Entropy and Unavailable Energy J. N. Spencer and E. S. Holmboe Franklin and Marshall College. Lancaster, PA 17604 The piston engine of automohiles converts ahout 25% of the heat produced from the combustion of gasoline into work to move the automobile. Coal-fired elect,ric power plants can convert about 47% of the total heat produced in their hoilers to electrical work. The fuel cell reaction H2(g)+ 'b2 02(g) HrO(1) liberates 68 kcal mol-' of heat hut only 57 kcal of heat may be utilized to drive electrons through an external circuit. Even if mechanical loss of heat could he eliminated, the complete conversion of heat into work is not possihle for these pro&sses. That part of the total heat not available for production of mechanical work is called unavailahle heat or unavailable energy. The first law of thermodynamics is the principle of the conservation of energy applied to processes involving the evulution or absorption of heat. The first law requires only that for a chemical process carried out at constant temperature a certain amount of heat he given to the surroundings to reestahlish the initial temperature and that this amount of heat must he ahsorbed when the reaction is reversed. No exact definition of temoerature is eiven hv the first law and no re-
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supplied. The second law of thermodynamics relates the initial and final states of any process and shows whether it is possihle to pass from the initial to final state without leaving other bodies unchanged. The initial and final state must he fully specified. In addition to the chemical composition, physical conditions such as temperature and pressure must be known ( I ) . The origin of the unavailable heat must be found from a consideration of the second law of thermudynamics. The second law is, like the first law, a law of experience, and must he accepted as valid until some experience which contradicts it is fnund. The second law may he stated in several
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Journal of Chemical Education
wavs. For the oresent ouruoses the followine two statements . I rnt,.hnth~,.iIaolrk 111tt. Ihr t r I I I J ~I K C I mlrlctr. 11111. 111.11 ~ . t hc;d 1 1 1 1 ~wlrk n i III. ~In* .mt,lvtr." I'l.in~l; I I h,,, ntlrcd thar this statement is quite correct in certain specla1 cases hut does not express the essenttal feature^.^ This formulation is, however, Eonvenient for a discussion of entropy and available energy. ~ i ; s ta thermodynamic process will he examined on the microscopic level. Consider the reversible adiabatic compression of an ideal gas confined to a piston-cylinder container. For this process the entropy change of the system, AS,,,, = ASt,,t = 0. The distrihution of molecules among the system energy levels does not change as a result of the compression (2). The spacings between the energy levels do change, depending on the volume change. The energy of the eas must be greater after the cumuression than before the compression because the energy level spacings have increased hut the populations of the levels have not changed (see figure). The populations of the levels remain unchanged hecause there is sufficient thermal enerrv from the cumuression to maintain the populations of the le&. The second'law does not permit a decrease in the entropy of a gas as a result of an adiabatic compression but an entropy increase is allowed. Suppose that the gas is compressed adiabatically and irreversibly by placing a weight on the piston. The final volume is the same as for the reversible case. In this situation AS,",, I Certain orocesses do allow the total conversion of heat into work. Tne r & a s d e sotnzrmal exrrirns on ot :m deal gd5 oues perm t T I S to oe recoveren as ,.urn out t n r s,>lr.m :mu ,urrounu nq, arr not reitorzfl to me in !#a states A nrcwid) ma l c n to the ,Dove stalzmpnt is 'without producing a net change in the surroundings."
Ein arbitrary units. I1 An ideal gas is corn~ressed reversiblyand adiabatically. A s = 0.the papulahons of energy levels are unchan& 11) An ideal gas is compressed irreversibly and adiabatically. AS > 0 , the populations of the energy levels more upwards.
