A OVERVIEW OF THERMODYNAMICS
Here an overview of th erm odynam ics will be presented with the intent of defin in g an d describin g it in rather broad term s. Som e topics that are lightly covered here will be developed in m ore detail in later chapters. The intent of this chapter is to provide the con text for these later explorations. Many m odern-day textbook au thors take statistical m echan ics to be an integral part of therm odynam ics, bu t here the term therm odyn am ics will refer to classical or ph en om en ological th erm odyn am ics. On e of the objectives of this work is to exam in e th erm odyn am ics and its relation ship to statistical m echan ics.
1 .1 Th e St ran gen ess of Th erm odyn am ics.
Am on g th ose form ally exposed to th erm odynam ics, few wou ld disagree that wh ile th ere is no dou bt of its u sefu lness, there is also no dou bt of its stran gen ess. Lest this be taken as a defect in ou r intellectu al powers, it is both appropriate and com fortin g to qu ote two Nobel lau reates in physics wh o have pu blicly ackn owledged th eir discom fort with the su bject. Max Born , one of the fou nders of qu an tu m m ech an ics, freely adm its his m ystification1
"I tried h ard to u nderstan d th e classical fou n dation s of the two theorem s, as given by Clau siu s and Kelvin; they seem ed to m e wonderfu l, like a m iracle produ ced by a m agician's wan d, bu t I cou ld not fin d the logical and m athem atical root of these m arvelou s resu lts."
And Percy Bridgm an, a respected therm odynam icist, is bothered by th e strange h u m an -scen ted qu ality perm eatin g th erm odynam ics2
My Life: Recollections of a Nobel La urea te,
Max Born , Taylor and Fran cis Ltd.,
London, 1978, p. 119.
The Na ture of Thermodyna mics
1961, p. 3.
P.W. Bridgm an , , Harper Brothers, New York, 1-2
"It m u st be adm itted, I think, that th e laws of therm odyn am ics have a different feel from m ost of the other laws of the physicist. There is som eth in g m ore palpably verbal abou t them — th ey sm ell m ore of their hu m an origin. The gu iding m otif is strange to m ost of physics: n am ely, a capitalizin g of the u n iversal failu re of hu m an bein gs to con stru ct perpetu al m otion m achines of eith er the first or secon d kind. Wh y sh ou ld we expect Natu re to be interested either positively or n egatively in th e pu rposes of hu m an beings, particu larly pu rposes of su ch an u nblu sh ingly econom ic tinge?"
Th e m ost n otable aspect of this strangeness is the m athem atics em ployed by therm odyn am ics. While m any therm odyn am ic qu antities are expressed as partial derivatives, m an y of th e equ ations relating these qu antities are algebraic an d bear little resem blan ce to th e differen tial equ ation s of classical physics. Also, the partial derivatives of therm odynam ics are different in th at they are adorned with su bscripted variables. The reason for this is th at th ere is an excess of variables in th erm odynam ics. A sim ple system is defined wh en two variables are determ in ed, bu t th erm odynam ics does not specify which two am ong th e m any variables sh ou ld be ch osen . Th u s, if the system were a qu an tity of gas, the two
TPVvariables cou ld be chosen from am ong tem peratu re, , pressu re, , volu m e, ,
UHSintern al energy, , en th alpy, , or entropy, . The choice of variables is indicated from the appearan ce of th e partial derivative. For exam ple, the partial derivative
( / )P in dicates th at an d
V T TPwere ch osen to represent the volu m e while
V T V T TS
( / )S in dicates th at an d were chosen. The sim ple notation / wou ld be
∂∂∂∂am bigu ou s.
On th e oth er h an d, th e other laws of classical physics are expressed in term s of differen tial equ ations wh ich, when integrated for a given set of bou n dary condition s, yield algebraic equ ations with tim e and position coordinates as th e independen t variables. Th ese laws specify the independent variables an d there is n o n eed to su bscript partial derivatives. For exam ple, th e equ ation of contin u ity of flu id m ech an ics is a partial differen tial equ ation expressing flu id den sity, , as
ρta fu n ction of tim e, , an d position coordin ates an d there is no dou bt as to the iden tity of th e in depen den t variables held constan t wh en the partial derivative t
/is written. Again , con trasting the m athem atics of therm odyn am ics with
∂ρ ∂ that of classical ph ysics, we n ote th at the defined therm odynam ic variables ,
1-3 an d can n ot be expressed as absolu te valu es3 and are always written as
, and . Th is featu re derives from the ∆∆∆changes between two states:
cen tral idea of th erm odyn am ics — the con cept of a state and a state variable.
