physical chenistry

5/24/2018 Physical Chenistry

1/146

PhysicalChemistrySECOND EDITION

R. STEPHEN BERRYThe University of ChicagoSTUART A. RICEThe University of Chicago

JOHN RossStanford University

New Y ork OxfordOXFORD UNIVERSITY PRESS2000


2/146

Oxford University PressOxford New YorkAthens Auckland Bangkok Bogota Buenos Aires CalcuttaCape Town Cbennai Dares Salaam Delhi Florence Hong Kong IstanbulKarachi Koala Lumpur Madrid Melbourne Mexico City MwnbaiNairobi Paris So Paulo Singapore Taipei Tokyo Toronto Warsawand associated companies inBerlin IbadanCopyright 2000 by Oxford University Press, Inc.Published by Oxford University Press, Inc.,198 Madison Avenue, New York, New York 10016http://www.oup-usa.orgOxford is a registered trademark of Oxford University Press.All rights reserved, No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any m eans,electronic, mechanical, photocopying, recording, or otherwise,without the prior p ermission of Oxford University Press.

Library of Congress Cataloging-in-Publication DataBerry, R. Stephen, 193 1-Physical chemistry.-2nd ad. FR. Stephen Berry, Stuart A. Rice, John Ross.p. cm. (Topics in physical chemistry)Includes bibliographical references and index.ISBN 0-19-510589-3 (acid-free paper)1. Chem istry, Physical and theoretical. . Rice, Stuart Alan, 1932- II. Ross, John,1926- 11 1. Title. IV. SeriesQD453.2 .848 200054 l.3dc2 I 0-024923

987654321Printed in the United States of Americaon acid-free paper


3/146

CHAP[ER 12.k t~la WW _ i

The Perfect Gas at Equilibriumand the Concept of TemperatureWe begin the study of the properties of bulk matter withthe simplest type of matter: the dilute gas. It is simplebecause its molecules are tar apart and hardly interact withone another, so that the macroscopic properties are directlyrelated to those of the individual molecules. As indicatedin the introduction to Part II, we approach the same prop-erties from the viewpoints of statistical mechanics andthermodynamics.In both cases we deal in this chapter not with a real gasbut with an ab straction kno wn as a "perfect gas" (also calledan 'ideal gas"). In the microscopic theory the perfect gas isa model with no interaction between the molecules. We con-sider the properties of a collection of such mo lecules, in par-ticular the pressure they exert on the walls of a container.and develop an equation of state connecting the gas's pres-sure, volume, and energy.In thermodynamics the perfect gas is an abstraction fromthe limiting behavior of real gases at low pressure Here theequation of state is experimental, as embodied in the famil-iar ideal gas law, and involves the nonm echanical concept oftemperature. We shall devote an extensive discussion to themeaning of temperature, which is much less simple than onemight imagine.Finally, comparing the two equations of state, we shallbuild our first bridge between microscopic and macroscopicproperties. The two equations have the same form if tem-perature is proportional to average mo lecular kinetic energy.By this correspondence we can say that the "perfect gases"of statistical mechanics and thermodynamics are the same,so that the results of each theory can he used to throw lightupon the other.

12.1 The Perfect Gas:Definition and Elementary ModelAccording to the kinesic hypothesis, the individual mole-cules of which matter is composed are continually inmotion, regardless of whether or not the piece of matter as a

whole is moving. These individual motions take place in alldirections and with a variety of speeds, so that, as far asgross motion is concerned, the contributions of individualmolecules tend to cancel. From the macroscopic viewpoint.however, the molecular motion has two major conse-quences: (I) The kinetic energy of the molecules contributesto the internal energy of the material; and (2) the impacts ofthe mov ing molecu les on the container wall contribute to thepressure exerted by the material on its surroundings.Although the molecular motion is also presumed to deter-mine all other properties of the medium, for our present pur-poses it is sufficient for a dilute gas to consider only theinternal energy and the pressure.We must first define what we mean by a dilute gas. Weassume for simplicity that the molecules of the gas aremonatomic (such as He. Ne, or Ar). to avoid having to con-sider the relative motions of the atoms in a polyatom ic mol-ecule. We also assume that the molecules are independentand exert no forces on one an other, except during occasionalbinary collisionscollisions in which only two moleculescome sufficiently close together to exert a significant forceon each other. In an elastic collision the translationalmomentum and kinetic energy of the pair of molecules areconserved. For monatomic molecules we can safely neglectany internal energy changes. e.g., those due to electronicexcitation, and regard any collision as elastic. A gas that sat-isfies all these conditions will he called a pefrct ,as. Themodel is a fairly good approximation for most real gases atmoderate pressures (as in the atmosphere). and such gasescan he called 'dilute?'We m ake the additional assumption that the gas is macro-scopically at equilibrium. Equilibrium, as we use the termhere, means that the properties of the system do not changemea surably with time. This is a purely macro scopic concept,since the positions and velocities of individual molecules arecontinually changing at all times: yet experiment shows thatmacroscopic properties such as pressure and temperaturecan and do become time-independent in a real gas. Supposethat we have a quantity of dilute gas in a container of fixed

353


4/146

354 Matter in Equilibrium: Statistical Mechanics and Thermodynamicsvolume, which we can isolate in such a way that no interac-tion of any kind OCCUS between the gas and the rest of theuniverse) No matter what the initial state of the gasevenif all the gas is initially in one corner of the container, withsome of it hotter than the restwe find that in time a stateis reached in which the properties of the gas are uniformthroughout the container and thereafter do not vary measur-ably. We shall see in this chapter that such quantities as pres-sure and temperature can be related to averages over themolecular velocities. Although these averages also changewhenever a molecular collision occurs, the number of mol-ecules is so enormous that these changes can heand atequilibrium must he.-.-negligibly small fluctuations in thetotal quantities. Thus. we shall assume that at equilibriumthe velocity distributionthe relative numbers of moleculesin different velocity rangesremains invariant with time 2

Additional properties of the equilibrium state can bededuced from the observation that, if there is no externalfield acting on the gas, the gas becomes uniform throughoutits container (see Footnote I). Measurements of such prop-erties as density and temperature then give the same resultsif performed at any point within the gas. The gas is alsofound to be at rest relative to the container, with no macro-scopic flow,or current,: Any flows that may have been ini-tially present die away with time. What are the conse-quences of these conditions for the equilibrium distributionof molecular velocities and positions? Clearly the density ofgas molecules must he the same in every volume elementlarge enough to perform a measurement on, that is, in everyvolume element containing very many molecules. Thekinetic energy per unit volume (which we shall see is pro-portional to temperature) must also he the same in everysuch volume element: for practical purposes, this is possibleonly if the distribution of velocities among the molecules isthe same everywhere. If the gas as a whole is at rest, theaverage velocity of all the molecules must be zero,

=0 and thus Vi=0. 12.1)(Remember that velocity is a vector, so that velocities inopposite directions cancel one another: obviously the aver-age molecular speed is not zero.) Furthermore, at any pointin the gas there must be equal numbers of molecules travel-ing in every possible direction, since any imbalance wouldconstitute a net flow of gas: this must hold not just overallbut within each range of speeds, if the velocity distribution

We can not actual ly isolate a g as 1mm gravi ty, and the Earth'sgrav i ta t iona l f i e ld p roduces a densi ty grad ien t w i th he ight . but this lack ofuni formi ty is ord inar i l y too sma l l to measure o ver laboratory distances.Beca use the binary col l is ions are infrequen t, a m o l e c u le c a n move al ong dis tance a long a l inear t ra jecto ry wi thou t in te r rupt ion . For thepurpose of calculat ing the propert ies of he gas at eq ui l ibrium, theco l l is ions ca n he neglected. Howeve r. the existence nI col l isions permitsus to descr ibe qua l i tat ively h o n a g a s arr ives at an e qui l ibr ium state .

is to remain the same from point to point. All of the preced-ing can be summarized as follows: In a perfect gas at equi-librium the distributions of molecular velocities and posi-tions must be homogeneous (uniform in space) and isotropic(uniform in all directions).

A word here about the types of "densities" we shall use.The ordinary density p, or mass density, is simply the massper unit volume. The number density n is the number of mol-ecules per unit volume. If there are N molecules, each ofm a s s m, in the volume V we have

Nn= and p=nin. )2.2)One can also define a kinetic energy density (kinetic energyper unit volume) and other quantities of the same type.

Having defined our terms, let us consider first a very ele-mentary description of the relationship between the pressureexerted by a gas and the kinetic energy of its molecules. Theargument is essentially one of dimensional analysis. butyields a result of the right order of magnitude. Suppose thata perfect gas is contained in a cubical box with edge length/, and define a Cartesian coordinate system with axes normalto the walls. Imagine that the walls of the box have the prop-erty of reflecting molecules elastically: that is, when a mol-ecule strikes a face of the box, its motion perpendicular tothat face is reversed and its motion parallel to that face isunaltered (see Fig. 12. I). Thus, if a molecule initially has avelocity v with components v, v, z.., its velocity at anylater time must always he some combination of the compo-nentsv, v, u-. Let S be a face of the box normal to thez axis, with area 1. Between two successive collisions withS. a molecule must move to the opposite face (which is a dis-tance1 away from S) and back. Since the molecul&s veloc-ity component perpendicular to S is v, he time intervalbetween successive collisions with S is 2//v.., and the fre-quency of these collisions is v/21. At each collision the mol-ecule undergoes a change of momentum in the z direction ofamount 2,,,v. from +m v, to -mv. an equal and Oppositemomentum is transferred to the wall. We conclude that the

Figure 12.1 Elastic reflection of a particle from the wall of box.The initial velocity is V. with components u perpendicular to thew a l l a nd v, parallel to the wall. After reflection the velocity is V,with components = i and u


5/146

The Perfect Gas: A General Relation between Pressure and Energy 355total change per unit time in the z component of a singlemolecule's momentum due to collisions with S isI(mv 12.3)[ t S 2/We now assume the simplest possible velocity distribution.We imagine that all the molecules have the same velocitycomponents v, u, u along the x, y, z axes. Then, ifthere are N molecules in the volume V = /, the total changein momentum per unit time arising from all collisions withS is

I i=1vd(mv)j1 ,nv' nVmv =nml2 v?, (12.4)dt

where n is the number density. Now, by definition, the pres-sure on a wall is equal to the force exerted per unit area. Inturn, by Newton's second law, the force exerted is equal tothe rate of change of momentum. Hence, if p is the pressureexerted by the gas, p (the total force on S) is the rate atwhich m omentum is exchanged by col l is ions wi th S. and wehave

pfl,flV2 . 12.5)But we know that the pressure in a gas at equilibrium is thesame in all directions (Pascal's law), so we must have

V2 vt =v =+(v +U2 +V) 2 , 12.6)where v is the common speed of all the molecules. Wefinally obtain

p=4nmv2. 12.7)Since mv 2/2 is the translational kinetic energy of a singlemolecule and n is the number per unit volume, we concludethat the pressure exerted by the gas equals two-thirds of thekinetic energy density.

