General Definitions of Work and Heat in Thermodynamic Processes Eric A. Gislason University of Illinois at Chicago, Chicago, IL 60680 Norman C. Craig Oberlin College, Oberlin, OH 44074 There are difficulties with the concept of work (1-9), particularly for irreversible processes. A thoughtful writer on that topic has said (9)"For mechanically irreversihle volume chances, we usuallv cannot calculate the work from thermodynamic considerations." We believe that conclusion is incorrect. In this paper we give definitions of work and heat that overcome earlier difficulties. Both quantities are defined in terms of experimental quantities. The definitions are simple and general. They apply equally well to reversible and irreversible processes. Using these definitions, we are able to resolve several important problems as well as to obtain fresh insights into the First Law. Some of the results are surprising. They force a reevaluation of one's thinking about work and heat. Our orimarv emuhasis is uoon an ooerational definition of work, but an iperaiional definition oiheat arises in a natural way once work is defined. By operational definition, we mean a definition that includes a specification of how that quantity is to be measured (10). There is no difficulty defining work and heat in this way for irreversihle processes. We begin with a review of the common definitions of work and heat, which can be found in most textbooks on the subject. For any process the First Law of thermodynamics, equates the change in the internal energy U of a system to the flows of heat q and of work w. The usual definitions of heat and work are as follows. Heat is an energy transfer measured hy means of a temperature change in a calorimeter. Heat is sometimes defined as a transfer of energy induced by a thermal gradient. By comparison, work is an energy transfer caused by the action of external forces on the svstem. Alternativelv. .. work is defined as the result of the system acting against external forces. Work is viewed as eauivalent to lowerine (or raisine) a weight in the field of &avity. If the discussion is restricted to pressure-volume work, for an infinitesimal process the work is where dV is the infinitesimal change in the volume of the system and P,,, is the pressure exerted by the surroundings on the system. The symbol Dw is written to emphasize that this quantity is an inexact differential. That is, Dw is not defined by variables of the system, or, equivalently, Dw depends on the path. For a finite process eq 2 can be integrated to obtain
If P,,, is known a t all points along the path, it is straightforward to compute w. Various problems with these definitions are described in the following section Problems with Earlier Definitions of Thermodynamic Work Consider the apparatus shown in Figure 1. This example is 660
Journal of Chemical Education
Catch #2
Catch #I
'
Cylinder
Figure 1. The system (the gas) is contained inside the cylinder and beneath the piston, which is held initially by catch #I at height h,. When catch #1 is removed, the pressure of the gas lins the piston to catch #2 at height h2. The piston has mass m. Thespaceabove the piston andaisooutside thecylinder is assumed to be in a vacuum.
simple and yet shows the fundamental problems involved in defining the thermodynamic work as the integral of some force times the differential of distance, as in eq 3. The system is the gas contained in the cylinder below the piston. Everything else belongs to the surroundings. The piston has various weights on top of it; the total mass of the piston plus weights is denoted m. The space inside the cylinder above the piston is evacuated. For any real apparatus, frictional forces act between the piston a i d the &linder. (Some authorities avoid the problem of frictional forces by idealizing the apparatus. unfortunately, the resulting definitions of work cannot be applied to any real process.) In this section we consider forces rather than pressures because frictional forces are difficult to picture as pressures. Nevertheless, the analysis could be recast in terms of pressures by dividing each force by the area A of the piston. The three forces that, in general. act on the uiston are described in Table 1. The process under Eonsideration involves an expansion of the eas, which raises the bottom of the piston from height hi to h? A well-known theorem of classical mechanic; (llj states that the total work W,,, . . done on the uiston (plus vwightsi I,? n i l d f h e rurcei during rhe proce%sequal.ithe net inrreaseof th,, kinrricmrruv of rhe pisrun. (Uote rhar II',. , is not the thermodynamic work w.) ~ & twe , consider the ease where the piston is stationary a t the start and at the end of the process. Then, W,,, is given by
We define the integral off along the path to be -O. Then
It seems reasonable to let w = O, since -O appears to be the decrease in the internal energy U of the gas due to the force the gas exerts on the piston as the process is carried out. The problem with this choice, however, is that the frictional force Ft, acting between the piston and the cylinder transforms mechanical enerev into thermal enerpv. This thermal energy may end up in t h e surroundings, in-which case it is heat rather than work ( 6 8 ) .Alternatively, if the cylinder or the piston conduct thermal energy, some of this energy may return to the system, in which case it is neither heat nor work. We conclude that O cannot he w (unless the process is reversible). Furthermore, neither f nor F f , is known (6), except in special cases. Consequently, O cannot be computedor measured for any irreversible process. Many textbooks (12) state that the thermodynamic work is the integral over the path of the force exerted on the svstem hv the niston (that is.. hv. the surroundings). However, asshown in'l'iilh I, this t'urce ijsimply -/,SO the integral is ! I . \Vt: hnw seen that I 1 is not nn acce~tablecandidate for w. The process shown in Figure 1 raises a mass in the earth's gravitational field. In the next section we shall argue that w = -mg(hz -hJ, the negative of the potential energy gained by the piston. Of course, this quantity does represent the integral of a force times the differential of distance. However, -mg(hz - hl) is the integral of the conservative force -mg exerted on the piston, which is one part of the surroundings, by the earth, which is another part of the surroundings. As we have seen, the force -mg is not the force exerted bv the surroundines on the svstem or vice versa. unless the"piston is at rest. w e concludk that if the piston is stationary a t the start and end of the process, the thermodynamic work may be written as an integral of force times the differential of distance. hut the force involved is not the one commonly given. If the piston is moving a t the start or the end of the
Table 1.
