welcome back in this lecture we will introduce the central equation of electrochemical kinetics which is the butler warmer equation there are different approaches to introducing this equation the approach we would take is the one that relates most to experimental chemical kinetics this can be quantified mathematically but many of the important parameters is obtained from experiments what is that we want to describe here we show the possibility of two electrochemical reactions so when you vary the potential there is a possibility that oxygen can evolve if the potential is varied in the anodic direction if the potential is varied in the cathodic direction hydrogen can evolve the measure of the rate of evolution of oxygen and hydrogen is the current that is subsur the direction of current across the electrode electrolyte interface will change depending upon whether we are in this region or in this region there is also a region where there is no current uh passing through or negligible current passes through so we want to describe this kind of scenario so you have seen this in chemical kinetics also what is the typical variable the typical variable is temperature as the temperature is varied the rate of chemical reactions can increase in an exponential manner okay so let's try to build all that analogy here the equivalent of temperature is the electrochemical potential as the potential is vary beyond a particular point the rate which is measured by current is varying in an exponential manner so varying exponential manner here and varying exponential manner can we describe this system okay so we had introduced a relevant quantity to further describe this electrode potential based on the over potential right supposing we associate an equilibrium potential for this reaction to occur so the potential which we apply on the electrode this is an exotherm endothermic reaction so we have to apply a potential on the electrode that potential is above that of the equilibrium potential that is called the over potential ok so as we keep moving here we are increasing the over potential there seems to be an exponential change in rates of oxygen evolution as the o potential is increased so this is the kind of experimental observation that is measured in a variety of electrochemical systems what is the underlying mathematical framework is the agenda for the for this lecture there are two approaches to introducing this butler warmer equation to broad approaches as i mentioned in this lecture we will take an approach that is closest to phenomenological chemical kinetics what do we mean by this word this word refers to what can be experimentally measured things are going to be quantified but certain important parameters are going to be obtained from experiments right it has certain advantages but it also has certain disadvantages the main disadvantage is that there are there is another approach which has greater interpretational value if there's an approach to introducing butler warmer equation wherein the potential energy surface involved in an electrochemical reaction is introduced and that has certain value it seems to give a greater mechanistic insight into the problem but this interpretation is not unambiguous it's not straightforward so so so we are not going to first adopt that approach we are going to first use this approach in the next lecture we will introduce this approach in addition i would like to stress that we are going to take the simplest case the simplest case being a single electron transfer reaction okay so that is the system we are going to introduce uh most of the electrochemical reaction involved multiple electrochemical reaction but this simplicity of single electron transfer reaction makes the analysis more straightforward on much of the analysis we make using single electron transfer can be extended to multi-electron transfer reactions so with these preambles we will get on with our analysis what is that we are trying to analyze we are trying to look at an electrochemical reaction so when we say there is an electrochemical reaction occurring that means electrons are part of the reactants or the products okay so there is a reduced species in solution which loses an electron and becomes an oxidized species and there is an electron in the metal these two are in the um electrolyte most common electrolyte being the aqueous electric the water is the most common liquid so an example of such a system is this has been studied very well this is a model system in electrochemistry in this case the reduced and oxidized species both are ionic species there they have charges here typically what is getting reduced and oxidized to get to understand that you should look at the oxidation number of the cation the catheters oxidation number would change but both react this while these two are in the chat state that need not always be the case even when species are not charged it can still be in the solution so these are some of the systems which we are looking at and what are the central features here there's a single electron transfer okay so there's one electron that is a stoichiometric coefficient here is one which is not always the case in electrochemical reaction but as i said all the analysis which you are going to be making in this lecture can easily be extended so let's understand further what is the challenge in modeling these systems as i said what is it empirically observed is that the rate is proportional in some region in it may appear to have a linear dependence and beyond a particular o potential it can have an exponential dependence to construct a scientific theory for this system there are certain challenges what are the challenges there are two main challenges a lot has been discussed about chemical equilibrium chemical thermodynamics and in some cases even statistical thermodynamics but all that has been discussed in chemical thermodynamics and statistical thermodynamics is about systems in equilibrium okay so we know the thermodynamics of chemical equilibrium the statistical thermodynamics of chemical equilibrium is also reasonably straightforward however when you have a chemical reaction it is not an equilibrium process okay there is a finite rate of a chemical reaction so we are inherently describing a system not in equilibrium okay so the science of systems that are not in equilibrium is not well