Thermodynamic Inefficiency of Conversion of Solar Energy to Work Arthur W. Adamson, James Namnath, V. J. Shastry, and Vida Slawson University of Southern California. Los Angeles, CA 90089 I t might be thought that photon energy should be pure energy, that is, should have no entropic content, and hence that solar energy should in principle be entirely convertible into useful work. A more conservative guess might be that since solar energy relates to radiation from a 6000 K source, a Carnot efficiency factor of about ((6000 - 298)/6000) = 0.95 might apply. Neither of the preceding statements is correct for a quantum or threshold solar device, hut i t is true that there is a thermodynamic limitation to the efficiency with which light energy can he converted into work. The limitation depends on the intensity of the radiation and its wavelength and can be substantial. Chrmists practicing in the solar enerm firld seem generally not to nnv m w h attention to this limitation. Its treatmenr is hut analyses seem not to have appeared in not n e b easily accessible chemical literature. A useful review, however, is by Bolton and co-workers (5). In addition, the various treatments have been couched in terms of black-body radiation fluxes or semi-conductor solar cell analyses, and the language has been more that of physics than of chemistry. In this paper, we present a more chemical and less sophisticated approach.
energy. The approximation of equating energy and free-energy differenres is discussed later in this paper. As we discovrred some years ago (8), [R~(hpy)3~+]* is a good reducing agent, good enough to reduce water. The product, Ru(hpy)z3+, is a good oxidizing agent, good enough t o oxidize water. The simple scheme,
+ H30+ [R~(bpy)3~+]* Ru(bpy)a3++ 3/2 Hz0
+
-
Ru(bpy)j3++ %H2 + Hz0 -AGozsa = conga '10.84V (3)
Ru(bpyh2++ % 0 2 + H30' -AG0298= c029g = 0.03V (4)
gives, in combination with eqn. (2), the net reaction
i14),
Einstein Did Us In
The thermodynamic limitation is usually mingled with another kind of efficiency limitation, and i t seems well to separate the two aspects. This other limitation is a consequence of the quantum aspect of nature, as embodied in Einstein's law of photochemical equivalence (6, 7). Light energy comes in quantum packets of value hu, where h is Planck's constant and v is the freauencv. Photochemistrv involves putting a molecule in an excited state of some defmiG enerev E* above the eround state. (In the case of a semiconduc& solar cell, there is a minimum energy E* of promotion of an electron into the conduction band.) Einstein's law affirms that unless the energy hv is at least equal to E* there can be no absorption of light energy. Pursuing this last aspect, suppose that we have the photochemical system
where the electronic excited state A* is to be the source of chemical or electrorhemiral energy. Nowadays, for rxample, manv inorganic photochemists are exploring the case of
2
A
G '1E* ~ = 2~ . 1 ~~ (2)~
where bpy denotes 2.2'-bipyridine, and Go is standard free
(the potentials are from ref. (9)).In effect, Ru(bpy)?+ acts as a catalvst wherehv. lieht .. enerev -.is used t o split water. The producedhydro~enrould be used as a fuel t i r u n an engine. Thrreare mans difiiculties with thesimplescheme ahove, but a more compliiated version has shown some success (10). The svstem is cited here merely to illustrate how the energy conk e d in an excited state may he used to produce useful work. We do not concern ourselves here with the many complexities and inefficiencies of actual conversion schemes. Rather, we are after some basic limitations upon which no scheme can improve. Returning t o the type reaction of eqn. (I),if the transition were between two specific states, we would see a single sharp line absorption spectrum of A. In this case, only light of frequency corresponding to hu = E* would be absorbed, neglecting line broadening inherent in the uncertainty principle. In nractice. however. we are dealine with at least diatomic and usually with polyatom~cmolecules and the electrmic excited state A* will have adifferent hond leneth in rhr d~otomiccase, and different bond lengths and bond angles in thepolyatomic eeometricallv distorted relative to the one. Thus. A* will he " ground state A. The situation is ~llustratedin Figure 1 The Franck-Condon vrincide tells us that noa~~preciahle nuclear motion occurs during'the absorption of a Eght quantum, that is, that the transition is a "vertical" one. The consequence, shown in Figure 1,is that light absorption produces a distribution of vibrationally excited A* molecules. The experimental observation is that the absorption bands are broad, as illustrated for the case of R ~ ( b p y ) ~ in 2 +Figure 2. Our energy E* is now actually that of the transition from the u = 0 level of A to the u = 0 level of A*, or of the so-called zero-zero transition (where u denotesvibrational quantum number).