12.2 The Perfect Gas:A General Relation betweenP re s s u r e a n d E n e r g yThe argum ents just given are too restrictive. In general, mo l-ecules are not reflected elastically from a vessel w all (whichis, after all, also composed of molecules); the molecularspeeds are not all the same, but are distributed over a widerange; the directions of motion are random; and the vesselneed not be cubical. We now proceed to give a more generalderivation of the relationships between the distribution ofmolecular velocities in a gas and the pressure and energy ofthe gas. We shall see that for many purposes we need not

know the detailed form of the velocity distribution, but onlysome of its properties.Consider a perfect gas at equilibrium, with N moleculesin a volume V We expe ct that the m olecules wi ll move withdifferent velocities. Let the number of molecules per unitvolume with velocities between v and v +dv (that is. with xcomponents between v and v + dxv , etc.) be denoted asf(v) dv. The function f(v) is known as the velocity distribu-tion function, and it plays an important role in the kinetictheory of gases. Because the total kinetic energy of allN molecules is finite, there must be some maximum speedof molecular motion that cannot be exceeded. Thus, weexpect that f(v) *O as I v I -+0Furthermore, since1 ( v ) dvis the number of molecules per unit volume with velocitiesin a given range dv, if we sum over all possible velocityranges we must obtain the loud number of molecules perunit volume, which is just the number density n. This lastcondition can be formalized by writing the integral

51(v) dv=JJJf(v) dv dv, dv.. =n. 12.8)Now, we have already specified that the molecularmotion at equilibrium is isotropicthat equal numbers ofmolecules in any given speed interval must travel in everydirection. Thus f(v) must be independent of the direction ofv and can be a function only of its magnitude, the speed uIt is therefore useful to introduce a new distribution func-tion: We define the number of molecules per unit volumewith speeds between vand v+ dv as f(u) dv. Suppose thatwe represent the molecular velocities by points in a "veloc-ity space;' a Cartesian space whose coordinates are v-, v,t;n this space dv (=dv dv dz.) i s a volume element , and

v is a radial coordinate. The molecules with speeds betweenvanddv then occupy a spherical shell of radius vand thick-ness dv: the formulasof Appendix 2A show that the "vol-ume" of this shell is 4,rv 2 dv. Thus, the relation between thetwo distribution functions must bef(v)=4rv2f(v). 12.9 )

(See Section 28.1 for a more systematic proof.) Although w eshall not at this time derive the form of f(u), the results ofsuch a d erivation (cf. Chapter 19 ) are il lustrated in Fig. 1 2.2.Recall also that at equilibrium the g as is homog eneou s, each

V0

Figure 12.2 The distribution of mo lecular speeds, f( v). The speedZA J. which is the most probable, is the mode of the distribution.


6/146

356 Matter inEquilibrium: Statistical Mechanics and Thermodynamicsvolume element being the same as every other volume ele-ment. so ( v) must be independent of position within thegas. A remarkable feature of the functionf(v.which will bedemonstrated in Chapter 21. is that it is the same for all sub-stances for which the energy of interaction of the moleculesis independent of their velocities.

We can now formally calculate the internal energy ofthe gas arising from the molecular motion. A moleculewith mass in and speed v has kinetic energy in v2/2. Toobtain the total kinetic energy of the gas, we multiply thekinetic energy of a single molecule with speed Vby thenumber of molecules per unit volume with speeds betweenv and v + dv. then sum contributions from all possiblespeeds and from all possible volume elements in the totalvolume. For the perfect monatomic gas (neglecting theinternal structure of the atoms) the kinetic energy is thesame as the total internal energy. which we shall call UThis internal energy is thus

U=5 .i'Ivf(v)dvdV. 12.1(1)2But since .f( v) must be independent of position. Eq. 12.10can be immediately integrated over volume to give

u=..YJ"u2f(v) di) 12.11)for the internal energy of the perfect monatomic gas. Othertypes of gases (which we consider in Chapter 2 H may con-tain polyatomic molecules, or may have significant interac-tions between molecules. In these cases there are also con-tributions to the internal energy U from molecular vibrationand rotation, or from the intermolecular potential energy.

We wish now to obtain the relationship between internalenergy and pressure, corresponding to Eq. 12.7 in the sim-pie model. As we mentioned before, the pressure on the con-tainer wall is the force exerted by the gas per unit area ofwall or, equivalently, the rate at which momentum is trans-ferred to a unit area of wall. But we need not restrict our del-inition to the wails: By the assumptions of homogeneity andisotropy, the pressure at equilibrium must he the same any-where within the gas (this again is equivalent to Pascal'slaw). Picture a plane surface S fixed at an arbitrary positionwithin the gas: we define a coordinate system with the z axisnormal to S (Fig. 12.3). We assume that gas molecules col-lide elastically with S, in the same way as shown in Fig. 12.1.Thus whenever a molecule whose z component of momen-tum is ni v strikes S. an amount of momentum 2mv. is trans-ferred to S. The pressure on S is the total momentum sotransmitted per unit time per unit area of S. and at equilib-rium must be the same on both sides of S.

Now suppose that the real surface S is replaced by animaginary plane in the same position. The moleculesthat would rebound from a real surface are just those thatin this model cross S. But a molecule that would transfer

F igure 1 .2 .3 The p lane sur face S is f i xed a t an a rb i tr a r y pos i t ion ina con ta iner o f gas . ThexYz coord ina te sys tem i s de f ined so t ha t t hec axis k normal to S .

Figure 12.4 All t he mo l e cu l e s w i th i n t he i n c l i n ed p r i sm t ha t ha v eve loc i t y v in the d i r ec t ion 9w i l l s t r ike the a rea A wi th in t ime x . Thev o l ume o f t he p r i sm is flut cos

z-momentuni 2mv to the real S carries only niv. (half asmuch) across the imaginary S. We define the pressure to bethe same in both cases, thus obtaining a definition inde-pendent of the presence of a wall: The pressure normal to animaginary plane S is twice the momentum transported acrossS per unit time per unit area of S. by molecules coming fromone side of S (since the two sides of a real surface are dif-terent areas).

How do we evaluate this rate of momentum transport?The molecules we shall consider are those crossing S in thepositive z direction. Let 9 he the angle between a givenmolecule's trajectory and the z axis (Fig. 12.4). We defineF(O. v) dO du as the number of molecules with speeds be-tween vand v + d that cross S per unit area per unit timeat angles between 9 and 9+ dO. Each such molecule has amomentum of magnitude ni v. with z component inv cos 0Only the z component of momentum contributes to the pres-sure. Because of the gas's isotropy, the other componentsadd to give a null result; that is, for a given range of Othereare equal numbers of molecules with x components +invand -inv r with y components+rn v and in v, so that thenet transport of these components across S is zero. Thus thecontribution to the pressure from the molecules in the rangesdv, d9 is the number per unit time per unit area of such mol-ecules multiplied by twice the z component of momentumtransported per molecule, or

dp=2nzvcos GF(O,v) dOdv. 12.t2)


7/146

Some Comments about Thermodynamics 357Integration over all possible values of & and v that is , overall molecules crossing S in the p ositive z directiong ives atotal pressure of

rp=Jf:

01 2 2mvcos 9F(t9,v)dGdv 12.13)u=0=

(note the limits of integration for 9).To use Eq. 12.13 we must evaluate F(& v). The expres-sion F( v) d& dv may be thought of as the product of(I) the number of molecules per unit volume with speedsbetween v and v + dv moving at angles between 0 and0+ dO, which we call f(9 v) d9dt and (2) the volumeoccupied by all such mo lecules capable of crossing unit areaof S in unit time. The latter volume is easily evaluated: Asshown in Fig. 12.4, it is simply a prism inclined at an angleOfrom the n ormal to its base. For base area A and timer, thevolume of such a prism is A yr cos 9; since our calculationrefers to unit area and unit time, the volume we require isvcos 0 We can thus writeF(0,v) d0dv=ucos Of(0,v) dO dv. 12.14)

Remember tha t f( v) dv is the total number of molecules perunit volume with speeds between v and v + dv, or f(9, v)integrated over all values of 9. Since the molecular motionis isotropic, f(9, v) dO dv must bear the same relation tof(v) dv that the range of solid angles between &and 9+ dObears to 4ir(cf. Appendix 2A). By Eq. 2B.7, the solid anglebetween Oand 0-i- dO is 2 sin 0 dO, so we havef(0,v) dO dv = 2r sin &d& = I dO. 12.15)f(v ff

Substituting Eqs. 12.14 and 1 2.15 into Eq. 12.13, we findthe pressure to bef ' 2 1P=J J (2mv cos 0)(v cos 0)(sin 9d9 If(v) dvv=0 8=0 .2

=mI v 2 f(v)dv I o cos2 & sin 9d&f .o v 2 f(v) dv. 12.16)But, comparing this result with Eq. 12.11. we obtain for aperfect monatomic gas