Description of Forces Acting on Plston in Figure 1
F,,, the total force on the pismn, is given by Ft0,= f - mg
+ Fh.
fis the instantaneous force exerted by the system (the gas) on the Piston. Note that f > 0. That is, f is directed upward. f is not a conservative farce unless the piston is stationary. if the (b) piston has a velocity v, f is larger when v < 0 than when v > 0 (at a fixed value ot h).Here v < 0 means the piston movesdownward: v> 0 means it moves upward.
(a)
By Newton's third law. - f is the force exerted by the piston on the system.
(c)
process, the analysis is more complicated. Kivelson and Oppenheim (7) were the first to discuss this case in detail. We denote the initial kinetic energy of the piston by K1 and the final value by Ki. Then, by the aforementioned theorem of classical mechanics, the total work done on the piston is
In the following section we shall argue that the thermodynamic work for this process is given by w = -mg(h2 - hl) (K2- K1). The first term was discussed before. The second, which is the negative of the net increase in the piston's kinetic energy, can also be converted into lifting a weight, such as the piston, in the gravitational field (7,13-15). Using eq 6 we can write:
The work is straightforward to determine experimentally usine ea 7. Eauation 8 shows that w can be written as the integralof a fdrce times the differential of distance. However, w cannot be calculated from this integral since neither f nor F f , is known for general processes (6). Two conclusions can be drawn from the analysis in this section. First, the presence of frictional forces makes the definition of work difficult. Since frictional forces accompany any irreversible process, however, they must be treated properly. The only correct way to write w as an integral for the irreversible process shown in Figure 1 is given in eq 8. Other integrals do not give the proper result in every case. For exam~le.ea. 4 eives .. the correct value of w for a reversible process but nut ior an irrrvrrsihle m e . Similarly, the integral of \ - r n # ) o w r the height change 111 equals L only u h r n K ? = Kt. T h e second conclusion is t h a t t h e correct integral representation of w given in eq 8 is of no value as an operational definition of work, because the forces f and Ff, cannot be measured during an irreversible process. This point was made earlier by Canagaratna (6). Because of the problems described in this section, some authors take the point of view that work can only be defined for "quasi-static" processes (16). Others have further restricted the definition to reversible processes. The definition of work need not be restricted to these special cases. Certainly, it is not desirable to do so. For example, the well-known theorem that maximum work (-w) is done by the system in a reversible process is not very meanineful if work is onlv defined for reversible ~rocesses! More g&erally, the importance of the First Law as given by AU = a w is that Q and w can be measured. Consequentlv, us to determine changes in U, ;hang& the irk Law that would not he measured in any other way. Since all real processes are irreversible, the First Law could not be developed and related to experiments unless work and heat could be defined for irreversible processes. In the following section we present our general definitions of work and heat. Both quantities are well defined for irreversible processes.
+
(d) if the system is at equilibrium and the piston is at rest, f = PA. where A is the area of the piston and Pis the system's pressure.
(el The quantity f/A is sometimes referred to as P,.*, the instantaneous pressure exerted by the system on the surface of the piston. -mg is the (conservative) force exerted downward gravity.
on the plston by
F* includes all nonmnservative (frictional) farces exened on the piston by other parts of the surroundings. Far example, it would include any frictional forces between the piston and the cylinder. F,, can be positive or negative. Normally F, 0 when v < 0, and 6,< 0 when v > 0.
>
(5)
The forces fand
F,. are normally not known unless the piston is at rest.
General Deflnltlons ot Therrnodynarnlc Work and Heat We take the point of view that work and heat are quantities to he measured as changes in the surroundings (2) when a process takes place. There should be no need to examine the system itself to determine q and w. Rather, it should be sufficient to carry out measurements on the surroundings at the start and at the end of the process to obtain the heat and work. Our definitions are consistent with this approach. We assume that temperature has been defined and can he meaVolume 64 Number 8 August 1987
661
sured using an ideal gas thermometer. We also assume that heat capac-itiesare known ior common suhstanceli.l \\'e seek operational definitions of 11. and o. We shall d p fine thermodynamic work w for a process as the negative of the energy change in t h e surroundings during the process that is potentially conuertible into lifting (or lowering) a mass in the earth's grauitational field. Similar definitions have been given before (2, 13, 14). To complete the definition, however, it is necessary to clarify exactly what types of energy are convertible into raising a mass. T o do so, we assume that the surroundings can be divided, a t least conce~tuallv.into various Darts. oiston. a . . such as a cvlinder.. a . large voiiime of water serving as a calorimeter, etc. Each is a one-uhase, chemicallv homoaeneous substance. We also assume that the total energy Ei of part i of the surroundings can he decomposed as (17)
Here KEj is the macroscopic kinetic energy of part i. This term is restricted to the energy associated with the translational motion of the center of mass of the part and any rotational motion of the part as a whole about its center of mass. Similarly PEi is the total macroscopic potential energy of part i. A number of types of potential energy can he considered here. These types include the potential energy associated with the position of the part in a gravitational or electric or magnetic field, the potential energy associated with increasing the surface area of a liquid or the length of a spring, and the potential energy associated with electrically charging the part (assuming i t is a capacitor, for example). This list is not meant to be exhaustive. Finally, U; in eq 9 is the remaining (microscopically diverse) energy of part i. This function is the internal energy of part i. The various parts of the surroundings are not allowed to undergo chemical reactions or phase chanaes during the ~ r o c e s s . ~ One is tempted at this p;int to define u-as the negative of rhechanges in PE, and KE. for varlous part.; dthesurroundingsanrl to define q as the negativeof the sumof the chnngen in I.:. However, the resulting definitions would not be satisfactory. In some cases part of the change in U; must he assigned to w. If part i is a pure substance, Ui will be a function of two variables. sav V; and T;. If V: remains constant during a process, i t seems reasonable that any change in Uj due to a temperature change will contribute only to q.
