**5. Thermodynamic scale of the efficiency of chemical action of solar radiation**

Radiant energy conversion has a limit efficiency in natural processes. This efficiency is lower in solar, biological and chemical reactors. With the thermodynamic scale of efficiency of chemical action of solar radiation we will be able to compare the efficiency of natural processes and different reactors and estimate their commercial advantages. Such a scale is absent in the well\_known thermodynamic descriptions of the solar energy conversion, its storage and transportation to other energy generators (Steinfeld & Palumbo, 2001). Here the thermodynamic scale of the efficiency of chemical action of solar radiation is based on the Carnot theorem.

Chemical changes are linked to chemical potentials. In this work it is shown for the first time that the chemical action of solar radiation **S** on the reactant **R** 

$$\mathbf{R} + \mathbf{S} \leftrightarrow \mathbf{P} + \mathbf{M} \tag{9}$$

is so special that the difference of chemical potentials of substances R and P

$$
\mu\_{\mathbb{R}} - \mu\_{\mathbb{P}} = f(T) \tag{10}
$$

becomes a function of their temperature even in the idealized reverse process (9), if the chemical potential of solar radiation is accepted to be non-zero. Actually, there are no obstacles to use the function *f*(*T*) in thermodynamic calculations of solar chemical reactions, because in (Kondepudi & Prigogin, 1998) it is shown that the non–zero chemical potential of heat radiation does not contradict with the fundamental equation of the thermodynamics. The solar radiation is a black body radiation.

Let us consider a volume with a black body R, transparent walls and a thermostat T as an idealized solar chemical reactor. The chemical action of solar radiation S on the reactant R will be defined by a boundary condition

Let us leave the discussion of technological matters relating to the manipulation of antenna processes aside for the time being. We will devote a subsequent publication to this subject. Let us only remark here that the conversion of solar energy involving the participation of antenna molecules figures in the description of photosynthesis in biology. Every chlorophyll molecule in plant cells, which is a direct convertor of solar energy, is surrounded by a complex of 250- 400 pigment molecules (Raven et al., 1999). The thermodynamic aspects of photosynthesis in plants were studied in (Wuerfel, 2005; Landsberg, 1977), yet the idea of antenna for solar cells was not proposed. We hope that the notions of the antenna and working states of an absorber particles will make it possible to attain very high efficiencies of the radiant energy convertors, especially in those cases when solar radiation is not powerful enough to make solar cells work

This leads us to conclude that reemission of radiant energy by absorbent particles can be considered a quasi-static process. We can therefore hope that the concept of an antenna process, which is photon absorption and generation, can be used to find methods for attaining the efficiency of solar energy conversion close to the limiting efficiency without

Radiant energy conversion has a limit efficiency in natural processes. This efficiency is lower in solar, biological and chemical reactors. With the thermodynamic scale of efficiency of chemical action of solar radiation we will be able to compare the efficiency of natural processes and different reactors and estimate their commercial advantages. Such a scale is absent in the well\_known thermodynamic descriptions of the solar energy conversion, its storage and transportation to other energy generators (Steinfeld & Palumbo, 2001). Here the thermodynamic scale of the efficiency of chemical action of solar radiation is based on the

Chemical changes are linked to chemical potentials. In this work it is shown for the first time

 **R** + **S**↔**P** + **M** (9)

becomes a function of their temperature even in the idealized reverse process (9), if the chemical potential of solar radiation is accepted to be non-zero. Actually, there are no obstacles to use the function *f*(*T*) in thermodynamic calculations of solar chemical reactions, because in (Kondepudi & Prigogin, 1998) it is shown that the non–zero chemical potential of heat radiation does not contradict with the fundamental equation of the thermodynamics.

Let us consider a volume with a black body R, transparent walls and a thermostat T as an idealized solar chemical reactor. The chemical action of solar radiation S on the reactant R

μR – μP = *f*(*T*) (10)

**5. Thermodynamic scale of the efficiency of chemical action of solar** 

**4.8 Antenna processes in plants** 

**4.9 Conclusion** 

**radiation** 

Carnot theorem.

invoking band theory concepts.

efficiently yet suffices to drive photosynthesis in plants.

that the chemical action of solar radiation **S** on the reactant **R** 

The solar radiation is a black body radiation.

will be defined by a boundary condition

is so special that the difference of chemical potentials of substances R and P

$$
\mu\_{\rm IF} - \mu\_{\rm IF} = \mu\_{\rm IF} - \mu\_{\rm S} = f(T),
\tag{11}
$$

where μm is a chemical potential of heat radiation emitted by product P. Then the calculation of the function *f*(*T*) is simply reduced to the definition of a difference (μm–μS), because chemical potential of heat radiation does not depend on chemical composition of the radiator, and the numerical procedure for μm and μS is known and simple (Laptev, 2008). The chemical potential as an intensive parameter of the fundamental equation of

thermodynamics is defined by differentiation of characteristic functions on number of particles *N* (Laptev, 2010). The internal energy *U* as a characteristic function of the photon number

$$\mathcal{U}(V,\mathcal{N}) = (2.703\,\text{Nk})^{4/3} / (\text{\textdegree}V)^{1/3}$$

is calculated by the author in (Laptev, 2008, 2010) by a joint solution of two equations: the known characteristic function

$$\mathcal{U}(\mathcal{S}, V) = \sigma \text{ } V (\Im \mathcal{S} \mathcal{Y} 4 \text{ or } V)^{4/3} \text{ }$$

(Bazarov, 1964) and the expression (Couture & Zitoun, 2000; Mazenko, 2000)

$$\mathbf{N} = \mathbf{0}.\mathbf{370} \text{ or } \mathbf{T} \circ \mathbf{V}/\mathbf{k} = \mathbf{S} / \mathbf{3.602} \text{ k},\tag{12}$$

where *T*, *S*, *V* are temperature, entropy and volume of heat radiation, σ is the Stephan– Boltzmann constant, *k* is the Boltzmnn constant. In total differential of *U*(*V*, *N*) the partial derivative

( *U N* )*<sup>V</sup>* ≡ heat radiation = 3.602*kT* (13)

introduces a temperature dependence of chemical potential of heat radiation (Laptev, 2008, 2010). The function *U*(*S*, *V*, *N*) is an exception and is not a characteristic one because of the relationship (12).

