Henry A. Bent University of Minnesota Minneapolis
I I
The Second l a w of Thermodynamits introduction for beginners at any level
There is something fascinating about science," Mark Twain once remarked. "One gets such wholesale returns of conjecture out of such a trifling investment of fact." Of course he was not entirely serious. Not everyone finds thermodynamics fascinating, for example. And yet in thermodynamics one has in Poincare's view the ideal facts for serious study; namely, a set of facts that reveals unsuspected kinships between other facts, long known, but perhaps wrongly believed to be strangers to one another. Nonetheless, to many students a course in thermodynamics is like quinine-a hitter pill that is alleged to have medicinal virtues. This distaste for such a splendid subject would seem to support Keynes's belief that the most difficult th'mg that teaching has to do is to convey a worthy sense of the meaning and scope of a great idea. The task is often complicated by a language barrier. The complaint of an eminent biologist that the whole subject can be and usually is developed as a kind of mathematical double fugue bespeaks the heartfelt anguish experienced by many students confronted for the first time by the strict mathematical rules of classical thermodynamics. It also echos the sentiment expressed some years ago by a young woman who in her house cleaning one day saw open on her cousin Josiah Willard Gibbs' desk a manuscript titled "On the Equilibrium of HeterogeneousSubstances." To a friend Based on a paper read as part of a Symposium on Teaching Thermodynamics sponsored by the Division of Chemical Education at the 141st Meeting of the ACS, Washington, D.C., April, 1962. Attendance at this symposium was made possible by a grant from the Department of Chemistry of the University 01 Minnesota.
she wrote, "It looks full of hard words and signs and numbers, not very entertaining or understandable looking, and I wonder whether it will make people wiser or better (I)." Two distinguished physicists have commented to somewhat the same effect. In the introduction to their book on atomic spectra, Condon and Shortley (93) observed that when a physical scientist is desirous of learning new theoretical developments in this suhject, one of the greatest harriers is that it generally involves new mathematical techniques with which he is apt to be unfamiliar. We can do this person a real service, they say, if we can minimize the amount of mathematics he must learn in order to penetrate a new field. Gihbs himself once said, "If I have had any success in mathematical physics, it is, I think, because I have been able to dodge mathematical difficulties (I)." I would add that the same may be said about teaching thermodynamics to beginners. Success in this endeavor critically depends, I believe, on one's ability to dodge mathematical difficulties. This is not always easy to do. It may sometimes be necessary for the teacher to sacrifice-at least temporarily-mathematical tools and techniques that he has labored long and hard to acquire for devices whose immediate appeal is less obvious, much as Matisse and Picasso in pursuit of their art found it necessary to sacrifice draftsmanship a t which they were highly skilled for effects they sincerely believed to he more important. But, you ask, can one really take the sting of mathematics out of the second law? Can one really dodge the mathematical difficulties and still teach the true substance of the suhject? Can one in Oppenheimer's words he both humane and robust (S)?
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I knew a wise old professor who had made his mark in both pure and applied mathematics whose answer t o these questions would surely have been, "Young man, there is only one way to find out. Try it and see." And to this he would probably have added a word of encouragement, for he had himself labored a lifetime to bring knowledge from the ivory tower t o the market place, and he was fond of quoting Matthew Arnold t o the effect that it is after all a person's job as a teacher to divest knowledge of all that is harsh, uncouth, difficult, abstract, professional, exclusive; a teacher's job is to humanize knowledge, to make it efficient outside the clique of the cultivated and learned. The remainder of this talk will be devoted to a description of an effort that has been made to follow this high charge. Necessarily, the description given here of this effort is more illustrative than exhaustive.
This is because the weight lower-surroundings warmer process results in an enormous net increase in the microscopic disorder of the universe, which we may here take to be simply the weight plus its thermal surroundings, see Figure 2. The arrows in Figure 2 are intended t o indicate the random thermal motions of the individual atoms and molecules of the universe. This random thermal motion is greater in state 2 than in state 1. State 2, the state of greater microscopic disorder, is intrinsically much more probable thau state 1 and passage from the less microscopically disordered state 1 to the more microscopically disordered state 2 can occur spontaneously. The reverse process does not occur, for the microscopic disorder, or e n t r ~ p yof, ~the universe never (well, hardly ever) decreases. (For example, transcription of Shakespeare's complete works
General Features
Our discussion begins by recognizing the fact that most students have an intuitive feeling for the first and second laws of thermodynamics. They know, for example, that objects cannot change altitude without some other change occurring in the universe; i.e., they know that processes whose net effects are equivalent to merely the raising or lowering of a weight are impossible (Fig. 1). As we often say, energy is conserved. When a man jumps out a window, for example, energy equivalent to the decrease in the potential energy of the system (man plus earth) appears ultimately as heat (Fig. 2) (4).'
Figure 2. A spontaneous process The conversion of potential energy into random molecular motion. The reverse process is imporrible. The energy of rondom molecular motion can never be converted completely into potential energy. In the i ~ o n t m e o u r roce err remerented b v mrsaae from lhe state labeled 1 dn the left to ihe rtote labeled 2 on the &ht, the increase in the entropy of the universe-i.e., in the entropy of the weight and its thermal surroundings (denoted here b y the word " f i o o r " k i r equal to mgh/T. In this expression, rn represents the moss of the weight, g the acceleration due to the force of grovity (980 cm/recz a t tea levell, h the ~nmalheight of the weight above the fioor, ond T the temperature o f the system. The chonse the reverse process will occur i s 1 in e9h'x'; thir is normally negligible. For example, at 27'C the chmce a one gram weight might rise rpontoneoutly one centimeter into the air at the expense of the thermal energy of its surroundings is 1 in 1 0 to the power 10'9
. ..