'
It is not obvious that heat capacities can be definedand measured for various substances without invoking the First Law of Thermodynamics. A number of writers have argued that it is possible; the argument is as follows (12). One first defines the heat capacity of a common substance such as water and then carries out experiments placing other substances at one temperature in thermal contact with water at a second temperature and measuring the final common temperature. A series of these experiments establishes that one can assign to every system an extensive property called heat capacity under both constant oressure and constant volume conditions. In fact,there is n'o need to exclude phase transitions in any part of the surroundings.For this case the heat capacity of that part includes a delta function at the transition temperature. Similarly, one could allow the surroundings to contain a reactive mixture of components, provided that this mixture is at equilibrium (includingchemical equilibrium) at the start and end of the process. It would be necessary to determine the heat capacity of the mixture in the standard way; that is, by measuring the temperature change 6Twhen a known quantity of heat 69 is added to the mixture. The fact that the temperature change 6Tmight also beaffectedby theenthalpy of reaction is irrelevant. One simply determines C = 6q/6T, which is the "heat capacity" of the mixture. We cannot, however, allow reactive mixtures in the surroundings that are not in chemical equilibrium at the start and end of the process. In the text we refer to this by forbidding "irreversible chemical reactions in the surroundings." To simplify the presentation we exclude in the text further consideration of phase transitions and chemical reactions in the surroundings.
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Journal of Chemlcal Education
However, if Vi changes during the process, that change is potentially convertible into lifting a weight. As an example, consider a process where part i is held a t constant pressure PE)j = 0. In this case, AH; = and, in addition, A(KE Cp,iATj = AU; PiAV;, where AVj is the volume change of part i. For simplicity we have assumed that Cp,;, the constant-uressure heat cauacitv . . of Dart i, is indeuendent of temp?-rature.The result can he rewritten as 31'. (.p,,l\T, P A V , . It is reasonable toassign the firit term in AC. tu o. and the secmd trrm t o u . \Ye conclude that, in general, parr of AU, must be alisirnrd to q and parr to ro. Unfortunately. the allocation betwein q and; cannot hemade for the completely general case where Tj, Vj, and the pressure Pj of part i all change. We can, however, make the allocation for the cases of constant volume and constant pressure. Consequently, we can define wand q for these cases. Each part i of the surroundings may contribute to both the work and the heat. We define w and q as
+
+
=
where the sum runs over all parts of the surroundings in direct or indirect contact with the system, and W;and Qj are computed according to the following rules. If part i is held a t constant uolume during the process,
+
where A(KE PE)i is the change in the macroscopic kinetic and potential energy of part i, Cv,j is its constant-volume heat capacity, and dTj is its differential temperature change. If, on the other hand, part i is held a t constant pressure P; during the process,
We shall refer to the term -PjAV; in eq 14 as the energy of compression of part i. This energy could be used to lift a weight if part i were returned reversibly and isobarically to its original volume. Note that this term is positive if AVj is negative. A numher of comments can he made about these definitions. First, we have not used any of the laws of thermodynamics in our definitions. This restriction is in keenine with the standard development of thermodynamics that dlfines wand a before introducina the First Law. Second. if both the volume and pressure of a& part of the surroundings change during a urocess, we eenerallv cannot define work and heat. ~ortunacely,this reskction-is not a severe one. Later we shall show that W,and Q, can also he computed for part i if i t is held at constant temperature during a process. This latter case is of limited interest because we have to use the First and Second Laws to obtain the results. Third, application of eqs 1 2 and 13does not require thevolume of part i actually to be constant throughout the process. I t is sufficient that the initial and final volumes he the same. Similarly, for eqs 14 and 15 only the initial and final values of P, need he the same; the pressure of part i need not he well defined during the process. Thus, our definitions apply equally well to reversible or irreversible processes. Fourth, the system itself is not restricted to a constant pressure or constant volume process; any type of process is allowed. The system, for example, may undergo an irreversible chemical reaction or phase change. Fifth, some parts of the surroundings may he
held a t constant volume while others are held at constant pressure. Sixth, the definitions in eqs 10-15 are clearly operational definitions because i t is straightforward to measure every quantity in the equations. ( ~ l c a l that l we have assumed that heat capacities are known.) Seventh, our decision to insert minuisigns into eqs 10 and 11was somewhat arbitrary. We could have omitted them and simply changed the signs of all terms on the right-hand side oi' eqs 12-15. However, with the sign choice used in this paper we shall be able to write (see below) AEi = Qi W; as one form of the First Law. The alternative choice of sign would give AEi = -Qi - W i ,which would be correct but unnecessarily confusing. To the best of our knowledge, the definition of Wi in eq 12 for the constant volume case has never been presented in this form before. However, this definition is consistent with many well-known examples of work (usually presented for reversible vrocesses): in a -gravi. . for example, . . raising- a piston tational field, stretching a spring, running an electric motor, etc. The definitions of Qi in eqs 13 and 15 are standard. In both cases heat is identified by temperature changes in the surroundings. We believe that our definition of Wi in eq 14 for the constant pressure case represents the first time that a definition of work has considered the energy associated with compressing or expanding any part of the surroundings during the process. Other definitions implicitly assume that the various parts of the surroundings are held a t constant volume.3