The Sun is a total radiator with the temperature *T*S = 5800 K. According to (13), the chemical potential of solar radiation is 3.602*kT* = 173.7 kJ/mol. Then the difference *f*(*T*) = μm – μS is a function of the matter temperature *T*m. For example, *f*(*T*) = –165.0 kJ/mol when *T*m = 298.15 K, and it is zero when *T*m = *T*S. According to (13), the function *f*(*T*) can be presented as proportional to the dimensionless factor:

$$f(T) = \mu\_m - \mu\_S = -\mu\_S \text{ (1-}T\_m/T\_S\text{)}\dots$$

According to the Carnot theorem, this factor coincides with the efficiency of the Carnot engine ηC(*Т*m, *T*S). Then the function

$$f(T) / \ \mu\_S = -\text{\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tag{14}}$$

can characterize an efficiency of the idealized Carnot engine-reactor in known limit temperatures *T*m and *T*S.

In the heat engine there is no process converting heat into work without other changes, i.e. without compensation. The energy accepted by the heat receiver has the function of compensation. If the working body in the heat engine is a heat radiation with the limit temperatures *T*m and *T*S, then the compensation is presented by the radiation with temperature *T*m which is irradiated by the product P at the moment of its formation. We will call this radiation a compensation one in order to make a difference between this radiation and heat radiation of matter.

Photons as Working Body of Solar Engines 379

For example, water evaporation at 298.15 K under solar irradiation is caused by the

Δ*G* 298.15 = –228.61 – (–237.25) + (–165.0) = –156.4 kJ/mol.

The changes of the Gibbs energies calculated above have various signs: Δ*G*298.15 < 0 and Δ*G*<sup>0</sup> 298.15 > 0. It means that water evaporation at 298.15 K is possible only with participation of solar radiation. The efficiency ζ of the solar vapor engine will not exceed ζ(18) = 5.2%. There is no commercial advantage because the efficiency of the conventional vapor engines is higher. However, the efficiency of the solar engine may be higher than that of the vapor one

exchange, nor chemical conversion of solar energy cause any chemical changes in the system at this temperature. The processes (19) and (20) are demonstrative. Nevertheless, at the temperatures when Δ*G* becomes negative, the chemical changes will occur in the reaction mixture. So, in solar engines–reactors there is a lower limit of the temperature *T*m. For example, in (Steinfeld & Palumbo, 2001) it is reported that chemical reactors with solar

The functions Δ*G*(*T*) and ζ(*T*) describe various features of the chemical conversion of solar energy. As an illustration we consider the case when the phases R and P are in thermodynamic equilibrium. For example, the chemical potentials of the boiling water and the saturated vapor are equal. Then both their difference (μP – μR) and the efficiency ζ of the chemical action of solar radiation are zero, althought it follows from Eq. (21) that Δ*G*(*T*) < 0. Without solar irradiation the equation Δ*G*(*T*) = 0 determines the condition of the

The thermodynamic scale of efficiency ζ(*T*) of the chemical action of solar radiation presented in this paper is a necessary tool for choice of optimal design of the solar engines\_reactors. It is simple for application while its values are calculated from the experimentally obtained data of chemical potentials and temperature. Varying the values of chemical potentials and temperature makes it possible to model (with help of expressions (17), (21)) the properties of the working body, its thermodynamic state and optimal conditions for chemical changes in solar engines and reactors in order to bring commercial

It is known that solar energy for glucose synthesis is transmitted as work (Berg et al., 2010; Lehninger et al., 2008; Voet et al., 2008; Raven et al., 1999). Here it is shown for the first time that there are pigments which reemit solar photons whithout energy conversion in form of heat dissipation and work production. We found that this antenna pigments make 77% of all

298.15 = (μP – μR) = –228.61 – (–237.25) = 8.64 kJ/mol.

difference of the Gibbs energies

At the standard state (without solar irradiation)

Δ*G*<sup>0</sup>

if the condition Δ*G*298.15 < 0 and Δ*G*<sup>0</sup>

takes place is an illustrative example. If the condition is Δ*G*298.15 < 0 and Δ*G*<sup>0</sup>

advantages of alternative energy sources.

chemical action of solar radiation. If Δ*G*298.15 > 0 and Δ*G*<sup>0</sup>

radiation concentrators have the minimum optimal temperature 1150 K.

thermodynamic equilibrium, and the function ζ(*T*) loses its sense.

**6. Thermodynamic efficiency of the photosynthesis in plant cell** 

Δ*GT* = (μP – μR) + (μm – μS). (21)

298.15 > 0 is fulfilled. The plant cell where photosynthesis

298.15 < 0 the radiant heat exchange replaces the

298.15 > 0, then neither radiant heat

The efficiency of heat engines with working body consisting of matter and radiation is considered for the first time in (Laptev, 2008, 2010). During the cycle of such an engine–reactor the radiation is cooled from the temperature *T*S down to *T*m, causing chemical changes in the working body. The working bodies with stored energy or the compensational radiation are exported from the engine at the temperature *T*m. The efficiency of this heat engine is the base of the thermodynamic scale of solar radiation chemical action on the working body.