developed it is still one of the biggest open fields okay wherein there are a lot of open questions in uh this field okay so the thermodynamics or statistical thermodynamics of systems not in equilibrium okay so have you if you think about uh whether you've looked at systems not in equilibrium if based on your engineering education most of the most of you are coming from let's say a chemical background or a materials pattern or at the most physics platform you might have looked at description of rates okay transport processes for example in momentum transfer heat transfer and mass transfer you have looked at rates but such theories are continuum theories okay so these are macroscopic theories we do not really introduce how to describe the rate processes involving molecular systems okay so that is the issue okay so we know certain theories where rates steady states and rates are well described consistently and compared against experiments however these are macroscopic theories okay they are not readily transferable to molecular systems okay so we are somewhere in between okay we are having a molecular system this is a molecular system for which we know the mathematical equations underlying chemical equilibrium and there is this other extreme where the rate processes the equations the underlying rate processes are known but that is of macroscopic systems so let us see how we are going to introduce what we know to develop equations that is relevant to electrochemical kinetics okay system that is not in equilibrium but yet we would use certain principles of chemical equilibrium to describe the system let's move on i would like to emphasize that if we want to model non-equilibrium systems we have to understand the equilibrium of such systems we are going to use these principles principles at equilibrium to model the system that is not in equilibrium so here we are looking at electrochemical kinetics that are cell assisted not in equilibrium but we are still going to use some principles of chemical equilibrium we'll see how we are going to use it so this is the reaction which we are thinking about so we can define a net reaction that is dominantly controlled by the electrode potential okay so that is the dominant variable that is going to change the net rate of the reaction we are going to rate write the net rate of the reaction as we write in chemical kindness a forward reaction minus the backward reaction this is indicative of the forward reaction so when the forward reaction proceeds you get oxidized species and this is the backward reaction when backward reduce reaction proceeds you get the reduced reaction okay so the net reaction overall reaction is the rate of forward reaction minus the rate of backward reaction rate in electrochemical system the signal the rate is measured a current which varies as a function of potential can be related to the net rate this is a straightforward relationship which we have seen it is the central characteristic of an electrochemical reaction this this there is a relationship between rate and potential is non-owned okay um as opposed to what you've seen in your high school in simple systems in analogy with your chemical kinetics we can write the rate of forward reaction in this in this manner you can associate a rate constant which is a function of potential it's a constant at a particular potential no uh mind you that constant at a particular potential times the activity of the reduced species in solution okay so this is r indicates reduced species s indicates solution so this is a constant at a particular potential so therefore the rate of forward reaction can be written in this manner it is proportional to activity of the reduced species in solution and that constant proportionality constant at a particular it's a constant the particular potential is this number okay which is a number at a particular potential it is a constant we can further rewrite this activity in this manner using activity coefficients which we have introduced in the earlier lecture and in your courses relevant to chemical thermometers we can combine all these constants and write this equation as this constant times the concentration of the reduced species in solution so this is the electrochemical reaction rate constant relevant to the forward reaction which generates an oxidized species in a similar manner we can also write the rate of the backward reaction we get another constant which is a constant a particular potential is the concentration of the oxidized species in solution putting this together we can write the current signal at a particular potential as sum of two terms as mentioned here so the challenge here is how do we get a mathematical expression for the reaction rate constant as a function of potential if this is known if this is known and if this is known then our mathematical description of this entire reaction is complete so the central goal of this reaction is to get a mathematical expression of this quantity this and this quantity is similar we would undertake that procedure how do we frame this expression in a mathematical form as i said we are going to use some concepts of equilibrium at equilibrium there is no net rate of a reaction so at a particular potential called the null potential n indicates null potential the net rate is zero therefore the rate of forward reaction is the same as the rate of backward reaction therefore we are indicating this the bulk concentration here uh solvent near the surface okay so more precisely this is the concentration in the solution but the solution is near the surface so it's better to indicate uh this yes this is to be interpreted as the concentration near the surface of the element okay this and this is the concentration near the surface of the electrode okay so just to i'll correct that i mean i indicated earlier that this s refers to what is present in solution concentration solution but the more accurate uh measure is the concentration near the surface of so as opposed to this scenario when there is no reaction the concentration near the surface is the same as the concentration in the body okay so there's one on the same because there is no reaction uh overall reaction so at equilibrium the net rate is zero and this can be written in this map the overall reaction can be written at this time but we know something about this equilibrium condition