+
Interesting Consequences of the Deflnitlons of Heat and Work Our definitions of heat and work have two interesting consequences. These are best described in the context of an example; the one we use is similar to Problem 7.1 of Pitzer and Brewer (18). This problem summarizes the imvortant elenie~itsof the.loule merhaniral-equi\.alent-ot-heatexperiment I I.{). Cnnsider a steel hall of mass m located at a height h , ; it sitso5,er a large liqni~l-waterculorimettr whose hottom iint height h?. Assumethe ball isco\.ered bya substanre that does not conduct thermal energy; thus, the ball is thermally isolated from the calorimeter. The hall and calorimeter are initially a t the same temperature T. When the ball is dropped into the calorimeter, we assume the potential euergy decrease mg(h2 - hl) is converted totally into thermal energy of the calorimeter, whose temperature rises a small amount. However, the temperature of the ball remains the same. Both the ball and the calorimeter are at 1atm pressure. To analyze this problem we use the First Law in its most general form. As indicated in eq 9 a system can have macroscopic potential and kinetic energies ( P E and KE) in addition to internal energy, and each of these energies can change as a consequence of a chemical process. The First Law as given in eq 1 does not allow for changes in P E and KE. T o take these into account the First Law must be written as
where each quantity in the first line is defined as in eq 9 but refers to the system, and q and w were defined in the previous section. This form of the First Law is widely known (9, 19). First, we consider the ball to be the system. From simple thermodynamics and our definitions of q and w we have AE = A(PE) = mgAh AS = 0 AA=AE-TAS=mgAh w=o q = mgAh
(17)
(18) (19)
(20) (21)
where Ah = h2 - hl is less than zero. Here AS = 0 for a change in a purely mechanical system (13). Note that -w 5 -AA as required by the First and Second Laws. If we now reverse the description and call the calorimeter the system, a similar analysis gives w = -mgAh (22) q=O (23) AE=AU=q+w=-mgAh (24) (25) AS = -mgAhlT AA = 0 (26) Note that q = 0 because the temperature of the ball does not change. The calculation of AS requires comment. We assume the calorimeter is very large, so that the change in temperature AT is very small. Then, we can write
which gives the result in eq 25. A simpler but less rigorous derivation (20) . . of AS(calorimeter) can be eiven bv treating the bull as the system. Then because the surroundings con= lS(cnsists of a larre heat bath we have AS~surroundinr;~ lorimeter) -qlT = -mgAhIT. wehave taken; from eq 21 because the ball is the svstem. Examinine the results of eas 22 and 26, we see that 5 -AA, as expected. Also, the sum of the two entropy terms in eqs 18 and 25 is positive, as required by the Second Law for the total entropy change in an irreversible process. The startlingconsequence of our definitions of q and w, as exemplified by this problem, is that q is not necessarily zero, even though t h e system and surroundings are thermally isolated from each other. It is often stated that thermal isolation of the svstem is sufficient to enarantee that a process is adiabatic.; This idea is clearly iicorrect. The resilt is a conseauence of the frictional force which the water exerts on the insulated ball as the ball comes to a stop. An important corollary is that if q is nonzero, it does not necessarily follow that there was a direct flow of thermal energy from the svstem to the surroundings or vice versa. - The other interesting consequence of our definitions concerns the hypothetical equalities (27) ~(systern)= -w(surroundings) q(system) = -q(surroundings) (28)
=
-w
Whether or not these equalities are valid has been discussed previously (1, 3, 12). First we clarify what ~(surroundings) and q(surroundings) mean, since we did not define them in the previous section. The most reasonable way to define these quantities is to reverse the identification of the system and the surroundings as we did in the example and to determine ~(surroundings)and q(surroundings) from changes in the properties of the original system. If we examine the results in eqs 20-23, we find that the identities in eqs 27 and 28 are not valid. This voiut has been made before (3.12). The .. explanation for the general nonvalidity of the identities is the presence of frictional forces during the process. If there are no frictional forces acting during a process, then the identities in eqs 27 and 28 are valid. The absence of frictional forces, however, implies that the process is reversible (16). In a reversible vrocess, it is possible to talk about heatflowing from the &stem to the-surroundings or vice versa. Conversely, if the system is thermally isolated from the surroundings during a reversible process, then q must be zero. Both Bent ( 13)and Craig ( 14)have explicitly pointed this out. Their work anticipates the present result in eq 14. For an adiabatic process 9 = 0.In fact, we prefer to restrict the definition somewhat further. We define an adiabatic process as one for which 0,= 0 for each part i in contact with the system during the process. Volume 64
Number 8 August 1987
863
One final remark concerns our straterv of treatine" the same process from different points of view, first choosing the hall to be the svstem and then choosine the calorimeter as the system. This approach gives, in our opinion, useful insights. The development of Bent (13, 14), who treats all subsystems equally, is similar to this; we shall discuss Bent's approach further below.