:l.t\i-i
8
I...'til.
,
I
L ..----J 2
L ..--...
I
Figure 1. This figure is intended to illurtrate the f a d that any process whore net sffed i s equivalent to the raising or lowering of a weight i s impossible. For this would violote the first law of thermodynamic% which asserts thot in ony natural process the energy of an iroloted system-r "univene" as we rhall call such a system in thir article-is conserved. In other words, energy can be neither created (rpontoneous porroge fmm state 1 to stole 21 nor destroyed (spontmeous p w r o g e from state 2 to = constant; stole 1). Other rtotements o f the Rnt law are; Eu.iuer.s ( E u o i w d z = (EuDirereJl; AEmnirer8e= 0.
They know, moreover, that the process depicted in Figure 2 is not reversible. Inanimate objects do not of themselves spring spontaneously into the air a t the expense of the thermal energy of their surroundings. In other words, processes whose net effects are equivalent to the raising of a weight and the cooling of its surroundings are impossible--or, a t most, "highly improbable." This means that events so labeled never have been and in all probability never will be observed. These authors cite the exknple of a ball that e m roll spontmeously down an inclined plane. We would say the reason that the ball eventually coma to rest at the botlom of the indined plane inclined plane thermal suris that when the universe ball roundings hns reached this state (ball motionless at the bottom, dl parta slightly warmer than before) its entropy (Sasll Simlin.d PIS. Stbarrnal.urroundiwd is as lawe as it can be, subject to the condition that its energy (Eb.11 Einolined .I.~. Ea.,,, di,.) and its volume (Vb,n Vi.,~i.,a V,hc.-, .u.l oundiw.) are constant.
+
.,.,
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fifteen quadrillion times in succession without error by a tribe of wild monkeys punching randomly on a set of typewriters is a more likely event thau is the conversion a t room temperature of one calorie of thermal energy t o work.) Students know, too, that warm ice melts spontaneously and that this process is not reversible; warm water does not freeze (Fie. 3). The reason warm ice a It is convenient to have a name for eomething as important as microscopic disorder. As Stephen Leacock has written, "All physicists sooner or later say, 'Let us call it entropy,' just as a man says, when you get to know him. 'Call me Charlie.'"
Figure 3. Changer in disorder in the universe when ice melts. In thir figure melting corresponds to pmroge from state 1 to rtote 2. Besavre this ir an endothermic process, the entropy o f the surroundings decrearev as the entropy of the water increases. There two eRe& balance each other at OaC and a t that temperotvre there is no net change in the entropy of the universe when ice meltr to form pure woter (Table 1). Above O S C melting produces o net increase in the entropy of the universe ITable 1) ond the reverse process Iwoter freezing at, ray, 1 'CI i s imporsible.
melts is that a t high temperatures (T > 273OK) the decrease in entropy of the surroundings during the endothermic melting process (1/T times the heat of fusion) is more than balanced by an increase in entropy of the water as it changes from the relatively ordered solid t o the more disordered liquid. They know, however, that cold water can freeze and that this process, when it happens, is irreversible (cold ice does not melt). This is because a t low temperatures (T < 273'K) the increase in entropy of the surroundings in the exothermic freezing process more than balances the decrease that occurs in entropy of the water as it solidifies. The numerical values of the entropy changes in the universe when ice melts are given in Table 1. Table 1.
Temp.
("C.) -10 0 C10
-
Entropy Changer in the Universe When Ice Melts. HxO (solid) 1440b cal mole-I H.0 (liq)
+
A S S$;o-S!$
f5.3 +5.3 t5.3
Assd AEe/T= -1440/T -5.5 -5.3 -5.1
* + ASe)
ASun:
(AS,
-0.2 0.0 t0.2
Comments Won't melt Could melt Should melt
The entropy changes in the universe when water freezes are equal in mwnitude but opposite in sign to the values listed in this table in the seoond. third. and fourt,h columns. All - entronv values are exaressed in callder-mole. . 80 osl g-I X 18 g mole-' = 1440 oal mol-1. 'The meaninp: of the svmbols a and 9 is indicated in Fieure 3. A "
is 'At low temperatures (large ! / T ) , the sign of AS,.i,.. determined by the sign of AS#; thm mgn is positive for exothermic reactions and negative (as here) for endothermic reactions. At high temperatures (small 1/T) the signof AS,.i,.,. 1s determined by the sign of AS.; this s i g i i s positive (as here) if S..duGt.> S.,,tamt. and negative if S,.d..t, < S The forms of matter stable a t low temperatures are low energy forms (EpdUct. < E n reaction reactants products exothermic), whereas the forms of matter stable at high temperatures tend to be highentroav forms. This association of chemical stahilitv at low temp6;atures with low energy and at high temperat6res with hig6 entropy can be neatly expressed by introduiing into thermodynamics a function called the free energy, E - TS (see below and (4)).