Assume that the reactant R at 298.15 K and solar radiation S with temperature 5800 K are imported in the idealized engine–reactor. The product P, which is saving and transporting stored radiant energy, is exported from the engine at 298.15 K. Limit working temperatures of such an engine are 298.15 K and 5800 K. Then, according to relationships (10), (11), (14), the equation

$$(\mu\_m - \mu\_s) \;/\; \mu\_s \mathbf{s} = -\mathbf{r}\_{\mathbb{R}} \left( T\_m \; T\_{\mathbb{S}} \right) \tag{15}$$

defines conditions of maintaining the chemical reaction at steady process at temperature *T*m in the idealized Carnot engine–reactor.

According to the Carnot theorem, the way working body receives energy, as well as the nature of the working body do not influence the efficiency of the heat engine. The efficiency remains the same under contact heat exchange between the same limit temperatures. The efficiency of such an idealized engine is

$$\text{tr}\_{\mathbb{I}0} \left( T\_{m\nu} \, T\_S \right) = \text{ (1} - T\_m \, / T\_S \text{)}.\tag{16}$$

Then the ratio of the values ηC and η0 from (15), (16)

$$\mathbf{r}\,\mathcal{J} = \mathbf{r}\_{\mathbb{R}} / \,\,\mathbf{r}\_{\mathbb{R}} = (\mu \,\, \rho - \mu \,\, \mathbf{a}) \,\, / \,\, \left[\mu \,\, \mathbf{s} \,\, (1 - T\_m / T\_S)\right]. \tag{17}$$

is a thermodynamic efficiency ζ of chemical action of solar radiation on the working body in the idealized engine–reactor.

We compare efficiencies ζ of the action of solar radiation on water in the working cycle of the idealized engine-reactor if the water at 298.15 K undergoes the following changes:

$$\rm H\_2O\_{water} + \rm S\_{solar\ rad} = H\_2O\_{vapor} + M\_{heat\ rad,v\text{vapor}} \tag{18}$$

$$\rm H\_2O\_{water} + S\_{solar\,rad} = H\_{2gas} + 1\%O\_{2gas} + M\_{heat\,rad,\,\rm H\_2} + 1\%M\_{heat\,rad,\rm O\_2} \tag{19}$$

$$\text{H}\_{2}\text{O}\_{\text{water}} + \text{S}\_{\text{solar rad.}} = \text{H}^{+}\_{\text{gas}} + \text{OH}^{-}\_{\text{gas}} + \text{M}\_{\text{heat rad..}}\text{H}^{+} + \text{M}\_{\text{heat rad..}}\text{OH}^{-}.\tag{20}$$

Chemical potentials of pure substances are equal to the Gibbs energies (Yungman & Glushko, 1999). In accordance with (17),

$$\mathcal{J}\_{\text{(18)}} = \begin{bmatrix} -228.61 \text{-(-237.25)} \end{bmatrix} \text{ / 173.7 / 0.95 = 0.052}$$

$$\mathcal{J}\_{\text{(19)}} = \mathcal{J}\_{\text{(18)}} \begin{bmatrix} 0 + \updownarrow \ 0 \ - \text{(-228.61)} \end{bmatrix} \text{ / 173.7 / 0.95 = 0.052 \text{x} \newline 1.39 = 0.072$$

$$\mathcal{J}\_{\text{(20)}} = \mathcal{J}\_{\text{(18)}} \begin{bmatrix} 1517.0 - 129.39 \ - \text{(-228.61)} \end{bmatrix} \text{ / 173.7 / 0.95 = 0.052 \text{x} \cdot 9.79 = 0.51$$

So, in the engine–reactor the reaction (20) may serve as the most effective mechanism of conversion of solar energy.

In the real solar chemical reactors the equilibrium between a matter and radiation is not achieved. In this case the driving force of the chemical process in the reactor at the temperature *T* will be smaller than the difference of Gibbs energies

The efficiency of heat engines with working body consisting of matter and radiation is considered for the first time in (Laptev, 2008, 2010). During the cycle of such an engine–reactor the radiation is cooled from the temperature *T*S down to *T*m, causing chemical changes in the working body. The working bodies with stored energy or the compensational radiation are exported from the engine at the temperature *T*m. The efficiency of this heat engine is the base

Assume that the reactant R at 298.15 K and solar radiation S with temperature 5800 K are imported in the idealized engine–reactor. The product P, which is saving and transporting stored radiant energy, is exported from the engine at 298.15 K. Limit working temperatures of such an engine are 298.15 K and 5800 K. Then, according to relationships (10), (11), (14), the

 ( *<sup>m</sup>* – *<sup>S</sup>*) / *<sup>S</sup>* = – ηс (*Тm*, *ТS*) (15) defines conditions of maintaining the chemical reaction at steady process at temperature *T*m

According to the Carnot theorem, the way working body receives energy, as well as the nature of the working body do not influence the efficiency of the heat engine. The efficiency remains the same under contact heat exchange between the same limit temperatures. The

is a thermodynamic efficiency ζ of chemical action of solar radiation on the working body in

We compare efficiencies ζ of the action of solar radiation on water in the working cycle of the idealized engine-reactor if the water at 298.15 K undergoes the following changes:

Chemical potentials of pure substances are equal to the Gibbs energies (Yungman &