temperature a t the start and end of the process. A second example occurs when the final state of part i can he reached from the initial state via a reversible adiabatic' path. In this case
A More General View of Work and Heat
However. it is well known that two thermodvnamic states .. usually cannot he connected by a reversihle adiabatic path (21). Consequently, this last result is of limited usefulness. At this point we can go no further. Our analysis has shown that W; and Qi must have the form given in eqs 30 and 31. In addition, we have chosen to define W; and Q; as changes in state functions of part i. These two conditions toeether hold only for a limited number of cases; namely, thosecases considered in eqs 12-15 and 33-36. The problem is that the differential TdS is not the differential of a state function. Put another way, the integral of TdS is path dependent. For example, if part i goes from an initial state (TI, S1) to a final state (T2, S2). one obvious reversible path gives Qi = Ti (Sz S1) and another gives Qj = TdS2 - SI). Similar considerations hold for -PdV, We conclude from this that Wiand Qj (and, consequently, w and q) cannot be defined for every conceiuable path in the surroundings. Nevertheless, it is very satisfying that they can he defined for the three common cases of constant volume, constant pressure, and constant temperature (in the surroundings). Previously, we have defined work in words as the negative of the energy change in the surroundings during the process that is potentially convertible into lifting a mass in the earth's gravitational field. A comparable definition of heat is more difficult to give. For constant-volume and constantpressure processes in the surroundinas, eqs 13 and 15 indicate thatheat is proportional to the tempkrature change in the surroundings. However, eq 34 shows that the heat term can be nonzero euen when the surroundings are held a t constant temperature. (A well-known example of this phenomenon is the ice calorimeter.) The general expression for Q;, eq 31, demonstrates that heat is related to both the temnerature and the entronv . , of the various narts of the aurrcu~dingsi n 3 nuntri\,ii#lu,ay. Resause oi thrse rmsiderarions the unls rrnerdl dtfinition oi heat would be as firllows: heat is the negative of that part of the energy change in the surroundings during the process that cannot be assigned to work. This simple definition recognizes that the macroscopic, mechanical energy changes associated with work are more easily identified than are the thermal energy changes, which are microscopic in nature, associated with heat. While i t is difficult to give a general definition of heat in words, operational definitions of heat are straightforward for the four cases where heat and work are well defined (see eqs 13, 15, 34, and 36). The equation
U"
~d
In eqs 10-15 our definitions of work and heat were restricted to the cases where each part of the surroundings was held at either constant volume or constant pressure. Commonly, thermodynamic experiments are carried out in these ways. From the definitions the full range of thermodynamics follows including the three laws and the functions H, S, A, and G. Thus. our definitions are consistent with the standard development of thermodynamics. An important aspect of our definitions of work and heat is that they involve only changes in certain properties of the surroundinas durina the nrocess. The contribution of each i surroundin~sIO 11. and y does not depend u p m part ( ~the the path which tlnr part takrs in roinr from its initial 10 its finai state. To use the language ofihe&odynamics q a n d w are measured as chances in thermodvnamic state functions of thesurroundings.~lthoughthe macroscopic kinetic and potential energies of a body are not normally viewed as state functions, they clearly since changes in these energies depend only upon the initial and final values and not upon the path. This view of work and heat is implicit in the writings of (13, 14). We emphasize that w and q are not changes in thermodynamic state functions of the system; this point is discussed further below. At this stage we attempt to define q and w for processes other than constant pressure andlor constant volume in the surroundings. We require that each part of the surroundings, as well as the system, be in a well-defined state a t the start and end of the process. We assume that q and w represent changes in state functions of the parts of the surroundings; consequently, we can calculate them along reversible paths in the surroundings. If we combine eq 9 with the differential form of the combined First and Second Laws, d U = TdS - PdV, we obtain for part i
+
dE, = d(KE PE), + TidSj- P;d V,
(29)
We then must partition dE;between the differentials of W; and Q;. Since Dq,,, = TdS, the only possible way to define W; and Q; is W, = A(KE + PE), -,fP,d V, Q;= JT,dS;
(30) (31)
where the integrals are taken along the same reversihle path (in the surroundings). We see that these definitions are consistent with eqs 12-15 for the constant-volume and constant-pressure cases. We also see that the equality (32) AE;= W; + Q, is satisfied. We can compute Wj and Q; from eqs 30 and 31 for two important cases other than constant volume and constant pressure. If the part is held a t constant temperature T during the process, then
-
W, = A(KE + PE), + A& - TAS; (33) Q; TAS; (34) where AS, is the change in the entropy of part i. Note that both Wi and Q; are given by changes in thermodynamic functions. For example, Q; equals A(TS). Both W; and Q; are independent of the path that part i took during the actual process. We emphasize that part i need not he a t constant temperature throughout the process but only have the same 664
Journal of Chemical Education
~~
.
~
~
~
~
may he confusing. In this expression, w, which is known to be path-dependent, is equated to the sum of terms W;, which, we have argued, are "independent of path". (In this latter cast. a(:mt,;k that W, is indbprndent ofthe path whichporr i takes in going - from its initial state to its final ctate.1 in fact. there is no contradiction. The value of w will vary, depending upon the path the system takes to go from its initial to final state. Each different path which the system follows will give different final states of the various parts of the surroundings. Consequently, the W; and Q; values will he different. As stated earlier, we have defined w and q in terms of changes of state functions of the surroundines. Conseauentlv. " .. for our purposes, a process is uniquely specified by the initial and final states of thesystem, and the path is uniquely specified
by the initial and final states of each part of the surroundings. The path will depend upon how the system interacts with the surroundines. Putting the developments of this section into the perspective of the rest of the paper is useful. In doina so we emphasize that the definitiom'of w and q in eqs 10-15 were made before the First Law was introduced. At that stage, q and w could be defined only for constant volume and constant pressure processes in the surroundings. (From the point of view of thesystem the process was not restricted in any way.) Fortunately, these situations are the common ones. With these definitions all of thermodynamics can be developed. After this development constant temperature processes (in the surroundings) may be included in the definitions of w and q. However, this extension requires knowledge of the Second Law and the entropy function S. Consequently, this case could not have been considered earlier. We include it here to permit the widest possible range of processes to be analyzed in terms of work and heat.
