BioSystems, 11 (1979) 65--76
65
© Elsevier/North-Holland Scientific Publishers Ltd.
E N T R O P Y OF RADIATION IN BIOLOGY E. BRODA
Institute of Physical Chemistry, Vienna University, Wiihringer Strasse 42, A-1090 Vienna, Austria H. PIETSCHMANN
Institute of Theoretical Physics, Vienna University, Boltzmanngasse 5, A-1090 Vienna, Austria (Received June 8th, 1978) (Revised version received January 3rd, 1979)
The concept of the entropy of electromagnetic radiation and the relationship between entropy and probability for radiation fields ;are explained. Equations for the variations of entropy and temperature in the reversible and irreversible volume changes of black radiation fields are given. Following Boltzmann and SchrSdinger, it is pointed out that living matter, e.g., in a stationary state, struggles for the chance of acquiring the needed free energy by converting low-entropy solar energy radiation into high-entropy terrestrial energy radiation. The amount of energy available from a Carnot process with utilization of high-temperature (low-entropy) radiation and the influence of radiation scattering are computed. The considerations are applied to photosynthesis. Technically useful amounts of hydrogen could in principle be obtained through water photolysis b y means of artificial, simulated, photosynthesis. F o r the purpose, the membrane principle has central importance, i.e., by means of asymmetric, vectorial, membranes separation o f the primary products of photolysis is sought.
1. Introduction
The intuitive grasp and consistent use of the concept of entropy, however important it is, is not easy. This applies with even more force to the transfer of the concept from the customary systems, consisting of substances, to electro-magnetic radiation fields. According to our present views these fields also consist of particles, but o f particles very different from substances, namely, of light quanta. In view of the biological importance of the effects of radiations, especially in photosynthesis, we shall n o w a t t e m p t to make the entropy of quantum radiation more intuitive. We shall deal only with non-ionizing radiation, i.e., visible, ultraviolet and infrared light; X- and 7-radiation will not be considered.
2. E n t r o p y and probability
Max Planck (1910a) stated that it is the
most important feature of a physical process whether it is reversible or not. According to the Second Law o f Thermodynamics in irreversible processes the entropy of the entire system, which takes part in a process, increases. By definition the system is isolated. If this were n o t true, i.e., if energy at least could be exchanged with the environment, the systems considered would be smaller than the whole system that takes part in the process. Of course, strictly speaking no terrestrial system nor even a stellar system is entirely isolated. However, in many cases isolation applies in sufficient approximation, or at least it is possible to approach it infinitely well by a limiting process in a thought experiment. Reversibility is approached in processes by economizing driving force, e.g., no larger difference in temperature or pressure between the "driving" and the "driven" partial system than necessary is applied; in the limiting case the difference goes to zero. It is true that the reduction of the difference is bought at the
66 price that the processes slow down more and more, and in the limiting case the velocity also becomes zero. Thus strictly reversible processes can be obtained only by idealization of real processes. Now reversibility implies constancy of entropy. This concept has been created, i.e., invented, by Rudolf Clausius, and has been named in 1865 when he taught in Zurich. It has acquired central importance and constitutes one of the most precious achievements of physics. This is already seen from the statement by Planck that was quoted before. Yet power of abstractio~ and ample practice a r e needed for the incorporation of entropy into the instinct system of the scientist. But every effort in this direction is justified. In particular, the bioenergetic processes cannot be understood without the concept of entropy. Entropy (S) is a so-called state function. Its numerical value is unambiguously given by the present state of the given material system, and is independent of its history. Although the absolute values of entropies of systems can be calculated at least in principle and often in practice, we are more interested in the entropy difference between 2 states of the same system, i.e., in the values of AS. According to Clausius, the entropy of an isolated system remains constant as long as all processes inside it are reversible: zAs
= 0
(1}
This applies for instance to every one of the 4 steps in an ideal Carnot cycle. The entropy changes of the working substance and of the heat bath, which together form the isolated system, are always opposite and equal. Hence ~ A S = 0 in every step. If processes are at least partly irreversible, entropy increases. Clausius defined the entropy changes connected with reversible processes by the quotient of the heat absorbed, QR, and the absolute temperature. In organisms all processes are essentially isothermal; T is
constant. Hence AS = Q R / T
(2}
The interpretation of the concept of entropy on the basis of atomistics has been given by Ludwig Boltzmann, who recognized in entropy a measure of the "thermodynamical probability" W of the system. In the form given by Planck (see Planck, 1913a) the relation is S = k In W
(3)
where the "Boltzmann constant" k is given by R / N (R universal gas constant, N Avogadro's
number} with the numerical value 1.38 × 10 -23 J . grad-'. By thermodynamical probability we mean the number of the p o s s i b i l i t i e s for the realization of a macroscopic system of given properties by different (micro-)states of its (micro-)components. In substances, these components are the atoms (and molecules}. F o r example, gas molecules in a given volume at given temperature and pressure can assume a large variety of different micro-states. These different micro-states are distinguished by differences in the spatial configuration or in the state of movement of the components, or both. The numerical value of W is always very large. This is in contrast to probability in the usual sense, the "mathematical probability"; by" definition the sum of all mathematical probabilities of the different possible states of a system equals unity. But thermodynamical and mathematical probability are proportional. In order to work o u t the probabilities, some additional hypothesis is necessary: for instance, the assumption of "molecular disorder" (Boltzmann, 1896; see Planek, 1913b) states that there is no correlation between velocity and location of a molecule (or atom}. Mathematically, it means that the probability of finding a molecule (or atom} at a certain location with a certain
67 velocity is the p r o d u c t of the 2 respective probabilities. Estimates of numerical values of 8 from eqn. (3) are instructive. Metallic iron is a typical solid. At 273 K its entropy is 24 J/degree.tool. Hence In W = 24/1.38 X 10 -: 3 1.74 X 1024 , and W ~ 1 0 7 s × 1 ° 2 3 , a n enormous number. Similarly, the e n t r o p y of H~ (at 298 K) is 131. Hence W = 109.s x I 0 ~4
It is possible to assign directly a definite energy content not only to a multitude of particles, but also to a single particle, although this energy c o n t e n t may depend on interactions with neighbouring atoms as far as they exist. In contrast, entropy is a statistical concept (see Planck, 1910b). A definite numerical value can be given without further discussion only for a multitude of particles. But this value can also be referred to a single particle by calculation, i.e., we divide, for instance, the entropy of 1 mol by N, and obtain an average value/mol. Similarly, entropy can be referred to unit volume. The statistical interpretation of entropy is not only very fruitful, b u t also, because of its intuitive nature, very popular. Yet it should not be overlooked that quite often we lack knowledge of numerical values of W. Then we must introduce into calculations the values of S obtained empi:dcally from the measurement of heat effects. For instance, z~S can be obtained on the basis of eqn. (1) by dividing by temperature the heat absorbed in the reversible transil~on from the initial to the final state of a system, QR. Serious errors would be committed if Q/T measured in arbitrary conditions (the "reduced heat") were used instead of QR/T. This is obvious ,e.g., in the expansion of an ideal gas into a vacuum after removal of a dividing wall. The reduced heat, Q/T, is clearly zero. But the entropy of the gas has increased as the uniform distribution of a gas over the whole space is more probable than its restriction to a partial space. Q R c a n be measured by reversible, isothermal, expansion of the gas in contact with a large external heat
bath with performance of mechanical work; the absorbed h e a t , QR, equals the work performed, and QR/T is the entropy difference.