the
= AS,,,, > 0. (The entropy of the weight does not change hut its energy does.) The increase in entropy occurs for the irrewrsihle &oct..\ herause t he temperat uri and hence thr tnerrv o f t he ras iz greater ;lifer the cumpre.;siun than in the case or' the re;ersihre process. This extra thermal energy causes the energy level populations to shift upward (see figure). Part of the thermal energy produced by the irreversihle process is not necessary to maintain the population uf the energy levels the same before and after compression. That part of the thermal energy necessary to maintain the populations of the levels is the energy that must he supplied so that AS,,,, 2 0. The work required for the irreversihle compression is greater than that required for the reversihle compression. Thus, the final temperature of the gas is greater for the irreversible orucess than for the reversible vrocess. That is, the irreversihle pnwess produces mart, h w t than is neressarv I ~ I prevent a viohtion o t t h e s e n d I;arr. The same final trmprrature o d d he x h i w e d for hoth procesirs b y rarrvinl: out an additional a t w ; tht. nac could hc r w l r d foll~u'inrthe irrrversihle process,or foliowing the reversihle proceis, the gas could he heated, either step occurring at constant volume. The implications of this are developed the following analysis. Now consider the process in thermodynamic detail. The gas is compressed adiabatically and irreversibly to temperature T I and volume V. For this process AS,,,,,, = AS,,,, ;,,, > 0. Suppose the process is carried out reversihly to achieve T Iand V. This could he accomplished by first reversihly and adiabatically compressing to T2 and V where T I > T2. For this step of the reversible process AStOt,l = AS,,,, = 0. Next the gas is reversibly heated a t constant vulume by a large heat reservoir from T? to T I ;then ASt,,,:z = AS',,,,,, + AS,,,, = 0, where AS,,,, is the entropy change of the reservoir, AS,,,, = AEJT,, where T , and AE, are the temperature of the reservoir and the change in energy of the reservoir due to removal of heat, respectively. The first law requires that AE,,,t + AE, = 0. For the two steps of the reversihle process
The gas has now been taken from thesame initial conditions to the same final conditions by a reversihle and an irreversihle process. Because the physical conditions of temperature and ;olume are the same for hoth processes: ASqystm" + AS'8yntre"= AS-yat,,,- > 0
(2)
Then for the reversible process from eqns. (1) and (2): AS ,,,
= AS,,,t,,.
A& t -= 0
T.
or
-AE, AS8yntxrmv = -- AStot,,..
T,
rnwvt
=
,o
T,
T o achieve the same final conditions realized in the irreversible nrocess. the reversihle orocess must. hv suitahle arrangement of mechanical devices, extract energy as heat from a reservoir. This energy is transformed into work through the mechanical devices. In the irreversible process this energy did not anoear as work. that is. this work is lost as a result of the irrev&ihle process. The energy that is lost for work in the irreversible process is T,AS,,,,,,-. The previous process was carried out in a system which did not exchange energy or mass with the surroundings. The applicability of the second law is not, however, limited to adiabatic enclosures. T o apply these arguments to a system which can undergo exchange with the surroundings, it is only necessary to include in AS,,,, all of the environment which exchanies mass, work, or energy with the system. This global formulation allows the system to he treated as closed and then AS,,, = AS,,, AS,,,,, > 0 for any process occurring with the closed environment. T h e results for the adiabatic compression just considered can he generalized to T,AS,,.,,,,, = AE where AE is the heat energy which is a consequence of all irAS ,,,; and T , is the reversible processes; AS, 0 and hence for the spontaneity of this process. If the two phases could coexist a t different temperatures, the higher temperature molecules would he restricted to the water phase and the lower temperature molecules to the ice phase (8). In order to reach thermal equilibrium, one of the species undergoes chemical change, i.e.. melts. T h e tempeiature eradient, T H . , ~ Ti,.,, is degraded. Because the entropv ehange for hody a t a b w e r temperature is greater than foi a hody a t a higher temperature ( A S = AEIT) for the same absorption of ihermal energy, the positive entropy change for the ice is greater than the negative entropy change for water. The process is driven by the favorahle entropy change for ice melting. As the melting proceeds, the temperature gradient. is degraded, and a t equilibrium the temperatures of the two phases are the same. If it were not for the degradation of the thermal gradient by ahsorption of heat from the surroundings, the system could not reach equilihrium. The first law would allow the process to occur in either