-
+
AS8 = AEE/T 68,000 eal mole-' = 300°K = 227 e d mole-' deg-1
When PHZ= POZ= 1 atm Asr = Sprd""he- Srew'& = 16.7 - [31.2 (1/2)49.01 eal mole-' deg-1 = -39.0 eal molecLdeg-I
+
Therefore, under these conditions, Su.i,
= ( -39
+ 227) eal mole-'
deg-'
Note that a t very high temperatures AS0 would be numerically much smaller. Hydrogen and oxygen can in fact be made too hot to explode. That is t o say, a t high temperatures (about 10,OOO°C) water dissociates into hydrogen and oxygen. Every student knows that a warm-blooded person cannot warm himself at the expense of the thermal energy of an icicle. Although nothing in the lirst law alone prohibits the transaction icicle colder-man warmer, it is nevertheless a well known fact that energy transactions of t,his kind never occur. On the other hand, energy can flow spontaneously in the reverse direction: from a warm-blooded person to a cold icicle. This is because by "warm" we mean a substance that acts as a relatively good donor of thermal energy, i.e., one whose entropy decreases relatively little for a given loss of energy (AE/AS large); whereas by "cold" we mean a substance whose entropy increases by a relatively large amount for a given addition of energy (AE/AS small) and which therefore acts as a relatively good acceptor of thermal energy. Hence the flow of thermal energy from a hot object to a
....
They know, in addition, that hydrogen and oxygen at room temperature and one atmosphere pressure will explode (if given a little activation energy) and that this process is not reversible a t low temperatures (Fig. 4). This is because a t low temperatures the enormous increase in entropy of the surroundings 0 in the very strorgly exothermic process Hz(g) '/202(g) = Hz0 (liq) 68,000 cal easily balances the decrease tbat occurs in the entropy of the chemical system n and
+
makes the change in the entropy of the universe (the chemical system c plus its surroundings 0) positive. Thus
+
Figure 4. Bomb calorimeter for measwing a t room temperature the heat Ilr091g, 1 mtml = of the spontaneous, irreversible change Hnlg, 1 otml H1O(ll 68,000 cd. Spontaneous passage of the universe lbomb plus contents plus thermal surroundings) in the opposite direction is inconceivable. Even if octivoted momentarily with on electric ,pork or a match, 0 mole of liovid water at 2 7 - C will not soontaneouslv ablorb 68.000 cdoriol of thermal energy from its surroundings and dissociate into hydrogen ond oxygen gar a t 1 d m . The maximum thermal energy thot could be absorbed during dissociation a t 27'C I30O0Kl to form gmseous products a t 1 d m is bee text) 3 0 0 X 3 9 col/mol+ of water dissociated. Any more than this would constitute o violotian of the second law. Tho remoinder of the energy reqoired to make up the difference between ( E m '/lEorl and Eem comesto(68.000 3 0 0 X 391 = 56,300 c a l o r i o r i e . , in general, to -1AEs TASCIT.~c a l o r i e r a n d must b e supplied in some other fodion. for examole.. eleetricollv. At the rots of 2 faradow conwmed per mole of water dirrocioted, this electrical energy would have to be supplied ot a minimum voltage of
+
+
-
+
-
.
56,300 r o l 2
X 96,500 coulombs
X 4.1 -8 ioules -- = 1.? roltr 1cd
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colder one always increases the entropy of the cold ohject more than it decreases the entropy of the hot object and the net result is an increase in the entropy of the universe (Fig. 5).
does not depend on its potential energy; i.e., on its height above sea level). Often the first law is expressed in the less symmetrical form a = & - W
where aE = AE,
............................
Q W L
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
J
Figure 5. A spontaneous process. The spontaneous, irreverrible Row of 4 0 0 calories of thermal energy from the 400'K-ebiect to the 300°Kobject decreases the entropy of the 400PK-object 1 colldeg, increases the entropy of the 3OO0K-objed b y l'/red/deg, and results, iherefore,in o net cal/deg. Note that the increase in the entropy of the universe of expression Q/T may be written QIAEIAS, i.e., or (ASIAEbQ and read: the change in S per unit chmge in E time. the change in E; this produd giver thetotalchangein S.
'Iz
This brief discussion of spontaneous, irrevenible, entropy-producing processes illustrates our general approach to the second law. The general features of this approach may be listed as follows. First, generous use is made of specific examples, particularly the downfalling habits of weights, the melting of "warm" ice and the freezing of cold water, the explosion of hydrogen and oxygen, and the flow of heat from hot to cold. The reason for doing this is that generalization, as Polya has pointed out (6),starts naturally with the simplest, the most transparent particular case. To start by stating generalizations is unnatural and may merely intimidate the neophyte (6) and understandably does not often succeed in promoting in him an early acceptance and appreciation of the second law. As Carl Becker has said, "The thin vision of things in the abstract rarely reaches the sympathies of the beginner (7)." Second, the point of view adopted is always that of a person looking at the entire universe of any particular problem. The inspiration behind this procedure stems partly from Br@nstedlscriticisms of classical thermodynamics (8)and partly from a remark once made by Professor Giauque to the effect that thermodynamics is not hard if you can keep in mind precisely what it is you are talking about. The word universe as used here signifies everything that might conceivably suffer a significant change in the problem under consideration. Very often in applications of thermodynamics to chemical problems the universe may be regarded as composed of these three parts: a chemically reactive system u, its thermal surroundmgs 8, and a weight (Fig. 6). The symbol 6 is used for the thermal surroundings because the only thing important about that part of the universe is its temperature. The weight is introduced to represent through changes in its potential energy any form of useful mechanical (or electrical) work produced or consumed by the remainder of the universe, with which it is assumed to be in mechanical (or electrical) but not thermal contact. For any change that occurs in this universe the first and second laws state that the following will be true. AE".i,.,
=
AE.