3. Entropy o f radiation fields The utilization of radiation energy by organisms has tremendous importance for bioenergetics. Jan Ingen-Housz from the Netherlands had demonstrated the need of light for the plants by his famous experiments carried o u t in London in 1773. But it was only Julius Robert Mayer, one of the founders of the First Law of Thermodynamics (conservation of energy), who recognized in 1845 that light serves in photosynthesis as a source of energy. This idea was suggested to him by the First Law of Thermodynamics. On the basis of the Second Law of Thermodynamics, the question o f the role of entropy in photosynthesis presents itself. It will be seen that this role is decisive. (Light is also required for further functions of organisms, including phototaxis, the regulation of differentiation in plants by the p h y t o c h r o m e system, and animal vision. In these cases, light is not a source of appreciable amounts of energy, b u t it is used as a signal. Possibly the concept of entropy will sooner or later become i m p o r t a n t in these areas of biophysics, too.} Simple considerations show that radiation .fields must carry n o t only energy but also entropy, and that the entropy is a state function for fields as well as for substances. Let us imagine an isolated system. A h o t solid ("radiator") is separated by a perfect vacuum from a wall that is an ideal mirror, and therefore does not absorb radiation. Then the evacuated space fills up with radiation, but no energy can be transferred to the wall. Thereafter let us increase the distance of the wall from the radiator, so that the additional volume of the vacuum can likewise fill with radiation coming from the solid. By application of an external heat bath we keep
6t~ the process isothermal. Clearly the process is reversible: in an inverse process, reducing the distance between well and radiation, we can transfer radiation energy from the field to the solid. Because of the reversibility the entropy content of the whole system solid + heat bath + vacuum remains constant. As an overall effect, the heat bath has lost heat (QR) and therefore entropy (AS) during the emission of radiation. As in an isolated system entropy cannot decrease, the lost entropy must be found in the radiation field. Ideas on the entropy of radiation fields were developed first by Bartoli (1876) in Florence, and later, in more precise form, by Boltzmann {1884) in Vienna, who referred to Bartoli. Boltzmann used them for the theoretical derivation of the law that had been found experimentally by his admired teacher Josef Stefan, and which has since been known as the Law of Stefan-Boltzmann. According to this law, the energy in a vacuum {volume V) in equilibrium with a " b l a c k " body increases with the fourth power of temperature*: E = aT 4 V
(4)
According to Planck (1913c) the numerical value of the constant a can be derived from universal constants, although in practice it is obtained from experiments. The value was given by Planck as 7.39 X 10 -~s erg • degree -4 . em-3. Black bodies are defined by the feature that they completely absorb incident light of any wave length. In a field of " b l a c k " radiation, light of any wave length is in t h e r m a l equilibrium with the " b l a c k " radiator. Hence the temperature of all light in the field is the same. For simplicity, in the following we limit ourselves largely to black bodies and to their radiation fields. The energy n o t only of a substance, but also of a radiation field can be referred to unit mass or volume. This applies also, as has been • T h e e q u a t i o n s of this p a r t M. P l a n c k ( 1 9 1 3 ) , i.c.
are readily f o u n d in
explained, t o entropy. According to Boltzmann, a radiation field in equilibrium with a black body carries the entropy S=
(5)
4-aTaV. 3
The proportionality of the entropies as well as the energies with the volumes (or masses) of given substances or fields expresses the fact that S and E are "extensive" state functions. From (4) and (5) we obtain S/E
-
4
1
3
T
,
(6)
i.e., in a field of a black body the entropy content of energy decreases with increasing temperature. The zero point energy of a field the energy of a field with the temperature zero -- is zero; there is no radiation at absolute zero. The zero point entropy is also zero. As a c c o r d i n g to the equation of Boltzmann-Planck any a m o u n t of entropy corresponds to a definite thermodynamical probability, the a m o u n t of entropy in unit volume of a radiation field must be correlated to a definite probability of the distribution of the components ("particles"). In this respect the situation in a radiation field parallels that in a substance. In the t h o u g h t experiment outlined before, the radiation must be in equilibrium with the radiator, in the given case with a black solid. Of course this is only possible if this radiation is also in internal equilibrium. Therefore this distribution of the energy within the radiation field has maximum probability. Any other distribution would have smaller probability and therefore smaller entropy. Note, however, that substances or radiations out of equilibrium can also be ascribed definite value of entropy. This is evident, as states in non-equilibrium as well as in equilibrium are characterized by unambiguous probabilities. A radiation field with energy c o n t e n t / u n i t volume equal to t h a t of a field in equilibrium, but with a less probable distribution of the energy over the particles, is quite possible. -
-
69 Clearly it has less entropy than the field in equilibrium. But a field of this kind cannot be obtained in the way described in a system radiator + heat bath + vacuum + mirror, as the formation of such a field would require an entropy decrease, in the system as a whole. An answer to the question of the nature of the components of the radiation field in the vacuum was not given, but at least suggested by Planck in his momentous lecture in December, 1900 (see Planck, 1901; Planck, 1910c). Other outstanding physicists had worked hard at the problem of the dependence of the radiation density above a black body on temperature and wave length, without obtaining full success. Planck showed that the correct law is obtained on the basis of the assumption that the radiator can absorb and emit energy only in the form of energy quanta, i.e., discontinuously. A further step, to postulate quantisation of energy also within the radiation field, was taken later by Albert Einstein (1905). Accordingly the radiation field consists of quanta, or " p h o t o n s " , i.e., particles, to which, just as to particles within substances, energy and m o m e n t u m are assigned individually. As on the other hand the well known phenomena of the diffraction and the interference of electromagnetic radiation clearly contradict an interpretation on the basis of particles, we have arrived at the well known dualism between particles and waves. Nowadays it is hard to imagine how difficult it was to assume the quantum structure of radiation. At first, Planck could not follow Einstein in this step. As late as 1913, in connection with the election of Einstein to the Prussian Academy of Sciences, he wrote in an evaluation, which he drafted and signed along with Walther Nernst, Heinrich Rubens and Emil Warburg (see Kirsten and KSrber, 1975): "He should not be blamed too much for occasionally exaggerating his speculations, for instance in his hypothesis of the light quanta. Not even in the most exact science can real innovations be introduced without venturing risks".
The energy of the quanta of electromagnetic relation is hv or, equivalently, hc/~,, h is Planck's constant (6.62 × 10 -3 a j . s), k the wave length, v the frequency and c the velocity of light. Accordingly the quanta of radio waves (k large) are poor in energy, those of -/-rays (v large) are rich in energy. The quanta of infrared, visible and ultraviolet light (k decreasing in this order) are intermediate. The quanta richest in energy are now made in the laboratory by mutual annihilation of pairs of subatomic particles and antiparticles. Each of these quanta may have 101 s times more energy than radiowaves in the centimeter range.
4. Volume changes of radiation fields In a certain sense the photons of the radiation correspond to the atoms of the substances. N e v e r t h e l e s s there is a fundamental difference between atoms and photons: the former exist in every substance from the beginning, while the photons are created only during their emission into the radiation field. While the atoms are in practical conditions indestructible, this is n o t true of the photons which are n o t s u b s e q u e n t l y conserved. They may be absorbed again. Also, the energy can be redistributed within a radiation field though as a whole the energy must be conserved. In redistribution, the numbers of photons with different energies may change. Thought experiments on the fate of photons in radiation fields during volume changes are instructive (Planck, 1913d). It is possible to expand a radiation field at constant pressure and temperature with performance of work against a mobile piston, just as it can be done with an (ideal) gas that is replenished by evaporation of a liquid (Boltzmann, 1884; Planck, 1913e). During the process heat is converted into work, but isothermal conditions may be maintained by surrounding the liquid or the radiator by a
70 heat bath of large capacity. In the case of radiation, too, the work done is p a y , where p is the radiation pressure. According to Boltzmann {1884), in his derivation of the Stefan Law, a
p = -- T 4 3
(7)
The constant a appeared already in eqn. (5). The energy is furnished directly by the radiator, and indirectly by the heat bath. Of course, energy and entropy of a given volume of a radiation field are also conserved if it is separated, without any other change, from the radiator. This can be carried o u t by insertion of an ideally reflecting mirror between t h e radiator and the field. Subsequently the radiation field is so-to-speak torn away from the parent radiator, and must from n o w onwards depend on itself. Thereafter it is, always in a thought experiment, possible to expand the radiation field reversibly and "adiabatically" with performance of work (Planck, 1913d). By definition, in adiabatic processes no heat is exchanged with the environment. There is no heat bath. The entropy of the expanding radiation, just as the entropy of an expanding substance, does n o t increase; it is clear that adiabatic processes, if reversible, are "isentropic". The work must be performed at the expense of the energy c o n t e n t of the expanding radiation itself. As in the case of gas, decrease of the energy content of the radiation also means decrease in temperature. A c c o r d i n g to Planck (1913d) black radiation can be maintained black during a reversible adiabatic process. For the purpose in the thought experiment a speck of carbon may be added that is infinitesimally small and ideally black. We can neglect its volume, energy and entropy, but it catalyzes the establishment of the radiation equilibrium (Planck, 1913f). Thus within the field we have stable equilibrium and well defined temperature at any instant of the process. According to eqn. (6) the energy E of the
radiation is given by E = ~ TS
(8)
and decreases in proportion to the temperature during the expansion, while S remains constant. Clearly the loss of energy equals the work performed in this reversible process. With S constant, according to eqn. (5) T3V = const.
(9)
i.e., the temperature of the radiation decreases with the cubic r o o t of the reciprocal volume. Again S/E is inversely proportional to T. Incidentally a refined analysis by Wilhelm Wien and Planck showed that in the thought experiment described -- b u t not, e.g., in an irreversible adiabatic process, n o w to be mentioned -- the speck of carbon can be omitted (Planck, 1913g). The expansion of the isolated radiation field dan be carried o u t irreversibly, without performance of work (Planck, 1913h), by opening a previously radiation free volume by removal of the dividing wall, just as an ideal gas can be admitted into a vacuum. During such "dilution" the radiation must conserve its energy. Yet it cannot be assigned the original temperature as it is n o t any more in equilibrium with the parent (black)radiator; it is t o o dilute. Nor would the radiation be in equilibrium with a black b o d y of lower temperature, as it still has the spectral distribution characteristic of the higher temperature. Consequently after expansion the radiation field has no unambiguous temperature at all any more. In contact with a black b o d y (speck of carbon) the isolated radiation field would within immeasurably short time after dilution adopt the temperature corresponding to the existing energy density (Planck, 1913h). Loss of energy cannot be a consequence of the spectral change. Therefore E'= E
(lo)
71 and
T '4 V' = Ir~ V
(ll)
and T' = T (V/V')
TM.