Volume 61 Number 3 March 1984
221
This transition is not totally forbidden by the Franck-Condon principle and can, in fact, sometimes be seen as a weak feature on the long wavelength tail of an absorption band. In the case of Ru(hpy)sz+,E* is about 50 kcal mole-', corresponding to a A* of 570 nm. Einstein's law tells us that light ofwilvelenyh longer than A* will not be absorbed ineelectine the small tail of the ahsorption band beyond A*). ifow, however, light of wavelength shorter than AX will be absorbed because of the broad nature of the absorption hand (for example, 450 nm radiation in the case of Fig. 2). The produced A* is vibrationally excited, but this vibrational energy is ordinarily very quickly lost to the solvent medium (within picoseconds). We thus end up with non-vibrationally excited, that is, A* species. Moreover, this is apt to be the situation even if we are populating some higher excited state by using light of yet shorter wavelength (such as light of below 300 nm in the case of Fig. 2). Usually (although not always), crossing from one excited state system to another occurs so that we still end up with A*. A further, and important point is that we actually are dealing with a collection of A* molecules in thermal equilib-
-DISTORTION-
Figure 1. Energy level diagram showing the eflect of excited state distortion. Straight arrows represent a radiative process and wavy arrows non-radiative processes.
rium with their surronndines. This collection is. in effect. a thermodynamic ensemble having entropy and hence free enerw as well as enerw. This excited state ensemble has been callrza thermnlly e q ~ k h r a t e dexcited state; we have u s ~ dthe SII~&I twm ihcxi srof? 111) A thexi state is a eocd chemical species; i t is, in effect, an isomer of the ground state. Thexi states. as noted above. have well-defined enerm, entrow, and free energy. They have standard oxidatioiand red&tion notentials; thev have a definite structure, reaction chemistrv, absorption spe>trum, etc., all of which may be different fro& the corres~ondingmound state properties. One special feature of a thexi;tntr is ihat it may he alhe to return 10 the gnrund for state A hy emission of light. This is true for [Hu(hpy)~~+]', example, and the emission spectrum is includcd in Figure " z . We are now a t the ~ o i nwhere t the limitation imoosed bv Emstein's law can he examined. The problem is that shar light covers a wide enerev beine essentiallv black-hodv ... soectrum. . radiation. A standard intensitysprcthn for light incident on the earth'ssurfacr isshown in Firure 3. The situation is that all light of wavelength greater th& A* is wasted since it cannot he absorbed at all. Because of the typical broadness of absorption bands, we can assume that all light of smaller wavelength can be completely absorbed. However, the energy content in excess of E* per mole of light quanta is also wasted. This excess energy goes into vibrational excitation of A*, an excitation which is promptly lost as heat to the surroundings. We will limit ourselves to the case of an isothermal system, so that this heat is unavailable to do useful work. The above qualitative exposition can be made quantitative in the form of the integral,
where I(A) is the intensity of s o h radiation in incident quanta per warelenrth interval and E(AI is riven bv hrlA. TheroefAcient f,y is ;he efficiency factor resilting &om this working of Einstein's law. This integral has been evaluated as a function of E* (see Ref. (5),for example). It maximizes a t f,y = 0.44 for A* about 1100 nm. (There is a second-order dependence on whether we are talking about the solar spectrum reaching the earth's surface, or that just outside the earth's atmosphere.) It is prohably no coincidence that the chlorophyll of plants absorbs in the red, so that f~ is relatively large. An interesting point is that while f~ is thus made large, light of such long wavelength does not carry enough energy per quantum to do the needed photochemistry. Plants solve the problem by means of a complex chain of molecular events whereby two or more photons are used to provide the energy needed for the
I
I .m.
