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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.

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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.

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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.

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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.