(12.17)3Vwhere U/V is the internal energy density.Thus, the hypotheses we have introduced lead to the con-clusion that the pressure in a perfect monatomic gas is justequal to two -thirds of the internal energy den sity. This resultis the same as that obtained in Eq. 12.7, from analysis of our

original, very crude model. Note that we have been able toobtain this interesting relationship without specifically cal-culating f( v). Subsequent calculations, especially in Part 11 1of this book, will require knowledge of the form of f(v),which we shall derive in a subsequent section. Later in thepresent chapter (Section 12.6) we shall examine criticallythe analysis just given, but first we begin our study of themacroscopic theory.12.3 Some Commentsabout ThermodynamicsAs we explained in the introduction to Part II, thermody-namics deals with relationships among the macroscopicproperties of mattermore specifically, among a particularclass of macroscopic variables that we must define. Theserelationships can be derived from the laws of thermodynam-ics, which are inferred from observations of the behavior ofbulk matter. The laws of thermodynamics are analogous tothe principles of quantum mechanics (Chapter 3), in thesense that both are inferred (guessed, if you like) from awide variety of observations, and ultimately justified by theagreement of the conclusions drawn from them with exper-iment; and both are extraordinarily fruitful in the range ofsuch conclusions that one can draw. Therm odynam ics in factprovides an accurate and useful description of the equilib-rium properties of any piece of matter containing a verylarge number of molecules, and it leads to an interpretationof, and a method of calculating the changes associated with,spontaneous processes. Thermodynamics goes beyondmechanics in providing an interpretation of "the arrow oft i ine"speciflcaily, why irreversible processes occur.

The methods ordinarily used to formulate the laws ofthermodynamics make no assumptions about the micro-scopic structure of matter. An internally consistent and log-ically complete theory can be constructed without referenceto the existence of atoms and molecules. Naturally, the spe-cific properties of real matter must appear in the theory, inthe form of material constants such as heat capacities orcompressibilities, but thermodynamics concerns itself onlywith the relations among these quantities, not with why theyhave the values they do; to answer the latter question onemust resort to microscopic theories of the properties andinteractions of molecules. By considering the same quanti-ties from the viewpoints of both microscopic and macro-scopic theories, one can gain significant insights into themacroscopic consequences of molecular behaviorand themicroscopic implications of macroscopic measurements.As a concrete example of how the theories can be inter-related on various levels, the pressure of a real (imperfect)gas can be expressed empirically as a power series in thedensity n: the coefficients of the pow ers of n are determinedfrom experimental data. Thermodynamics gives the rela-tionships between pressure and other macroscopic quanti-ties (internal energy, entropy, etc.), and thus enables one to


8/146

358 Matter in Equilibrium: Statistical Mechanics and Thermodynamicscalculate these quantities in terms of the coefficients in thedensity expansion of the pressure. All this is on the macro-scopic level: what of the microscopic approach? Here webegin with the concept of a potential energy of interactionbetween molecules, the V(R) described in Chapter lO.Using the methods of statistical mechanics, one can averageV(R)overall the molecular pairs in a gas, over all the triplesof molecules, and so on, obtaining average quantities thatcan be identified with the coefficients in the density expan-sion of the pressure. Combining the two theories gives onea complete chain of connections between the molecularinteractions and the thermodynamic properties. In principleone could obtain V(R). and by extension the macroscopicproperties, by solving the SchrOdinger equation for themolecular Interaction, In practice, as we know. this isextraordinarily ditlicult, and one ordinarily works in theopposite direction. deducing V(R) from macroscopic Incas-urements of the pressure or other properties, or moredirectly l'rom molecular scattering.But we have a long way to go hefre we can draw suchconnections as these: they will he developed in Chapter 2 I.Let us now get down to actually developing the principles ofthermodynamics. on what must initially be a much moreabstract level. We begin by examining some basic notions.starting with definitions of what it is that thermodynamicsstudies.A s vsrem is that part of' the physical world which is underconsideration. It may have fixed or movable boundaries ofarbitrary shape and may contain matter, radiation, or both.All the rest of the universe is defined to be the surroundings.A system is said to he open if matter can be exchangedbetween the system and the surroundings. A closed .cvxi'em.which does not exchange matter. may still be able toexchange energy with its surroundings. An isolated xvstenrhas no interactions of any kind with its surroundings.To describe the properties of a system, one needs to heable to measure certain of its attributes, in thermodynamicsthe system is described by a set of macroscopic variables or"coordinates." the determination of which requires onlymeasurements over regions containing very many molecules.carried out over periods long enough for very many molecu-lar interactions, and involving energies very large comparedwith individual quanta. The thermodynamic properties of' thesystem are assumed to be entirely described by the set ofmacroscopic coordinates. To anticipate a bit, the coordinatesto which we refer are most often the temperature, pressure,and volume. When a macroscopic coordinate is fixed in valueby the boundary conditions defining the system, there is saidto be a constraint on the system. There can be other forms ofconstraints, such as one allowing no deformation of the sys-tem. Many of these, and the roles they play in thermody-namics, will be discussed later, in Chapter 19.The classical theory of' thermodynamics deals with theproperties of' systems at equilibrium. This has essentiallythe meaning already outlined in our treatment of the perfectgas: A thermodynamic system is in equilibrium when none

of its thermodynamic properties change with time at ameasurable rate. (This should not he taken to include a.s'teadv state, in which matter or energy flows steadilythrough the system but the properties measured at any givenpoint are time-independent. An equilibrium state is a stateof rest.) The equilibrium state of a system depends on theconstraints placed on it, which must be specified. In gen-eral, there is only one true equilibrium state for 'a given setof constraints. However, many systems have states thatshow no change with time, and thus can he regarded forpractical purposes as equilibrium states, even though theyare not the true equilibrium state. An example is a roomtemperature ni sture of H2 and 02, for which the reactionH2 + -- O, - HO is so slow (in the absence at a catalyst)that the mixture can be considered nonreactive for mostpuracs. Such a condition is called a metasiable equilib-item. lhcrmodynamies can also be used to describe someat the properties of systems not at equilibrium: this aspectof the theory is still under active development.The thermodynamic state of a system is assumed to be

uniquely determined when a complete set of' independentmacroscopic coordinates is given. But what constitutes a"complete' set of thermodynamic coordinates? Given thatthe mass of the system is known. they are just those vari-ables that must he independently specified for one to repro-duce a given thermodynamic state. as th'iermineei by exper-iment. It should he noted that the number of thermodynamiccoordinates necessary to specify the stale of a system is verysmall (frequently, for it system with fixed mass, the mini-mum number is only two), whereas the total number ofmicroscopic coordinates defining the same system is of theorder of' the number ol atoms present.The number of variables required [or a complete set dif-fers from system to system. Let us concentrate on the prop-erties ot aJluiil. a form of matter that assumes the shape ofits container (i.e., a gas or liquid). In general. the shape isfound to be of no importance in determining the hulk prop-erties of the fluid. Experiment shows that, for a given massof pure fluid (fluid containing only it single chemicalspecies) in the absence of external fields, the specification ofthe pressure and volume uniquely defines the thermody-namic state: for a fluid mixture one must also specify thecomposition.3 For example. given the pressure and volumeof a certain mass of oxygen. one finds that all equal massesof oxygen whose pressures and volumes are adjusted to bethe same are in identical thermodynamic stales. That is, forthe given mass of oxygen there is a unique relationshipamong pressure, volume, and any other thermodynamic

Of course. should one be concerned with. say. the electric or magneticproperties of a system, other variable must also he used. There are alsocertain anomalous cases, the best known be i ng that of' liquid water justabove its I'reei.ing point: The density increases from U to 4"C, thendecreases again, so that, say. ' = I arm and u = I .()(tOl(l cmVg cancorrespond L u either 2C or 6C. 'tItus in this region p , v do not constitutea complete set of variables (but p. 1 do.


9/146

Temperature and the Zero-th Law of Thermodynamics 359variable; this relationship can be expressed by an empiricalequation of state,

f(p,V,X)=0, 12.18)where X is any third variable. In the next section we shallconsider the properties of one such variable, the quantity wecall temperature.12.4 Temperature and the Zero-thLaw of ThermodynamicsWe wish to describe the properties of matter, beginning withthe dilute gas, from a thermodynamic point of view. Oneconcept important in this description is the thermodynamiccoordinate known as temperature, the nature of which wemu st now clarify. Our goal is to formu late a quantitative def-inition of temperature which is free of ambiguities associ-ated with intuitive notions. Although everyone is familiarwith the qualitative notion that temperature is associatedwith the "hotness" of a body we can easily fool ourselves inthis regard: For example, try placing both your hands in abowl of tap water after one has been in ice water and theother in very hot water. This experiment shows that we can-not trust our simple intuitive or physiological reactions toprovide a quantitative measure of temperature.4We shall develop our definition of temperature using onlythe variables required to define the macroscopic state of asystem and how those variables are related when particularexperiments are carried out. Consider again a mass of purefluid, the thermodynamic state of which depends on thepressure and volume. We can define a temperature 0 bywriting an equation of state in the form

f(p,p,O)=0, 12.19)where w e have replaced the volume by the density pto makethe equation independen t of the particular mass of fluid.- ' Weemphasize that Eq. 12.19 is to be interpreted as a definitionof 9: We choose some function of pressure and density andcall it the temperature. How do we decide what function tochoose? Well, certainly C should be well behaved (finite,continuous, and single-valued); it ought to bear some rela-tionship to the intuitive notion of temperature (increasing Cshould make a system "hotter"); and it should be mathemat-ically convenient to use. But there are still infinitely manyfunctions that satisfy these conditions, For example, we

Despite the different sensations generated by tap water after exposure ofone's hands to hot and cold water, one can train oneself to identify thetemperature of an object to within about 1C. touching the object to betested to the inside of the forearm or wrist.