rn
1 m
*"V~L%wo...M
Figure 2. Room temperature absorption (-.-)and emission I-) aqueous R~(bpy)~Cl~. 222
Journal of Chemical Education
spectra of
Figure 3. The specha1distribution of solar energy withthesun at 60' fromtha la.(The dips in tksp-aare due toabswption by variws atmospheric and solar atmospheric species.) : zena (
reduction of COz. Man-contrived chemical solar energy conversion systems, however, are more likely to take the mechanistically simpler route of doing the job with just one photon per primary reaction event. There is a trade-off, however, in that the required photon energy is increased, with a corresponding reduction in f ~ T.o take the Ru(hpy)a2+ system as an example, A*, is now 570 nm, and f~ has dropped to 0.20.' The Thermodynamic Limitation We come now to the additional limitation that is the focus of this paper. Any means of reversible generation of work would do, hut for convenience of exposition, let us assume the following. We have the cell 1 Solution of AX ] concentrations are I Pt/C(s) (Af.,) and (At*,) j !
I
Solution of AX under
irradiation. Concentrations (AC'). I We take species A as actually to be a cation, A+, so we can he dealing with an electrolyte; the anion, X-, is purely a spectator species. C i s a one-electron-reduced form of A+. We suppose that the couple C(s)/A(solution) is reversible at a Pt electrode, and that the electrode M is such that the couple C(s)/ A+*(solution)/M is reversible while the couple C(s)/A+(solution)/M is not, that is, that electrode M is polarized with the . ~ dashed vertical line in the respect to the second p r o ~ e s sThe cell denotes a liquid junction. The right-hand solution is under steady irradiation of wavelength A*, and is of sufficient concentration that the incident light is completely absorbed by A+. The irradiation builds un a concentration of A+* t o some value (A+*) and a t this poiit the cell is allowed to operate, drawing j& s h c i e n t current to maintain (A+*) a t a steady state value. The whole system is at constanttemperature and, under the steady state condition, light energy is being converted to useful work. A potential problem is that A+* can in general disappear by other processes than the cell reaction. A+* can, for example, return to A+ by light emission or by non-radiative relaxation. These are first order processes with rate constants k , and k,., respectively, and the sum, k = k, k,,, determines the rate a t which A+* would ordiuarilv disamear. The requirement here is that the cell reaction must keep (A+*) small enough that k(A+*) is negligible compared with the rate of the photochemical reaction (see Example below). We want our idealized cell to be one which can operate under steady state conditions. T o accomplish this, suppose that essentially all of the current across the liquid junction is carried by A+."he cell processes per faraday are as follows. The right-hand compartment gains one mole of A+ by transference, loses one mole of A+ by photoexcitation to A+*, and this mole of A+* is reduced to C(s). The left-hand compartment gains one mole of A+ by oxidation of C(s) and loses one mole of A+ by transference. There is thus no net change in either compartment. The net cell reaction is just A+* = A+, and the corresponding emf is Ime (A+) and
+
where a denotes activity. In the left-hand solution A+* is in equilibrium with A+, and for the process A+*., = A+.,, we have AG" = GDn.- Go*+= -RT in K, K = on+,lan+.,
(8)
Further, AG = -96 and AGO = -$Go, and we can replace AGO by its equivalent from eqn. (8). We thus obtain
where w., is the reversihle work. The activitv coefficients of A+* and A+*., can be taken to be the same (bdth solutionswill he very dilute in A+*), as can those of A+, and A+ (the total concentration is the same in the two comp&ments). eqn. (9) simplifies to w,
= RTln[x'(l
- x*,)l(l
- x*)x*,]
(10)
where x* is the fraction present as A+*. For any system likely to he of interest, x*,, will he very small, and likewise x* (see Example below). Equation (10) reduces to w,
