V ∂P N,T Notice the explicit - sign in the definition of κ.All stable materials have positive κ (things get smaller when you squeeze them). The factor of 1/V makes κ intensive,and therefore easier to tabulate. The isothermal compressibility,becomes 1/P for the ideal gas, or more generally for the vander Waals liquid or gas - 1 - 1 ⎡ ⎤ ∂P RTV 2a ⎛ ⎞ ⎛ ⎞ κ =- V = -,(3.22) ⎢ ⎥ 2 2 ⎝ ⎠ ⎝ ⎠ ∂V (V - b) V T ,N ⎣ ⎦ cf. Eq. (3.15). Similarly,for small temperature changes, ΔV ≈ α ΔT,(3.23) V where the (isobaric or constant pressure) thermal expansion coefficient, α,isdefined as 1 ∂V ⎛ ⎞ α ≡.(3.24) V ⎝ ∂T ⎠ N,P Like κ,the factor of 1/V makes α intensive.Howev er, unlike κ,the thermal expansion coeffi- cient can be positive orneg ative (e.g., it vanishes for liquid water at 4C and 1 atm). Forthe ideal gas, α = 1/T,while for the van der Walls gas or liquid - 1 ⎡ ⎤ (V - b) RTV 2a ⎛ ⎞ α = -,(3.25) ⎢ ⎥ 2 2 ⎝ ⎠ R (V - b) V ⎣ ⎦ which is obtained by differentiating the van der Walls equation of state with respect to tempera- ture keeping pressure constant, using the chain rule, noting that (∂V /∂T ) = Vα,and the solving P the resulting equation for α . Forso-called ideal solids, these being crystalline materials with roughly harmonic inter- atomic interactions, one can go considerably farther in calculating mechanical quantities like κ and α,aswell as elastic constants, heat capacities, energies, electronic properties, etc.; this is well beyond the scope of this course. Liquids are less tractable than solids having the same com- plications arising from the molecules being close together without the simplifications associated with having an underlying periodicity or crystal lattice. At present, complexliquids are studied theoretically by using brute force methods likeMonte Carlo or molecular dynamics computer simulations. However, see, e.g., M. Born and K. Huang, Dynamical Theory of Crystal Lattices,(Clarendon Press, 1962). 2015, Fall Term