required if the volume, pressure, and temperature of o change in a process. Bent's practice of dividing the universe exclusive of the system into clearly identified and distinct mechanical surroundings and thermal surroundings is a very helpful device. Our work has emphasized the commonly used concepts of heat and work in developing the First Law, but this treatment can be recast into a form similar to that of Bent. Using eqs 10,11, and 16, we can write the First Law as
Comparison with Formulatlon of Bent The treatment of work. heat. and the First Law of thermodgndmir given here is similnr in spirit ro the de\.elnpnlenr e i v e ~h\.~ Bent r.1.7.1.1) . . a n~unherof wars am,. - C'omwirinpuur development with his is instructive. Consider a universe made up of N mutually interacting subsystems. The word "subsystem" singles out one part of the universe, without specifying whether it is the system or a part of the surroundings. The most general form of the First Law can he written
which is equivalent to eq 37. T o determine a particular change A E k , our procedure is somewhat more general than Bent's. The subsystem h may interact with as many other subsystems as is desirable experimentally. However, each of the other subsystems is restricted to either constant volume or constant pressure conditions. In addition, irreversible chemical reactions are not permitted except in subsystem k. Then, analogously to eqs 39-41, we can write
-
However, from eq 32 we know that AEi = Qj can be rewritten as
+ W,,so eq 42
+ PE), + JCv,,dTi AEj = A(KE + PE); - P,AVj AEi = A(KE
+ JCpjdTj
where E, = PE,
+ KE, + Ub
(38)
All of the terms have been defined earlier. In this form each subsystem is treated on an equal basis. For every subsystem k the functions Ek, PEk, KEk, and Ua are each thermodynamic state functions. To use eq 37 there must he rules which permit experimentalists tomeasure the A E k values for all but one subsystem ( 1 4 ) . The develo~mentof the First Law bv Bent beains with ea 37. He goes or; to develop all of thermbdynamic& a picture that treats all subsvstems on an equal footina. Further details of this elegant treatment of thermodynamics from a "elobal point of view" can be obtained from his hook and rzated work (13, 14). T o measure AE, for a particular change of state of a reactive suhsvstem denoted "a", he uses a mechanical energy reservoir (weight) denoted "wt" and a thermal enernv reservoir (calorimeter) denoted "8". He assumes that thk weight reservoir and calorimeter have constant volumes and that the weight subsystem is linked to the reactive subsystem by a connector of negligible heat conductivity. Then, the First Law for the process can be written AE,+AEE,+AEo=O
(39)
AEw, = mgAh
(40)
AE, = CvAT
(41)
where the various terms have been defined earlier. Of course, one may immediately identify w = -AEWt and q = -AE,,hut this equivalence is secondary to Bent's development. His treatment deals only with the total energy change A E k for each subsystem; thus, i t avoids possible ambiguities, including algebraic signs, that might arise from the use of the energy transfer (or energy flow) terms q and w. Rules for determing AE for the one chemically interesting subsystem are given in terms of measuring AE's for other suhsystems. In fact, two such additional subsystems are the minimum
(45)
(46)
Here the sum over iQ) includes all subsystems which remain a t constant volume (pressure). This general formulation of the First Law makes absolutely no reference to heat or work. And, unlike Bent's result in eq 39, we cannot single out one subsystem that measures "work" and another that measures "heat". All of the subsystems can contribute in both areas. At this point we compare further the two versions of the First Law given in eqs 16 and 32:
The first equality is always true but is only meaningful when q and w can be measured; this requires that each part of the surroundings undergo either a constant pressure or constant volume or constant temperature change. Similarly, the second equality is true, provided that there is no irreversible chemical reaction going on in subsystem h.2 However, the second equality is only meaningful when Qk and IVk can be measured; this requires that the system undergo either a constant pressure or constant volume or constant temperature process. In cases where a. ,. w .. Qh. .... and W , can all be measured, the two equalities represent complementary ways of viewina the same process. We have seen in the discussion followingeq 28 that h # Qk and w # Wh in general. We shall refer to Wk and Qk as "self-worY and "self-heat" because they are defined solely within the subsystem of interest, without reference to any other part of the universe. In our opinion, much of the confusion concerning work and heat bas arisen because the quantities Wk and Qk have not clearly been distinguished from w and q. (Certainly adding to the confusion is the fact that they are equal for a reversible process; that is, w,, = W k and q,,, = Qk.) An interesting question is whether or not thermodynamics could be formulated solely in terms of the self-work and the self-heat, without any reference to changes occurring in the surroundings. Volume 64
Number 8 August 1967
665
In our opinion, the answer is no, because i t is not possible to define Wh and Qk for a process in a subsystem where an irreversible chemical reaction takes place. Consequently, there would he no way to determine AU for any chemical reaction. These considerations show that work and heat must he defined in terms of energy changes in the surroundings, as we have done in this paper. Any definition which speaks of energy changes in the system will be, at best, misleading and, a t worst, incorrect. It is satisfying that our definitions of work and heat can he reformulated in such a way that the First Law can he written consistently with the workof Bent. We have provided definitions which allow us to interchange various suhsystems, first calling one the system and then another. This "equivalenceof-subsystems" approach was pioneered by Bent and is, in our opinion, a fruitful way to view thermodynamics. This point of view avoids the awkward asymmetry between system and surroundings in conventional thermodynamics, especially when treating a process which involves two or more similar subsystems, where it is not ohvious which subsystem should he designated as the system. Another similarity between Bent's treatment and ours is the use of finite differences to define q and w or AEs and AEWt,rather than the use of differentials. He made this choice to make thermodynamics more readily understandable to students. Our choice arises out of the difficulty of defining work in terms of force times the differential of distance as in eq 2 and out of the recognition that q and w must he expressible as changes in state functions of the surroundings. Two