Spontaneity, Accessibility, Irreversibility, "Useful Work" The Availability Function, the Helmholtz Function, and the Gibbs Function R. J. Tykodi
University of Massachusetts Dartmouth, North Dartmouth, MA02747 Engineers often carry out processes i n which a closed svstem interacts with its environs (the atmos~here,etc.) so that the surroundings of the system may bethought of a s a medium of essentially fmed temperature T, and fixed pressure P,. I n dealing with such processes, they find i t convenient to define a n availability function
function, AA, and changes in the Gibbs function, AG, to the concepts spontaneity, irreversibility, and useful work. l b reach the key concerns of the paper, i t will be necessary to cover a lot of intermediate ground and to be very explicit about sundry thermodynamic operations. I t will be best, therefore, to start with a rather extensive glossary.
u(mteml - T ( 8 n m u n d i n p ~ 1 s p t e m 1+ P i s u m ~ d i n g r l ~ s ~ ~ t e m l that d e ~ e n dboth s on the n r o ~ e r t i e ofthe s svstem (U.S. V) and on'the properties of held in place by srop,,P, > P, Frrr th+ pla..n frum rhr imps, nnd the p~*tcmwll tvtnturll~, 0
(6)
The Heat Gained by the Surroundings, Qr Let Q, be the heat gained by the surroundings. For this paper, i t is neater to catalog effects in the surroundings and to infer from them the effects in the system (8). The First Law Let Au(snmoundinw~ =
Qr
(7)
- Wr
and (8)
AUicompwiw + AUisurnundingsl= 0 Then = -AUisurronndingg~ AU 0 or d(m) . . > 0 and is a natural Drocess. Suppose we doncentrate attention just on the composite system and look for processes that can "go all by themselves" in the system. These processes are called s-spontaneous. A process that is not s-spontaneous requires a positive work contribution from the surroundinas. But the surroundings can also make positive workcontributions in ways that do not store work equivalents in the system. A boulder rolling downhill iB an s-spontaneous process. Suppose, now, I run alongside the boulder and push i t faster. I thus hasten the s-spontaneous process. Suppose that I drag a heavy sled along as I hasten the rolling of the boulder. So, part of W, can be used to hasten an s-spontaneous process, and part of i t can simply be dissipated (as the work against friction of dragging the sled). Therefore, for the processes investigated in this paperall of which are subject to restrictions 2 to 4--I define an s-spontaneous process as one for which < TSmY
and (my > 0
(11)
The case of a battery discharging across a load (&G)T,P, < 0) is a n example of an s-spontaneous process. The case of
would be -P+iV~,m,,,i,l. Similarly, W: i s t h e net useful work done by the surroundings on the composite system, and W:' i s the net work done in unlocking and locking the constraints. Also, Q, is the net heat gained by the surroundings.
a battery being recharged by a source of higher voltage (4G)T3, > 0) is an example of a process that is not s-spontaneous; but the process is, of course, s&r-spontaneous.
Restrictions
Accessibility
The processes considered in this paper will each be subject to an appropriate subset of the following general set of restrictions.
The restrictions associated with a process initiated from a given state limit the spectrum of states accessible to the system from the given initial state. For the system-surroundings complex, that is, for a n isolated system, the states accessible to the complex from a given initial state are those of equal or greater entropy.
1. The volume of the composite system remains constant. 2. Closed system 3. Diathermal walls 4. The composite system is in contact with a thermostat of temperature T,. 5. Cyclical operation of constraints If a process starts with the unlocking of an effectiveconstraint, then when the process terminates the constraint is to he restored to its original locked position. 6. The composite system is in contact witha harostat ofpressure P,. 7.