+ AEe + AEwt = 0
where always (see below) ASo = AEe/T and where normally ASw, = 0 (a substance's microscopic disorder 494
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energy change of the system o "heat" absorbed b y the swtem frmn its surrounding6 e = AEwr = work done by the system rr (strictly speaking by o 8 ) a the weight, Wt =
-AEe
= =
+
Third, as indicated above, the first two laws of thermodynamics are introduced into problems through simple, easily remembered, perhaps even almost selfevident statements about the total energy and total entropy of the universe: AE,.,, = 0, AS,,, 2 O.$ It has been found that focusing attention on the universe's total entropy-rather than on, say, the free energy of one part of the universe--encourages students beginning a study of the microscopic states of the universe and chemical equilibrium to utilize fully from the outset what they already know about the structure of matter and "heat" and spontaneous processes. Fourth, attempts to see a t a glance the reasons for macroscopic events in terms of models of microscopic events are emphasized. This always seems to be a very natural procedure to adopt when one is talking to young chemists. Fifth, frequent use is made of the fact that the absolute temperature T represents the value of AE/AS (strictly speaking, @E/~S)V.), and vice versa. This relation can be put to many uses, some of which will be discussed in a moment. The relation is moreover a reasonable one. Both T and AEIAS are intensive properties; both have large values for hot objects (small A S for given AE) and small values for cold objects (large A S for given G); both increase as thermal energy is added to a body; and both approach the value zero as the thermal energy approaches zero. .\nd sixth. an wrly +urvry is mulr oi molar entropies. It i.: Irsnird that as a rnle hard ~ I I ~ s I ~havr I ~ smull (.~s entropies and soft substances have large entropies. Diamond and ice, for example, two isoelectronic substances, have entropies a t O°C of 0.5 and 9.8 cal./deg.mole, respectively. Liquid water and water vapor ~
~~
"Die Energie der Welt ist constant. Die Entropie der Welt streht einem Maximum zu." Clausius.
L...
....................2
Figure 6. A partition of the universe thot is frequently useful in applicotiow of thermodynamics to chemical problems. (See text for definitions of symbolr.1 A constant external pressure on the chemical system r can be represented by placing on it a weight. Correrpondingly, a term should be added to the statement of the Rrrt low to ollow b r the fact that the weight's potential energy will chonge whenever there is o change in the volume of m This chonge in potential energy may be written a. ImgIAh, where m is the mass of the weight which rests on an effective area A; hence (by definition) P(..t.,.,~ = mg/A ond AVO = A&. The additional term is therefore equal to IPAIAh = PAVo and the first law reods: AEc PAVg AEe AEwi = 0. Using the fact from the second law that AEe> - TASm (see text1 and the abbreviation W for AEwE, one obtains PAVg - TASs W 0. from thisthe well known expression: AEo
+
+
+
+
+
2
have entropies a t 25'C and 1 atm of 16.7 and 45.0 cal/deg-mole, respectively. This explains why endothermic reactions such as the melting of ice and the evaporation of water become spontaneous reactions a t elevated temperatures (see footnote e to Table 1). A more careful comparison shows (in accordance with the fundamental properties of the wave equation and statistical ensembles) that the larger the volume involved in the random thermal motions of the constituent particles of matter and the larger the mass of these individual particles, the larger the entropy. This explains why the translational entropy of a gas increases so dramatically as its partial pressure decreases. Indeed, by continually decreasing the partial pressure of a gas a point can always be reached where its entropy is sufficiently great to make evaporation of the corresponding liquid or solid, however, cold, a spontaneous (although not necessarily rapid) process. Snow will evaporate, even on cold days, if the air is very dry. Uses of T
= (bE/bS)v
Two of the most engaging qualities of the relation T (bE/bS), are its power to make new things familiar and familiar things new.4 Among many illustrations that might be given of this fact are the following. I t will be seen from these illustrations that most of the fundamental equations of classical and statistical thermodynamics are a direct consequence of the thermodynamic definition of temperature. I n point of fact, with beginners, we base the logical elaboration of the second law of thermodynamics entirely upon the relation T = (bE/bS),. It is believed that this represents a logically sound and heuristically valuable method for develo~ina . - the fundamental equations of thermodynamics. 1. Heat Engine Eficiency. The absolute temperature of an object is the thermal energy that must be added to or taken from it to change its entropy by one unit. (Again, we have in mind large objects whose temperatures are not sensibly altered by the addition or removal of 300 or 400 calories of thermal energy.) Take 400 calories from a 400°K-object and its entropy decreases by one unit. Add 300 calories to a 300°1G object and its entropy increases by one unit. The entire difference, (400-300) calories, could be used to do useful mechanical work, such as the lifting of a weight, without violating the second law, which requires that the entropy of the universe, the 400°K-object plus the 300°K-object plus the weight, remain the same or increase. The efficiency of this process in converting thermal energy to work is (400 - 300)/400 = ' / 4 . 2. Third Law-Entropies. At every stage of a reversible warmup (at constant volume) dS = (l/T)dE. The area beneath the continuous curve formed by plotting the value of 1/T during warmup against the corresponding value of E is, therefore, the change in entropy of the warmed-up object (Fig. 7). I n this treatment of warmup data, which to our knowledge has not hitherto been considered in the literature and which might be useful with modern adiabatic calorimeters, reversible first-order transitions such as melting and vaporization and lambda points where the heat capacity =
6
A paraphrase of Ben Jonson's statement about authors.