(12)
V' being larger than V, the temperature must have decreased. Because of the irreversibility of the process, the entropy of the isolated radiate.on field must have increased. From eqn. (5) we have S ' / S = T '3 V ' / T 3 V > 1
(13)
The entire entropy change occurs in 2 steps, before and after addition of the speck of carbon, as. each of the 2 steps in itself is irreversible. In step 1 we have expansion into the vacuum, in step 2 establishment o f internal thermal equilibrium by spectral change: Already after step 1 the radiation field can be assigned a definite value of entropy. We recall that non-equilibrium as well as equilibrium states are characterized by unambiguous probabilities. If the radiation, after being diluted in the described way as a consequence of irreversible expansion, is again brought into contact with the original ("parent") radiator, the volume is filled up with radiation until temperature, density of energy and density of entropy have recovered their normal values. This applies before or after spectral change. It is seen that isolated radiation fields can be ascribed n o t only definite energy and entropy, but al:;o, provided they are in internal equilibrium, definite temperature. The fields in equilibrium have reached the state of maximum entropy, and therefore they contain black radiation. The customary concept of chemical thermodynamics apply to the reversible transitions between different states of radiation fields. This was to be shown by the preceding considerations. Clearly the transition to black radiation, where radiation of any wave length has the
same temperature, need n o t occur as long as no catalyst, a b o d y at least partly black, is present. Then the radiation field is n o t really in a stable state. But thermodynamics can nevertheless be applied, just as it is true for metastable substances, for instance, for a mixture of hydrogen and oxygen. Changes in the radiation field are described by values of AE and AS, without taking into account the fact of the continuing disequilibrium. In the statistical analysis of isolated non-black radiation fields, i.e., of fields n o t in internal equilibrium, light of different wave lengths (colour) must be considered separately, because the mixing ratio is indefinite a priori. Lights of different colour must be ascribed different temperatures, according to the temperature of the black body in equilibrium with them. In practice this leads to enormous complications. Here we shall not deal with this further. In this context we may refer to the radiation that has been discovered recently and that apparently fills the Universe (Penzias and Wilson, 1965). Measurements of the spectral distribution have shown that the radiation is black with a temperature of 3 K. This radiation is considered to be a remnant of the "big bang", in which our Universe arose many gigayears ago. Because of the expansion of the Universe, the radiation, which originally had been very hot, was cooled down to 3 K without losing its character of black b o d y radiation.
5. The terrestrial entropy balance So far reference was made mainly to fields in equilibrium. While the discussion of such equilibria is important for an intuitive understanding of the possibilities of radiation to do work, also in living nature, it is unrealistic. In nature there are no containers or pistons with ideally reflecting walls, and they cannot be manufactured. Therefore radiation fields of arbitrary composition cannot be freely handled. Equil!brium fields
72 exist only within closed containers in interaction with the walls. The Earth is n o t in equilibrium, b u t in a more or less stationary state. This is true not only for the substances, but also for the radiation. Radiation flows from the hot Sun to the cold Earth and later from the Earth into space, far colder still. Yet broadly the energy and the entropy contents of the Earth have remained constant for several gigayears, as shown by geology and palaeontology. In a good approximation, absorption or emission of solar energy is the only appreciable interaction of the Earth with the environment in respect to energy or entropy. In the h o t interior of the Earth the equilibria either are already established or their establishment is so slow that changes are unimportant. Hence also the flow of energy out of the Earth is tiny (0.02% of the energy flow from the Sun to the Earth}, and much the same must be true for the entropy flow. Consequently in practice the terrestrial balance of energy and of entropy involves only a thin surface layer. Because entropy is produced all the time in irreversible processes in this layer, in the stationary state more entropy is emitted than absorbed here. It is also possible to speak of an inflow of "negative e n t r o p y " (SchrSdinger) or simply " n e g e n t r o p y " (Brillouin) instead of an outflow of entropy. A constant inflow of negentropy evidently goes n o t only into the terrestrial surface layer as a whole, b u t also to any particular part of the surface layer provided it is in a stationary state. This applies also to the biosphere and within it to the totality of the organisms, as expressed in 1888 by Boltzmann (see Boltzmann, 1905) in majestic words: "Therefore the general struggle of the organisms is n o t a struggle for the elements, nor for energy, which is present in large amounts in every body in the form of heat, unfortunately unchangeable, b u t the struggle for entropy, which becomes available in the transition from the hot Sun
to the cold Earth. To exploit this transition as far as possible, the plants spread o u t the immeasurable areas of their leaves and force solar energy in a manner as y e t unexplored, until it sinks down to the temperature level of the surface of the Earth, to carry o u t chemical syntheses of which we have no inkling as y e t in our laboratories. The products of this chemical kitchen are then the object of the struggles of the animal world." Thus the p h e n o m e n o n of life as a whole has been made possible by consumption of negentropy. What can n o w be said a b o u t the flow of energy and entropy? Generally within a homogeneous radiation field the flow of energy (specific intensity, brightness) K is related to the energy density e by K= eq/4n
(14)
where q is the velocity of light (Planck, 1913i). Similarly the entropy flow L is related to the entropy density (Planck, 1913j)
by L = sq/4~.
(15)
In the vacuum, c can be substituted for q. When a hole is drilled into the thin limiting wall of a homogeneous radiation field a diverging pencil of radiation escapes, within which the intensity of the radiation decreases with increasing distance. However, during its way across (empty) space the radiation maintains n o t only its energy (per unit solid angle), b u t also its entropy (likewise per unit solid angle}. The increasing distance from the source has no effect; this is seen already from the absence of a distance term in eqns (14) and (15}. Thus the process is reversible. That the straight "centrifugal" propagation is a reversible process, which does n o t increase entropy, is evident also from the fact that the ray8 may be reflected back into the source without loss and independently of distance by
73 means of an :imagined ideal spherical mirror. On the basis of the Boltzmann-Planck Law i t follows from the constancy of entropy that the state of order of the radiation field does not decrease in the propagation of the radiation, or, equivalently, that the probability of the state does not increase. Consequently the solar radiation meets the top layer of our atmosphere with its original energy and e n t r o p y / u n i t solid angle. The conservation (constancy) of the ratio of energy and enl~ropy implies, as has been se~n, conservation of the temperature of the radiation (Planck, 1913k). Therefore the situation in the straight propagation of light rays in space is entirely different from that in the diffuse flow of disordered homogeneous radiation into a vacuum, as discussed before. In the latter process the entropy of the radiation does iincrease. The maintenance of the quality of the solar radiation during its long journey through space can also be recognized directly from the fact that it can be concentrated again on Earth by means of lenses or mirrors. In the ideal case the temperature of the source can nearly be reached again. ("Nearly": because of the influence of our atmosphere). In a famous research laboratory at Odeillo in the French Pyrenees, concentrating mirrors are actually used for the production of very high temperature for the investigation of materials ("solar furnace").
6. Work from solar radiation Black radiation being characterized by a definite temperature, the maximum work W in the transition of radiation energy between different temperatures can be c o m p u t e d by means of the equation for the Carnot cycle, i.e., the equation for the work done by a gas in a reversible thermal process. The higher the temperature Tt of the heat source and the lower the temperature T2 of the heat sink, the larger the efficiency of the heat engine~i.e.,
the better the Carnot factor 7, the yield, 71 = W / Q
= (T2
- T2 )/TI .