w;=o
The Process-by-Itself Generated Entropy When a constraint is operated cyclically, the net work of unlocking and locking the constraint, W,: gets dissipated a n d eventually shows u p a s a contribution to Q, a n d thereby a s a contribution to the overall generated entropy (m). Let
Then (m)'is the entropy generated by the actual process, exclusive of the entropy generated by the unlocking and locking operations on a constraint; I shall call (mythe (process) generated entropy. Spontaneity Aspontaneous process can "go all by itself". The composite system together with all those other systems that in any way interact with i t are called the system-surroundings complex. Then, with respect to the system-surroundings complex, a process that goes all by itself is called an
(m) = ~1S,,m,,,iw1
+
~k"rn""di",,
20
(12)
If (yrl) = 0, the process was reversible; whereas if (yll)> 0 the process was irreversible. For special subsets of the general set of restrictions, we shall find that there are appropriate state functions that act a s indicators of the states accessible to the composite system upon the unlocking of a constraint. Names for -Wr In dealing ..with work interactions between the com~osite syst(m and the surroundings, we ;ilw;lys tin this piiptJr, deal a i t h , and catoloa, the effects ofthe the forces that thc surroundings exert on the composite system. By means of such work interactions, energy passes either to or from the composite system. When energy, in the form of a work interaction, passes from the composite system to the surroundings, we often write -W, > 0 rather than W, < 0. Now whereas W, is the work done by the surroundings on the composite system, there is no fully satisfactory name for -W,. Lacking anything better, I shall refer to -W, as the work extracted from the svstem or as the work delivered to the surroundings. (In a manner of speaking, -W, i s the work done bv the svstem against the forces exerted bv the surroundings.) Marching On Now that matters of terminology and notation have been settled, let us explore the relationships between the work exchanged between a closed composite system and its surroundings for changes of state occumng via suitably re-
Volume 72 Number 2 February 1995
105
stricted processes and the corresponding changes in the Helmholtz function (AA),the availability function (AD), and the Gihhs function (AG) for the composite system. The Helmholtz Function
The work delivered hy the composite system to the surroundings, -W,, is thus less than or equal to
IT,
Consider a closed composite system with diathermal walls that interacts with a medium of fixed temperature T,and fixed pressure P,and that undergoes a change in state via a prore& suhject to the restrictions 2 4 in ~he-~lossary If the proctss W:Minitiated bv unlorkingnn d%cti\v: nmstr:lint, the initial state is the preprocess state with the constraint in the locked position; and the final, postprocess state is a state accessible from the initial state via a process suhject to restrictions 2-6, with the constraint again in the locked position. First Law Write the first law for the composite system as ~~lC,,,,,itel
= 4+ w r
(12)
If the container is of the piston-and-cylinder type and the piston is allowed to move during the process by freeing i t from one set of stops and letting i t move to a second set of stops (cyclical unlocking and locking of a constraint), then the piston is stationary in both the initial and final states. The kinetic energy temporarily accumulated by the piston ultimately shows u p a s heat gained by the surroundings. The same is true of the work of unlocking and locking the first set of stops: W:' ultimately shows up as heat gained by the surroundings. In the expression 4, + W, the work of imparting kinetic energy to the piston is neutralized by an equivalent gain in Q,.Likewise, the work of unlocking and locking the stops is neutralized by an equivalent gain in Q,.So the combination -Q,+ W, does correctly represent AUl,,,p,.it,l, a s i t must (eq 9). Second Law Write the second law a s 0 < (Vl) = a ~ s u m o u n d i n g s ~Gicompositer +
It follows from eq 13 that Qr + T@~cmposiw = T r ( ~ l2) 0
(14)
In the initial and final states, the temperature of the composite system matches that of the surroundings (restriction 3): T,= T,.
that Thus, -AAl,,m~tel is the maximum work, -W,, can he delivered by the composite system to the surroundings for the given change of state if that change in state is carried out via processes subject to restrictions 2-6. Requirements I n eqs 12-19 the only requirements on the temperature are that the external temperature T,remain fured and that in the initial and final states T,= T,.Nothing is said or implied about the temperature inside the composite system during the transition from the initial (preprocess) state to the final (postprocess) state. I t is quite legitimate for the composite system to develop inhomogeneities in temperature during its transit from initial to final state, provided that those inhomogeneities vanish in the final equilibrium state. It will be convenient to abbreviate the expression uniform and matching with respect to ... in both the initial and final states by U&Mw.r.t ..... I&F, The reauirementsforthevaliditv ofrq 19; therefore, are that thc ch:ingt, in state becanicd &t ria 3 pn)cess suhiert to resmctioni 2 4 and that the stivulation -form and matching with respect to T i n both tge initial and final states (U&M w.r.t.T, I&F) hold. Special-Case Application Aspecial-case application of eq 19 is a process that takes place under constant-volume conditions. Let
,,,,, ite, = constant V= V,, for a process subject now to restrictions 1-6. When restriction 1 i s i n force, restriction 6 becomes ineffectivechanges i n P, having no influence on what takes place inside the container. I t is therefore immaterial whether we consider our restrictions to he 1-6 or 1-5; let us, however, c a n y along all six restrictions so a s to treat the present case a s a suhcase of the previous case. For processes subject to restrictions 1-6, W, reduces to W: + W:', and eq 17 becomes ~1,mpSite1
I , v - (W:+ w;? = ( Q r+ T@compositej)
= -TSYI) 5 0
(20) where we have drawn on eq 14. Rearrange eq 20 to give
Change in the Helmholtz Function Consider now the change in the Helmholtz function ~li,,,,,O,it,,
- T~SIC~,,,~,,,,
for the composite system brought about by the process in question.