also becomes large, though not infinite, cause no particular problems. At a lambda point, for example, the curve merely levels off slightly. 3. The Free Energy. For any change that occurs in the universe depicted in Figure 6, it was shown previously that AS, A& 2 0, where ASs = AEs/T (it is assumed here that T, = T. = T), and where AEs = -(AE* A E d ; i.e.,
+
+
or, since T
> 0 and AE,,
W W 5 -(AE.- - TAS.) =
The combination of terms E - TS is called the free energy. A spontaneous reaction is one that in principle can be harnessed to do useful work (W > 0); it is, therefore, a reaction in which the free energy of the products is less than the free energy of the reactants. It has been assumed here that the initial and final states of o have identical volumes. This is a valid assumption for reactions that occur in bomb calorimeters. Many reactions, however, occur in vessels open to the atmosphere. Under these conditions, P,, and not necessarily, V,, is constant. The significance to thermodynamics of this observation lies in the fact and A(S.)T.V that in general (AE,),., # (AE,),., f (AS.)T,P, although particularly for reactions that occur at moderate pressures in condensed systems where volume changes (P, constant) are small, differences from strict equality are small and practically speaking unimportant. Diamond synthesis and a host of other high pressure geochemical processes constitute notable exceptions, however. It can be shown that the relation between the mark of a spontaneous reaction( A - TAS,),., negative--and the quantities (AE.),., and (AS,),., may be expressed in the following manner. (AE,- TAS,)r,v = (AE. - TASC PAT'.-)r,p
+
For a reaction to be spontaneous, these expressions must be negative. 4. Isothermal Expansion of An Ideal Gas. Consider an ideal gas that is confined to a chamber of large heat capacity (henceforth called a thermostat) by a piston
Figure 7. The (1/71-E diagram. A plot of the instontaneour valve of the reciprocal of the obrotute temperature during warm-up against the corresponding valve of the odded thermal energy. The flr* steep portion of the curve (moll E and small T; large 1/71 corresponds to warm-up of the d i d . The horizonto1 portion of the curve represents melting of the solid; the are. beneath this portion of the curve i 3 equal to AEruaioo/Tiusion. The remainder of the c v n e represents warm-up of the liquid.
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that is coupled mechanically to a weight. For convenience designate these three parts of the universe as v (the gas), 9 (the thermostat), and Wt (the weight, which includes the mass of the piston), and consider what happens to their energy and entropy when the gas expands reversibly (ASeS., = 0) and isothermally (AT. = 0) from VI, PI to Vz, Pz (V* > VI; Pz < PI). The energy of the gas does not change (AE, = 0; for E. depends only upon T,, which by supposition does not change; cf. part 9 below). But as the gas expands, work is done on the weight and the potential energy of the weight increases. The first law states, however, that for any process AE,,,,, = 0; i.e., AB, AEe AE,, = 0. Therefore the energy of the thermostat must decrease by exactly the amount by which the energy of the weight increases (A& = - AEwt). And by calculus and the equation P V = nRT we know
+
+
Hence AEg = nRTln (P2/P1). This is a negative quantity whenever, as here, P2 < PI. AS for the entropy, we know that as the thermal energy of the thermostat decreases (AEe < 0) its entropy must also decrease. We know, also, that the entropy of the weight does not change. The second law states, however, that for any reversible process ASt,w = 0 ; i.e., AS, ASe aS,, = 0. Therefore the entropy of the gas must increase by exactly the amount by which the entropy of the thermostat decreases (AS. = ASs). And by the definition of T and the results already obtained we know that ASb = AEe/T = nRln(P2/PI). Hence AS, = - nRln(PdP1). Briefly put, the argument leading up to this result went as follows: from AV. to AEWIto AEe to AS0 to AS.. Writing AS, as S,, - SF,, we have for one mole of an ideal gas
+
I n this last expression the superscript zero stands for "at one atmosphere" and P stands for the actual pressure (or partial pressure) of the gas in atmospheres. 5. Chemical Eouilibrium. For a universe that is a t equilibrium with respect to the process reactants products,
+
- Xrlmlnl.X where AX, = XDroduOta example, for the reaction
Therefore where AS.'
/
T = AE. -
As,
A small addition of a second compment to the liquid phase has generally a relatively small (we shall assume zero) effect on the molar energy of the solvent and no effect whatsoever on the molar energy and entropy of the solid if the substance added is insoluble in the crystalline solid, as is often the case. The entropy of the solvent, however, is increased. Thus, of the four quantities E"q, Eralidt Sliq, Sp"lid, only SIiq is seriously affected. Let 6T and 6(AS,) represent the changes that occur in the transition temperature and the entropy of transition on the addition of a second component that changes the value of the mole fraction of the solute from an initial value zero to the value N2, where N2 is small so that changes in T and AS, from their values for the pure solvent are small. Then by calculus
+
+
496
6. Freezing Point Depression. At every point along a freezing point curve
=
P?7%0< PZ~o.