With the exception of heat an d work, all th erm odynam ic variables are state variables.4 To im prove ou r u n derstan ding of a system we wou ld like to know how the system m oved between initial an d a final states, bu t therm odyn am ics does n ot provide th at inform ation . Also, therm odynam ics has nothing at all to say abou t the tim e requ ired for th e system to m ove between th e two states; tim e is n ot a therm odynam ic variable. While the silen ce of therm odynam ics on th ese points is an obviou s disadvantage to the state approach, there are com pensation s — n am ely, flexibility and econom y of description.
T, P, V, U,
Th e state of a gas is defin ed when an y two of the state variables
, or are specified. Th is m eans th at the valu es of all of the other state variables are fixed an d th at between an y two su ch specified states the ch anges in th ese variables are fixed regardless of th e actu al path taken by the system . Becau se of this, changes in state variables can be evalu ated by any convenien t path . Th u s, we can ch oose variables that are con ven ient and we n eed not be concerned with m echanistic particu lars wh en applying therm odynam ics. Unfortu nately, these advantages often cau se discom fort for the n eophyte or casu al observer, for it wou ld seem th at th erm odynam ics revels in in difference an d arbitrariness.
Th ere are two oth er factors that con tribu te to th e au ra of stran geness su rrou n din g th erm odyn am ics — the reversible process and th e stan dard state.
Th e reversible process is an idealization n ecessary for the developm en t an d im plem entation of th erm odynam ics, an d it im poses conditions th at can be approach ed in practice. While one can u su ally visu alize th e physical conditions requ ired to approach reversibility, the concept of a stan dard state, so u sefu l in dealin g with chem ical reaction s, often in volves conditions that can not be ph ysically realized. Both of these concepts im part a certain am ou n t of artificiality an d rem oten ess to th erm odyn am ics.
The case of th e third law an d absolu te entropy will be discu ssed in Ch ap. 3.
Th e state approach , first applied to therm odynam ics by Gibbs, is also u sed in qu antu m m ech an ics with con siderable su ccess. 1-4
1 .2 Th e Met h od of Th erm odyn am ics.
There are th ree m ajor application s of therm odynam ics:
1. Th e first an d second laws can be applied to the calcu lation of heat and work effects associated with processes. Th is wou ld inclu de fin din g the m axim u m work obtainable from a process, th e m inim u m work to drive the process, or wh ether a proposed process is possible. Th e an swer to the last qu estion is a perm issive yes if neith er law is violated, bu t an em phatic no for an y violation .
2. Th e establish ed n etwork of therm odynam ic equ ations can be u sed to determ in e relation sh ips am on g the state variables of a system . Th ese relation ships can be u sed to calcu late valu es of variables that are difficu lt to m easu re from valu es of variables th at are easier to m easu re.
3. Special fu n ction s defined by therm odynam ics can be u sed to m ake calcu lation s in volving phase and chem ical equ ilibriu m .
Actu ally, th e property chan ges needed for the first application m u st be calcu lated from experim ental data via the therm odyn am ic network. Th erefore, in all its application s th erm odyn am ics requ ires experim ental inform ation. Often it is easy to m iss th is in tim ate experim ental dependence when solving textbook problem s in which th e steam tables are u sed or wh ere the su bstance is conven ien tly specified as an ideal gas with a specified heat capacity. Wh en the origin of th e data is not recogn ized, therm odyn am ics is disassociated from its experim en tal con text and rendered a lifeless and m eaningless set of equ ation s.
Th e experim ental con text su ggests the followin g definition. It is intended to be heu ristic rath er th an rigorou s and shou ld be su itable for the discu ssion wh ich follows.
Thermodyna mics ma y be broa dly defined a s a mea ns of extending our experimenta lly ga ined knowledge of a system or a s a fra mework for viewing a nd correla ting the beha vior of the system.
Accordingly, it wou ld be better to speak of the therm odyn am ic m ethod rather 1-5 thermodyna mic than therm odyn am ics as an area of science; the adjective is thermodyna mics. preferable to the n ou n
One is certainly ju stified in protesting that th is defin ition is so gen eral that it cou ld be applied to alm ost an y area of scien ce or to science in gen eral. Nevertheless, becau se it will aid ou r u nderstandin g of th erm odynam ics, this non-exclu sive definition will be u sefu l.
Th e th erm odyn am ic m eth od is qu ite general and is capable of treatin g any system which can exist in an observable an d reprodu cible equ ilibriu m state and wh ich can exch ange h eat an d work with the su rrou n din gs. In addition to flu ids, chem ically reacting system s, an d system s in phase equ ilibriu m , therm odyn am ics h as also been su ccessfu lly applied to stressed solids, system s with su rface effects, and su bstances su bjected to gravitational, centrifu gal, m agn etic, and electric fields.5
A diagram illu strating th e developm ent of the therm odyn am ic network and the application of the th erm odynam ic m ethod is shown on Fig. 1-1. The sequ en ce of rectangles con nected by heavy arrows traces the developm ent of th e laws, fu nction s, and relation ships th at constitu te the therm odyn am ic network
— the tools of th erm odyn am ics. The sequ ence of circles conn ected by light arrows shows th e steps in applying th erm odynam ics to the solu tion of practical problem s, an d th e dash ed arrows indicate where the variou s th erm odynam ic tools are em ployed in th e problem -solving process.