(16)
Q is the heat absorbed at the upper temperature. T2 would in a steam engine be the temperature of the cooling water. As explained, no capacity for work (temperature) is lost in the straight propagation of radiation through space. Consequently the ideal efficiency of a (radiation) engine does n o t depend on distance from the Sun. Focussing the solar rays on a working substance with a mirror or lens, the substance can be heated to the temperature of the primary source in the Sun (6000 K). With a heat sink of the temperature of the terrestrial environment (300 K), excellent yields could in principle be obtained with a "radiation engine": ~7 = (6000 - 300)/6000 % 0 . 9 5 It follows from the independence o f the efficiency of the magnitude of the captured energy flux, i.e., of the dimensions of the engine, that a mirror or a lens should always still work as a collector, however small. That means that in the limiting case one could omit the collector altogether. But naturally the power supplied by the system would decrease with decreasing energy flux collected. Moreover, thermal insulation is more difficult with smaller systems. The directional correlation within the radiation can be destroyed by diffuse reflexion at a surface that is ideally white, but not a mirror. Such a surface is approximated, for example, by a layer of magnesia. By definition ideally white surfaces, like ideal mirrors, do not absorb any light at ~ll. Therefore white surfaces, like mirrors, do n o t change the energy content or the spectrum of the radiation. But after diffuse reflexion the radiation cannot be concentrated any more by lenses or mirrors. The effect is the same as if radiation had expanded, without doing work, into a vacuum. This is seen in a simple
74 thought experiment again. A pencil of solar radiation is introduced into a white hollow sphere. After equilibration, the radiation, still with the same energy and spectrum, fills the spherical space uniformly, but the radiation density per solid angle has dropped. After being made diffuse, the radiation is not in thermal equilibrium any more with the source. As in the case of expansion into a vacuum, it has now no defined temperature any more, but increased entropy. A definite (lowered) temperature, with entropy further increased, is established only after intern, al equilibration of the radiation. This two-step irreversible process must lead to a decrease in the capacity for work. A radiation engine with destroyed directional correlation would have decreased efficiency, especially after i n t e r n a l equilibration. Of course, the temperature of the radiation in internal equilibrium still much exceeds that of the ideally white reflector, to which by definition no energy was transferred. No equilibrium obtains between the substance of the reflector, which may be at the temperature of the environment, and the radiation field. According to computations by Spanner (1964), carried out in connection with the efficiency of plant photosynthesis, the temperature of solar radiation after complete diffuse reflexion on the surface of the Earth is only 1350 K. By the Carnot equation this corresponds, with a temperature of the heat sink of 300 K, to a maximum efficiency in the performance of work of 77%. T h e diminished capacity of cooled radiation to do work must, because of the general validity of thermodynamics, also be expressed in photochemical reactions. One would expect that on illumination in the case of endergonic reactions chemical states with high contents of free energy are favoured compared with states with less free energy. The higher the input of light energy, the further the reaction is driven. In thought experiments chemical actions could be carried out in analogy to ordinary (light-independent, "dark") reactions by
means of a reaction box of Van't Hoff, as described in the standard textbooks of physical chemistry. With the reaction box, one derives the affinity, i.e., maximum work, from the equilibrium constant in dark reactions. In photochemical reactions, one would conversely use the relationship to compute the equilibrium constant, i.e., the yield, from the input of free energy by means of photons. True, semipermeable walls (membranes) would have to be imagined that let pass excited, but not non-excited molecules, and vice versa. While there is no t h e r m o d y n a m i c a l objection against the assumption of such walls, the considerations would be formalistic rather than intuitive. This applies already to non-directional, isotropic, radiation. Thought experiments with radiation with directional correlation (less entropy/energy} would be even less intuitive. Moreover, in realistic considerations it would be necessary to take account of the quantization of the energy states of the reactants and products along the real reaction paths. Existing knowledge for all this is insufficient.
7. Applications to photosynthesis The light reaction in plant photosynthesis consists essentially in the photolysis of water (Van Niel, 1962). It is Crue that in physiological conditions elementary oxygen, but no elementary hydrogen is produced. It would escape uselessly. But a different strong reductant is generated, namely, the reduced form of a well defined iron-sulfur protein, ferredoxin (see Arnon, 1971). This is subsequently used for the production of biomass from CO2 and H20. The standard redox potential of the couple reduced/ oxidized ferredoxin is practically equal to that of the couple hydrogen gas/neutral water. H e n c e from the photochemical and thermochemical point of view the achievement of the plant is not less than if it actually produced free hydrogen.
75 For simplicity we therefore speak simply of water photolysis. Thermodyn,'unically we can imagine an idealized syst~m within an ideally white sphere that consists of the following components: (1) Black radiation (1350 K); (2) Water (300 K); (3) Arbitrary substances and devices (300 K). Component 3, 'which is not supposed to suffer any permanent change, can, in an expanded sense, be called a system of catalysts or simply as " a " catalyst. Now we carry through as a thought e~:periment, in the way usual in chemical thermodynamics, a reversible process differentially, i.e., to a sufficiently small extent that neither the quantities of the components nor the temperatures change perceptibly. The process consists in the performance of work on the basis of the temperature difference between 1350 and 300 K and, coupled with this process in a loss-free way, in a photolysis of an equivalent amount of water to give H~ + 0.5 O~, as the gases. In standard conditions this photolysis requires a free enthalpy AGo = 238 kJ/mol water. In principle there is no reason why the process should n o t be carried out reversibly, without entropy increase. Hence the work performed is limited only by the Carnot factor. This is, as explained, about 77% f o r diffuse solar radiation. A challenge to indicate a concrete possibility for the realization of the ideal l o s s - f r e e process can be rejected by thermodynamics as this is not concerned with m e c h a n i s m s . At the same time, the biophysicist admires the plant for obtaining, in optimum conditions, chemical energy from light with 33% efficiency in the form of biomass. From the point of view of efficiency optimum conditions apply when long-wave light (small quant~a) is used that nevertheless is still just fully active photosynthetically: red light of 680--700 nm. Biomass production clearly cannot occur with maximum
thermodynamical efficiency, as is true also for all chemical processes in industry. Already the need for sufficient speed, without which the process would be useless for the plant, leads to irreversibility. The fact that the sacrifice in efficiency made by the plant is not larger is all the more remarkable, as the entire process of photosynthesis up to the stage of carbohydrate includes 20--30 simple steps (Bassham and Calvin, 1957; Arnon, 1971), and surely every step contributes to loss. Within the reaction chain the 2 light reactions (Hill and Bendall, 1960) have key positions. In plant photosynthesis, they are connected in series and make possible the tremendous achievement of water photolysis. The ancestors in evolution of the plants, the photosynthetic (purple and green) bacteria were still satisfied with 1 light reaction (see Broda, 1975, 1978a). For this reason they cannot split water, and they need as substrates hydrogen compounds with less bond strength, typically H~S. Therefore they cannot set free oxygen, but only sulphur and the like. Incidentally, the principle of the utilization of several light reactions that follow in an ordered sequence, with each of the reactions taking place in a different absorbant partial system, is known only from living nature. So far it h a s never been developed independently, or even imitated successfully, in science or technology. The principle could be called the "superposition" principle. The energy problem of mankind could be solved if the cheap photolysis of water by means of solar radiation were possible on a commercial scale (Broda, 1975a, 1976, 1978b,c). The hydrogen would be introduced into a hydrogen e c o n o m y and serve for the supply of heat or electricity, of chemical and metallurgical reductant and even of biomass. Hydrogen can be cheaply stored and transported, and in its combustion it does not pollute the environment. A small part of the 170 X 10 ~2 kW of light energy that constantly flows towards the Earth could cover all energy requirements of mankind.
76 The best chance for abiotic photolysis may consist in the application of the "membrane principle", likewise invented by Nature. Artificial asymmetric membranes should be constructed that in "vectorial" photoreactions set free primary reductant only on one side, primary oxidant only on the other side. In this case, a recombination of the primary products to water and their loss would be prevented. All photosynthetic organisms have made use of the membrane principle for at least 3 gigayears (see Broda, 1975b, 1978a).
References Arnon D.I., 1971, Proc. Natl. Acad. Sci. USA 68, 2883. Bartoli A., 1876, Sopra i movimenti prodotti dalla luce e dal calore (Le Monnier, Firenze). Bassham J.A. and M. Calvin, 1957, The Path of Carbon in Photosynthesis (Prentice Hall, Englewood Cliffs). Boltzmann L., 1884, Wied. Ann. 22, 31,291. Boltzmann L., 1896, Vorlesungen fiber Gastheorie, Vol. 1, p. 21. Boltzmann L., 1905, Populate Schriften (Barth, Leipzig), p. 40. Broda, E., 1975a, Naturwiss. Rundsch. 28,365. Broda E., 1975b, the Evolution of the Bioenergetic Processes (Pergamon Press, Oxford). Broda E., 1976, Photochemical Hydrogen Production Through Solar Radiation "by Means of the Membrane Principle, UNESCO/WMO Solar Energy Symposium, Geneva 1976. Broda, E., 1978a, The Evolution of the Bioenergetic Processes (Pergamon Press, Oxford) Revised edition.