= RTln(x*lx*,)
(11)
Equation (11) is essentially eqn. (16) of Ref. (5)and eqn. (6) of Ref. (21, obtained by a more aeneral route. The simple assumption is thatthe excited state energy,E*, is entirely available for work, and we now want to compare this amount of work tow,,,. We can structure this assumption more sperifirally as follows. First, we say that A* is converted entirely m AT' so that thr energy producing process A" A + is between species at thes:unt. concrntration, that is, there is noentronv of dilution difference. We can tnke the renditions of A+ and A+* to be their respective standard states. Next, we neglect anv internal entroov difference between A+ and A+*. that is, we take AS' to hkzero4 and equate energy and en: t h a l p ~With . ~ these assumptions, E * = -AG0. The thermodynamic efficiency factor, f ~ can , now be written as
-
f~ = w&-AGO
= ln(x*/x*,)An K
(12)
or f~ = 1- In(z')/ln(~*,~)
(We have approximated CA+,,/CA+*,
(13)
by 1 / ~ * , ~ )
A Numerical Example
We can build an example around the Ru(hpy),Zf case. We take A* to he 570 nm or E* = 5 0 kcal per mole of light quanta.
' h readlng articles on solar energy conversion. It is well to rememDer that authors generally do not Include I, In lhelr efflclencystatements. Rather, the effmency reported is !hat tor complete aosorpt on ot llght of optimum wavelength, or around A'. (For example, Graetzel and co-workers report this type of efficiencyas -0.1 for their R u ( b p ~ ) ~ ~ + system ( I D ) . ] Such an efficiency statement is useful, of course, in evaluating the chemical performance of the particular system: it is misleading, however, in that if the I, factor were to be included, the practical interest ot the system might be greatly diminished. Remember that A+' is an isomer of A+ which can have quite diC ferent properties fromthose of A+. As an example. A+ and A+' might have acidic protons, but the pK, of A+' might be very different from that of A+, so that in a given solution, the degree of dissociation would be very different. This difference might allow electrode M to be polarized with resoection to the reduction of A+ but reversible with resoect to that of A". Such selectivty is a matter of kmetcs, and wnile the condition might De difficA to acnieve in practice, it is a thermoaynamically permissible one. The condition of near unitary transference number for A+ might be achieved, for example, by using a cation exchange membrane to separate the two compartments. This could be a dangerous assumption. One might guess that A 9 is apt to be negative on the grounds that some bond weakening should occur in the excited state, with concomitant reduction in fr. This amounts to neglecting any partial molal volume difference between A+ and A+'. Volume 61 Number 3 March 1984
223
If this is taken also to he -AGo, then 1/K at 25OC is 2.13 X 10-3, This is also the value of x',,:~ The average solar power a t noon In the general latitude of ~ 0.20, about 150 the U S . is about 750 watt m-2 (12). With f = watt m-2 might be available to produce [R~(bpy)3~+]*. The molar extinction coefficient of Ru(bpyh2+is quite large (note Fig. 2), and we can suppose that a layer 1mm thick of 0.001 M solution will be totally ahsorhing. One liter of solution will thus cover 1m2 and in this solution (150)/4.184) (50.000) = 7.17 X mol sec-I of [ R ~ ( h p y ) 3 ~ +will ] * be produced. We can now estimate a practical value of x*. The rate of production of [R~(b;py)3~+]*, is 7.17 X 10-4 molehiter-sec, and is also to be the rate of its removal in our hypothetical cell. This rate must comfortably exceed the combined rate of the >> various diss.ipative processes. That is, we want 7.17 X k(A+*). The minimum value of k is given by 117, where 7 is the emission lifetime in the absence of specific quenchers or excited state reactants. This lifetime is about 600 nsec at 25'C and in an aqueous medium, giving a minimum k value of 1.67 X 106 sec-I. Our requirement is that (A+*)