This can be done only for those properties, such as temperature, that arein fact independent of the total mass. We shall discuss this distinction inSection 13.4, on intensive and extensive variables.

shall see that for an ideal gas a convenient tem perature func-tion is 8= PIP; but (p/p)2 or log(p/p), or either of these (ormany other combinations of the same variables) times a con-stant multiplier, would do equally well. We can express thisgenerality by "solving" Eq. 12.19 for temperature in theform(12.20)

where g(0) is any of the alternative possible temperaturefunctions and a(p, p) is the function of pressure and densitythat defines i t . Which of the functions we choose as the tem-perature is strictly a matter of convention, as we shall see inthe next section. But first we must introduce some conceptsthat apply to all temperature scales.We begin by defining the properties of two idealizedwalls. Consider two systems I and 2, initially isolated andseparately at equilibrium. Let the two systems initially con-tain the same pure fluid at different temperaturesthat is,have values of the pressure and density such that, for a givenchoice of temperature function (Eq. 12.20), we have(p .Pi)# ( P2 'P2). 12.21)

Suppose now that the isolation of the two systems from eachother is reduced to their separation by a rigid barrier B, bothremaining isolated from the rest of the universe (Fig. 12.5).If the thermodynamic states of the two systems subse-quently remain unchanged from the initial states, then thebarrier B is called an adiabatic wall. In less precise lan-guage, an adiabatic wall is a perfect insulator across whichheat cannot flow ("adiabatic" means uncrossable), so thatthe temperatures of the two systems remain different . We areall familiar with real approximations to adiabatic wallsforexample, the silver-coated, evacuated, double glass walls ofa Thermos bottle, or a thick, rigid sheet of asbestos. Clearly,the concept of a perfect adiabatic wall is an abstraction, butone that is linked to the observed behavior of a number ofmaterials. Obviously, an isolated system must be surroundedby adiabatic walls.There is anotherand more likelyoutcome of theexperiment. Suppose again that the two systems, initiallyisolated and at different temperatures, are separated fromone another on ly by a rigid barrier B. The usual result is that

Figure 12.5 The two systems labeled I and 2 are separated by arigid wal B. Both systems are isolated from the remainder of theuniverse.


10/146

360 atter in Equilibrium: Statistical Mechanics and Thermodynamicsthe thermodynamic states of the two systems change withtime. eventually approaching a limit in which they are time-independent. Such a harrier is called a diathermal wall. andin the final time-independent state the two systems are saidto be in thermal equilibrium with each other. In ordinarylanguage, a diatheririal wail (din, "through," + therme,"heat") allows heat to pass through until the two systems areat the same temperature. All real material walls are diather-mat, although in some cases the rate at which heat can crossis very small, so that the approach to thermal equilibriumcan be very slow.

The definitions of adiabatic and diathermal walls are notlimited to fluid systems, nor to systems of the same chemi-cal composition. In general. the thermodynamic stales of thesystems may he defined by more variables than p and p. butthe principle is the same: If the stales of Iwo otherwise iso-lated systems change when they arc separated only by arigid harrier B, then B is diatliermal: if both stales remainunchanged. then B is adiabatic (except fir the trivial casethat the systems are in thermal equilibrium to begin with).The requirement that the harrier he rigid is necessary to pre-vent the perftirmance of work. which we shall discuss in thenext chapter.

We said that the original isolated systems were separatelyin equilibrium, the equilibrium defined by the constraint ofisolation. When the constraint of isolation is replaced by theconstraint of corn act through it d iat hernia I wall, the syste inmoves towards, and ultimately reaches, it new equilibriumstate. The overall system is initially out of equilibrium withrespect to the new constraint. but eventually approaches theequilibrium state. The way in which this state is reached isimmaterial at present; only the iinal cquilihriuni itself is ofconcern to us. Experiment shows that, once two systemshave reached thermal equilibrium with each other, one can-not change the slate of system I without simultaneouslychanging the state of system 2. Thus the overall system atthermal equilibrium is not described by separate equationsof state like Eq. 12.18. hut by some single relationshipinvolving the independent coordinates of both systems. Forsimplicity we shall restrict ourselves again to fluid systems.for which this single relationship has the fruii

( ii 'Pt .12 P2 )=0. 12.22We need not know the precise form of the function F 2,which will depend on the nature of the two fluids and theirindividual equations of state.

At this point it is appropriate to introduce the first postu-late of thermodynamic theory. which is known as the zero-ti,

1 ( 1 1 4 of t is called "zero-ui" because,although it was formulated after the first, second, and thirdlaws, it is logically prior to them in a rigorous developmentof the subject. The zero-th law is stated in the form:

Two systems, each separate/v in th'rntal equilibrium with athird sv.riem, at-e in i/,e,mal equilibrium with each al/let:Like the other laws of thermodynamics. this statement isderived from the results of observations of many systems. Itclearly embodies everyday experience: For example, if eachof two systems is separately Round to "have the same tern-perature' with respect to, say, a mercury-in-glass ther-mometer. we do not expect to observe any change when thetwo systems are brought into contact with each other. Butwhy do we even need to state what seems so obvious it fact?Thus far we have only examined how to define a temper-ature for it particular system, by some such means asEq. 12.19. But what we wish to obtain is a universal defini-tion of temperature applicable to all systems. For such it un i -versal temperature to exist, there must he some property thatall systems "at the same Leiiiperature' t i.e., in thermal equi-libriuni with one another) have in common. Since we cannotactually investigate all systems, for logical completeness wemust assitmc' that this is true: the ,.ero-th law has been cho-sen as the simplest way of lirniulatiiig this assumption. Weshall now see just how the zero-th law leads to a unique def-inition of, temperature.

Consider three lluids. I. 2, and 3. for each of which p andp are the only independent variables. When fluid,,. I and 2are in mutual thermal equilibrium, their thermodynamicstates are related by an equation of the frrii 12.22, whereasfor equilibrium between fluids I and 3 we have the equation

F1 1 (p 'Pt ,p,p)=-0. 223)Equations 12.22 and 12.23 both contain the variables p andp. n principle, each of these equations can he solved for,say, the pressure of fluid 1. As a result of solving these twoequations. /'ican he expressed as some function 012 of thevariables im,p2.p, r as some I unction w1 I of the variablesp m , p. p1Pm = w 2 (Pt .Th .P:) r p=VI (Pm . p .p). (12.24)

But since piis the same physical quantity in both of theseequations, we can immediately write

02(P,p2.2)Wi3 (Pm .p .p1). 12.25)Equation 12.25 is an alternative representation of the condi-tions placed on the stales of systems 1. 2, and 3 by therequirements that I and 2 he in mutual equilibrium and thatI and 3 he in mutual equilibrium

We now introduce the new element required by thezero-th law. According to this law, ii fluids 2 and 3 are sep-arately in thermal equilibrium with fluid I, they are also inthermal equilibrium with each other. Thus, the relationshipbetween the states of fluids 2 and 3 must he completelydescribed by an equation of the form

"The f i rs t fo rmu la t ion o f the zero-ch law n t ihcr rno ,Jvnan i ics wa spree ntefJ by H. von H elmho ltz. Crel le 's Journa l 97. 154 (I8SJI. 23(I'2 'P2 'P3 .p) =0. 12.26)


11/146

Empirical Temperature: The Perfect Gas Temperature Scale 361But Eq. 12.25 contains the density Pt. whereas Eq. 12.26does not. If the two equations are to hold simultaneo usly, asthe zero-th law requires, the variable pi must drop out ofEq. 12.25. That is , the functions w12 and w must be of suchi t form that we can wri te a new equat ion

W7 (P2 , P2) =W3 (P.3 , P3), 12.27)in which each of the functions w2 and w3 depends on thestate of a single system only. By an obvious extension o f theargument we obtain the more general equation

W1 (Pt ,Pi) w( P 2 , P2)= w (P3, P.A 12.28)which can be further extended to any number of fluids inmutual thermal equilibrium. Note that the w (pt, p) may bequite different functions o f pressure an d den sity for differentfluids; i t is only their numerical values that m ust be the sameat equilibrium.Example: These abstract arguments are clarified by aconcrete example. Consider the ideal gas, a hypotheticalsystem that obeys the equation of state pV = nRT we shallexplain this equation in the next section, but for now take itas a given. If we put ideal gases I and 2 in thermal equilib-rium and determined the relationship between their states,the form of Eq. 12.22 that would obtain is

p M 1 p,MF 2 (p ,p ,p ,p ) 0. 12.22')Pt 2where M is the molecular weight. Similarly, we have forideal gases I and 3,

F 3 (p .p ,p,p) p M1 M= 0. 12.23')P1 3Solving both equations for p,we obtain

Pt = Wt (PI 'P2 ,P2) p 2 M 1Pi =w13(p1 1p3M3 , 12.24')p M

and thus.2 p2 M 2 = p1p3M1 I225')p2 M 2 3 M1

The zero-th law tells us that we must also haveP2M2 P3M3(P2 ,P2.P3,P3) 0. 12.26')P2 3

which does not contain Pt. By the reasoning in the text, thismeans that Pt must drop out of Eq. 12.25', as it obviouslydoes, givingW2 (P2, P 2 ) P2 IW2 =w3 (p .p ) I2.27P2 3