Addltlonal Examples of Work and Heal To show the generality of our definitions we consider two more examples of thermodynamic processes and calculate w and q in each case. The first example involves 1 mol of an ideal gas confined in acylinder by a frictionless piston which exerts a pressure P on the gas. The volume above the piston is evacuated. We assume C, = 4R(Cv = 3R) for the gas and the heat capacities of the piston and cylinder are negligihle. The initial temperature of the gas is T. The cylinder is plunged into a very large calorimeter whose temperature is T/2, and the entire apparatus is allowed to reachequilibrium a t a final temperature of essentially T/2. T o calculate the values of q and w for this irreuersible, constant pressure process we first compute the changes in the thermodynamic functions of each subsystem separately. Clearly, AV = 0 for both the piston and the calorimeter, and AV = -RTI2Pfor the gas. In addition, A(KE PE) must he zero for both the gas and the calorimeter. If we denote the area of the piston by A, then its height change must he Ah = -RT/2PA. Since PA = mg at the start and end of the process, for the piston A(PE) = mgAh = -RT/2. The internal energy changes are zero for the piston and CVATfor the gas. Then, AU(ca1) is obtained from the First Law expression AE(gas) AE(piston) AE(cal) = 0 with AE = A(KE+PE) AU for each of the subsystems. Clearly, AS(gas) = CJn (TdT1) for this constant pressure process. The entropy changes for the piston and the calorimeter are computed following the arguments below eq 26. In particular, AS(ca1) = AU(cal)l(T/2). The results are summarized in Tahle 2. We then, in turn, treat the gas, the piston, the calorimeter, and the pistonplus-calorimeter as the system and calculate q and w from the changes that occur in the other parts (which are the surroundings) using eq 14 and 15. Note that C,,iAT = -2RT, 0, and 2RT for the gas, the piston, and the calorimeter, respectivelv. The values of q and w are also in Tahle 2. ~ ' i & d l t~h ,c sum of AS for t h e three components of the apparatus exceeds 7eru as expected for this irreversilde pro.. cess. It is instructive to compare the w and q values for the gas with those for the combined piston-plus-calorimeter suhsys-
+
+
666
+
Journal of Chemical Education
Table 2.
Heat and Work lor an lrreverslble Constant Pressure Process8
Choice of System
Pmpertf
A(KE+ P O -PAV
AU AS w 0
a
m
Gas 0 %RT -3RTl2 -4Rin 2 XRT -2RT
Piston T
0 0 0 -XRT 0
Calorimeter
piston and Calorimeter
0 0
-XRT
2RT 4R
2RT 4R -%RT 2RT
0 2RT
0
fir* tour mws give the change in state propenies of me various companent me last column eesn me orston-du~calortmetsras a combined system.
SU~SVJ~B~S,
tem. If we call the gas the system, then
where q and w for the system are taken from column two and for the surroundings from column five in Tahle 2. These results stand in contrast to those for the process described earlier, where q(system) # -q(surroundings) and system) # -w(surroundings). The results in eqs 47 and 48 surprised us initially, because the process is not reversible, and there must he frictional forces between the gas and the piston. However. these frictional forces do not result in anv net conversion of mechanical energy into thermal energy. The mechanical energy loss -RT/2 of the piston is converted into a compression energy -PAV = -RT/2 of the gas which can he com~letelvrecovered as mechanical enerev. The irreversibility bf the process is due to the direct flow of thermal enerav gas to the low-tempera-.from the high-temperature . tnre calorimeter. The second examvle is similar to a problem first proposed by Bauman ( I ) . A sntisfnctory soluriun has not heen preicnted beforr.'l'ht*exveritnental arrangement is shown in Figure 2. Two portions bf an ideal gas 2 temperature T inside a closed cylinder of constant volume are separated by a massless, frictionless piston held by a catch. We assume the piston has negligible heat capacity and can conduct thermal energy between the two subsystems. This conductivity guar~
~
+
Figure 2. To the lee of the piston is one male of ideal gas at 2 atm pressureand temperature T; to the right is one mole of the same gas at 1 atm and T. When the catch is released the piston moves to the right until the volumes are equalized. The piston can conduct thermal energy, so the two subsystems have the same final temperature T.
Table 3. Heat and Work for Compression of One Gas by Anothera Choice of Property
High-Pressure Gas
field.) After that the specific example in Figure 1 could be treated. The expression for w is
System Low-Pressure Gas
'The high-pressure gas is initially at 2 atm: me lowpressure gas is initially at 1 elm. The final equilibrium pressme Is 413atm.The format dthistable isdesnlbed inmaedetail in me loomme ol Table 2, The experimwai arrangement is shown in Figure 2.
antees that the two gases have the same final temperature. Because no enerav is exchanaed with the surroundings beyond thegasea, th;. initial andiinal remperaturesofthe-gases are the same (nilniels, 7'1.We assume there is 1 mnl of pas on each side of the one gas has an initial pressure of 2 atm and the other 1 atm. When the piston is released, i t oscillates until equilibrium is reached with both gases a t 413 atm. The prohlem is to calculate how much work the high pressure gas does on the low pressure gas. This problem cannot he done using eq 3 because the gas pressures are not well defined during the process. Similarly, we cannot use q = TAS, because this expression is only valid for a reversible nrocess. However. we can use ea.33,. which is valid for this irreversible constant temperature process. In Table 3 we summarize the chanaes in U. S. and A for each aas. We also treat each gas in turn as thd system and compute w and q from changes in the other gas using eqs 33 and 34. The results are also given in Table 3. For this very simple case where the surroundings consist of only one part we have for each choice of the system w = -AU
,., ,.,,
,,,+ TAS
= -AA
s = -TAs,,, where AA,,,, and AS,,, are changes in thermodynamic functions of t h e surroundings. Regardless of which gas is assumed to be the system, we have -w < -AA as required by the Second Law for an irreversible isothermal process. The work done by the high pressure gas (the system) on the low pressure gas is -w = RT in (413). This process can be analyzed in terms of q and w only by using the Second Law. Note that q Z 0 even though the temperatures of both system and surroundings do not change. Also, q(system) f -q(surroundings) and system) # -w(surroundings) for this mechanically irreversible process. Presumably this problem has not been solved before because q and w , which are normally used to compute AU, can only be calculated here after AU (and AS) have been obtained. The simplicity of our analysis illustrates the power of our definitions. ..