ζ (18) = [–228.61–(–237.25)] / 173.7 / 0.95 = 0.052

ζ (19) = ζ (18) [0 + ½ ·0 – (–228.61)] / 173.7 / 0.95 = 0.052х1.39 = 0.072

ζ (20) = ζ (18) [1517.0 – 129.39 – (–228.61)] / 173.7 / 0.95 = 0.052х 9.79 = 0.51 So, in the engine–reactor the reaction (20) may serve as the most effective mechanism of

In the real solar chemical reactors the equilibrium between a matter and radiation is not achieved. In this case the driving force of the chemical process in the reactor at the

temperature *T* will be smaller than the difference of Gibbs energies

η0 (*Тm*, *ТS*) = (1–*Tm / TS*). (16)

ζ = ηс/ η0 = ( *<sup>P</sup>* – *<sup>R</sup>*) / [ *<sup>S</sup>* (1–*Tm / TS*)]. (17)

Н2Оwater + Ssolar rad.= Н2Оvapor + Mheat rad.,.vapor ; (18)

Н2Оwater + Ssolar rad.= Н2gas + ½О2gas + Mheat rad., Н2 + ½Mheat rad.,О2; (19)

Н2Оwater + Ssolar rad.= Н+gas + ОH <sup>⎯</sup> gas + Mheat rad., Н+ + Mheat rad., ОH⎯. (20)

of the thermodynamic scale of solar radiation chemical action on the working body.

equation

in the idealized Carnot engine–reactor.

efficiency of such an idealized engine is

Glushko, 1999). In accordance with (17),

conversion of solar energy.

the idealized engine–reactor.

Then the ratio of the values ηC and η0 from (15), (16)

$$
\Delta G\_T = (\mathfrak{\mu}\_P - \mathfrak{\mu}\_R) + (\mathfrak{\mu}\_m - \mathfrak{\mu}\_S). \tag{21}
$$

For example, water evaporation at 298.15 K under solar irradiation is caused by the difference of the Gibbs energies

$$
\Delta G\_{298.15} = -228.61 - (-237.25) + (-165.0) = -156.4 \text{ kJ/mol.}
$$

At the standard state (without solar irradiation)

$$
\Delta G^{0}{}\_{298.15} = (\text{μ} - \text{μ}\_{\text{R}}) = -228.61 - (-237.25) = 8.64 \text{ kJ/mol.}
$$

The changes of the Gibbs energies calculated above have various signs: Δ*G*298.15 < 0 and Δ*G*<sup>0</sup> 298.15 > 0. It means that water evaporation at 298.15 K is possible only with participation of solar radiation. The efficiency ζ of the solar vapor engine will not exceed ζ(18) = 5.2%. There is no commercial advantage because the efficiency of the conventional vapor engines is higher. However, the efficiency of the solar engine may be higher than that of the vapor one if the condition Δ*G*298.15 < 0 and Δ*G*<sup>0</sup> 298.15 > 0 is fulfilled. The plant cell where photosynthesis

takes place is an illustrative example.

If the condition is Δ*G*298.15 < 0 and Δ*G*<sup>0</sup> 298.15 < 0 the radiant heat exchange replaces the chemical action of solar radiation. If Δ*G*298.15 > 0 and Δ*G*<sup>0</sup> 298.15 > 0, then neither radiant heat exchange, nor chemical conversion of solar energy cause any chemical changes in the system at this temperature. The processes (19) and (20) are demonstrative. Nevertheless, at the temperatures when Δ*G* becomes negative, the chemical changes will occur in the reaction mixture. So, in solar engines–reactors there is a lower limit of the temperature *T*m. For example, in (Steinfeld & Palumbo, 2001) it is reported that chemical reactors with solar radiation concentrators have the minimum optimal temperature 1150 K.

The functions Δ*G*(*T*) and ζ(*T*) describe various features of the chemical conversion of solar energy. As an illustration we consider the case when the phases R and P are in thermodynamic equilibrium. For example, the chemical potentials of the boiling water and the saturated vapor are equal. Then both their difference (μP – μR) and the efficiency ζ of the chemical action of solar radiation are zero, althought it follows from Eq. (21) that Δ*G*(*T*) < 0. Without solar irradiation the equation Δ*G*(*T*) = 0 determines the condition of the thermodynamic equilibrium, and the function ζ(*T*) loses its sense.

The thermodynamic scale of efficiency ζ(*T*) of the chemical action of solar radiation presented in this paper is a necessary tool for choice of optimal design of the solar engines\_reactors. It is simple for application while its values are calculated from the experimentally obtained data of chemical potentials and temperature. Varying the values of chemical potentials and temperature makes it possible to model (with help of expressions (17), (21)) the properties of the working body, its thermodynamic state and optimal conditions for chemical changes in solar engines and reactors in order to bring commercial advantages of alternative energy sources.