By extension the function pM/p has the same value for allideal gases in thermal equilibrium; in the next section weshall see the significance of this fact in defining a tempera-ture scale.We have shown that, given the zero-th law, there mustexist a set of functions w, each depending only on the stateof the ith fluid, which all have the same value for any num-ber of fluids in thermal equilibrium. For this to he true, thew must all depend on some single property that all fluids inthermal equilibrium have in common, regardless of theirnatures. This single property is defined to be their commontemperature. Furthermore, the argument is not restricted tofluids: If an arbitrary system is described by the set of inde-pendent variables X . Y , Z1.....hen the F12 ofEq. 12.22 becomes F (X1. Y . Z.....X. X2 . Z , . .and the rest of the derivation proceeds along exactly thesame lines Thus it is possible to define a common tempera-ture for all systems of wh atever kind in thermal equilibrium.The functions Wj, i) are clearly equivalent to the(p, p) of Eq. 12.20, which says that for any fluid one candefine a temperature function g(9) on the basis of that par-ticular fluid's equation of state. What we are now saying isthat each such g ( 0) is a function o f the property that all sys-tems in thermal equilibrium have in common. This tells ushow to obtain a universal temperature scale: We choose aparticular system as a standard, select one of that system'spossible temperature functions ,g(Th, and define the numer-ical value of this g(0) as the temperature 0; the samenum erical value of 9 is assigned to any o ther system in ther-mal equilibrium with our standard system. Such a standardsystem is an example of wh at we call a thermometer and thefunction used to define a tem perature is called a thermomet-ric property; in the next section we shall see how such a sys-tem and property are chosen.12.5 Empirical Temperature:The Perfect Gas Temperature ScaleThe arguments of the preceding section seem a long wayfrom our intuitive understanding of "temperature." What isthe relationship between these abstract considerations and ausable temperature scale?Any system property that varies with temperature in awell-behaved way can be used to define a temperature scale.The choice among the various possible thermometric prop-erties is arbitrary and based essentially on conv enience. On eimportant requirem ent is that significant changes in the the r-mometric property be accurately measurable in a small sys-tem (a thermometer should be small enough to come into


12/146

362 Matter in Equilibrium: Statistical Mechanics and Thermodynamicsthermal equ i l i br i um w i th a s y s tem w i thou t cha ng ing the s ys -tem*,, temperature significantly). The best-known thermo-metric property is the volume of a liquid, most commonlymercury: the volume changes are magnified by being mea-sured in a narrow tube attached to a larger reservo ir. The firstthermometers i n ven ted were of this liquid-expansion type.with temperature scales defined as linear functions of theliquid column's length. Among the other thermometricproperties in common use are the electrical resistance of afine metal wire (usually platinum); the electromotive forceg e n e r a te d b y a thermocouple (the junction between twowires of different metals); and the intensity of the radiationemitted by a hot body, assume d to obey the black-body radi-ation l aw (Section 2.6), as measured with a pyrometer Butnone of these properties is actually used as the basis of ourtemperature scale; rather, instruments of all these types arecalibrated with respect to the scale we shall now describe.A particularly elegant and useful temperature is thatbased on the low -pressure prope rties of a very dilute gas, towhich we now finally return. Since the easily measuredproperties of a gas are its volume (or density) and pressure,we seek a convenient way of relating temperature to theseproperties, as in Eq. 12.20. The m ost straightforward proce -dure is to define the temperature as proportional to one ofthese properties, with the other held constant. The two alter-natives lead to constant-volume and constant-pressure gasthermometers, and instruments of both types in litet exist;but as we shal l see, both temperatures converge to the samelow-pressure limit.Suppose that we measure the pressure of a fixed volumeof gas in thermal co ntact with an equilibrium m ixture of ice,liquid water, and water vapor (i.e., water at its triple point)and call this pressure P3 . We use the pressure p of the samevolume of gas under other conditions as a thermometricproperty. defining a constant-volume temperature scale bythe equation

0(p)= V =constant). 12.29)P3

where 6~j is an arbitrary constant, the temperature of thetriple point of water. B y c o n v e n t io n (for reasons that willbecome apparent later) we set the value of 0 1 equal to273.16 K, the "K" standing for the temperature unit, thekelvin (formerly degree Kelvin. K). Alternatively, we canmeasure the volume Vi of a given mass of gas at the triple-point temperature and gas pressure p, then use the volume atthe same pressure to define the constant-pressure tempera-lure scale

(p = constant). 12.30)The empirical temperature scales defined in Eqs. 12,29and 12.30 are functions of both the mass of gas in the ther-mom eter and the nature of the gas. Experimentally, however,

as one low ers the quantity of gas (and thus the value of p) ina constant-volume thermometer or the constant pressure p ina constant-pressure thermometer, both temperatures arefound to approach the same limiting value,lim0(p)= li m O(V)=9*, 12.31)

which is, furthermore, the same for all gases. This limitingprocess is illustrated in Fig. 12.6. The limit 9* is called theperfect gas temperature.Although the elaborate l imiting procedure described hereis not itself convenient for ordinary temperature measure-men ts, it can be u sed as a primary standard to calibrate prac-tical thermometers of the various types we have mentioned.The International Practical Temperature Scale of 1968(IPTS-68) is a set of reproducible fixed points' and interpo-lation form ulas that have been reliably me asured in terms ofthe perfect gas tempe rature scale. The IPTS-68 is defined interms of a platinum-resistance thermometer from 13.81 K to903.89 K: of a platinum/platinumrhodium thermocouplefrom 903.89 K to 1337.58 K; and of Planck's black-bodyradiation law above 1337.58 K. These can in turn be used tocalibrate thermometers for ordinary measurements, so that aperfect gas temperature can be determined, even if indi-rectly, for any system of interest.Equation 12.31 is a direct consequen ce of the experime n-tal observations em bodied in Boyle's law and Gay Lussac'slaw (the latter of which is often called Charles 's law), whichwe can now formulate as follows: If the temperature 0 of amass of gas is held fixed and its pressure p is lowered, thenthe product pV/n. where V/n is the volume per m ole (cf. Sec-t ion 1.6), approache s for all gases the same limiting value,

lim pV/n =6(0*)(12.32)

where the common limit ,#(O*) is a function of the temperature 9* only (Boyle's Jaw). Furthermore, the limits ob-tained at two different temperatures are related by theequation - 12.33)I3(0)The fixed point' of the IPTS-68 are as follows ("boiling point" and

"freezing point" imply a pressure of ) aim):13.81K (-259.34 'C) t r ip le po in t o f equ i l ib r ium hy drogen (see C hap te r 21) ;1 7 . 0 4 2 K (-256.1 08C) liquid-vapor equilibrium point of equilibriumhydroge n at 33330.6 N1m 2 pressure :20.28 K (-252.87'0 boiling point of equilibrium hydrogen:27.102 K (-246.048C) boiling point of neon:54.361 K (-2)8.789C) triple point of oxygen:90.188 K (-182.962C) boiling point of oxygen;273.16 K (0.01C) triple point of water;373.15 K (l(X).OtYC) boiling point of water:692.73 K (4)9.58C) freezing point of zinc;235.08 K (961.93C) freezing point of silver:1337.58 K (1064.43C) f reez ing po in t o f go ld .


13/146

Empirical Temperature: The Perfect Gas Temperature Scale 363Ne(p0) -He(VaIi

He lp 0 ) . _N eW 0 ) j150.000149.9951

49.990

149.980149.985

H2 (Vol\N2

50 .020 I8

4,'50.015

V0),Ne(p050.010 2(pa1']50.0055000C 0.2 .4 .6 .8 .0

P0 (cm Hg)He (pg )and Ne(V0)-Figure 12.6 As the p ressu re is l owe red i n a c o ns t an t -v o l ume t h e r -mom ete r [8 (p) l . o r a cons tan t -pressu re the rmome te r rnV)], bo t hi ii p i ri ca l tempe ra tu res approach a c ommo n va lue , name ly , t he per-eel gas temperature 9. In the figure po a n d V 0 are , res pec t i ve l y .lie p r e ssu r e a nd v o l ume o f t he ga s i n t he t he r mome te r a t t he i c epo int. F rom F . H . C raw f o rd , H ea t , Therenodvna,rtjcs and Statistical/'lovics, H arc ou r t B rac e J ova nov i c h . New Yo rk . 1 963 .

where 9* is the perfect gas temperature defined inI '.q. 12.31; that is. B( 9*) is directly proportional to 9* (GayI .ussac's law).

The temperature scale was formerly defined in terms ofIwo fixed points, the temperatures of melting ice and boilingwater (each under a pressure of I atm). The Celsius (centi-grade) scale was obtained by dividing the interval into100 degrees, the melting-ice and boiling-water points beingdesignated as 0C and 100C, respectively. For a constant-volume gas thermometer, then, one could define (in the per-Ic as limit) the Celsius temperature

f*=urn 100(ppo) V = constant). 12.34)('.911 / l o o AAt that time the absolute temperature & was defined by theequation

(12.35)

where O*,the ice-point temperature, had to he chosen suchthat Eq. 12.33 would be satisfied: the experimental valuewas about 273.15. corresponding to 273.16 for the triplepoint 9*.To eliminate the experimental uncertainty, aninternational agreement in 1954 made 273.16 K bydefinition, as indicated above. The melting point of ice, 8o

is then still 273.15 K. but the boiling point of water changesfrom 373.15 to 373.146 K. Since the Celsius degree is nowdefined as equal to the kelvin. Eq. 12.35 still holds as a def-inition of the Celsius temperature t.but Eq. 12.34 is nolonger exactly valid.8

We shall show in Chapter 16 that it is possible to estab-lish a thermodynamic temperature scale independent of theproperties of any real or hypothetical thermometric sub-stance. However, it will also he shown that this thermody-namic temperature T is in fact equal to the perfect gas tem-perature 9*. To simplify our notation, we shall anticipatethis result by regularly using T to designate the perfect gastemperature.

We may as well also say something here about the unitsof pressure. The ST unit (adopted in 1971) is the pascal(Pa), or newton per square meter: closely related are thedyne per square centimeter (=0.1 Pa) and the bar (ml05 Pa).However, most scientific measurements are still expressedin terms of the atmosphere (atm), which has the definedvalue

I aim ml.01325xl05 Pa,chosen to fall within the range of actual atmospheric pres-sures near sea level. We shall see in subsequent chapters thatthermodynamic quantities are commonly defined in relationto a standard state at I atm. For lower pressures one com-monly uses the torr defined as - tm: this is for practicalpurposes the same as the conventional "millimeter of mer-cury" (mm Hg)."