Replacing AU,,,,p,,i,, (eq 121, we get
by its equivalent from the first law
where we have drawn on eq 14. I t follows that
A.4 I,~,,
106
eaitelI~-wr~o
Journal of Chemical Education
(18)
using eq 10. Thus, a n d -AAlmmpositel 1 T , , ~i s t h e m a x i m u m useful work, -Klm,,l, t h a t can be extracted from a composite system that undergoes the given change of state via processes subject to restrictions 1-6 (or 1-5). Examples of the Relations To illustrate the relations we have been discussing, let us consider a change in state for a n ideal gas via processes subject to restrictions 1-6 (see Fig. 1). Let given amounts of the same ideal gas fill two bulbs, bulb 1 and bulb 2, of equal size. The bulbs are connected by a capillary tube containing a stopcock. The bulbs also communicate with the opposite sides of a gas turbine (engine) e; the shaft of the turbine can he braked and locked in place.
By construction V1 = V2, SO i t follows that
If we write A, (...) for a molar value of the Helmholtz function for the ideal gas, then for the change in state (eq 24) we get the following equations because the solid parts of the apparatus are rigid (15).
Figure 1. Glass bulbs 1 and 2 contain the same ideal gas at pressures p, and p, with p, > p2. The pressures can be equalized by opening a stopcock in the connecting capillary shunt or by sending the gas through the turbine (engine) e. The entire apparatus is immersed in a medium of temperature T, and pressure p,..
x,)
+x2(lnrz)+In 2 (28)
where The pressures exerted by the gas in the bulbs a r e p l and p2, with p l > p2. The entire apparatus is immersed i n a medium of temperature T, and pressure P,, which acts as a constant-temperature heat bath. The volume Vi represents the inner volume of glass bulb i plus the volume inside the capillary tube up to the stopcock plus one-half of the free space inside the turbine; by construction VI = Vz. The amounts of gas a t the two pressures are ~ l V l
n1=-
RTr
and
By manipulating the stopcock and the brake on the turbine shaft (by unlocking and locking the two constraints), we can bring about the change i n state (24) (Tr,~~= . ~(Tr.p~1 2) via a process subject to t h e restrictions 1-6. In other words, by transferring gas from bulb 1 to bulb 2 via the capillary shunt or the turbine, we can equalize the pressures i n the two bulbs, the final pressure being p. In the initial state, both constraints are effective, whereas in the final state both are ineffective. The two constraints are to be returned to their locked positions in the final state (restriction 5 ) . Varying the State and Process Constant T,a n d Constant V Let all solid parts of the apparatus be rigid; the process then takes place a t constant T, and constant V where
v = VI,,~,,! Under the given circumstances, we have (eqs 21 and 22)
I T,,V - W:= -Tr(im)'
M1,,,,,i,)
0
-
~
~
~
I ~,.v=-W:~maxi ~ m ~ ~ ~
i
t
e
The evaluation of AA,,, in eq 28 is equivalent to taking nl moles of gas i n an unconstrained equilibrium state a t T,,pl to another unconstrained equilibrium state a t T,p via a sequence of unconstrained equilibrium states, with a similar treatment of n2 moles of gas going from T,pz to T,p, via a sequence of unconstrained equilibrium states (quasi-static reversible process). Constrained Equilibrium States I t i s also possible to create a sequence of constrained equilibrium states (quasi-static irreversible process) and to integrate dA(,.,, along that sequence. For the apparatus described in Figure 1,keep the brake on the turbine shaft in its locked position. Start in the initial constrained equilibrium state (Tr,pl,pz). Open the stopcock for a short interval of time and then close it again, allowing the system to come to a new state of constrained equilibrium (with restrictions 1-6 in force). Keep repeating the operation until the stopcock constraint becomes ineffective and the system reaches the final equilibrium state (T,,F,F). Return the stopcock constraint to its locked position in that final equilibrium state. The procedure generates a sequence of constrained equilibrium states for the composite system. With respect to two neighboring states a differential distance apart in the sequence ~ ~ m m p o s i t e ldA~ga~! =
(29) because the solid parts of the apparatus are rigid. Treat the two bulbs a s open systems, and write dAi for dA (gas occupying the space VJ. Then
(251
dAi = -SidTr - pidVi + pidni = pidni
~(261
because T, is constant and the sizes of bulbs 1 and 2 remain fixed. So
and
-W:~
and
where Wr is the useful work done by the surroundings on the shaft of the turbine.