Th e first an d second laws h ave been fashion ed in m athem atical langu age
USfrom statem en ts of experien ce and have defin ed th e fu nctions and . These fu nctions, previou sly u n kn own, were shown to be state fu nctions. By way of the Clau siu s in equ ality, the com bin ation of th ese two laws leads to the fu ndam ental equ ation of th erm odyn am ics. And application of the m ethods of th e calcu lu s to this equ ation yields th e n etwork of relation ships am ong the state properties.
Th u s, th e laws wh ich defin e an d lead even tu ally to the prescriptions for their evalu ation from experim ental data.
A new an d u sefu l state variable, the ch em ical potential, can be iden tified wh en the fu n dam en tal equ ation is exten ded to inclu de m ixtu res. Th e application of this n ew fu n dam en tal equ ation leads to the specification of the conditions of equ ilibriu m , wh ich requ ire that the tem peratu re, pressu re, and ch em ical
For a gen eralization of the therm odynam ic m ethod see Sec. 5-9 of this textbook.
1-6 poten tial be u niform throu gh ou t the system . The ch em ical poten tial an d other fu nctions derived from it form the basis of the therm odyn am ic treatm ent of ph ase and ch em ical equ ilibriu m and the equ ation s relatin g these fu nction s are also a part of th e therm odynam ic network.
As we h ave seen , the en tire th erm odynam ic n etwork derives from th e first an d second laws. Th e laws are u nqu estioned, bu t their acceptance rests m ain ly on th e su ccessfu l application of the n etwork they su pport; the experim ental per se eviden ce for th e laws is n ot overwhelm in g. The application of therm odynam ics is represen ted in Fig. 1-1 as a three-step process requ iring the inpu t of experim en tal inform ation. While the problem -solving process sh own in
Fig. 1-1 is certain ly not u n iqu e to th erm odynam ics, th ere are certain featu res that distingu ish therm odyn am ic problem s. Th e m ost obviou s featu re is th at the U, H, S defin ed th erm odynam ic variables (e.g.,
, and ch em ical poten tial) do not
Road m ap for th e con ceptu al developm en t an d application of th erm odyn am ics.
Figu re 1 -1 . 1-7 appear in eith er th e problem statem en t or th e solu tion . These variables are sim ply u sed to solve th e problem and th erefore h ave no u ltim ate valu e beyond this application ; in a sen se they are du m m y variables. Con trast this situ ation with th at of flu id m ech anics, for exam ple, where th e variables velocity, tim e, and position are part of th e problem statem ent, are the variables in wh ich th e th eory is form u lated, an d are the variables in which the solu tion is expressed. In addition to n ot bein g in trin sic to th e problem statem ent or solu tion , the defined therm odynam ic variables often requ ire com plicated, m u lti-step path s for their evalu ation. Som etim es th e path inclu des h ypothetical steps an d requ ires several types of experim ental data. Th e th erm odynam ic m ethod is th u s ch aracterized by an em ph asis on inform ation processing.
In both stru ctu re an d application, therm odyn am ics resem bles m athem atics; both are self-contained an d confiden tly u sed an d both con tain m u ch th at is never u sed. J u st as m u ch of m athem atics has fou n d no application, th ere are m any u nu sed relationships in th e th erm odyn am ic n etwork. An d wh en u n acceptable resu lts are obtained from the application of either of th ese tools, an d no errors are fou nd in the execu tion , it is the conceptu al description of th e system th at m u st accom m odate.
To u n derstan d this state of affairs it is necessary to distingu ish the therm odynam ic n etwork, which derives solely from the first and second laws, from the equ ation of state u sed to describe the particu lar system . The equ ation of state is an algebraic equ ation th at relates the state variables of the system : the ideal gas law or Cu rrie's law for param agnetic system s. Often the equ ation of state will h ave a th eoretical basis (e.g., the ideal gas law or the virial equ ation of state), bu t its ju stification or the determ ination of its param eters is always em pirical. Wh en the relation sh ips of th e network are u sed with an equ ation of state, resu lts specific to th at system are obtained. Thu s, if the th erm odynam ic m eth od were applied to a gas with u nsatisfactory resu lts, the efficacy of the equ ation of state wou ld be qu estioned and a m ore descriptive, an d u su ally a m ore com plex, equ ation of state wou ld be em ployed.