We can now combine Eqs. 12.32 and 12.33 to obtain thefamiliar perfect gas equation of state,'()

j)V =nRT 12.36)

where n is the number of moles of gas in the volume V Theproportionality constant R is known as the (universal) gasconstant, and can he evaluated by measuring the limit

R=lim (2.37)jO nT0 The I967 General Conference on Weights and Measures adopted thename "kelvin." symbol 'KS' for the ( m u of thermmidvnaniic temperature.with "degree Celsius:' symbol "('." as an alternative unit for atemperature interval. Note the omission of the word "degree" and thedegree symbol from the kelvin scale.To he precise. the "millimeter in mercury" is defined as the pressure

exerted by a l-mnm column of a fluid with density 13.5951 glctn (equalto that o1 mercury at OC) at a place where the acceleration of gravity isLf l tO( ibS mIs.For the record we now that the torr is named after'flirrieelli, who was a prominent Renaissance scientist.iu This is also and perhaps more commonly known as (he ideal gas law.One can make a formal distinction between a ,nrfei': gas, which has themicroscopic propel-ties described in Section 12. 1, and an ideal gas , o newhich satisfies Eq. 12.36. But to the extent that both have the samelimiting properties, we can identify the two and call both perfect gases.


14/146

364 Matter in Equilibrium: Statistical Mechanics and Thermodynamicswhich is the same for all real gases. It has the value

R = 0.0820568 liter atm. / mol K=8.31441 Jfmoll> m and v>> v ,, the rateof gain of kinetic energy of the gas is

2Zmvv,dtwhere Z is the numbe r of coll isions per seco nd of al lmolecules w ith the piston. Interpret this result.

27. Show from the result of Problem 2 6 that the gain ofkinetic energy by the gas in some infinitesimal timeinterval is just the wo rk

= - p dv,

28. In Problem 25 we modeled an adiabatic expansion orcompression. To model an isothermal expansion orcompression, imagine that a frictionless piston isbetwe en two p erfect gases, and subjected to collisionsfrom both (see diagram). As a result of collisions

gas asthermostatsurrounds rictioneascontainer iston

between the piston and the m olecules of gases I and 2,the piston jiggles back and forth. When it is balancedbetween the tw o gases its average velocity, (vu ) is zero,but its mean square velocity (v), s not zero. The jig-gling of the piston is the mechanism by w hich energy istransferred betwe en gases 1 and 2.The isothermal com pression is imagined to occur asfollows. Suppo se that there is a very small imbalance offorces on the piston (say, gas 2 pushes on gas 1), sosmall that the resultant piston velocity is very, verysmall, indeed sensibly zero. Suppose further that thesurroundings of the container are a thermostat, so thatthe temperatures of gases I and 2, and the piston, arefixed througho ut the compression. N ote that this condi-tion implies that the kinetic energy of the p iston cannotchange during com pression. Show by direct calculationthat the kinetic energy of gas 1 is not increased as aresult of compression. Use the same simple model formolecular mo tion as in the preceding problem s.


38/146

(1HA1TER 14Thermochemistryand Its Applications

We now pause in our development of thermodynamics to seewhat use can he made of what we have already learnedmainly, the first law. In short, we shall come down fromthe heights of abstraction and deal for a while with appli-cations- In this chapter we are concerned with the energyand temperature changes that accompany physical andchemical processes of various kinds. The measurement andinterpretation of such changes constitutes the science ofilier,noclxemisirv.

To obtain meaningful interpretations of thermochemicalmeasurements, the conditions under which the variousprocesses occur must be carefully defined. We anticipate.for example, that a given chemical reaction will release dif-ferent amounts of heat when it proceeds under different con-straints, since the heat and work associated with any processdepend on the path. The first part of the chapter is thereforedevoted largely to definitions of the many specific kinds ofenergy change, and of the standard states relative to whichthey are defined.

The remainder of the chapter deals with the molecularinterpretation of thermochemical data. This is an extremelyrich field, in which we finally have the opportunity to com-bine microscopic and thermodynamic knowledge in a usefulway. Although still more can he learned with the help of thesecond law of thermodynamics, we shall see that the firstlaw alone provides a wealth of fascinating information aboutsuch chemically significant topics as bond energies, theshape of molecules, and the stability of oxidation states.

Before starting our exposition. which uses physicalprocesses as examples. we need to point out that there hasbeen a subtle difference of focus in the applications of ther-modynamics to physical processes and to biologicalprocesses. Traditional thermodynamics grew out of theanalysis and optimization of heat enginesmachines thatconvert heat into other forms of energy. especially mechan-ical energy. For that reason considerable attention is paid tothe production of heat in various processes and to the effi-ciency of the conversion of heat into work. Living systems,for the most part. convert energy from one form to another

without going through an intermediate transformation toheat. For example, muscles convert stored chemical energydirectly into mechanical energy and membranes maintainconcentration gradients without using heat energy; theseprocesses are driven by coupled cycles of chemical reac-tions. This difference of focus is slowly vanishing as mod-ern technologies such as conversion of sunlight into electri-cal energy via solar photoelectric cells and conversion ofchemical energy into electrical energy in fuel cells stimulatedevelopments in engineering thermodynamics that deal with"directprocesses" in which energy is converted from oneform to another without an intermediate transformation LOheat. At the same time. the interpretation of how living sys-tems function is turning more and more from analyses thatdeal only with structure to analyses that include processes.These developments have given rise to concepts such as"nioleculai' motors," which have been applied from theorganisma and celular leves al the way to the(macro)molecular level. Indeed, the tools of thermodynani-ics. the most general of all sciences, are becoming increas-ingly useful in the analysis of' biological function and bio-logical processes.

14.1 Heat Capacity and EnthalpyWe begin by introducing some new thermuodynainic func-tions. including those that are particularly applicable todescribing energy transfer, the heat capacities. We have hadoccasion to mention heat capacity before (as early as Sec-tion 2.9), but now we are ready for a precise definition. Letus approach this by way of an idealized thermochemicalexperiment.

Consider a vessel with adiabatic walls, completely filledwith fluid and containing a thermometer and an electricheater. Suppose that the walls are rigid, so that the volumeof the vessel is fixed. The system is thus isolated except forthe heater. At equilibrium in the initial state the temperatureof the fluid is T . f some current is passed through the dee-

388


39/146

inc heater for a specified time period, a new equilibrium w illbe established at a higher tem perature T2; we define AT T2Ti. The electrical energy supplied is dissipated as a quan-tity of heat q. The limiting ratio of q to AT as both quantitiestend to zero,

lim 14.1)aT-Ol, AT),

is called the heat capacity at constant volume) But by thefirst law of thermodynamics, Eq. 13.18, in the processdescr ibed w e have(14.2)

since no work can be performed by the fluid (i.e., W V = 0).We therefore find that Cv is the rate of change of internalenergy with temperature when the system is maintained atconstant volume:CV 1- . 14.3)dT

But Cv is not the only kind of h eat capacity. Suppose thatwe perform the same experiment as before, except that onewall of the vessel is a freely m oving (bu t stil l adiabatic) pis-ton. The volum e of the fluid will no longer be the sam e afteraddition of heat. However, we can instead maintain theexternal pressure Papp fixed; if the heat is added so slowly asto approximate a reversible process, the fluid pressure pcan also be assumed fixed. The limiting ratio for such aconstant-pressure process,urn (_.2._) 14.4)ATO\, AT

is called the heat capa city at constant pressure. (In what fol-lows, pressure-volume work processes should be assumedreversible unless otherwise stated; thus we can consistentlytake p = Papp)Although one can define many other heat capacities,2 Cvand C,, are the only ones with which we now need to concernourselves. Both Cv and C,, as we have defined them areextensive quantities, proportional to the mass of fluid pres-ent. It is customary to refer them to a specific amount ofmaterial, usually the mole, in which case they are calledmolar heat capacities. (The term "molar specific heats" isalso used, but it is best to restrict "specific heat" to the mean-ing of heat capacity per gram.) They can be given in suchunits as joules per degree (C or K) per mo le. The molar heat

Heat Capacity and Enthalpy 389capacities are state functions characteristic of the particularsubstance, and we shall see in later chapters how they can b erelated to the microscopic energy spectrum. For notationalconsistency we shall use capital letters for extensive func-t ions, for example, U for the total internal energy of an arbi-trary mass of substance, and lower case letters for the corre-sponding molar functions, for example, u for the internalenergy per m ole. However , because the usage is so common,we also use capital let ters for the changes in energy , enthalpy,and so forth per mole of product in a chemical reaction.When the fluid in the constant-pressure experiment isallowed to expand, work is done by the system on the sur-roundings (w,, Cv.For a constant-volume proc ess we can rew rite Eq. 14.3 inthe differential form

(dU) = Cy dT. 14.5)Does there exist a comparably simple relationship for aconstant-pressure proc ess? Using the first law in differentialform, Eq. 13.19, we have

(dU) =C,, dT p dV 14.6)for a reversible constant-pressure process in a system that canperform only pV work. The heat added at constant pressure,C,, d7 thus equals the quantity (dU + p dv),,. This suggeststhat we should int roduce a new thermodynamic var iable,

HU+PV, 14.7)since we can then write

(dH),, =(dU+pdV),, =C,, dT, 14.8)analogous to Eq. 14.5; the analog of Eq. 14.3 is thus

CP=(dH) 14.9)dT P

As in the partial derivative notation, we use a subscript (here 1) todenote a variable that is held constant in the process described.2 For example, in a system whose properties depend on the electric heldE, one might need to use C5 =urn (qhT)s.An exception occurs if the fluid contracts, rather than expanding, whenheat is added at constant pressure (qp > 0, wi,>0); then we have (U),,>(ESL/)y and thus C,, < Cv. This anomaly occurs in liquid water between WCand 4'C.