dAya,, = M I + dAz = w
h + wdnz
(30)
(31)
and Volume 72 Number 2 Februaly 1995
107
4 %as,
=
4
l PI^ l w z h z +
I321
"2
"1
where n l and n2 represent the initial amounts of the gas, and n; and n i represent the final amounts. From the nature of the process, it follows that Use integration by parts for each of the integrals in eq 32.
!+
PI
(34)
with the same conventions for & and pj as for n: and n,. I n the final state
-
pXT,,P) = PI(T,,F) u'(T,P)
(35)
n.dp. =n.V . d .- Vdp. L I L m(oP,r I
(36)
Also,
where V,,,, is the molar volume of the gas i n hulh i, and Vj remains constant. We may now write eq 34 a s -
-
P
P
MI,.,
= (n; + nil@'- n1F1 - nzp, -
Vldpl- Iv~dpz
PI
PZ
where we have made use of eqs 27 and 33. Equation 37 is the same a s eq 28 and may he put in any of its equivalent forms. If all the gas passes from bulb 1 to hulh 2 via the capillary shunt (Wr= 01, the process is as irreversible as it can get under the given circumstances. Then the (process) generated entropy is
(m)' = -~14c0mp,.i~1 I Tr,v + (x21nx2)+ In 2
0 (38) 1 As we send more and more of the gas thraugh the turbine = ~ l ~ l (+p2) p l (xlln rl)
(
>
rather than through the capillary shunt, we use more and more of -AA for useful work, and the (process) generated entropy gets less and less.
I T ~ V -(-K)2 0 Trl~)'~-M~,mpmitel
139)
-K -Kim,, = - ~ i c O m p o 3 i I~T,~.V
140)
If we send all t h e gas through t h e turbine, -W: approaches -W&, a s closely a s possible. We can make -W: approach -WrImi,,, even more closely by improving on the performance of the engine e, in the following ways. Increase the number of vanes. Let the vanes contact the encompassing wall in a gas-tight, frictionless fashion. Decrease the friction in the bearings supporting the turbine shaft. 108
Journal of Chemical Education
For the situation described in Figure 1, we pass between the initial and final states (eq 24) via a series of processes (each suhiect to restrictions 1-61 in which we varv the Droportions of the gas passing through the capillary shunt a n d through the turbine. The (~rocess)generated entrow varies from process to and decreases as we &tract more and more useful work from the process. If we now consider general processes in closed composite systems (diathermal walls) a t constant T,and constant V, with W: = 0 (i.e., processes subject to restrictions 1-7), we find that
I have tried to think of a reversihle process for such a situation, a process for which MI,,, it,, I T,.V= 0 with restrictions 1-7 in force. The only nearly reversible process (i.e., a process for which (yrl)' = 0) that I could think of is the one described below. Perhaps the reader can think of a better example. A Nearly Reversible Process (W = 0) Place i n a thermostat maintained a t the triple-point temperature for water (273.160 K) a triple-point cell containing pure water and having rigid walls. Attached to the wall inside the cell, high u p in the vapor region, is a small, M aqueous sugar soclosed "huhhle" containing a 1 x lution. The "huhhle" is equipped with a capillary delivery tip closed by a needle valve that can he operated from outside the system. Let the triple-point cell contain 500 mL liquid water, 200 mL ice, and 100 mL water vapor. Now manipulate the needle valve on the "huhble" so that a drop (0.05 mL) of sugar solution falls into the 500 mL of pure water (i.e., unlock and lock the needle-valve constraint). The presence of the 5 x lo-" mol of sugar in the liquid phase will slightly unbalance the phase equilibria inside the cell. Ultimately, all the ice will melt, and there will be a readjustment of the amounts of water in the liquid and vapor phases. I n other words, the process 500 mL(l),200 mLls), 100 mL(g),0.05 mLll x 10"~),611Pa, 273.160 K + 683.9 mL(7 x lo-" MI, 116.1 mLlg), =611 Pa, 273.160 K (42) will take place in a nearly reversible fashion ((@' striction 16 in force), and we shall have
= 0,
re-
Spontaneity, Accessibility, Irreversibility Consider again the situation i n Figure 1.Any process for
which the stopcock in the capillary shunt is open is clearly a n s-spontaneous process and satisfies the conditions for s-spontaneity leq 11).
