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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
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.
Professor: David Ronis
Ofﬁce: Otto Maass 426
(Help my e-mail client direct your email;
Please put CHEM 223 somewhere in the
Ofﬁce: Otto Maass 25
Lectures: Tuesday and Thursday 11:35 - 12:25
Friday 11:35 - 12:25
Location: Otto Maass 217
Course Web Site: https://ronispc.chem.mcgill.ca/ronis/chem223
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
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.,
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
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
3. R. Kubo, Thermodynamics (Physics orientation, excellent, but somewhat advanced with fewer
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
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 www.mcgill.ca/stu-
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
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
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 .
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 ,
or,taking into account the equation of state, we have
= RT ,
u = c P/R.
Forair at atmospheric pressure,
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.
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
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’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
performed using twoﬁxedpoints: the ice point T K and the boiling point T + 100 K of pure
water under 1 standard atm (= 101325 Pa). Wecan measure T by a gas thermometric method.
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
If we put T = 0, we get the thermodynamic Celsius temperature scale. Hence, - T C means
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
hundred determinations of T were performed until 1942. Among them, the results of W.Heuse
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
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
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-
namic Celsius temperature differs by about 0.0001 from the ice point.
Forordinary purposes, the differences in these newand old scales are quite negligible.
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
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)
SI Units for some common quantities arising in the study of gases.
PPressure (Pascale) Pa kg/(msec)
NNumber of moles. mol moles
TAbsolute Temperature K Kelvin
RGas Constant 8.314442 J/(Kmol)
N Av ogadro’sNumber 6.0225×10 molecules/mol
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
V ≡ V /N = RT /P = 0. 0224 m /mol,ormore commonly as 22. 4 liters/mol.
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.,
P = N = P,(3.2)
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).
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
of state. As Dalton showed, this is not correct. To see why, consider the following experiment.
N , N , N , ...
1 2 3
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
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
right reads P ,the partial pressure of the permeable component. Thus, in establishing its equilib-
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
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-
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 )
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
be used to ﬁlter/selectively pass gases.
2015, Fall Term
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
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
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,
Z = =,(3.4)
where V = RT /P is the molar volume an ideal gas would at the same temperature and pres-
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)
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,
G. W.Castellan, Physical Chemistry,3rd ed.,(Benjamin Pub.Co., 1983), p. 266.
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.
2015, Fall TermChemistry 223 -15- Ideal&Non-Ideal Materials
B = B, B = C,etc.. In general, the virial coefﬁcients are intensive functions of temperature
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 ,
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.
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
where K is the equilibrium constant for the reaction, and is very small for van der Waals dimers.
where A, etc., denote molar concentrations. Since A is conserved in the reaction,
A ≡ A + 2A(3.8)
is constant, and can be used to eliminate A from Eq. (3.7), which becomes
2KA - A + A =0. (3.9)
This quadratic equation has one physical (positive)root, namely,
- 1 + (1 + 8KA )
A = .(3.10)
By using this in Eq. (3.8) we can easily ﬁnd A , and ﬁnally,
A + A 4KA - 1 + (1 + 8KA )
total total total
N ≡ A + A = = (3.11a)
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
Where the last result was obtained by noting that (1 + x) 1 + x/2 - x /8+... for x1.In
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
P = -,(3.12)
V - b
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:
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
1 + x + x + x +...,
1 - x
which when used in Eq. (3.13) shows that
a 1 b b
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
Z = 1 + b - + + +... . (3.14)
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
RT V V V
Thus, B = b - a/ RT and B = b ,for i ≥3. The higher order virial coefﬁcients are simply
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
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
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
∂P RT 2a ∂ P 2RT 6a
⎛ ⎞ ⎛ ⎞
0 = =- + and 0= = - ,(3.15)
3 2 4
⎝ ∂V ⎠ (V - b) ⎝ ⎠ (V - b)
V ∂V V
This assumes that a and b don’tdepend on temperature.
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cf. Eq. (3.12). These can be solved for a and b, giving
b = and a = 3P V,(3.16)
V = 3b, T = ,and P =,(3.17)
C C c
where we have used Eq. (3.12) to get the critical pressure P .
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
π = -.(3.19)
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-
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-
≈- κ ΔP,(3.20)
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
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
The isothermal compressibility,becomes 1/P for the ideal gas, or more generally for the
vander Waals liquid or gas
∂P RTV 2a
⎛ ⎞ ⎛ ⎞
κ =- V = -,(3.22)
⎝ ⎠ ⎝ ⎠
∂V (V - b)
cf. Eq. (3.15).
Similarly,for small temperature changes,
≈ α ΔT,(3.23)
where the (isobaric or constant pressure) thermal expansion coefﬁcient, α,isdeﬁned as
V ⎝ ∂T ⎠
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
(V - b) RTV 2a
α = -,(3.25)
R (V - b)
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
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
However, see, e.g., M. Born and K. Huang, Dynamical Theory of Crystal Lattices,(Clarendon Press,
2015, Fall Term