40/146

390 Thermochemistry and Its ApplicationsThe function H is called the enthalpy of the system. Since Uand pV are both unique functions of the thermodynamicstate of the system, H must also be a state function: it has thedimensions o f energy. and its zero has the same a rbitrarinessas that of U.The introduction of the function H may seem arbi t rary oreven w himsical. This is not so, how ever. When the system isa simple fluid that can perform only pV work, the naturalvariables defining the internal energy. those in terms ofwhich the function has the simplest form, are T and V . Butmany processes are more conveniently carried out at con-stant pressure than at constant volume 4 and are thusdescribed m ore com pact ly in terms ofenthalpy changes thanin terms of energy changes. For example, the heat trans-ferred in a constant-pressure pro cess is just (Mi),. = qp' 5 Thepassage from Eq. 14.6 to Eq. 14.8 is best regarded as achange of independent variable from V to p. If the samechange of variable is carried out for a constant-volumeprocess , in which (H)y =qv, we obtain

(dH) dU) +ld(pV) V =C dT+Vdp. 14.10)In Section 14.3 we shall see how equ ations such as these canbe used to obtain Ml for arbitrary changes of state.The apparatus actually used in experimental thermo-chemistry is not very different from the idealized modelsalready described. One uses an insulated vessel, called acalor:meier in which heat is delivered to a sample undercontrolled conditions, usually constant pressure or constantvolume. A lthough our discussion so far has been in terms ofa f luid sample, s imilar measuremen ts can of course be m adeon sohds. The science of calorimetry has advanced to thepoint where very small energy changes and very smallamounts of a substance can be studied. We shall not go intothe fascinating experimental details: it is sufficient to notethat particular calorimetric studies can involve a wide rangeof subjects, e.g., sophisticated analyses of the temperaturescale coupled with development of special techniques toreach very low or very high temperatures and the use of a

An obvious example is any process open to the atmosphere.11 is important to note that (A ),, = q, in a constant pressure p r ocess

whether that process is reversible or irreversible. To see that this is so,imagine a chemical reaction carried out in a cylinder with diathcrmzilwalls and closed by a movable piston. Let the fixed external pressureacting on the piston be efore the reaction starts. ihe system consistsof only reactants at pressure p, and alter it is complete the pressure of theproducts is pl. We require that p,,,, = p1 = p. During the reaction, thesystem pressure will vary and the piston will move, always against theconstant external pressureP.P. Since All = U + pV). where p is thepressure of the system, we can write AH= q + w + (pjV, - pV,) = q + WI-p(Vj V1). But, by definition. iii , _JprrdV_ppp(V1 VsoAH=qa t t h e Cons t an t app l i ed p r essu r e' See J. P. McCullough and D. W. Scott (Eds.). Experimental7her,nodvnan,u-s, Vol. I (Plenum Press. New York, 1968): B. LeNeindreand B. Vodar (Eds.). Experimental Tliermodvna,njcs, Vol. II(Butterworths. London, 1975); Precision Measurement and Calibration:'Ji'mperature (N.B.S. Special Publication 300. Vol. 2. 1968).

bolometer (a form of calorimeter) to measure the intensityof a molecular beam.The immediate results of calorimetric measurements arevalues of the heat capacities. Given a sufficient amount ofsuch data, one can ob tain C,, and C' as functionsof temper-ature and pressure (or density) over the whole range ofinterest. The various equations above can then be integratedto determine A U or Al- for any process; we shal l give exam-ples of such calculations in the following sections. Note,how ever , that to obtain (AU),. or (A H)v one must determinepAV or yAp, respectively, for the process in question; thisrequires either measurement or a knowledge of the fluid'sequation of state.Table 14.1 gives data on the heat capacities of varioussubstances. For the same reason that H is more useful thanU. most such tables give values of ci,, rather than C ,.For the same substance, C,. and C v are of course relatedto one another. One can easily find the relation between thetwo fo r the perfect gas. Given Eq. 12.42, U = nRT we cansee that (9UM T),. = (U /dTy: we thus have for the perfectgas

C Cv(-(lT ) PT )..HU+pV)1 _()Ti,. aTJ,.

_.F'p)1 +,1(nRT)] =nR, (14.111IT i,. 1T ,.where w e have subst ituted pV = nRT That is, the molar heatcapacities of a perfect gas should differ by the rather sizablequantity R, or about 8.3 J/K m ol. The difference is much lessin a solid or liquid, where the molar volume is only about0.001 as great as in the gas: as a result, pV is only a smallfraction of nRT so that H and U. and thus G,, and Cv, arenearly the same. Later, in Chapter 17, we shall derive anequation giving the relationship between G,. and C v for thegeneral case.Some interesting regularities can be obtained from the datain Table 14.1. For example. near room temperature we haveell = - R (20.79 J/K mol) for monatomic gases and c(29.10 J/K mol) for diatomic gases: in Chapter 21 we shallsee how these values can be obtained from microscopic the-ory. In general, the value of c, tends to increase with thenumber of atoms in a co mpound o r , more precisely, with thenumber of vibrational modes in the molecule. In most solidmetals we have to a good approximation c,., 3R (25 J/Kmol). This is the Law of Dulong and Petit, which we intro-duced in Section 2.9; it can also be derived theoretically, aswe shall see in Chapter 22. Similarly, for monatomic-ionsalts such as the alkali halides, we find c = 6R (50 J/K mol)or 3R per ion.The temperature dependence of the heat capacity isshown for a variety of substances in Fig. 14.1. Note that for


41/146

Heat Capacity and Enthalpy 391Isble 14.1 Heat Capacit ies of Some Substances at 25Cand 1 atm

Substance c, (i/K mol) Substance c, (i/K mol)GASES: SOLIDS:lie 20.79 C (graphite) 8.53Ar 20.79 C (diamond) 6.12Xe 20.79 S (rhombic) 22.60Ik 28.83 S (monoclinic) 23,64N 2 29.12 1 2 54.4407 29.36 Li 23.64('12 33.84 K 29.51lId 29.12 Cs 31.4CO 29.15 Ca 26.28112S 34.22 Ba 26.36N H 3 35.52 Ti 25.00CH4 35.79 Hf 25.52CO2 37.13 Cr 23.35SO2 39.87 W 24.08PC1 3 72.05 Fe 25.23C2H5 52.70 N i 26.05n - C 4 H 1 0 98.78 Pt 26.57

Cu 24.47LIQUIDS: Au 25.38Hg 27.98 A l 24.34Br2 75.71 Pb 26.82H 2 0 75.15 U 27.45CS 2 75.65 NaCl 49.69N 2 H 4 98.93 NaOH 59.45CC11 131.7 NaNO3 93.05H 2 SO4 138.9 LIF 41.90CI-I6 136.1 Cs 51.87nC6H1 4 195.0 AgCI 50.78C 14 50H 111.4 BaSO4 101.8(C 2 H5 )20 171.1 K4Fe(CN)6 335.9CH3 COOH 123.4 CB r4 128.0

SiO2 (quartz) 44.43B203 62.97VO 127.3C10 - g (naphthalene) 165.711 -C33H68 912

all substances c,, vanishes at absolute zero, rising in mostcases to a "plateau" (often the Dulong-Petit value) at highertemperatures. Discontinuities occur at the melting and boil-ing points, with the liquid usually having a higher heatcapacity than the other phases. Some solid-solid phase tran-sitions are also shown: an ordinary discontinuity in N2, butsharp peaks in CH4 and FeF2; the latter are called "lambdapoints," from the shape of the C(T curve, and represent

30 f70- H20(1J

N 2 (s,)/N2 ) -6 050 - r(l)

Ar(g)

1 10) I I0 50 100 150 200 250 300 350 400 450 500T(K)

(a)

o so 100 150 200 250 300 350 400 450 500T (K)(b)

Figure 14.1 Temperature dependence of the heat capacity forvarious substances. (a) Room-temperature gases and liquids.(b) Room-temperature solids.

second-order phase transitions (see Chapter 24). Within agiven phase, the temperature dependence is usually welldescribed by an equation of the form c,, = a + bT + cT-,where b and c are typically of the order of lO- i/Kmol andI i/K mol, respectively. The pressure dependence of theheat capacity is slight in the condensed phases,7 but can beappreciable for the gas, as shown in Fig. 14.2.

In liquids, c , , typically decreases by about 10% in 2000-3000 atm, thenincreases more slowly; in metals there is an almost negligible decrease, ofthe order of I 0 5/atm.


42/146

392 Thermochemistry and Its Applications

I p 3 0016 -260 K4

4 4 200K 160K2 lOOK

'0 -1 . .6 00K -250 K4 00K -400 K500 K2 I0 000 000 000 000 000

Pressu re (a tm)

Figure 14.2 Constant-volume heat capacity ofgaseous argon as a function of pressure and tem-perature. From F. Din (Ed.), ThermodynamicFunctions of Gases, Vol. 2 (Butterworths, Lon-don, 1962) .

14.2 Energy and Enthalpy Changesin Chemical ReactionsOne of the most important applications of the first law ofthermodynamics is to the study of the energy changes whichaccompany chemical reactions. Suppose that we consider ageneral chemical reaction

aA+bB+=1L+mM+. 14.12)For the present purposes, this is more conveniently writtenas an algebraic equation with all the terms on one side,

1L+mM+-..aAbB-....=vj X =0, 14.13)

where the symbols X1 stand for the r species A, B, .and the v are the stoichiometric coefficients a b.....By convention we always subtract reactants from products,so the v, must be positive for the products and negativefor the reactants. To make this more specific, consider thereaction1H2++Cl2 =HC1,

for which we have vticl = 1, VH2 =-41V,=fn thenotation of Eq. 14.13.How do we go from an equation in terms of chemical formulas to one involving the thermodynamic properties ofsubstances? Consider the process by which we balance achemical equation. What we actually do is apply the princi-ple that the atoms of each element are conserved in chemi-cal processes. But mass is also conserved, if we neglect rel-

ativistic effects.8 We can therefore write, for the change ofmass in a reaction,,6M

jMj =0, 14.14)

where M , is the molar mass (molecular weight) of sub-stance i.Now consider the internal energy and the enthalpy. Theyare state functions, and thus defined for any thermodynamicstate of a system. Furthermore, they are extensive functions,so that we can define an energy or enthalpy per mole. Wecan therefore write equations similar to Eq. 14.14 for thetotal changes in internal energy and enthalpy in a reaction.These changes will not in general be zero, unless the systemis isolated. We have

LW=vu 14.15a)and

r14.15b)

If u,and h, are the internal energy and enthalpy per mole ofcomponent i, respectively, then the LW and M-I defined by

Einstein's fundamental relation of equivalence between mass andenergy, E = me' (where c is the speed of light), implies that there must bea mass difference between reactants and products if there is an energychange in a reaction. For typical chemical reaction energies (say,40 ki/mol), however, the mass change is less than 10 g/mol, and canbe safely neglected.