Broda, E., 1978b, Int. J. Hydrogen Energy 3, 119. Broda, E., 1978c, Photochemical Production of Hydrogen from Water, 2nd World Hydrogen Energy Conference, Zurich. Einstein, A., 1905, Ann. Phys. 17,132. Hill, R. and F. Bendall, 1960, Nature (Lond.) 186, 136. Kirsten C. and H.G. KSrber, 1975, Physiker ~ber Physiker (Akademie-Verlag, Berlin), p. 201. Penzias, A.A. and R.H. Wilson, 1965, Astrophys. J. 142, 419. Planck, M., 1901, Ann. Phys. 4, 553. Planck, M. 1910a, Acht Vorlesungen fiber Theoretische Physik (Hirzel, Leipzig), p. 19. Planck, M., 1910b, Acht Vorlesungen fber Theoretische Physik (Hirzel, Leipzig), p. 50. Planck, M., 1910c, Acht Vorlesungen ~ber Theoretisehe Physik (Hirzel, Leipzig), pp. 45, 95. Planck, M., 1913a, Vorlesungen fiber die Theorie der W~mestrahlung (Barth, Leipzig), p. 118. Planck, M., 1913b, Vorlesungen ~ber die Theorie der W~rmestrahlung (Barth, Leipzig), pp. 114, 115. Planck, M., 1913c, Vorlesungen fber die Theorie der W~rmestrahlung (Barth, Leipzig), p. 165. Planck, M., 1913d, Vorlesungen fber die Theorie der W~rmestrahlung (Barth, Leipzig), p. 66. Planck, M., 1913e, Vorlesungen fiber die Theorie der W~irmestrahlung (Barth, Leipzig), p. 65. Planck, M., 1913f, Vorlesungen iiber die Theorie der W~mestrahlung (Barth, Leipzig), p. 48, Planck, M., 1913g, Vorlesungen fber die Theorie der W~rmestrahlung (Barth, Leipzig), p. 69. Planck, M., 1913h, Vorlesungen fher die Theorie der W~irmestrahlung (Barth, Leipzig), p. 67. Planck, M., 1913i, Vorlesungen fber die Theorie der W~mestrahlung (Barth, Leipzig), p. 23. Planck, M., 1913j, Vorlesungen iiher die Theorie der W~irmestrahlung (Barth, Leipzig), p. 94. Planck, M., 1913k, Vorlesungen iiber die Theorie der W~mestrahlung (Barth, Leipzig), p. 171. Spanner, D.C., 1964, Introduction to Thermodynamics (Acad. Press, London). Van Niei, C.B., 1962, Ann.Rev.Plant Physiol. 13, 1.
65
© Elsevier/North-Holland Scientific Publishers Ltd.
E N T R O P Y OF RADIATION IN BIOLOGY E. BRODA
Institute of Physical Chemistry, Vienna University, Wiihringer Strasse 42, A-1090 Vienna, Austria H. PIETSCHMANN
Institute of Theoretical Physics, Vienna University, Boltzmanngasse 5, A-1090 Vienna, Austria (Received June 8th, 1978) (Revised version received January 3rd, 1979)
The concept of the entropy of electromagnetic radiation and the relationship between entropy and probability for radiation fields ;are explained. Equations for the variations of entropy and temperature in the reversible and irreversible volume changes of black radiation fields are given. Following Boltzmann and SchrSdinger, it is pointed out that living matter, e.g., in a stationary state, struggles for the chance of acquiring the needed free energy by converting low-entropy solar energy radiation into high-entropy terrestrial energy radiation. The amount of energy available from a Carnot process with utilization of high-temperature (low-entropy) radiation and the influence of radiation scattering are computed. The considerations are applied to photosynthesis. Technically useful amounts of hydrogen could in principle be obtained through water photolysis b y means of artificial, simulated, photosynthesis. F o r the purpose, the membrane principle has central importance, i.e., by means of asymmetric, vectorial, membranes separation o f the primary products of photolysis is sought.
1. Introduction
The intuitive grasp and consistent use of the concept of entropy, however important it is, is not easy. This applies with even more force to the transfer of the concept from the customary systems, consisting of substances, to electro-magnetic radiation fields. According to our present views these fields also consist of particles, but o f particles very different from substances, namely, of light quanta. In view of the biological importance of the effects of radiations, especially in photosynthesis, we shall n o w a t t e m p t to make the entropy of quantum radiation more intuitive. We shall deal only with non-ionizing radiation, i.e., visible, ultraviolet and infrared light; X- and 7-radiation will not be considered.
2. E n t r o p y and probability
Max Planck (1910a) stated that it is the
most important feature of a physical process whether it is reversible or not. According to the Second Law o f Thermodynamics in irreversible processes the entropy of the entire system, which takes part in a process, increases. By definition the system is isolated. If this were n o t true, i.e., if energy at least could be exchanged with the environment, the systems considered would be smaller than the whole system that takes part in the process. Of course, strictly speaking no terrestrial system nor even a stellar system is entirely isolated. However, in many cases isolation applies in sufficient approximation, or at least it is possible to approach it infinitely well by a limiting process in a thought experiment. Reversibility is approached in processes by economizing driving force, e.g., no larger difference in temperature or pressure between the "driving" and the "driven" partial system than necessary is applied; in the limiting case the difference goes to zero. It is true that the reduction of the difference is bought at the
66 price that the processes slow down more and more, and in the limiting case the velocity also becomes zero. Thus strictly reversible processes can be obtained only by idealization of real processes. Now reversibility implies constancy of entropy. This concept has been created, i.e., invented, by Rudolf Clausius, and has been named in 1865 when he taught in Zurich. It has acquired central importance and constitutes one of the most precious achievements of physics. This is already seen from the statement by Planck that was quoted before. Yet power of abstractio~ and ample practice a r e needed for the incorporation of entropy into the instinct system of the scientist. But every effort in this direction is justified. In particular, the bioenergetic processes cannot be understood without the concept of entropy. Entropy (S) is a so-called state function. Its numerical value is unambiguously given by the present state of the given material system, and is independent of its history. Although the absolute values of entropies of systems can be calculated at least in principle and often in practice, we are more interested in the entropy difference between 2 states of the same system, i.e., in the values of AS. According to Clausius, the entropy of an isolated system remains constant as long as all processes inside it are reversible: zAs
= 0
(1}
This applies for instance to every one of the 4 steps in an ideal Carnot cycle. The entropy changes of the working substance and of the heat bath, which together form the isolated system, are always opposite and equal. Hence ~ A S = 0 in every step. If processes are at least partly irreversible, entropy increases. Clausius defined the entropy changes connected with reversible processes by the quotient of the heat absorbed, QR, and the absolute temperature. In organisms all processes are essentially isothermal; T is
constant. Hence AS = Q R / T
(2}
The interpretation of the concept of entropy on the basis of atomistics has been given by Ludwig Boltzmann, who recognized in entropy a measure of the "thermodynamical probability" W of the system. In the form given by Planck (see Planck, 1913a) the relation is S = k In W
(3)
where the "Boltzmann constant" k is given by R / N (R universal gas constant, N Avogadro's
number} with the numerical value 1.38 × 10 -23 J . grad-'. By thermodynamical probability we mean the number of the p o s s i b i l i t i e s for the realization of a macroscopic system of given properties by different (micro-)states of its (micro-)components. In substances, these components are the atoms (and molecules}. F o r example, gas molecules in a given volume at given temperature and pressure can assume a large variety of different micro-states. These different micro-states are distinguished by differences in the spatial configuration or in the state of movement of the components, or both. The numerical value of W is always very large. This is in contrast to probability in the usual sense, the "mathematical probability"; by" definition the sum of all mathematical probabilities of the different possible states of a system equals unity. But thermodynamical and mathematical probability are proportional. In order to work o u t the probabilities, some additional hypothesis is necessary: for instance, the assumption of "molecular disorder" (Boltzmann, 1896; see Planek, 1913b) states that there is no correlation between velocity and location of a molecule (or atom}. Mathematically, it means that the probability of finding a molecule (or atom} at a certain location with a certain
67 velocity is the p r o d u c t of the 2 respective probabilities. Estimates of numerical values of 8 from eqn. (3) are instructive. Metallic iron is a typical solid. At 273 K its entropy is 24 J/degree.tool. Hence In W = 24/1.38 X 10 -: 3 1.74 X 1024 , and W ~ 1 0 7 s × 1 ° 2 3 , a n enormous number. Similarly, the e n t r o p y of H~ (at 298 K) is 131. Hence W = 109.s x I 0 ~4
It is possible to assign directly a definite energy content not only to a multitude of particles, but also to a single particle, although this energy c o n t e n t may depend on interactions with neighbouring atoms as far as they exist. In contrast, entropy is a statistical concept (see Planck, 1910b). A definite numerical value can be given without further discussion only for a multitude of particles. But this value can also be referred to a single particle by calculation, i.e., we divide, for instance, the entropy of 1 mol by N, and obtain an average value/mol. Similarly, entropy can be referred to unit volume. The statistical interpretation of entropy is not only very fruitful, b u t also, because of its intuitive nature, very popular. Yet it should not be overlooked that quite often we lack knowledge of numerical values of W. Then we must introduce into calculations the values of S obtained empi:dcally from the measurement of heat effects. For instance, z~S can be obtained on the basis of eqn. (1) by dividing by temperature the heat absorbed in the reversible transil~on from the initial to the final state of a system, QR. Serious errors would be committed if Q/T measured in arbitrary conditions (the "reduced heat") were used instead of QR/T. This is obvious ,e.g., in the expansion of an ideal gas into a vacuum after removal of a dividing wall. The reduced heat, Q/T, is clearly zero. But the entropy of the gas has increased as the uniform distribution of a gas over the whole space is more probable than its restriction to a partial space. Q R c a n be measured by reversible, isothermal, expansion of the gas in contact with a large external heat
bath with performance of mechanical work; the absorbed h e a t , QR, equals the work performed, and QR/T is the entropy difference.