- 2 SONO,.
At equilibrium
Journal of Chemical Educafion
6T
-
RT . N ?
ASo
For a 1 molar aqueous sucrose solution
=
-1.83"
7. Boiling Poinl Elevation. The relation GT= (RT/ AS,)Nz is also applicable to changes in boiling temperature with small changes in solute concentration. I n this case T represents the normal boiling point and AS. = ~ w o -r S"4. This difference is often about 20 cal/deg-mole for non-associated liquids (Trouton's Rule). In these cases the value of the boiling point constant k, in the expression AT = k,Nz is approximately one-tenth the normal boiling temperature of the solvent on the Kelvin scale (Table 2 ) . -
5 This is equivalent to assuming that the solvent oheya Rmdt's Law (or to the assumption that the change in the partial molar entropy of the solvent with added solute is equal to the change in the pwtid moler configurational entropy of the solvent as calculated from a simple lattice model).
S or H. For
AS, = AS," - Rln = So~,o,
therefore
Table 2. Comparison of Normal Boiling Temperatures of Pure Substances with their Boiling Point Constants
Compound
10
ks (OK)
Chlorine Phosgene Ethyl ether Methyl sulfide Dichloromethane Carbon disulfide Bromine Hexane Ethyl nitrate Chlorabenzene Butvric rtcid
23.85 28.14 30.76 30.93 31.31 31.94 33.19 34 21 38.79 40.52 43.66
23.3 29. 29.2 30.4 31. 31. 33.0 34. 37. 42. 44.8
8. Mmwell's Relalions. These relations can be derived with the help o f calculus directly from the definitions T (bE/bS), and -P = (bE/bV),6 without recourse to the usual Legendre transformations. For example,
--
10. Boltzmann's Factor (10). A system composed of n distinguishable, loosely coupled particles of which no are in their ground state with energy eo and n, are in their first excited state with energy el and n2 are in their second excited state with energy rp and so forth has by Boltzmann's relation the following entropy:
Suppose, now, that u is in thermal contact with its surroundings 8 and that one of the particles in u that is in its ground state absorbs energy from 0 and lands in its first excited state. of a in (Particle gro"nd state ) + (%%?)
=
For an ideal gas (dP/bT), = nR/V (cf. part 4 above). 9. Equations of State. For any system in internal equilibrium
By Boltzmann's relation, S = ( R / N ) In 0 (9). (Our reason for using the symbol Q to denote the number of different quantum mechanical states accessible to a system in preference to the symbol W, which is usually interpreted "Wahrscheinlichkeit," is that usually one associates the word "probability" with a number between 0 and 1, whereas in fact the Q (or W ) of Boltzmann's relation is never less than 1 and is usually a very large number.) For a system of nearly independent point particles, Q = f(E,n).VnN. For such a system S nR in V R/N In f(E, n), and, therefore,
-
+
Conversely, one can use the thermodynamic relation (PIT) = (DS/bV),,, together with the expression given above for Q and the corresponding equation of state P V = nRT to derive the value of the constant of proportionality in Boltzmann's relation. Or, one can use the thermodynamic relation (PIT) = (dS/bV),.. and the equation of state P V = nRT to show that for an ideal gas
Since for any system in internal equilibrium
it follows that for an ideal gas
- ~-
IIf dE = TdS (aE/av)s = -P.
- PdV, dE,
= -PdVs;
ie..
(dE8/dVS)=
Particle of o in (first excited state)
This process alters the occupancy numbers in u from no and n, to no - 1 and nl 1 (the other occupancy numbers are unchanged) and decreases the thermal energy of 8 by the amount (el - eo). The entropy changes in the two parts of the universe (u and 8) are therefore the following.
+
ASe
el
-9
= - --
S i c e ASuni, = AS,
T
+ ASe, ASumi,> 0 if and only if
In other words, thermal excitation of a particle from the ground state to the first excited state can occur if and only if
Additional Topics
The topics discussed above are largely topics from classical thermodynamics. Other topics that fall in this category and that seem particularly well suited to discussions of thermodynamics with beginners are: graphical representations of phase equilibria through plots of the chemical potential against temperature (Fig. 8); the use of tabulated thermodynamic data to obtain equilibrium constants and cell voltages; applications of the approximation AFo(T)= AH0(298) - TAS0(298) (11); and the presentation of standard free energy data in tabular, easy-to-use, donor-acceptor form (11) through introduction of such transfer species as e- for oxidation-reduction reactions, H+ for Brgnsted acid-base reactions (Fig. 9), Ag+ for silver-ion reactions, C1+ for certain organic displacement reactions, 0 for ordinary flame reactions, and so forth. With students who have mastered calculus and who enjoy mathematics for its own sake, it has been found stimulating to devote some time to these topics: "other variables," particularly those variables appropriate to the physics and chemistry of surfaces and elastic solids (whose thermodynamic properties may be compared with the analogous properties of gases); the homogeneity condition (1%) E(XS, XV, An,, Ann, . . .) = XE (8, V, nl, 121, . . .) and the Gibbs-Duhem relation, with applications to black-body radiation; Volume
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497
the albegra of vector spaces and the problem of obtaining relations among thermodynamic derivatives (13); Lagrange's multipliers; and applications of the variable "the degree of advancement of a chemical reaction" (14) to the quantitative illucidation of Le Chatelier's Principle. But we would place before any of these topics a preliminary discussion of Boltzmann's relation, S = Ic in n, in which special emphasis is placed on: the plausible feastures of this relation: the statistical mechanical properties of small systems; the lattice model of solutions and the partial molal configurational entropy (15,19) of the solvent, solutes, and lattice defects; the thermodynamic properties of an Einstein solid, with an introduction to the properties and uses of the partition function (including its factorization in the case of an ideal gas for which rotation-vibration interaction can be neglected); and, lastly, in which special emphasis is placed on the factors that influence the 7 For example, klna is (like S) a quantitative measure of microscopic disorder; also, it is an extensive property and it increases in spontaneous processes. I t vanishes st O°K for perfectly crystalline substances, is never negative, and increases as E increases. It has the same units as S and, like S, is small for hard substances and large for soft substances. Its value usually falls in the range 1-100 cal/deg-mole; e.g., if 0 = 2N, klnn = 1.38 calfdeg-mole.