43/146

Energy and Enthalpy Changes in Chemical Reactions 393these equations refer to I mol of the stoichiometric reaction.lorexaniple, if the reaction is -H, +0 7 = HC1, then AUi s the energy change when 0.5 mol each of H2 and C12 reactto form 1 m ol of HCI.But w hat are the states for which u, and /t1 are defined? Tospecify AU and A ll for a reaction, we must clearly know n otonly the stoichiometric equation but the initial states of thereactants and the final states of the products. By "state." asusual, we mean the specification of the state of aggregation(solid, liquid, or gas) and the temperature and pressure; u1and h i in Eqs. 14.15 refer to the initial state for the reactantsand to the final state for the p roducts. It is simp lest to treat areaction that occurs entirely at the same T and p, but obvi-ously we cannot expect all real-life reactions to be this sim-ple. A com mon situation, howev er, is a reaction occurring ina thermostat subject to atmospheric pressure.An additional complication is the fact that changes inenergy and enthalpy occur even when substances are mixedwithout reaction. Since in practice chemical processes areusually directed to the production of pure substances, it iscustomary to take the initial state as consisting of pure reac-tants isolated from one another, and the final state as isolatedpure products. One can then imagine the overall reaction asa three-step process:1.Take the pure rea ctants, in their initial equilibrium states,and physically mix them under reaction conditions.2. Allow the reaction to proceed.3. Isolate the individual products from the reaction mixtureand bring them to their final equilibrium states.This is not how a reaction ordinarily takes place, of course;reaction begins w hile step I is stil l under way, and it is oftenuseful to start step 3 before the reaction is ov er. But for statefunctions such as U and H, the total change must be thesame for this idealized path as in reality, as long as the ini-tial and final states are the same. If an experimenter wishesto determine the energy change in step 2 alone, then, hemust be careful to account for the changes corresponding tosteps I and 3. Step 2 is of particular interest because it islikely to correspond to the model of any microscopic theoryof reaction.Once we have defined exactly what process we are con-sidering, what is it that Eqs. 14.15 tell us? The first of theseequations is in fact a statement of the conservation ofenergy: The total internal energy of the reactants plus theenergy absorbed by the system (AU) must equal the totalinternal energy of the products. That is, the differencebetween the total energies of the reactants and the productsgives just the amount of energy that must be absorbed orreleased for the reaction to occur under the given co nditions.If we have a table of the internal energy per mole as a func-tion of T and p for each of the species involved, we canobtain AU directly by subtraction. The same is true of theenthalpy change, which is more commonly used, and

expressions like Eqs. 14.15 can also be written for otherextensive state functions.The algebraic properties of chemical equations have ledus to Eqs. 14.14 and 14.15. Since a set of chemical equationscan be thought of as a set of mass balances, it follows thatdifferent equations can be combined by addition and sub-traction to obtain any desired mass balance. It does not mat-ter whether or not the final equation represents a real chemi-cal reaction, that is, one observed in the laboratory. For anexample of this process, consider the several reactions(1) C+02 =CO2 ,(2) H2-07=I-I20,(3) C,H 6 2CO2 +3H20;if we algebraically add 2(Eq. I) + 3(Eq. 2) - Eq. 3, multi-plying each equation by the indicated numerical factor, weobtain the composite reaction(4) 2C4-3H2 C2 1 1 6 .But w hat good is such m anipulation? The answ er is that i t isequally valid for extensive quantities other than mass. Untilnow we have thought of a substance's chemical formula asrepresenting the substance itself, or in stoichiometric con-texts a mole of the substance; but it can equally well repre-sent the internal energy, enthalpy, or any other extensiveproperty of a mole of the substance. This makes possible atremendous reduction in empiricism, since AU, All, andother such quantities can be obtained for reactions withoutactually performing them.Let us examine this. The enthalpy balance for Eq. Iabove, namely Eq. 14.15b, can be written as

hc+ho, zhco, -.or, in the more usual shorthand,

C+02 =CO2 All:similarly for the other equations, and with either All or AU.Since Eqs. 14.15 are of the same form as the mass balance,Eq. 14.14, the internal energy and enthalpy changes forEq. 4 can be computed by adding those for Eqs. 1-3 in thesame way as we added the equations themselves:

AU (Eq. 4)= 2AU(Eq. 1)+3AU(Eq. 2)AU(Eq. 3),AH (Eq. 4) = 2 AH (Eq. 1)+3A11(Eq.2)AH(Eq.3).

These equations are examples of Hess's law, which statesthat the internal energy and enthalpy changes of such com-posite reactions are additive, provided that all the reactionsare carried out under the same conditions (i.e., at the sametemperature and pressure). Like Eqs. 14.15 themselves,


44/146

394 Thermochemistry and Its ApplicationsHess's law is a direct consequence o f the first law. Observenow w hat we have shown: If All (or AU) has been measuredfor reactions 1, 2, and 3, the above equa tions enable us tocalculate All (or AU) for reaction 4 without eve r actuallyobserving that reaction. We thus beg in to see the practicalvalue of all the formal structure and laws of thermodynam -ics, in that they enable us to extend our knowledge beyondwhat has been m easured directly. To do this efficiently, how-ever, it is not enough to organize our know ledge in terms ofspecific chem ical reactions. As w e shall see in Section 14.4,a far greater reduction in em piricism is obtained when ther-modyn amic measurem ents are analyzed in terms of individ-ual substances.Hess's law is a con sequence o f the fact that U and H arestate functions, and therefore that AU and All are indepen-dent of path. It does not matter whether the reac tion sequence2(Eq. I) + 3(Eq. 2) Eq. 3 is an actual pathway or even aphysically possible pathway for reaction 4. As lon g as thenet reactants and products of the sequence are, respectively,identical in chemical composition and physical state to thereactants and products of the overall reaction, Hess's law isapplicable.Here is a num erical example of the application of Hess'slaw. Suppose that for the following reactions we kno w All(often called the heat of reaction, because at constant pres-sure it is the same as q), all at 25C an d I atm:(5) PbO(s)+S(s)+O2 (g)= PbSO4 (s),

AR= 39.56 U I mol;(6) Pb0(s)+H 7 SO 4 5H2 0(/) = PbSO 4 (s)+6H20(0,

LsR= -5.57 kJ/mol;(7) S01 (g)+6H2 0(/)= H2 SO 4 .5H 2 0(l),AR=-9.82kJ/mol;where the symbols in parentheses fol lowing the chemicalformulas indicate that the substances are in the solid (s), liq-uid (1), or gaseous (g) state. What is the heat of reaction for(8) S(s)+O (g)=SO 3 (g)at 25C and 1 atm? In this case, since the product S03 isformed directly from the elements S and 02, the enthalpychange is known as the enthalpy of formation (heat offor-mation), AHfi of S03(9). We can obtain Eq. 8 by simply sub-tracting Eqs. 6 and 7 from Eq. 5; the heat of formation ofS03(g) at 25C and 1 atm is thus

iXH j (SO 3 ,g)LiH(Eq. 8)=AH(Eq. 5)AH(Eq.6) A H (Eq . 7 )= [-39.56 (-5.57) (-9.82)] U I mol=-24.17 kJ/mol.

Note how easy the analysis becomes w hen the chemical for-mulas are regarded as algebraic symbols, each representing

a mo le of the corresponding substance. Ordinary algebraicmanipulations can then be used to simplify complicated setsof equations.Because of the sign convention for reactions embodied inEqs. 14.15, a negative value of AU means that energy is lib-erated in a reaction. This energy usually appears m ainly asheat; specifically, when only expansion w ork is p ossible, theheat liberated is L W at constant volume and All at con-stant pressure. In the above example, heat is liberated informing S03(9 ) from the elements at 1 atm. But in generalwork other than pressure-volume work may also be per-formed, and under some circumstances this can be the dom-inant effect. Indeed, both internal comb ustion engines andelectrochemical cells are designed to maximize the amountof work ob tained from a chemical reaction.In the preceding examples we have con sidered only reac-tions in which all the products and reactants are at the sametemperature and pressure. Of course, this will not always betrue, especially if large am ounts of energ y are liberated. Inpractice it is often imp ossible to keep T and p under controlduring a reaction (and sometimes a change in T or p isexactly what one wants to produce). What one can do to keepthe calculation tractable is to assum e that the reaction itselfoccurs at some constant T and p, and to consider the temper-ature and pressure chang es as separate steps in the overa llprocess. This w ill still give the right values o f AU and All, aslong as one does not change the initial and final states. But todo this one must know how to o btain the internal energy andenthalpy changes accompanying physical processes such asthe expansion of a fluid, or a phase chang e from liquid to gas.The next section is devoted to this sub ject.

14.3 Thermochemistryof Physical ProcessesLet us consider first the changes in internal energy andenthalpy when a fluid, originally at equilibrium in the statePi V1. T1, is brought to the new equilibrium state P2, V2, 12 .Since both U and H are state functions, all paths connectingthe initial and final states must give the sam e values of AUand All . For convenience we can then imagine the changeof state to be carried out in two steps, as shown inFig. 14.3a. First, the fluid is heated (or coo led) at constantvolume V 1 until the temperature T2 is reached. Second, withthe temperature 7 ' 2 held fixed in a thermostat, the fluid isexpanded (or contracted) until the volum e reaches V2; sinceonly two of the variables p, V, T are independent, this mustalso bring the pressure to P2. In short, we split the overallchange into an isochoric (constant-volume) process and anisothermal (constant-temperature) process. We are thusconsidering V and T as our independent variables, and weshall see that

physical chenistry

Documents

ithe perfect gas

dilute gas

perfect gas isa model

calledan ideal gas

equations of state

familiar ideal gas law

theequation of state

macroscopic properties