[MUSIC] The leaf is a remarkable machine, it uses extremely stable and easily available feed stock such as carbon dioxide and water and uses sunlight to convert it into a fuel like glucose. Humans then consume this fuel and use the energy derived from that to go about their daily activities. Now, the whole process is green with very, very minimum environmental footprint. Now, what can be learned from this process to engineer a new economy where we utilize sunlight and convert it into a fuel. Now, this fuel can then be consumed at a later time and convert it to electricity to power the world or convert to heat to give us warmth. In this video, you will learn how to think about an ideal leaf and how an ideal leaf would function and how we can use these ideas to develop and replicate in an engineered device. By the end of this module, you should be able to develop a thermodynamic model for the interaction of a radiation field with a photochemical system, for instance, sunlight and a plant. And now, using this thermodynamic model you should be able to evaluate the potential difference and the amount of work that can be derived as a result of photon absorption. The simplest picture for a system that can absorb photons from a source such as the sunlight is a two-level system. Now, let's consider two-level for the chemical system that consists of a collection of ground electronic states, termed as G, and excited electronic states, termed as E. Now, in systems where electrons are holes which can migrate easily, these collection of states are often referred to as the variance band on the conduction band respectively. Now, each of these electronic bands, usually consist of a number of vibrational sub-states. Now, when the rate of absorption of light quanta, causing excitations from the ground state G to the excited state E is rapid with respect to the thermal equilibrational of populations between the two bands. Then a transition back from the excited state E to the ground state G gives up some free energy which may be stored or used for chemical synthesis. The amount of work that can be done as a result of the absorption of each photon is limited by the product of the free energy change, and the quantum yield for the de-excitation pathway, which is coupled to work production. Now, there are two ways of viewing the excitations caused by the absorption of light. Now, the first view is the photoelectric view. In this view, the way we think about it is, the excitation can be considered as a way to increase the population of electrons in excited states in the excited state by a fixed number. Now, this is accompanied by a decrease in the population of electrons in another set of states, for instance, the ground state. Now this picture is adequate if the device operates primarily through electron migration. Now, a second view point is the photochemical view. Now, in this case the excitation maybe considered as producing an increase in the number of excited state molecular species. And a concomitant decrease in the number of ground state molecular species. Now, this picture's adequate if the light absorption can induce molecular rearrangement in a fast enough time scale. Now, in this video, we will take the photochemical view that is, we will analyze the change in the free energy of the light absorbing molecules to their ground state and to their excited state. Now, the action of light usually depletes the population of the ground state molecules only very slightly, altering the chemical activity of these species to a negligible extent. In the case of the potential difference arising between the bands, this is primarily due to the greatly increased population of molecules in the excited state. In order to evaluate the band to band potential difference mu caused by a radiation field in any given situation, we must first consider the conditions for the equilibrium between the band to band transitions and the radiation feed. Now, if we consider a reversible reaction, then this implies that there is no change in entropy accompanying the admission or absorption of radiation by the photochemical system at any frequency. Let's view this from the perspective of the radiation field, the entropy change corresponding to the loss of a photon of frequency mu from a radiation field may be evaluated by considering an equilibrium at that frequency mu between the field and a black body. Now, remember, the black body is in equilibrium with the radiation field at a frequency mu when the intensity of the radiation field obeys this relation. Now, this relation states that intensity scales as the inverse second power of the speed of light in that medium. It also has the occupancy factor given by the Bose–Einstein statistics relating to the Bosonic nature of photons. Now, the entropy gained by a by a black body upon absorption of a photon is given by energy absorbed divided by the temperature of the black body. In this case, that becomes the Planck's constant. H times the frequency mu divided by the temperature of the black body. Now, rearranging the equilibrium condition, we can find the entropy change of a black body upon loss of a photon from a radiation field to be given by. Now, assuming a canonical ensemble we can define a relationship between the potential difference mu and the change in entropy per photon observed using the relation. Now, lets shift our perspective to the point of the photochemical absorber. That is, a molecule that is absorbing sunlight for instance. Now we can assume that the photochemical absorber is isotropic in its interactions with the radiation field. Now, using this assumption we can integrate over the solid angle