spacing of quantum-mechanically allowed energy levels. It is shown at this point that thermal excitation of motion that involves a characteristic mass m, moving through a characteristic distance LCbegins to become important at a temperature that is equal to h2/km,LC2degrees Kelvin (Table 3). Table 3.
Characteristic Temperatures for Thermal Excitation of Quantized Motion
Type of motion
(%mu)
Translation Rotation Vibration Electronic Nuclear
50 10 10 1/1837 1
a
T2 =
LC
mr
hPl(km.L.')
("W
(em) 10 10'
lo-'' 2
3 X lo-' lo-' lo-"
2,000 40,000
lotL
T,is such that kT, = Ae, where As is the energy gap between
the ground and first excited state.
Ac
is equal, or t~pproximately
L.. ?he farm oi the bctional dependence bf T, on h, k, me, and L. can also be obtained by dimensional analysis or by fitting de Broglic wave lengths A = h/p into the available space L,.
Proton Donors Occupied Levels Bronsted A d d s
Proton Acceptors Vocont Level8 ~ r o n 8 t e dBales
P=0.001otm.
F Solid
'Solution vapor
-
T diagram. A plot against temperotvre o f the partial Figure 8. An F TS lor, as i t i s often called, p the "chemical molor free energy F = H At obof rolid, liquid, and gaseous water lopproximotel. solute zero, F = H. With increasing temperature, F d e c r e a ~ sa t o rote equal to the instantoneour valve of S. Since S incre~reswith T, the downward trend o f F with increoring T becomer more and more pronounced. The valve o f the molor free energy curve of liquid woter compared to thclt H)E :: but folk foster (SdxB> of the rolid begins higher (H&
-
>
SE$l and eventually crosser the free energy curve of the d i d . At m y temperature the most rtoble form ( d i d , liquid, gar) i s that form with the lowest molor free energy. At thore temperatures where two curves cross (points A, B, C, D, and El, the corre3ponding phases son coexist Adding an impurity to the liquid phase increases the portiol molar entropy of the solvent and, hence, decreases its partial molar free energy a t every temperature. This produces an extendon of the liquid range ot both ends. The freezing point i s lowered from C to B and the boiling point is m i r e d from D to E. (Point A represents the rublimotion point of the d i d when Plgor) = 0.001 atm.) Because the entvopy d i f f e r ence between d i d and liquid i, less than tho1 between liquid and gas, the slope, o f the curves are rvch as t o make the increase in the boiling point less L o n the decrease in the freezing point. Earlier i t war shown anolytic d l y that the values of there two colligative properties were given b y the expression 6T F-. RT/ASa.Nz, where ASo represents the change in partial molar entropy o f the solvent in going from the liquid to the solid or gar phase. For freezing AS* is a relatively rmoll negotive number; for boiling ASc is normally a large positive number.
498
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Journal of Chemical Education
Figure 9. Thermodynamic data in donor-acceptor form for proton-transfer reostionr (19). Proton, tend to fall from occupied levelr to lower vacant levels-i.e., from upper left to lower right. Transfers from level 1 to level 2, from 2 to 3, and from 1 to 3 (not shown) represent, respectively, neutroliration of HIOt b y A c , neutrdimtion o f ~ H A C b y OH-, and neutrolimtion of H 3 0 + by OH-. The reverse reactions, 3 to 2, 2 to I.m d 3 to 1 (not shown) represent, respectively, hydrolysis of Ac; ionization of HAG and self-ionization lor autopmtolyrirl of the solvent. The escaping tendency of protons from a level is given b y the expression to the right of that l e d ; i t is proportiond to the concentration of protons in thot level, ik., to the ratio of the concentration of occupied rites lthe m i d form) to the concentration of the vocmt rites (the bare form). For levels 1, 2, and 3, the constants of are, respectively, 1, KI, and K , The vertical distance between levelr 1 and 3 i. determined by the standard free energy chonge for the transfer o f protons between there two levels and i s equal to -RT In Ku. Similarly, the distance between levelr 1 and 2 is -RT In K., By subtraction the dirtonce between levelr 2 and 3 it found to b e - R T In I K ~ / K ~ Ithot ; i s to ray, When the . Khvd . = KdK.. escaping tendency of protons from two levelr is equal, equilibrium exists with respect to the distribution of protons between the two levels. When the escaping tendency o f protonr throvghovt the ryrtem i s uniform I201 this condition i s auicklv reached in w a t e r i t s volue from any level (ray level II, or the lo$aritl;m of its value, or the logarithm of its r&procd lthe "pH), moy be used ar a measure of the oddity of the ryrtem. The Rgure also shows the leveling effect of the solvent on acids stronger than HIOC and on bore. stranger thon OH-. The dotted orraws at the top and the bottom represent, respectively, the reactions HCI04 H20 + H1O+ C l O l and H10 f 0 - ( 0 s in CaO, No201 20H-.