3. Entropy o f radiation fields The utilization of radiation energy by organisms has tremendous importance for bioenergetics. Jan Ingen-Housz from the Netherlands had demonstrated the need of light for the plants by his famous experiments carried o u t in London in 1773. But it was only Julius Robert Mayer, one of the founders of the First Law of Thermodynamics (conservation of energy), who recognized in 1845 that light serves in photosynthesis as a source of energy. This idea was suggested to him by the First Law of Thermodynamics. On the basis of the Second Law of Thermodynamics, the question o f the role of entropy in photosynthesis presents itself. It will be seen that this role is decisive. (Light is also required for further functions of organisms, including phototaxis, the regulation of differentiation in plants by the p h y t o c h r o m e system, and animal vision. In these cases, light is not a source of appreciable amounts of energy, b u t it is used as a signal. Possibly the concept of entropy will sooner or later become i m p o r t a n t in these areas of biophysics, too.} Simple considerations show that radiation .fields must carry n o t only energy but also entropy, and that the entropy is a state function for fields as well as for substances. Let us imagine an isolated system. A h o t solid ("radiator") is separated by a perfect vacuum from a wall that is an ideal mirror, and therefore does not absorb radiation. Then the evacuated space fills up with radiation, but no energy can be transferred to the wall. Thereafter let us increase the distance of the wall from the radiator, so that the additional volume of the vacuum can likewise fill with radiation coming from the solid. By application of an external heat bath we keep
6t~ the process isothermal. Clearly the process is reversible: in an inverse process, reducing the distance between well and radiation, we can transfer radiation energy from the field to the solid. Because of the reversibility the entropy content of the whole system solid + heat bath + vacuum remains constant. As an overall effect, the heat bath has lost heat (QR) and therefore entropy (AS) during the emission of radiation. As in an isolated system entropy cannot decrease, the lost entropy must be found in the radiation field. Ideas on the entropy of radiation fields were developed first by Bartoli (1876) in Florence, and later, in more precise form, by Boltzmann {1884) in Vienna, who referred to Bartoli. Boltzmann used them for the theoretical derivation of the law that had been found experimentally by his admired teacher Josef Stefan, and which has since been known as the Law of Stefan-Boltzmann. According to this law, the energy in a vacuum {volume V) in equilibrium with a " b l a c k " body increases with the fourth power of temperature*: E = aT 4 V
(4)
According to Planck (1913c) the numerical value of the constant a can be derived from universal constants, although in practice it is obtained from experiments. The value was given by Planck as 7.39 X 10 -~s erg • degree -4 . em-3. Black bodies are defined by the feature that they completely absorb incident light of any wave length. In a field of " b l a c k " radiation, light of any wave length is in t h e r m a l equilibrium with the " b l a c k " radiator. Hence the temperature of all light in the field is the same. For simplicity, in the following we limit ourselves largely to black bodies and to their radiation fields. The energy n o t only of a substance, but also of a radiation field can be referred to unit mass or volume. This applies also, as has been • T h e e q u a t i o n s of this p a r t M. P l a n c k ( 1 9 1 3 ) , i.c.
are readily f o u n d in
explained, t o entropy. According to Boltzmann, a radiation field in equilibrium with a black body carries the entropy S=
(5)
4-aTaV. 3
The proportionality of the entropies as well as the energies with the volumes (or masses) of given substances or fields expresses the fact that S and E are "extensive" state functions. From (4) and (5) we obtain S/E
-
4
1
3
T
,
(6)
i.e., in a field of a black body the entropy content of energy decreases with increasing temperature. The zero point energy of a field the energy of a field with the temperature zero -- is zero; there is no radiation at absolute zero. The zero point entropy is also zero. As a c c o r d i n g to the equation of Boltzmann-Planck any a m o u n t of entropy corresponds to a definite thermodynamical probability, the a m o u n t of entropy in unit volume of a radiation field must be correlated to a definite probability of the distribution of the components ("particles"). In this respect the situation in a radiation field parallels that in a substance. In the t h o u g h t experiment outlined before, the radiation must be in equilibrium with the radiator, in the given case with a black solid. Of course this is only possible if this radiation is also in internal equilibrium. Therefore this distribution of the energy within the radiation field has maximum probability. Any other distribution would have smaller probability and therefore smaller entropy. Note, however, that substances or radiations out of equilibrium can also be ascribed definite value of entropy. This is evident, as states in non-equilibrium as well as in equilibrium are characterized by unambiguous probabilities. A radiation field with energy c o n t e n t / u n i t volume equal to t h a t of a field in equilibrium, but with a less probable distribution of the energy over the particles, is quite possible. -
-
69 Clearly it has less entropy than the field in equilibrium. But a field of this kind cannot be obtained in the way described in a system radiator + heat bath + vacuum + mirror, as the formation of such a field would require an entropy decrease, in the system as a whole. An answer to the question of the nature of the components of the radiation field in the vacuum was not given, but at least suggested by Planck in his momentous lecture in December, 1900 (see Planck, 1901; Planck, 1910c). Other outstanding physicists had worked hard at the problem of the dependence of the radiation density above a black body on temperature and wave length, without obtaining full success. Planck showed that the correct law is obtained on the basis of the assumption that the radiator can absorb and emit energy only in the form of energy quanta, i.e., discontinuously. A further step, to postulate quantisation of energy also within the radiation field, was taken later by Albert Einstein (1905). Accordingly the radiation field consists of quanta, or " p h o t o n s " , i.e., particles, to which, just as to particles within substances, energy and m o m e n t u m are assigned individually. As on the other hand the well known phenomena of the diffraction and the interference of electromagnetic radiation clearly contradict an interpretation on the basis of particles, we have arrived at the well known dualism between particles and waves. Nowadays it is hard to imagine how difficult it was to assume the quantum structure of radiation. At first, Planck could not follow Einstein in this step. As late as 1913, in connection with the election of Einstein to the Prussian Academy of Sciences, he wrote in an evaluation, which he drafted and signed along with Walther Nernst, Heinrich Rubens and Emil Warburg (see Kirsten and KSrber, 1975): "He should not be blamed too much for occasionally exaggerating his speculations, for instance in his hypothesis of the light quanta. Not even in the most exact science can real innovations be introduced without venturing risks".
The energy of the quanta of electromagnetic relation is hv or, equivalently, hc/~,, h is Planck's constant (6.62 × 10 -3 a j . s), k the wave length, v the frequency and c the velocity of light. Accordingly the quanta of radio waves (k large) are poor in energy, those of -/-rays (v large) are rich in energy. The quanta of infrared, visible and ultraviolet light (k decreasing in this order) are intermediate. The quanta richest in energy are now made in the laboratory by mutual annihilation of pairs of subatomic particles and antiparticles. Each of these quanta may have 101 s times more energy than radiowaves in the centimeter range.
4. Volume changes of radiation fields In a certain sense the photons of the radiation correspond to the atoms of the substances. N e v e r t h e l e s s there is a fundamental difference between atoms and photons: the former exist in every substance from the beginning, while the photons are created only during their emission into the radiation field. While the atoms are in practical conditions indestructible, this is n o t true of the photons which are n o t s u b s e q u e n t l y conserved. They may be absorbed again. Also, the energy can be redistributed within a radiation field though as a whole the energy must be conserved. In redistribution, the numbers of photons with different energies may change. Thought experiments on the fate of photons in radiation fields during volume changes are instructive (Planck, 1913d). It is possible to expand a radiation field at constant pressure and temperature with performance of work against a mobile piston, just as it can be done with an (ideal) gas that is replenished by evaporation of a liquid (Boltzmann, 1884; Planck, 1913e). During the process heat is converted into work, but isothermal conditions may be maintained by surrounding the liquid or the radiator by a
70 heat bath of large capacity. In the case of radiation, too, the work done is p a y , where p is the radiation pressure. According to Boltzmann {1884), in his derivation of the Stefan Law, a
p = -- T 4 3
(7)
The constant a appeared already in eqn. (5). The energy is furnished directly by the radiator, and indirectly by the heat bath. Of course, energy and entropy of a given volume of a radiation field are also conserved if it is separated, without any other change, from the radiator. This can be carried o u t by insertion of an ideally reflecting mirror between t h e radiator and the field. Subsequently the radiation field is so-to-speak torn away from the parent radiator, and must from n o w onwards depend on itself. Thereafter it is, always in a thought experiment, possible to expand the radiation field reversibly and "adiabatically" with performance of work (Planck, 1913d). By definition, in adiabatic processes no heat is exchanged with the environment. There is no heat bath. The entropy of the expanding radiation, just as the entropy of an expanding substance, does n o t increase; it is clear that adiabatic processes, if reversible, are "isentropic". The work must be performed at the expense of the energy c o n t e n t of the expanding radiation itself. As in the case of gas, decrease of the energy content of the radiation also means decrease in temperature. A c c o r d i n g to Planck (1913d) black radiation can be maintained black during a reversible adiabatic process. For the purpose in the thought experiment a speck of carbon may be added that is infinitesimally small and ideally black. We can neglect its volume, energy and entropy, but it catalyzes the establishment of the radiation equilibrium (Planck, 1913f). Thus within the field we have stable equilibrium and well defined temperature at any instant of the process. According to eqn. (6) the energy E of the
radiation is given by E = ~ TS
(8)
and decreases in proportion to the temperature during the expansion, while S remains constant. Clearly the loss of energy equals the work performed in this reversible process. With S constant, according to eqn. (5) T3V = const.