which we derived earlier that is the nernst equation the null potential can be written via the nernst equation in the following manner okay so actually the nernst equation has many aspects of chemical thermodynamics there's already the aspect of temperature and the other aspects of chemical potential already ingrained in that so we are going to use the nernst equation to develop our mathematical formulation of electrochemical kinetics let's this is the system we are involving we have written we are trying to describe and this is the nearest equation relevant to this system we are going to substitute this equation replace this by the rate constant and using these two equations we can write this equation again this is the reformulation of the nernst equation what is to be noted is for any concentration okay we whatever be the concentration of the reduced species and the oxidized species we can find a null potential so what is the null potential if you go back and think about it the null potential is the condition for chemical equilibrium that is the chemical potential of the reactants is the same as the sum of chemical potentials of the products okay because the potential couples to the chemical potential of the electron we are going to write the uh null potential as that condition wherein the net rate of the reaction is 0 because of this potential because of this condition on chemical potentials okay that is the chemical potential of the reactants is the same as the sum of the chemical potentials of the species involved in the products there are two species here this is one species and the electron is also one chemical species it's a rather electrochemical species so we will because if this equation can be written for any concentration of reductant and oxidant we are going to drop this subscript and just write it in general as e okay so the the notion is that even at slightly away from equilibrium the concentration is not going to deviate too much from chemicals we will look at these questions in the discussion session but for now we are just dropping this subscript and writing it in a form that you must already be familiar when we introduce the nernst equation i would like to again re-emphasize this notion of what is this formal potential this formal potential is the null potential when the concentration of this species and this species is equal to one okay in the appropriate units okay so formal potential is the null potential when there is no net reaction when these concentration are equal to one we are calling this the formal potential you might have also heard about the standard potential standard quotation relates to activity but the formal potential is more easily it's a practical potential which relates which is the null potential when these two concentrations are equal to one with that condition you would clearly say that this should be equal to this quantity and we are going to call these two quantity indicate that by this common quantity the oxidative rate constant and the reductive rate constant should be the same at the formal potential and that we are going to indicate by this quantity so i'm just putting this quantity expressing adding and subtracting i will get this equation putting this condition along with this condition i get this equation a simple manipulation of differentiation gives me this equation and i'm going to introduce two expressions that are central to the description of of electrochemical kinetics uh involving single contrast so these two quantities are alpha and one minus alpha they are called symmetry factors or charge transfer coefficients in in my opinion this is one of the most confusing aspects of aspect of electrochemical factors we'll elaborate these issues much later but we'll just give this terms a name why these names are i mean when you say symmetry factor okay that seems to have some aspects of symmetry which we are not going to elaborate we are just going to give names to these two quantities alpha and one minus alpha and we are going to emphasize two things one is that these two quantities add up to one of course alpha and one minus alpha add up to one and we are going to emphasize that we are going to measure these quantities from experiments okay even though a lot of theory deep electrochemical theory has been done to interpret alpha we are not going to utilize that interpretation here we are going to just emphasize that this can be measured from experiments we have just given them a name as symmetry factors and charge transfer position just a simple rearrangement using this and this we can write the reductive and oxidative rate constant in this following manner okay so what does it show that the reductive rate constant seems to have an exponential dependence on some quantities let's plot them when i plot them we have already introduced the quantity of formal potential formal potential is the potential is the null potential when the reductive rate constant and oxidative rate constants are equal okay so that is the null potential here and that is related to this quantity that is the same quantity appearing here here we have utilized two co values of alpha there's a lot of debate about the value of alpha the orders deep theory as i mentioned but for the scope of this lecture we are going to just measure them experimentally we have utilized two values of alpha alpha equal to 0.35 alpha equal to 0.5 and plotted 2 2 green curves and 2 red curves what do we see these two curves are curves related to this equation what do we see beyond a particular potential it exponentially decays below a particular potential this exponentially increases okay so this region is called anodic the reductive processes seems to decay beyond a particular when you go to a potential that is more positive than this formal potential so in the if you go to the anodic direction these reductive processes decrease in contrast to this rate constant this rate constant increases as you go in the anodic potential the oxidative reactions increases in an exponential manner and when the potential of the electrode is brought down below this formal potential these curves show an exponential d okay so we have just plotted this where are we going to use them we are going to use this expression for ultimately connecting our current signal to variation in the potential if i pluck it these two expressions