### **6. Thermodynamic efficiency of the photosynthesis in plant cell**

It is known that solar energy for glucose synthesis is transmitted as work (Berg et al., 2010; Lehninger et al., 2008; Voet et al., 2008; Raven et al., 1999). Here it is shown for the first time that there are pigments which reemit solar photons whithout energy conversion in form of heat dissipation and work production. We found that this antenna pigments make 77% of all

Photons as Working Body of Solar Engines 381

It is known (Laptev, 2009) that in the idealized Carnot solar engine–reactor solar radiation S

**R**reagent + **S**solar radiation ↔ **P**product + **M** thermal radiation of product

where μP, μR are chemical potentials of the substances, μS is the chemical potential of solar radiation equal to 3.602*kT*S = 173.7 kJ/mol. The efficiency of use of water for alternative fuel

Water is a participant of metabolism. It is produced during the synthesis of adenosine triphosphate (ATP) from the adenosine diphosphate (ADP) and the orthophosphate (Pi)

Water is consumed during the synthesis of the reduced form of the nicotinamide adenine

2NADP+ + 2H2O = 2NADPH + O2 + 2H+thylakoid (24)

 6СО2 + 6Н2О = С6Н12О6 + 6О2. (25) Changes of the Gibbs energies or chemical potentials of substances in the reactions (23)–(25)

The photosynthesis is an example of joint chemical action of matter and radiation in the cycle of the idealized engine–reactor, when the water molecule undergoes the changes according to the reactions (23)–(25). According to (22), the photosynthesis efficiency ζPh in

ζ(5) ×1/2ζ(6) × 1/6ζ(7) = 71%.

known in the solar cell theory (Wuerfel, 2005) as the efficiency of the joint chemical action of the radiation and matter per cycle. ζPh and the temperature dependence *η*L are shown in Fig. 12 by the point *F* and the curve *LB* respectively. They are compared with the temperature dependence of efficiencies *η*0*η*C*η*U (curve *CB*) and *η*0S*η*U (curve *KB*). Value *η*U is close to unity

We draw in Fig. 12 an isotherm *t*–*t*' of *η* values for *T*A = 300 K. It is found that *η*0S = 94.8% at the interception point *K*, *η*L = 93.2% at the point *L* and *η*0*η*C*η*U = 0.635% at the point *C*. The following question arises: which processes give the chloroplasts energy for overcoming the

First of all one should note that the conversion of solar energy into heat in grana has an efficiency *η<sup>g</sup>* smaller than *η*U of the radiant heat exchange for black bodies. From (26) follows

that the efficiency ζPh cannot reach the value *η*L due to necessary condition *η<sup>g</sup>* < *η*U.

ζ = (μP – μR)/[μS/(1 – *T*А/*T*S)], (22)

ADP + Pi = ATP + H2O. (23)

*η*L = *η*U – 4*T*0/3*T*S + 4*T*<sup>0</sup> *T*A3/3 *T*S4, (26)

)

produces at the temperature *T*A a chemical action on the reagent R

dinucleotide phosphate (NADPH) from its oxidized form (NADP<sup>+</sup>

are 30.5, 438 and 2850 kJ/mol, respectively (Voet et al., 2008).

The efficiency ζPh is smaller than the Landsberg limiting efficiency

point *C* and achieving an efficiency ζPh = 71% at the point *F*?

with efficiency

this model is

because (*T*A/*T*S)

4 ~ 10–5.

synthesis is calculated in (Laptev, 2009).

and during the glucose synthesis

pigment molecula in a photosystem. Their existance and participation in energy transfer allow chloroplasts to overcome the efficiency threshold for working pigments as classic heat engine and reach 71% efficiency for light and dark photosynthesis reactions. Formula for efficiency calculation take into account differences of photosynthesis in specific cells. We are also able to find the efficiency of glycolysis, Calvin and Krebs cycles in different organisms.

The Sun supplies plants with energy. Only 0.001 of the solar energy reaching the Earth surface is used for photosynthesis (Nelson & Cox, 2008; Pechurkin, 1988) producing about 1014 kg of green plant mass per year (Odum, 1983). Photosynthesis is thought to be a low– effective process (Ivanov, 2008). The limiting efficiency of green plant is defined to be 5% as a ratio of the absorbed solar energy and energy of photosynthesis products (Odum,1983; Ivanov, 2008). Here is shown that the photosynthesis efficiency is significantly higher (71% instead of 5%) and it is calculated as the Carnot efficiency of the solar engine\_reactor with radiation and matter as a single working body.

The photosynthesis takes place in the chloroplasts containing enclosed stroma, a concentrated solution of enzymes. Here occure the dark reactions of the photosynthesis of glucose and other substances from water and carbon dioxide. The chlorophyll traps the solar photon in photosynthesis membranes. The single membrane forms a disklike sac, or a thylakoid. It encloses the lumen, the fluid where the light reactions take place. The thylakoids are forming granum (Voet et al., 2008; Berg et al., 2010). Stacks of grana are immersed into the stroma.

When solar radiation with the temperature *T*S is cooled in the thylakoid down to the temperature *T*A, the amount of evolved radiant heat is a fraction

$$\eta\_{\mathsf{U}} = 1 - (T\_{\mathsf{A}}/T\_{\mathsf{S}})^4$$

of the energy of incident solar radiation (Wuerfel, 2005). The value *η*U is considered here as an efficiency of radiant heat exchange between the black body and solar radiation (Laptev, 2006).

Tylakoids and grana as objects of intensive radiant heat exchange have a higher temperature than the stroma. Assume the lumen in the tylakoid has the temperature *T*A = 300 K and the stroma, inner and outer membranes of the chloroplast have the temperature *T*0 = 298 K. The solar radiation temperature *T*S equals to 5800 K.

The limiting temperatures *T*0, *T*A in the chloroplast and temperature *T*S of solar radiation allow to imagine a heat engine performing work of synthesis, transport and accumulation of substances. In idealized Carnot case solar radiation performs work in tylakoid with efficiency

$$\eta\_{\mathbb{C}} = 1 - T\_{\mathbb{A}}/T\_{\mathbb{S}} = 0.948\_{\prime}$$

and the matter in the stroma performs work with an efficiency

$$\eta\_0 = 1 \text{--} \ T\_0 / \ T\_\text{A} = 0.0067.5$$

The efficiency *η*0*η*C of these imagined engines is 0.00635.