(9)
i.e., the temperature of the radiation decreases with the cubic r o o t of the reciprocal volume. Again S/E is inversely proportional to T. Incidentally a refined analysis by Wilhelm Wien and Planck showed that in the thought experiment described -- b u t not, e.g., in an irreversible adiabatic process, n o w to be mentioned -- the speck of carbon can be omitted (Planck, 1913g). The expansion of the isolated radiation field dan be carried o u t irreversibly, without performance of work (Planck, 1913h), by opening a previously radiation free volume by removal of the dividing wall, just as an ideal gas can be admitted into a vacuum. During such "dilution" the radiation must conserve its energy. Yet it cannot be assigned the original temperature as it is n o t any more in equilibrium with the parent (black)radiator; it is t o o dilute. Nor would the radiation be in equilibrium with a black b o d y of lower temperature, as it still has the spectral distribution characteristic of the higher temperature. Consequently after expansion the radiation field has no unambiguous temperature at all any more. In contact with a black b o d y (speck of carbon) the isolated radiation field would within immeasurably short time after dilution adopt the temperature corresponding to the existing energy density (Planck, 1913h). Loss of energy cannot be a consequence of the spectral change. Therefore E'= E
(lo)
71 and
T '4 V' = Ir~ V
(ll)
and T' = T (V/V')
TM.
(12)
V' being larger than V, the temperature must have decreased. Because of the irreversibility of the process, the entropy of the isolated radiate.on field must have increased. From eqn. (5) we have S ' / S = T '3 V ' / T 3 V > 1
(13)
The entire entropy change occurs in 2 steps, before and after addition of the speck of carbon, as. each of the 2 steps in itself is irreversible. In step 1 we have expansion into the vacuum, in step 2 establishment o f internal thermal equilibrium by spectral change: Already after step 1 the radiation field can be assigned a definite value of entropy. We recall that non-equilibrium as well as equilibrium states are characterized by unambiguous probabilities. If the radiation, after being diluted in the described way as a consequence of irreversible expansion, is again brought into contact with the original ("parent") radiator, the volume is filled up with radiation until temperature, density of energy and density of entropy have recovered their normal values. This applies before or after spectral change. It is seen that isolated radiation fields can be ascribed n o t only definite energy and entropy, but al:;o, provided they are in internal equilibrium, definite temperature. The fields in equilibrium have reached the state of maximum entropy, and therefore they contain black radiation. The customary concept of chemical thermodynamics apply to the reversible transitions between different states of radiation fields. This was to be shown by the preceding considerations. Clearly the transition to black radiation, where radiation of any wave length has the
same temperature, need n o t occur as long as no catalyst, a b o d y at least partly black, is present. Then the radiation field is n o t really in a stable state. But thermodynamics can nevertheless be applied, just as it is true for metastable substances, for instance, for a mixture of hydrogen and oxygen. Changes in the radiation field are described by values of AE and AS, without taking into account the fact of the continuing disequilibrium. In the statistical analysis of isolated non-black radiation fields, i.e., of fields n o t in internal equilibrium, light of different wave lengths (colour) must be considered separately, because the mixing ratio is indefinite a priori. Lights of different colour must be ascribed different temperatures, according to the temperature of the black body in equilibrium with them. In practice this leads to enormous complications. Here we shall not deal with this further. In this context we may refer to the radiation that has been discovered recently and that apparently fills the Universe (Penzias and Wilson, 1965). Measurements of the spectral distribution have shown that the radiation is black with a temperature of 3 K. This radiation is considered to be a remnant of the "big bang", in which our Universe arose many gigayears ago. Because of the expansion of the Universe, the radiation, which originally had been very hot, was cooled down to 3 K without losing its character of black b o d y radiation.
5. The terrestrial entropy balance So far reference was made mainly to fields in equilibrium. While the discussion of such equilibria is important for an intuitive understanding of the possibilities of radiation to do work, also in living nature, it is unrealistic. In nature there are no containers or pistons with ideally reflecting walls, and they cannot be manufactured. Therefore radiation fields of arbitrary composition cannot be freely handled. Equil!brium fields
72 exist only within closed containers in interaction with the walls. The Earth is n o t in equilibrium, b u t in a more or less stationary state. This is true not only for the substances, but also for the radiation. Radiation flows from the hot Sun to the cold Earth and later from the Earth into space, far colder still. Yet broadly the energy and the entropy contents of the Earth have remained constant for several gigayears, as shown by geology and palaeontology. In a good approximation, absorption or emission of solar energy is the only appreciable interaction of the Earth with the environment in respect to energy or entropy. In the h o t interior of the Earth the equilibria either are already established or their establishment is so slow that changes are unimportant. Hence also the flow of energy out of the Earth is tiny (0.02% of the energy flow from the Sun to the Earth}, and much the same must be true for the entropy flow. Consequently in practice the terrestrial balance of energy and of entropy involves only a thin surface layer. Because entropy is produced all the time in irreversible processes in this layer, in the stationary state more entropy is emitted than absorbed here. It is also possible to speak of an inflow of "negative e n t r o p y " (SchrSdinger) or simply " n e g e n t r o p y " (Brillouin) instead of an outflow of entropy. A constant inflow of negentropy evidently goes n o t only into the terrestrial surface layer as a whole, b u t also to any particular part of the surface layer provided it is in a stationary state. This applies also to the biosphere and within it to the totality of the organisms, as expressed in 1888 by Boltzmann (see Boltzmann, 1905) in majestic words: "Therefore the general struggle of the organisms is n o t a struggle for the elements, nor for energy, which is present in large amounts in every body in the form of heat, unfortunately unchangeable, b u t the struggle for entropy, which becomes available in the transition from the hot Sun
to the cold Earth. To exploit this transition as far as possible, the plants spread o u t the immeasurable areas of their leaves and force solar energy in a manner as y e t unexplored, until it sinks down to the temperature level of the surface of the Earth, to carry o u t chemical syntheses of which we have no inkling as y e t in our laboratories. The products of this chemical kitchen are then the object of the struggles of the animal world." Thus the p h e n o m e n o n of life as a whole has been made possible by consumption of negentropy. What can n o w be said a b o u t the flow of energy and entropy? Generally within a homogeneous radiation field the flow of energy (specific intensity, brightness) K is related to the energy density e by K= eq/4n
(14)
where q is the velocity of light (Planck, 1913i). Similarly the entropy flow L is related to the entropy density (Planck, 1913j)
by L = sq/4~.
(15)
In the vacuum, c can be substituted for q. When a hole is drilled into the thin limiting wall of a homogeneous radiation field a diverging pencil of radiation escapes, within which the intensity of the radiation decreases with increasing distance. However, during its way across (empty) space the radiation maintains n o t only its energy (per unit solid angle), b u t also its entropy (likewise per unit solid angle}. The increasing distance from the source has no effect; this is seen already from the absence of a distance term in eqns (14) and (15}. Thus the process is reversible. That the straight "centrifugal" propagation is a reversible process, which does n o t increase entropy, is evident also from the fact that the ray8 may be reflected back into the source without loss and independently of distance by
73 means of an :imagined ideal spherical mirror. On the basis of the Boltzmann-Planck Law i t follows from the constancy of entropy that the state of order of the radiation field does not decrease in the propagation of the radiation, or, equivalently, that the probability of the state does not increase. Consequently the solar radiation meets the top layer of our atmosphere with its original energy and e n t r o p y / u n i t solid angle. The conservation (constancy) of the ratio of energy and enl~ropy implies, as has been se~n, conservation of the temperature of the radiation (Planck, 1913k). Therefore the situation in the straight propagation of light rays in space is entirely different from that in the diffuse flow of disordered homogeneous radiation into a vacuum, as discussed before. In the latter process the entropy of the radiation does iincrease. The maintenance of the quality of the solar radiation during its long journey through space can also be recognized directly from the fact that it can be concentrated again on Earth by means of lenses or mirrors. In the ideal case the temperature of the source can nearly be reached again. ("Nearly": because of the influence of our atmosphere). In a famous research laboratory at Odeillo in the French Pyrenees, concentrating mirrors are actually used for the production of very high temperature for the investigation of materials ("solar furnace").