-
-
+
4-
Summary
The topics that comprise this introduction to chemical thermodynamics in order of appearance are: the first law and calorimetry, the need for a second law, characteristics common to T and AEIAS, selected applications of T = AE/AS, survey of molar entropies, entropy changes in the universe when ice melts and water evaporates, Boltzmann's relation, elementary statistical mechanics and quantum mechanics, the free energy, numerical computations, and further implications of the second law. Through "numerical computations" the discussion supposes no previous acquaintance with calculus and might best he described as "general chemistry from an advanced standpoint." But when all we can say has been said and when all we can write has been written, not all we can do for the student has been done. For teaching and learning are not the same thing. To be sure, teaching thermodynamics or writing a book about it is an excellent way to learn thermodynamics. But merely listening to such lectures and reading such books is not. For as Polanyi has observed (16), an expert always know more about his subject then he can express in words; a t the heart of every subject, even one as well codified as thermodynamics, lies a part that is ineffable and crucial. Hence a beginner cannot learn from words alone how to use thermodynamics. That is a skill that can be acquired only through an active apprenticeship. As Sophocles has said, "one must learn by doing the thing; for though you think you know it, you have no certainty until you try." And to try means to solve problems. Here it is that the teacher as connoisseur and critic can make perhaps his most significant contribution by posing problems for his students and by carefully reading and criticizing their solutions to these problems. The time this takes and, in the end, the money it costs (17), represents, in a sense, an expression of faith-of faith in the value to the students of being able to understand and to use the second law (18) and of faith in the ability of these students to justify in their productive years an expenditure by their professors and their school of a special effort on their education. Much of the material described in this paper has been used in recent years a t Minnesota with selected freshmen, ten to twenty-five in number, who have demonstrated during their first quarter a t the University superior intwest and ability in chemistry. Some
of the material has also been used with non-majors in larger sections of general chemistry. Our experience with these students strongly supports the view that has already been expressed in THIS JOURNAL (18) that the first year of college is not too early a time for young students to begin thinking about chemical and physical problems from the vantage point of the second law of thermodynamics. The author is indebted to his father Henry E. Bent, to his colleaeue Dovle Britton. and to the editor of THIS JOURNAL for gelpful crit,icisms and suggestions concerning this paper.
-
Literature Cited (1) WHEELER,L. P., "Jo~iahWillard Gibbs. The History of a
Great Mind," Yale University Press, New Haven, Connecticut, 1952. E. U. AND SHORTLET,G. R., "The Theory of (2) CONDON, Atomic Soectra." Cambridge Universitv Press. London. England, 1951. (3) OPPENHEIMER, R., Science, 123,397 (1956). (4) MACWOOD, G. E. AND YERHOEK, F. H., J. CHEM.EDUC., 38, 334 (1961). (5) POLYA,G., "Mathematics and Plausible Reasoning. Vol.
(6)
(7) (8) (9)
I. Induction and Analogy in Mathematics. Vol. 11. Patterns of Plausible Inference," Princeton University Press, Princeton, N. J., 1954. LEVIS, G. N. AND RANDALL, M. (revised by PITZER,K. S. A N D BREWER,L.), "Thermodynamics," McGrm-Hill Book Company, Inc., New York, 1961, the prefaee. BECKER,C. L., "The Declaration of Independence," Vine age Books, New York, 1958, p. 215. BRPINSTED, J. N., "Principles and Problems in Energetics," Interscience Publishers, Inc., New York, 1955. RUSHBROOKE, G. S., "Intr~duction to Statistical M e chanics," Oxford University Press, London, England,
1949. (10) GURNEY,R. W., "Introduction to Statistical Mechanics," McGraw-Hill Book Company, Inc., New York, 1949. See especially Chapter 1. (11) BENT,H. A,, J. Phys. C h a . , 61, 1419 (1957). (12) BENT,H. A,, J. Chem Phys., 23, 2199 (1955). (13) BENT,H. A,, J. Chem. Phys., 21, 1408 (1953). P., "Thermo(14) DE DONDER,T. A N D VAN RYSSELBERGHE, (15) (16) (17) (18) (19) (20)
dynamic Theory of Affinity," Stanford University Press, Staniord, California, 1936. BENT,H. A,, J. Phys. Chem., 60, 123 (1956). P o ~ w u r ,M., "Personal Knowledge," University of Chicago Press, Chicago, Illinois, 1958. KIEFFER,W. F.,J. CHEM.EDUC.,40, 109 (1962). KIEFPER,W. F., J. CHEM.EDUC.,38, 333 (1961). GURNEY,R. W., 'Tonic Processes in Solution," McGrawHill Book Company, Ine., New York, 1953. GIBBS,J. W., "Collected Works," Vol. I, Yale University Press, New Haven, Connecticut, 1948, p. 65.
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