into our rate expressions which is the current expression i am going to get this equation this is the form of butler warmer equation and i i've color coded them in a variety of uh with variety of colors there is black color green color blue color okay let's see what they are let's take the simplest quantities that is the current and potential why have they colored them blue this is for the following reason i am typically controlling e okay the electrode potential and i'm measuring i okay i can also do the same in the opposite that is i can control current and measure e okay so these quantities that are color coded in blue are either the control variables are the measured f is a constant okay so that i need not come an or p they are all constant at a particular temperature t is a constant r is a constant f is fast okay i'm not going to further come on these quantities that are mesh color coded in green are the concentration near the liquid surface okay typically these concentrations are not accessible in experiments it's very hard to measure but lot of effort is being made in contemporary research on how to measure the concentration near the electrode surface okay so these are typically not accessible to the experimentalist these quantities that are color coded in red are a constant or almost a constant for a particular experimental uh system they're very very little okay so that's the this this constant alpha and the formal potential for a given electrochemical systems they show a limited variation so there are three uh different quantities that are color coded in blue green and red so this is the bachelor walmart equation this uh quite a challenging equation because there are two exponential terms we'll further see what we can glean from this expression i'm going to rewrite this equation using a slightly different constant this constant is called the exchange current density i'm just going to just multiply few things and use this expression what is special about this expression we are referring to the bulk concentration of the reductant and oxidized species if b refers to bulk s refers to what is the constitution the electrical surface i'm going to use this expression this is just a constant right f is a constant this is a constant these are all constant in the bulk they're not going to change at this correct surface i'm going to use this constant and rewrite this expression and this is the central form of the butler warmer equation okay so it's really easy to use this and this and get this expression i'll give it to you i have again color coded something in the ring okay why is that this expression that is color coded in the red okay that is exponential minus f times e minus en divided by rt okay that appears here and here can be interpreted as the activity of the electron that are present in the metal remember the philosophy of this approach is to consider the electrons as a chemical species there are three species in the electrochemical reaction one is the oxidized species reduced species and the third species is the electron the philosophy here is to consider the electrons which are present in metal like a chemical species of your chemical kinetics and associate with the electrons and activity okay and the activity of the electrons is the expression in the red okay so that has a great advantage because all our electrochemical kinetics can be interpreted in a manner analogous to what we see in chemical kinetics so using this okay so these terms i am going to combine this term and this term and then rewrite another expression let us look carefully on how we got this expression we ascribe this exponential term to the activity of the electrons okay so we had this expression already but this had an exponential dependence because we are ascribing this to the activity of electrons we can write the current which is a function of voltage the current expression in this manner this manner is very similar to what we see in chemical kinetics as i said there are three species oxy oxidant reductant and electrons so we have associated activity to the electron and written down an expression which is just very similar to what you would have seen in chemical kindness okay so this is an important step please think through this step and let's discuss this further in the discussion session okay this is a it's a very beautiful step okay this remarkable uh evolution of concepts going from chemical kinetics to electrochemical kinetics that we can associate a activity to electrons electrons are present in the electrode metal electrode metallic electrode and the activity can be controlled by controlling the electrode potential with that i can write this phenomenological expression let us wrap up okay so first what was the need for butter warmer equation the need is that we have a current response to varying electrode potential so to describe this non-equilibrium phenomenon we need an electrochemical kinetic equation so that was the need for framing the butler roman equation how did we do it we did develop this electrochemical kinetic equation in a manner very similar to what we have done in chemical kinetics this a leap in concept is that electrons are also a chemical species and we can associate an activity to the electron we have utilized some principles of chemical equilibrium especially the nernst equation to develop the electrochemical kinetic equation so we have utilized the principles of chemical equilibrium to model systems not inequality overall we have arrived at an exponential form okay of electrochemical kinetics this is the butler warmer equation to make it more intuitive you have to think through what you saw in the argenius equation arrhenius equation also had uh exponential term there was an actual frequency and an exponential dependence on activation barrier and temperature so try to make connections between arrhenius equation and this expression this is the expression for butler roman equation we will discuss this connection further in the discussion session as well as in the subsequent lectures but you go back to your understanding of various equation try to see whether there are intuitive connection between arrhenius equation and this butler warner equation with that we will wrap up this lecture and we will again look at butter roma equation in from a slightly different approach in the next lecture thank you