6. Work from solar radiation Black radiation being characterized by a definite temperature, the maximum work W in the transition of radiation energy between different temperatures can be c o m p u t e d by means of the equation for the Carnot cycle, i.e., the equation for the work done by a gas in a reversible thermal process. The higher the temperature Tt of the heat source and the lower the temperature T2 of the heat sink, the larger the efficiency of the heat engine~i.e.,
the better the Carnot factor 7, the yield, 71 = W / Q
= (T2
- T2 )/TI .
(16)
Q is the heat absorbed at the upper temperature. T2 would in a steam engine be the temperature of the cooling water. As explained, no capacity for work (temperature) is lost in the straight propagation of radiation through space. Consequently the ideal efficiency of a (radiation) engine does n o t depend on distance from the Sun. Focussing the solar rays on a working substance with a mirror or lens, the substance can be heated to the temperature of the primary source in the Sun (6000 K). With a heat sink of the temperature of the terrestrial environment (300 K), excellent yields could in principle be obtained with a "radiation engine": ~7 = (6000 - 300)/6000 % 0 . 9 5 It follows from the independence o f the efficiency of the magnitude of the captured energy flux, i.e., of the dimensions of the engine, that a mirror or a lens should always still work as a collector, however small. That means that in the limiting case one could omit the collector altogether. But naturally the power supplied by the system would decrease with decreasing energy flux collected. Moreover, thermal insulation is more difficult with smaller systems. The directional correlation within the radiation can be destroyed by diffuse reflexion at a surface that is ideally white, but not a mirror. Such a surface is approximated, for example, by a layer of magnesia. By definition ideally white surfaces, like ideal mirrors, do not absorb any light at ~ll. Therefore white surfaces, like mirrors, do n o t change the energy content or the spectrum of the radiation. But after diffuse reflexion the radiation cannot be concentrated any more by lenses or mirrors. The effect is the same as if radiation had expanded, without doing work, into a vacuum. This is seen in a simple
74 thought experiment again. A pencil of solar radiation is introduced into a white hollow sphere. After equilibration, the radiation, still with the same energy and spectrum, fills the spherical space uniformly, but the radiation density per solid angle has dropped. After being made diffuse, the radiation is not in thermal equilibrium any more with the source. As in the case of expansion into a vacuum, it has now no defined temperature any more, but increased entropy. A definite (lowered) temperature, with entropy further increased, is established only after intern, al equilibration of the radiation. This two-step irreversible process must lead to a decrease in the capacity for work. A radiation engine with destroyed directional correlation would have decreased efficiency, especially after i n t e r n a l equilibration. Of course, the temperature of the radiation in internal equilibrium still much exceeds that of the ideally white reflector, to which by definition no energy was transferred. No equilibrium obtains between the substance of the reflector, which may be at the temperature of the environment, and the radiation field. According to computations by Spanner (1964), carried out in connection with the efficiency of plant photosynthesis, the temperature of solar radiation after complete diffuse reflexion on the surface of the Earth is only 1350 K. By the Carnot equation this corresponds, with a temperature of the heat sink of 300 K, to a maximum efficiency in the performance of work of 77%. T h e diminished capacity of cooled radiation to do work must, because of the general validity of thermodynamics, also be expressed in photochemical reactions. One would expect that on illumination in the case of endergonic reactions chemical states with high contents of free energy are favoured compared with states with less free energy. The higher the input of light energy, the further the reaction is driven. In thought experiments chemical actions could be carried out in analogy to ordinary (light-independent, "dark") reactions by
means of a reaction box of Van't Hoff, as described in the standard textbooks of physical chemistry. With the reaction box, one derives the affinity, i.e., maximum work, from the equilibrium constant in dark reactions. In photochemical reactions, one would conversely use the relationship to compute the equilibrium constant, i.e., the yield, from the input of free energy by means of photons. True, semipermeable walls (membranes) would have to be imagined that let pass excited, but not non-excited molecules, and vice versa. While there is no t h e r m o d y n a m i c a l objection against the assumption of such walls, the considerations would be formalistic rather than intuitive. This applies already to non-directional, isotropic, radiation. Thought experiments with radiation with directional correlation (less entropy/energy} would be even less intuitive. Moreover, in realistic considerations it would be necessary to take account of the quantization of the energy states of the reactants and products along the real reaction paths. Existing knowledge for all this is insufficient.
7. Applications to photosynthesis The light reaction in plant photosynthesis consists essentially in the photolysis of water (Van Niel, 1962). It is Crue that in physiological conditions elementary oxygen, but no elementary hydrogen is produced. It would escape uselessly. But a different strong reductant is generated, namely, the reduced form of a well defined iron-sulfur protein, ferredoxin (see Arnon, 1971). This is subsequently used for the production of biomass from CO2 and H20. The standard redox potential of the couple reduced/ oxidized ferredoxin is practically equal to that of the couple hydrogen gas/neutral water. H e n c e from the photochemical and thermochemical point of view the achievement of the plant is not less than if it actually produced free hydrogen.
75 For simplicity we therefore speak simply of water photolysis. Thermodyn,'unically we can imagine an idealized syst~m within an ideally white sphere that consists of the following components: (1) Black radiation (1350 K); (2) Water (300 K); (3) Arbitrary substances and devices (300 K). Component 3, 'which is not supposed to suffer any permanent change, can, in an expanded sense, be called a system of catalysts or simply as " a " catalyst. Now we carry through as a thought e~:periment, in the way usual in chemical thermodynamics, a reversible process differentially, i.e., to a sufficiently small extent that neither the quantities of the components nor the temperatures change perceptibly. The process consists in the performance of work on the basis of the temperature difference between 1350 and 300 K and, coupled with this process in a loss-free way, in a photolysis of an equivalent amount of water to give H~ + 0.5 O~, as the gases. In standard conditions this photolysis requires a free enthalpy AGo = 238 kJ/mol water. In principle there is no reason why the process should n o t be carried out reversibly, without entropy increase. Hence the work performed is limited only by the Carnot factor. This is, as explained, about 77% f o r diffuse solar radiation. A challenge to indicate a concrete possibility for the realization of the ideal l o s s - f r e e process can be rejected by thermodynamics as this is not concerned with m e c h a n i s m s . At the same time, the biophysicist admires the plant for obtaining, in optimum conditions, chemical energy from light with 33% efficiency in the form of biomass. From the point of view of efficiency optimum conditions apply when long-wave light (small quant~a) is used that nevertheless is still just fully active photosynthetically: red light of 680--700 nm. Biomass production clearly cannot occur with maximum
thermodynamical efficiency, as is true also for all chemical processes in industry. Already the need for sufficient speed, without which the process would be useless for the plant, leads to irreversibility. The fact that the sacrifice in efficiency made by the plant is not larger is all the more remarkable, as the entire process of photosynthesis up to the stage of carbohydrate includes 20--30 simple steps (Bassham and Calvin, 1957; Arnon, 1971), and surely every step contributes to loss. Within the reaction chain the 2 light reactions (Hill and Bendall, 1960) have key positions. In plant photosynthesis, they are connected in series and make possible the tremendous achievement of water photolysis. The ancestors in evolution of the plants, the photosynthetic (purple and green) bacteria were still satisfied with 1 light reaction (see Broda, 1975, 1978a). For this reason they cannot split water, and they need as substrates hydrogen compounds with less bond strength, typically H~S. Therefore they cannot set free oxygen, but only sulphur and the like. Incidentally, the principle of the utilization of several light reactions that follow in an ordered sequence, with each of the reactions taking place in a different absorbant partial system, is known only from living nature. So far it h a s never been developed independently, or even imitated successfully, in science or technology. The principle could be called the "superposition" principle. The energy problem of mankind could be solved if the cheap photolysis of water by means of solar radiation were possible on a commercial scale (Broda, 1975a, 1976, 1978b,c). The hydrogen would be introduced into a hydrogen e c o n o m y and serve for the supply of heat or electricity, of chemical and metallurgical reductant and even of biomass. Hydrogen can be cheaply stored and transported, and in its combustion it does not pollute the environment. A small part of the 170 X 10 ~2 kW of light energy that constantly flows towards the Earth could cover all energy requirements of mankind.
76 The best chance for abiotic photolysis may consist in the application of the "membrane principle", likewise invented by Nature. Artificial asymmetric membranes should be constructed that in "vectorial" photoreactions set free primary reductant only on one side, primary oxidant only on the other side. In this case, a recombination of the primary products to water and their loss would be prevented. All photosynthetic organisms have made use of the membrane principle for at least 3 gigayears (see Broda, 1975b, 1978a).
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