The product *η*0*η*C equals to the sum *η*0 + *η*C – *η*0S (Laptev, 2006). Value *η*0S is the efficiency of Carnot cycle where the isotherm *T*S relates to the radiation, and the isotherm *T*0 relates to the matter. The values *η*0S and *η*C are practically the same for chosen temperatures and *η*0S/*η*0*η*C = 150. It means that the engine where matter and radiation performing work are a single working body has 150 times higher efficiency than the chain of two engines where matter and radiation perform work separately.

It is known (Laptev, 2009) that in the idealized Carnot solar engine–reactor solar radiation S produces at the temperature *T*A a chemical action on the reagent R

$$\mathbf{R}\_{\text{reangent}} + \mathbf{S}\_{\text{solar radiation}} \leftrightarrow \mathbf{P}\_{\text{product}} + \mathbf{M}\_{\text{thermal radiation}} \text{ of product}$$

with efficiency

380 Solar Cells – New Aspects and Solutions

pigment molecula in a photosystem. Their existance and participation in energy transfer allow chloroplasts to overcome the efficiency threshold for working pigments as classic heat engine and reach 71% efficiency for light and dark photosynthesis reactions. Formula for efficiency calculation take into account differences of photosynthesis in specific cells. We are also able to find the efficiency of glycolysis, Calvin and Krebs cycles in different organisms. The Sun supplies plants with energy. Only 0.001 of the solar energy reaching the Earth surface is used for photosynthesis (Nelson & Cox, 2008; Pechurkin, 1988) producing about 1014 kg of green plant mass per year (Odum, 1983). Photosynthesis is thought to be a low– effective process (Ivanov, 2008). The limiting efficiency of green plant is defined to be 5% as a ratio of the absorbed solar energy and energy of photosynthesis products (Odum,1983; Ivanov, 2008). Here is shown that the photosynthesis efficiency is significantly higher (71% instead of 5%) and it is calculated as the Carnot efficiency of the solar engine\_reactor with

The photosynthesis takes place in the chloroplasts containing enclosed stroma, a concentrated solution of enzymes. Here occure the dark reactions of the photosynthesis of glucose and other substances from water and carbon dioxide. The chlorophyll traps the solar photon in photosynthesis membranes. The single membrane forms a disklike sac, or a thylakoid. It encloses the lumen, the fluid where the light reactions take place. The thylakoids are forming granum (Voet et al., 2008; Berg et al., 2010). Stacks of grana are

When solar radiation with the temperature *T*S is cooled in the thylakoid down to the

U = 1 – (*Т*A/*T*S)4 of the energy of incident solar radiation (Wuerfel, 2005). The value *η*U is considered here as an efficiency of radiant heat exchange between the black body and solar radiation (Laptev,

Tylakoids and grana as objects of intensive radiant heat exchange have a higher temperature than the stroma. Assume the lumen in the tylakoid has the temperature *T*A = 300 K and the stroma, inner and outer membranes of the chloroplast have the temperature *T*0 = 298 K. The

The limiting temperatures *T*0, *T*A in the chloroplast and temperature *T*S of solar radiation allow to imagine a heat engine performing work of synthesis, transport and accumulation of substances. In idealized Carnot case solar radiation performs work in tylakoid with efficiency

C = 1 - *Т*A/*T*S= 0.948,

0 = 1- *T*0/*T*A = 0.0067.

The product *η*0*η*C equals to the sum *η*0 + *η*C – *η*0S (Laptev, 2006). Value *η*0S is the efficiency of Carnot cycle where the isotherm *T*S relates to the radiation, and the isotherm *T*0 relates to the matter. The values *η*0S and *η*C are practically the same for chosen temperatures and *η*0S/*η*0*η*C = 150. It means that the engine where matter and radiation performing work are a single working body has 150 times higher efficiency than the chain of two engines where

radiation and matter as a single working body.

solar radiation temperature *T*S equals to 5800 K.

temperature *T*A, the amount of evolved radiant heat is a fraction

and the matter in the stroma performs work with an efficiency

The efficiency *η*0*η*C of these imagined engines is 0.00635.

matter and radiation perform work separately.

immersed into the stroma.

2006).

$$\zeta = (\mathfrak{\mu}\_{\text{P}} - \mathfrak{\mu}\_{\text{R}}) / [\mathfrak{\mu}\_{\text{S}} / (1 - T\_{\text{A}} / T\_{\text{S}})] \,\tag{22}$$

where μP, μR are chemical potentials of the substances, μS is the chemical potential of solar radiation equal to 3.602*kT*S = 173.7 kJ/mol. The efficiency of use of water for alternative fuel synthesis is calculated in (Laptev, 2009).

Water is a participant of metabolism. It is produced during the synthesis of adenosine triphosphate (ATP) from the adenosine diphosphate (ADP) and the orthophosphate (Pi)

$$\rm{ADP} + \rm{P\_i} = \rm{ATP} + \rm{H\_2O}.\tag{23}$$

Water is consumed during the synthesis of the reduced form of the nicotinamide adenine dinucleotide phosphate (NADPH) from its oxidized form (NADP<sup>+</sup> )

$$2\text{NADP\*} + 2\text{H}\_2\text{O} = 2\text{NADPH} + \text{O}\_2 + 2\text{H}^\* \text{thylakoid} \tag{24}$$

and during the glucose synthesis

$$\text{\textbullet{CO}}\text{\textbullet} + \text{\textbullet}\text{H}\_2\text{O} = \text{C}\_6\text{H}\_{12}\text{O}\_6 + \text{\textbullet}\text{O}\_2.\tag{25}$$

Changes of the Gibbs energies or chemical potentials of substances in the reactions (23)–(25) are 30.5, 438 and 2850 kJ/mol, respectively (Voet et al., 2008).