'The definirion of work given in eqs 10-15differs from that comtnonls ciren in teutbuuks (see eq 3). The auestion arises as to thr I,& method to present ou; definiri'n t u students. For the first discussion of work the svstem of urascontained beneath a piston shown in Figure 1 is commonly used. This example is easily understood and permits discussion of both reversible and irreversible processes. We suggest that it be explicitly stated that the kinetic energy of the piston is zero at the start and the end of the process. Then, thermodynamic work should be defined as the negatiue of the energy chanee chanee - in the mechanical surroundines. - .(An enerev u . in the mechanical surroundings is an energy change which is equivalent to raising a weight in the earth's gravitational
-
where Pmeeh = mglA is the pressure exerted on the piston by the surroundings, and m is the mass of the piston itself plus any added weights. The mass m need not he constant; it can be a function of the height of the piston (as it must be for a can be a function of reversible process). Equivalently, Pmem the volume V of the system. In either case, eq 49 gives the proper result for w. In general, Pmech # P , but the two pressures are equal whenever the piston is at rest or the nrocess is reversible. Note that Pmsrh ...--. . is also not the nressure kxerted by the piston on the gas unless the piston is at rest. The two pressures differ due to the frictional force between the piston and the cylinder when the piston is moving. That P m e r h is alwavs well defined and easilv measured should he emphasized. Once Pmeeh and eq 49 are presented, further discussion could center on Pme*.There would he no need to refer again to the mass m and area A of the piston. For example, a problem could illustrate the calculation of w when the sys= 1 atm tem expands against a constant pressure Pmech (which commonly is due to the atmosphere acting on the surface of the system). This presentation allows irreversible orocesses to be treated rieht from the start. From this point on the customary development of thermodvnamics is satisfactorv and can be left unchaneed. I t would hk worth emphasizing when discussing infinzesimal processes and using the expression
that the piston is understood to be at rest at the start and end of the infinitesimal process. Otherwise, the analysis is very complicated, because the differential kinetic energy change would have to be included in Dw (7). A parallel discussion of Bent's point of view (13,14) of thermodynamics early on also helps clarify concepts by giving a second perspective. The standard treatment in thermodynamics courses brines in tvoes -. of work other than nressure-volume work when the iree-enrrgy functiuns ure intruduccd. At this poinr the mure erncral definition c8fwork eiven in eas 10-13 could be presented. In our opinion, the students would have no difficulty comprehending this new material. We believe it would also be useful a t this point to emphasize the idea that our definition of w is an operational definition. That is, work in a process is defined by specifying how to measure it. For example, the work computed from eq 49 must equal -mgAh if m k constant. sincethe mass and-the change in height of the piston are easily measured, this interpretation of eq 49 reinforces the importance of operational definitions. We believe that a presentation of thermodvnamics which emnhasizes onerational definitions has advantages. For one thing such; treatment would relate the theory of thermodynamics as closely as possible to experiments. Consistent use of operational definitions can give additional insights. For example, consider the First Law in the form AU=q+w
(51)
+
where we have assumed that A(KE PE) = 0.The principal experimental content of the First Law5 is that the sum of the
There is an additional experimental component of the First Law that is not apparent in eq 51. This additional component is that the observed sum, q + w, is proportional to the amount of the material transformed in the process. Consequently, U is an extensive function. Volume 64
Number 8 August 1987
667
measurable quantities q and w is independent of path. Thus, the difference in the state function U is definable. "Conservationof energy" cannot be confirmed by experiment because AU cannot be measured independently. Rather, eq 51 is an operational definition of AU. The phrase "conservation of energy" is simply an interpretation of the significance of the First Law.
6. Canagaiatna, S. G. Am. J.Phy8. 1978.46.1241. I. J. Chrnn. E d u r 1966.43, 233. 7:Kivelron. D.:Oppe~~heim, R, ~ ~ t t R.: i Wmd. ~ , ~ S. E. T h ~ i r n u d v r r n n i iAn ~ ~ .I n h d u r l i u n : Academic: New York. 1968:~ h & 6. 9. L w i m I. N.Physicol Chen&lry. 2 n d d ; M c G r a w Hill: New Ynrk, 198S:p40. P; 10. Hdlim. G.: Holler. D. H. D. Foundoririnr 01 M , ~ P T , IPhysiiel S C ~ ~ UAddisonWesley: Reading. MA. 1958:p219. 11. Goldstein. H. Ciarrical Mmhonicr. 2nd ed.: Addison-Wesley: Resding, MA. 1981: p 3. 12. See. for example. Kirkwood. J. G . ; Oppenhdm. I. Chrniicai l'hrrmod~tianzirs: MrCraw-Hill: NevYurk. 1961:Chaotsr 3.
~~.
Acknowledgment
The authors would like to acknowledge the thoughtful cqmments of a referee.
16. Our definitions 0f"quari-atatic"and "reversih1c"processesare taken from Zemansky, i ~ ~ed.: , McGraw-Hill: New Yurk. 1966; pp SY. M. W. Hear a n d l ' h w m o d ~ ~ n a r n 5th ~~-
Literature Cited I. Rauman, H. 1'. J. Cheni. Edur. 1964,41,102,675.676, 2. Cherick. J. P. J . Chem. Educ. 1964.41.674. 1. K I I ~ PR..l ~ , J. Chenz. E d u c 1964,dJ. 675. 4. Mvsels,K. ,I. J. Ch