w: < T m ) ' and In the present case, we see that a correlate of s-spontaneity is that (eqs 25 and 44) AAlmmwite) I ~,,v=-T,I~rll' + W: = -(TATTI)'- W;) < 0 145j So in a n s-spontaneous process with restrictions 1-6 in force, the Helmholtz energy for the composite system must decrease. Another way of describing the situation is to say that, for a composite system undergoing processes that are subject to restrictions 1-7, the final (postprocess) states accessible
to the composite system from an initial (preprocess) state are states of equal or lesser Helmholtz energy (eq 41). At this point i t may be useful to distinguish between sspontaneity and irreversibility. All s&r-spontaneous processes are, by definition, irreversible processes ((m) > 0) and vice versa. (All natural processes are s&r-spontaneOUS.)All s-spontaneous processes are irreversible > 0, eq 111, but not all irreversible processes are s-spontaneous. In the situation depicted by Figure 1, let the composite system be in the state for which the stopcock constraint is ineffective. In other words, let the stopcock be open, and let the pressures in the two bulbs be the same: 5 and 5. Now unlock the brake constraint on the turbine shaft, and do work in the amount of W: on the turbine shaft so that gas is pumped from bulb 2 to bulb 1 (with restrictions 1 4 in effect a t all times). Then lock the brake constraint once again. The gas pumped into bulb 1will return to bulb 2 via the open stopcock, and the final equilibrium state will be the same as the initial state (T,,F,F). The composite system i n this case merely serves as an energy conduit, allowing the work W: to be dissipated and returned to the surroundings in the form of heat Q,. For the overall process,
where we have drawn on eqs 9 and 10. I t follows that
The useful work delivered by the composite system to t h e surroundings, -Wr, i s t h u s l e s s t h a n or equal to -%Omws,~~ I T.P.. Thus, -%omws,te~ T.P. is the maximum useful work -W& that can be extracted from the composite system for the given change of state (1,2 ) via processes subject to restrictions 2-6. Suppose we impose restriction 1 on our composite system. In other words, the composite system i s now to be of fixed volume while undergoing processes subject to restrictions 1-6. The composite system satisfies the uniform and matching with respect to temperature condition in both the initial and final states U&M w.r.t.T, I&F for the processes under consideration. For any one such process, we have (eq 51)
I
and
w;= T,(yq)'
>0
(46)
Although the process was irreversible ((my > 0) and s&rspontaneous, i t was not s-spontaneous: I t failed to satisfy the condition < T,(m)'. In the case shown in Figure 1, when the set of restrictions includes restriction 7, W: = 0,then s-spontaneity and irreversibility amount to the same thing (restrictions 1-7 are now to be in effect).
Under the given circumstances, if the process is irreversible, i t is also s-spontaneous: The first part of the sspontaneous requirement, < T,(m)', is automatically satisfied due to restriction 7, Wr = 0.Thus, irreversibility and s-spontaneity mutually imply one another in this case. The states accessible to the composite system from a given initial state via processes subject to restrictions 1-7 are states of equal (reversible process) or lesser (irreversible process) Helmholtz energy. The Availability Function Finally, let us look a t the availability function. Consider a composite system immersed in a medium of fixed temperature T,and fixed pressure P, undergoing processes subject to restrictions 2-6. For such a process, the work done by the surroundings on the composite system is (eq 5) Wr = -P,AVlmmiel
+ W: + W:'
u ~ c o m w s i t e-~TX+composiw P r v ~ m m p o s i w
the following. A&,,,,ite~
I T,?,
AVicomporiter = 0
I t is thus required that under U&M w.r.t.T, I&F conditions and with
for any change of state brought about by a process subject to restrictions 1-6. We have thus produced eq 22 as a special case of eq 51. Suppose that we are now dealing with a composite system that is uniform with respect to both temperature and pressure in both the initial and final states for processes subject to restrictions 2-6, and the initial and final temperatures and pressures match those of the surroundings (T,= T,, P. = P,). Refer to this set of requirements as the uniform and matching with respect to both T&P conditions (U&M w.r.t.T&P, I&F). For the given special circumstances ~ l c o m , , ~ tI T,P,. e~ reduces to AG~c,mpo,it,l I T.P.. SOin this w e cia1 case,
(48)
and we have, for the change in the availability function
D
because
= AUicomponte~ T@,composito~ +PldV