The photosynthesis is an example of joint chemical action of matter and radiation in the cycle of the idealized engine–reactor, when the water molecule undergoes the changes according to the reactions (23)–(25). According to (22), the photosynthesis efficiency ζPh in this model is

$$
\zeta\_{(5)} \times 1/2 \zeta\_{(6)} \times 1/6 \zeta\_{(7)} = 71\,\%.
$$

The efficiency ζPh is smaller than the Landsberg limiting efficiency

$$
\eta\_{\rm L} = \eta\_{\rm U} - 4T\_0/3T\_{\rm S} + 4T\_0 \, T\_{\rm A}^3/3 \, T\_{\rm S} \, \tag{26}
$$

known in the solar cell theory (Wuerfel, 2005) as the efficiency of the joint chemical action of the radiation and matter per cycle. ζPh and the temperature dependence *η*L are shown in Fig. 12 by the point *F* and the curve *LB* respectively. They are compared with the temperature dependence of efficiencies *η*0*η*C*η*U (curve *CB*) and *η*0S*η*U (curve *KB*). Value *η*U is close to unity because (*T*A/*T*S) 4 ~ 10–5.

We draw in Fig. 12 an isotherm *t*–*t*' of *η* values for *T*A = 300 K. It is found that *η*0S = 94.8% at the interception point *K*, *η*L = 93.2% at the point *L* and *η*0*η*C*η*U = 0.635% at the point *C*. The following question arises: which processes give the chloroplasts energy for overcoming the point *C* and achieving an efficiency ζPh = 71% at the point *F*?

First of all one should note that the conversion of solar energy into heat in grana has an efficiency *η<sup>g</sup>* smaller than *η*U of the radiant heat exchange for black bodies. From (26) follows that the efficiency ζPh cannot reach the value *η*L due to necessary condition *η<sup>g</sup>* < *η*U.

Photons as Working Body of Solar Engines 383

Schemes of working and antenna cycles are shown in Fig. 13. Working pigment (a) is excited by the photon in the transition 1 → 3. Transition 3 → 2 corresponds to the heat compensation in the chloroplast as engine–reactor. The evolved energy during the transition

When the antenna process passes beside the reaction centre, the photosystems make the reversible reemissions. Fig. 13 presents an interpretation of absorption and emission of photons in antenna cycles. The reemission 2 → 3 → 2 shows a radiant heat exchange. The reemissions 1 → 2 → 1 and 1 → 3 → 1 take place according to (27). Examples are the

According to the thermodynamic postulate, the efficiency of reversible process is limited. In our opinition, just the antenna processes in the pigment molecules of the tylakoid membrane allow the photosystems to overcome the forbidden line (for a heat engine efficiency) *CB* in Fig. 12 and to achieve the efficiency ζPh = 71% in the light and dark photosynthesis reactions. There are no difficulties in taking into account in (22) the features of the photosynthesis in different cells. The efficiency of glycolyse, Calvin and Krebs cycles in various living structures may be calculated by the substitution of solar radiation chemical potential in the expression (22) by the change of chemical potentials of substances in the chemical reaction. The cell is considered in biology as a biochemical engine. Chemistry and physics know attempts to present the plant photosynthesis as a working cycle of a solar heat engine (Landsberg, 1977). The physical action of solar radiation on the matter of nonliving systems during antenna and working cycles of the heat engine is described in (Laptev, 2005, 2008). In this article the Carnot theorem has been used for calculation of the thermodynamic

2 → 1 is converted into the work of electron transfer or ATP and NADPH synthesis.

efficiency of the photosynthesis in plants; it is found that the efficiency is 71%.

Fig. 13. The interpretation of energy transitions in the work (a) and antenna (b) cycles. Level

One can hope that the thermodynamic comparison of antenna and working states of pigments in the chloroplast made in this work will open new ways for improving technologies of solar

Thermal radiation is a unique thermodynamic system while the expression d*U*=*T*d*S*–*p*d*V* for internal energy *U*, entropy *S*, and volume *V* holds the properties of the fundamental

*1* shows the ground states, levels *2*, *3* present excited states of pigment molecules.

cells and synthesis of alternative energy sources from the plant material.

**7. Condensate of thermal radiation** 

pigments in chromoplasts.

Besides in the thylakoid membrane the photon reemissions take place without heat dissipation (Voet et al., 2008; Berg et al., 2010). The efficiency area between the curves *LB*  and *CB* relates to photon reemissions or antenna processes. They can be reversible and irreversible. The efficiencies of reversible and irreversible processes are different. Then the point *F* in the isotherm *t*–*t*' is the efficiency of engine with the reversible and irreversible antenna cycles.

The antenna process performs the solar photon energy transfer into reaction centre of the photosystem. Their illustration is given in (Voet et al., 2008; Berg et al., 2010). Every photosystem fixes from 250 to 400 pigments around the reaction center (Raven et al., 1999). In our opinion a single pigment performs reversible or irreversible antenna cycles. The antenna cycles form antenna process. How many pigments make the reversible process in the photosynthetic antenna complex?