direction. The second law shows the direction of the process and requires that heat pass from the hotter to the colder hody. Open Systems In an isolated system any temperature gradients in the system will decrease with time and the entropy will increase. T h e second law requires that if heat is converted into work, some heat must also be lost to a lower temperature. Only if a temperature gradient exists can heat-engine work be done. Yet such processes demade the temoerature gradient. When the temperature is everywhere thk same, the system has achieved maximum entropv. Fur processes dependent on concvntmti~~n rradicnts. ihe operiltion o f thew prucesws d e s t r ~ ~ vthe s (mwntration yrnd~ent.111 order to maintain t hrse yradienls. ;I swrce from outride the s\r;tem is necezwry. Sy.qlems uhwh inn exchange ma-s. mvrrv. or work wil h the surroundings are called open systems. The human body is often incorrectly described as a heat engine. The reason this is not accurate is that a heat engine requires a temperature gradient: the body, an open system, is a t a constant temuerature hut reauires a concentration gradient. dust as the temperature gradiknt of a heat engine can be degraded, the concentratiun gradient of the hody will degrade unless a supply of energy prcducin~compounds ic made availahle. For the enrine and the bodv, the deeradative processes lead to an increase in entropy and an approach to
equilihrium. dust as operation of a heat engine requires degradation of heat energy, the hody requires degradation of hixh energy compounds. Part ofthe heat extrarted for use in a heat engine must be evolved to a low temperature reservoir, part of the chemical energy of ingested compounds goes to the pnduction of heat. In the case of the heat engine, that portion of the heat transferred to the cold reservoir is "irrevocahly lost." For the body, part of the heat obtained from high energy compounds is spent by radiating heat to the surroundings and by work. Only about 10% of the energy ohtained by consumption of food is used by the body for building and replacing tissue (9).The heat engine maintains its temperature gradient by using an energy source to heat the hot reservoir. The bodv maintains its concentration gradient hy~. inrestion of fmld t u replenish the mt~letuli.sspent I)? c hrmical prcn.r.;*es. The nmversion of heat into work is an inefficient process. Uptake of energy by living systems does not depend on the direct conversion of heat to work. The energy received by plants from the sun is converted to chemical work before degradation to heat can occur. Simply heating the leaf of a plant is insufficient for biosynthesis. Only about 3% of the total incident solar radiation ends up as stored energy in olants ( 1 0 ) . In the human hodv. the stored enerev of ineested molecules is transferred to ATP; thus increasing'ihe efficiency over what could he obtained hv direct use o f t h e stored enerfy. I'he overall result of the citric acid cvcle is the combustion of acetic acid, CH:COOH(aa)+ 2 0 3 k I 2Hz0(11 t 2COd aq) for which AHo = -217.3 kcal m&l, ASo = -49.fi cal dee-lmol-I. and AGO = -202.9 kcal mol-I. Of the total heat. 211 kcal, the available heat is 203 kcal. Dickerson ( 7 ) has calculated that if the body ooerated as a heat ennine between 98.fi°F ( X ° C ) and room'te&erature (20°C) the efficiency would he 5.5%. Of the 217 kcal only 12 kcal would be "available." The heat is actually stored by ATP giving an overall efficiency of about 48%. All spontaneous processes are driven by a gradient. The gradients commonlv encountered are ones of temperature, pressure, or concentration. The quantitative meas"re of this cradient, the driving force for the process, is AS,.,,. Equilibiium is achieved byhestruction ofgradients. For a chemical reaction to occur spontaneously a t constant temperature and pressure, a gradient in free energy must exist. T h e first law requires that surroundings or system must change to maintain the temperature conditions or else a new gradient would develop. The second law points the direction of the orocess and soecifies how the chanees in svstem and suiriu;ldings must be related. The required change in the surroundines makes Dart of the heat available to the svstem unavailablefm work:~volution or absorption of heat ienerated hv.anv"orocess must he comoensated bv direct exchange . with the surroundings or by other processes such as work. It is the ootential for creating work. AS,,.,, . .. that should he considered in any efficiency analysis. Theauthors wish to thankdody Powell forcritical reading of the manuscript and for preparation of the figure.
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Literature Cited 11) I'lanrk. h l a x . " T r n l i s ~ ~ ~ n l ' h c m ~ ~ ~ d y r ! aInl c~wi m r IPubliri~timr. ." NPWymk. I!l4i.
Volume 60 Number 12 December 1983
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