and equating the entropies we can then find the rate of photon absorption as. Now this relation using the condition that the entropy change in the radiation field due to a loss of photon is equal to the magnitude of the entropy change for the photochemical absorber due to the gain of a photon. The term 1 in this relation is simply due to stimulated emissions and can usually be neglected. Now, this leads to an expression that is given as. Now keep in mind that this does not take into account the actual properties of the absorber. This intensity needs to be multiplied by the absorption cross-section of that particular material to evaluate the total rate of excitation and emission. And this is given by. Now, for simplicity, we may assume that the absorption cross-section is independent of the band to band potential mu. Although, this may not always be the case. Now, changes in temperature can largely be ignored. Then, by multiplying the absorption cross-section for band to band excitation to the frequency dependent factors of the intensity, we can find that the emission spectrum as a function of the frequency mu, can be given as. The Planck law of relationship between absorption and emission may be used to calculate the potential developed in a full chemical system whenever the absorption spectrum and the incident light flux are known. Now we have the setup, the background machinery required to analyze our system. Now, the rate of band to band excitations resulting from an arbitrary radiation field, Is, is simply given by an integration over the frequency range. Now, the developed expression for the emission spectrum, we can easily evaluate the rate of radiative decay for a photochemical system having a potential difference mu. Now, the abbreviated frequency integral as L, note that this depends on the properties of the absorption cross-section as a function of frequency for the photochemical absorber. Now, in the ideal limit, these two rates will be equal and can be used to evaluate the maximum possible potential derivable for a photochemical system, having an absorption cross-section, sigma of mu and illuminated by a radiation field Is. However, non-radiative band to band transitions are frequently a significant source of relaxation from the excited state E to the ground state G. Now, for simplicity, we assume that the rate of induced G to E transitions is large with respect to all spontaneous excitations. Then, we can specify that the total rate of decay from the excited state to the ground state is simply kappa times the rate of radiative decay alone. Now, by equating the incoming rate, and the rate of decay, we can then determine the overall potential developed in the presence of nonradiative relaxations. Now, as kappa is the reciprocal of the luminescence quantum yield, it may be frequently be determined experimentally. For the remainder of our discussion, we'll assume that kappa is independent of the band to band potential, mu. Although, it appears that this is generally true only for noninteracting excitations obeying Boltzmann statistics. Now, having derived all these expressions, we're now in a position to evaluate the power stored by light absorption. Now work is one of the most popular commodities that can result from photochemical absorption of light. So that the frequency one desires, to maximize the amount of power stored by such a system, is a useful quantity. The amount of power stored is simply a product of two quantities. One, the potential difference developed mu and the second one which is simply the difference between the incoming rate of excitations and the rate of transition from the excited to the ground state that is not coupled to the work storage process. Now, from the expression developed earlier, mu0 is the potential difference developed in the absence of work storage process. We can define the quantum yield for the last process using the following relation. Now, we can relate the potential difference mu as. Now, the amount of power stored can be rewritten using the following relation. Now, this expression shows that the power can be maximized by an appropriate choice of the chemical potential mu and phi loss. Now, the power storage process is approximately maximal when phi loss is equal to kBT divided by mu0. Now, a useful way to understand this relationship is to look at a concrete example. Now, let's consider, that for the chemical absorber that can develop a potential difference of about one wort. This is, for instance, the case for a silicon semiconductor which has a bandgap about 1.1 electron volt. Now, in this case, the loss associated with this device is about 0.1 volts about 10% of the overall available free energy. Hence, the analysis that we have developed is actually an extremely useful and important one that should be taken into account when designing practical devices that use photon as an incoming energy source. To summarize, in this lecture, we developed a relationship that permits a ready calculation of the maximum light-induced chemical potential difference, which can be developed by a photochemical system if we know the incident light intensity, and the absorption spectrum. Now, this immediately tells us how much useful work can be derived by that system. Now, if on top of this if we have knowledge of the quantum yield for luminescence, then we have developed expressions that can calculate the actual potential developed. Now knowledge of this potential may be useful in examining any photochemical system that uses light to generate any sort of thermodynamic potential gradient. And this has been particularly useful in analyzing the energetics of photosynthesis.