One can calculate the fraction of pigments performing the reversible antenna process if the line *L*С in Fig. 12 is supposed to have the value equal unity. In this case the point *F*  corresponds to a value *x* = ζ/(*η*L – *η*0*η*C*η*U) = 0.167. This means that 76.7% of pigments make the revesible antenna process. 23.3% of remaining pigments make an irreversible energy transfer between the pigments to the reaction centres. The radiant excitation of electron in photosystem occurs as follows:

$$\text{chlorophyll } a + \text{photon} \leftrightarrow \text{chlorophyll } a^\* + e^-. \tag{27}$$

The analogous photon absorption takes place also in the chlorophylls *b*, *c*, *d*, various carotenes and xanthophylls contained in different photosystems (Voet et al., 2008; Berg et al., 2010). The excitation of an electron in the photosystems P680 and P700 are used here as illustrations of the reversible and irreversible antenna processes.

Fig. 12. The curve *CB* is the efficiency of the two Carnot engines (Laptev, 2005). The curve *LB*  is the efficiency of the reversible heat engine in which solar radiation performs work in combination with a substance (Wuerfel, 2005). The curve *KB* is the efficiency of the Carnot solar engine\_reactor (Laptev, 2006), multiplied by the efficiency *η*U of the heat exchange between black bodies. The isotherm *t*–*t*' corresponds to the temperature 300 K. The calculated photosynthesis efficiency is presented by the point *F* in the isotherm.

Besides in the thylakoid membrane the photon reemissions take place without heat dissipation (Voet et al., 2008; Berg et al., 2010). The efficiency area between the curves *LB*  and *CB* relates to photon reemissions or antenna processes. They can be reversible and irreversible. The efficiencies of reversible and irreversible processes are different. Then the point *F* in the isotherm *t*–*t*' is the efficiency of engine with the reversible and irreversible

The antenna process performs the solar photon energy transfer into reaction centre of the photosystem. Their illustration is given in (Voet et al., 2008; Berg et al., 2010). Every photosystem fixes from 250 to 400 pigments around the reaction center (Raven et al., 1999). In our opinion a single pigment performs reversible or irreversible antenna cycles. The antenna cycles form antenna process. How many pigments make the reversible process in

One can calculate the fraction of pigments performing the reversible antenna process if the line *L*С in Fig. 12 is supposed to have the value equal unity. In this case the point *F*  corresponds to a value *x* = ζ/(*η*L – *η*0*η*C*η*U) = 0.167. This means that 76.7% of pigments make the revesible antenna process. 23.3% of remaining pigments make an irreversible energy transfer between the pigments to the reaction centres. The radiant excitation of electron in

 chlorophyll *a* + photon ↔ chlorophyll *a*+ + *e*–. (27) The analogous photon absorption takes place also in the chlorophylls *b*, *c*, *d*, various carotenes and xanthophylls contained in different photosystems (Voet et al., 2008; Berg et al., 2010). The excitation of an electron in the photosystems P680 and P700 are used here as

Fig. 12. The curve *CB* is the efficiency of the two Carnot engines (Laptev, 2005). The curve *LB*  is the efficiency of the reversible heat engine in which solar radiation performs work in combination with a substance (Wuerfel, 2005). The curve *KB* is the efficiency of the Carnot solar engine\_reactor (Laptev, 2006), multiplied by the efficiency *η*U of the heat exchange between black bodies. The isotherm *t*–*t*' corresponds to the temperature 300 K. The calculated photosynthesis efficiency is presented by the point *F* in the isotherm.

illustrations of the reversible and irreversible antenna processes.

antenna cycles.

the photosynthetic antenna complex?

photosystem occurs as follows:

Schemes of working and antenna cycles are shown in Fig. 13. Working pigment (a) is excited by the photon in the transition 1 → 3. Transition 3 → 2 corresponds to the heat compensation in the chloroplast as engine–reactor. The evolved energy during the transition 2 → 1 is converted into the work of electron transfer or ATP and NADPH synthesis.

When the antenna process passes beside the reaction centre, the photosystems make the reversible reemissions. Fig. 13 presents an interpretation of absorption and emission of photons in antenna cycles. The reemission 2 → 3 → 2 shows a radiant heat exchange. The reemissions 1 → 2 → 1 and 1 → 3 → 1 take place according to (27). Examples are the pigments in chromoplasts.

According to the thermodynamic postulate, the efficiency of reversible process is limited. In our opinition, just the antenna processes in the pigment molecules of the tylakoid membrane allow the photosystems to overcome the forbidden line (for a heat engine efficiency) *CB* in Fig. 12 and to achieve the efficiency ζPh = 71% in the light and dark photosynthesis reactions.

There are no difficulties in taking into account in (22) the features of the photosynthesis in different cells. The efficiency of glycolyse, Calvin and Krebs cycles in various living structures may be calculated by the substitution of solar radiation chemical potential in the expression (22) by the change of chemical potentials of substances in the chemical reaction.

The cell is considered in biology as a biochemical engine. Chemistry and physics know attempts to present the plant photosynthesis as a working cycle of a solar heat engine (Landsberg, 1977). The physical action of solar radiation on the matter of nonliving systems during antenna and working cycles of the heat engine is described in (Laptev, 2005, 2008). In this article the Carnot theorem has been used for calculation of the thermodynamic efficiency of the photosynthesis in plants; it is found that the efficiency is 71%.

Fig. 13. The interpretation of energy transitions in the work (a) and antenna (b) cycles. Level *1* shows the ground states, levels *2*, *3* present excited states of pigment molecules.

One can hope that the thermodynamic comparison of antenna and working states of pigments in the chloroplast made in this work will open new ways for improving technologies of solar cells and synthesis of alternative energy sources from the plant material.