Also, in applyin g the th erm odyn am ic m ethod it is necessary to know that the system is properly defin ed in term s of the state variables. The phase ru le provides th is gu idance for conven tion al applications, bu t there are som e system s that requ ire th e specification of addition al state variables su ch as field strength or su rface area. Wh en these system s are encou ntered it is n ecessary to revise the 1-8 fu ndam en tal equ ation to accou n t for the variou s ways in which the system can exch an ge work with th e su rrou ndings. The details of the therm odynam ic n etwork will ch an ge as a resu lt of this new form u lation, bu t the basic m ethod for workin g ou t th e details of the n etwork rem ains u nchan ged.
1 .3 Th e Nat u re of Th erm odyn am ics.
Th e th ree m ajor areas of classical physics are m echanics, electrom agnetic theory, an d th erm odyn am ics and it seem s to be a sou rce of em barrassm ent to m an y ph ysicists th at th erm odyn am ics does not fit too well into this triu m virate.
Th e reason s for th is lack of fit h ave already been discu ssed.
Statistical m ech an ics began as an effort to bring therm odynam ics in to closer con form ity with the rest of classical physics. Here the m ethods of m echanics were applied statistically to the astronom ically large n u m ber of m olecu les con stitu ting the average th erm odynam ic system . Qu antu m m ech an ics developed som ewh at later an d was shown to su bsu m e statistical m echanics.
Th is new field, n ow kn own as qu antu m statistical m echanics, has proven to be a valu able adju nct to th erm odynam ics by allowin g the evalu ation of therm odynam ic properties from m olecu lar param eters. However, despite this practical su ccess, th ere is still som e u n certainty in explain in g en tropy in m olecu lar term s. Th e two m ost com m on form u lations
S = k ln Ω
(1 -1 ) an d
S = ∑ ln
Pi Pi relate en tropy to th e nu m ber of accessible qu antu m states of an isolated system ,
, or th e probabilities, i's, of the variou s qu antu m states in an isotherm al
Ωsystem . If th e en tropy were an in trinsic property of m atter, an d not ju st a defined state fu n ction , one wou ld expect a m icroscopic form u lation to be expressed in physica l logica l term s of qu antities and qu an tities. In stead it is related to the Ω
i's wh ich do n ot refer to an y physical aspect of th e system , bu t to the m an ner in wh ich we ch oose to represent th e system . Early workers in statistical m ech anics referred to th e expressions on the right-han d side of Eqs. (1-1) and (1-2) as entropy an alogu es; n ow, they are called statistical entropy, or often, they are 1-9 sim ply called entropy. They provide th e basis for the pu tative interpretation of entropy as a m easu re of disorder.
Th e situ ation in regard to en tropy becam e m ore con fu sing wh en a term arising from com m u n ication th eory and resem bling the right-h and side of Eq. (1-
2) was also given th e n am e of entropy. Becau se it defines the inform ation con tent of a m essage, it is u su ally referred to as inform ation en tropy, however, the u nm odified term en tropy is often u sed. More recently, the confu sion has increased with th e assertion that there is an entropy ch an ge associated with m em ory erasu re in com pu ters — a com pu ting entropy. In chapters 4 an d 5 an attem pt will be m ade to clarify th is situ ation .
Th e im pressive su ccess of qu antu m statistical m echanics has prom pted m an y scien tists to state that th e second law is statistical in natu re. Som e even go so far as to su ggest that statistical m echanics su bsu m es therm odyn am ics.
Th is u ndou btedly h olds great appeal to th ose who are u n com fortable with th e lack of fit in th e triu m virate of classical physics. These qu estion s will be explored in chapter 9.
1 .4 A Ch em ical En gin eerin g Perspect ive.
Th ose wh o deal on ly with the fam iliar applications of the first an d second laws m ight be in clin ed to describe therm odynam ics as a consisten t, self-standing system of laws an d relation s wh ich govern processes or transition s wh ere en ergy is tran sform ed. Th ey m ay go so far as to visu alize the laws of therm odyn am ics in a Platonic sense as etern al an d im m u table laws of the u niverse which were discovered by hu m an inqu iry. This Platonic view, however, m ay not be so congenial to th ose wh o apply therm odynam ics to chem ical an d ph ase equ ilibriu m . These application s clearly involve contrived fu nction s (activity coefficien ts) an d artificial con ditions (standard states) wh ose sole pu rpose is to facilitate a desired calcu lation . When on e h as worked with th ese obviou s contrivan ces, it seem s n atu ral to regard the whole of therm odynam ics in this light. From th is perspective therm odynam ics seem s to be an inform ation processing system — an in ven tion of th e hu m an m ind serving a hu m an need to econ om ize on th e experim ental description of physical system s. Th is view is inclu sive en ou gh to in clu de all applications of therm odynam ics; it is clearly u tilitarian in n atu re.