Lecture 24: Fick's Second Law (FSL) and Transient-state Diffusion

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One announcement. There will be Quiz 9, the weekly quiz tomorrow based on the content of Homework 9 which is, I don't remember what it is, but you know what it is. I made it up, but I am already thinking about the next glorious event.

Now I remember. There is chemical kinetics and glasses. That's right, chemical kinetics and amorphous solids. That's right. Last day we started talking about diffusion which is solid state mass transport by random atomic motion.

And we were drawn to the paper that is up on the display which was published in 1855 by Adolf Fick which gave us the law that bears his name describing how matter diffuses through matter. Fick was a very talented individual, and I want to draw attention to something else.

Remember, he was a physiologist. He actually did some work in medicine as well. In 1870, he described what survives to this day as the Fick Principle for determining cardiac output and basically equates the amount of uptake of oxygen by the lungs with the amount that should be distributed in the blood.

But you have to take arterial oxygen pressure minus venal oxygen pressure to get the efficiency of the heart. And so I found this little cartoon off the Web that just briefly encapsulates the notion of the Fick Principle for measuring cardiac output.

His nephew, by the way, who was orphaned as a young child also has the name Adolf Fick. The young Adolf Fick, the nephew was strongly influenced by the uncle. He went on to invent the contact lens.

This family has been quite prolific in ways that influence many of us. How many in this room are wearing contact lenses? I think we are all thankful that there are some cardiac efficiencies. I think we are all beneficiaries.

Let's take a look, in more detail, at what Fick taught us. He said that if we have ingress of species i into some solid, the rate of ingress is given by the equation shown here which expresses the flux which is mass per unit time per unit cross-sectional area as proportional to the concentration gradient.

And the proportionality constant is the diffusion coefficient. And what we see here is a sketch of what that might look like. Some initial surface concentration held constant with ingress of material.

And since the flux is the derivative, you can see that the flux, but for some multiplication factor, tracks with the concentration. We have the steepest concentration gradient at the surface, and deep inside the specimen there is essentially no concentration gradient so the flux attenuates.

Flux shown here in fuchsia and concentration shown here. And when I say profile it means something as a function of distance. Profile, in this case, is the concentration profile or a flux profile. You can see those two.

And then we started looking at some of the atomistics and reasoned that there is some jumping involved. And with the jumping comes the notion of activation. We looked at this. And we concluded that there is an activation energy associated with jumping through that saddle point, an Arrhenius type behavior, natural log of d, linear in reciprocal of the absolute temperature.

Remember, this is not the Rydberg constant, this is the gas constant which is the product of the Boltzmann constant and the Avogadro number. And units will give you a clue as to which to use. If the activation energy is in units per mole then obviously use the gas constant.

Otherwise, you use the Boltzmann constant. And when we looked at the atomistics further, we reasoned that when we have diffusion by vacancy jump mechanism, this activation energy up here in the equation is the sum of two components, one being the enthalpy of vacancy formation, which is really the negative of the bonding energy, and then the energy associated with the atom migration which is the energy of squeezing through that saddle point shown in the figure.

In the case of interstitial diffusion, we do not have to form vacancies. There are enough vacant sites, by virtue of the fact that we have so much free volume, even in a close-packed solid. We are really just paying for the enthalpy of migration.

Lastly, we looked at how these two are related to the degree of confinement. In other words, the atom is restricted by the ease with which it can run down its appropriate raceway. And we reasoned that going through the bulk lattice is the most tortuous path and has the highest activation energy, lowest diffusion coefficient.

Grain boundaries have atoms with fewer nearest neighbors, lower coordination number, so the diffusion coefficient is higher in the grain boundary and is highest along the free surface. And we saw last day that the diffusion coefficient along the free surface of a solid approaches values equivalent to that of diffusion in a liquid state.

What I want to do today is step back to the continuum and use Fick's law to describe what is going on mathematically in the macro world. I want to first start with diffusion across a membrane. This is a membrane of some thickness L.

And I am going to put a gas to the left, P1, and I am going to hold it constant. And to the right, I am going to have a gas at P2 also being held constant. And, just so that we are all on the same page, I am going to make P1 greater than P2.

That means matter will diffuse from high pressure to low pressure, high concentration to low concentration. That means the flux will be left to right. And we can use the Universal Gas Law, pressure volume equals the product of mole number times gas constant times temperature in order to convert pressure units into concentration units.

And we know that the concentration of a species i is simply equal to the mole number i divided by the total volume of the system. This is moles per unit volume, so that is a concentration. And so that means then that if we go to the Universal Gas Law that will give us P over RT.

If you take a pressure, divide it by the product of the gas constant and temperature, it will move from pressure units to concentration units. And so what I would like to do is show what this system looks like after some period of time.

I am going to put the convention with zero on the left surface. This is distance moving from left to right. X equals zero is the left surface of the membrane. X equals L is the right surface of the membrane.

And now let's put concentration c1 on the left surface and c2 on the right surface. And what we know is we expect that after some period of time, if c1 is held constant and c2 is held constant then, after some period of time, we expect to have a profile that is linear.

And we should be able to infer that from Fick's First Law. We know that Fick's First Law tells us that J is equal to minus D, dc by dx. If I am holding the concentrations on both sides constant then it tells me then that the gradient is going to be constant.

Otherwise, I have some sources or sinks which we do not allow for. After this short time we have this profile which we term steady state. And it is termed steady state because it is invariant with time.

The flux is not a function of time. The flux entering the free surface on the left equals the flux exiting on the free surface on the right. And, therefore, the flux must be constant throughout the membrane.

Because if J is constant, and I am assuming that the diffusion coefficient doesn't change with time, then it means that this gradient must be constant. The slope must be the same all the way along, reinforcing the notion that we have a straight line profile.

Now, there is one assumption here, of course. And that is that the diffusion coefficient is independent of concentration. It could be. And we will return to that later and see what happens if we have the concentration depend on diffusivity.

We can describe at steady state everything we need to know about the behavior of this system with Fick's First Law alone. Fick's First Law alone is sufficient. What are the kinds of things that we might want to know? We might want to ask how much material diffuses over time, mass per unit time passing through the membrane.

All we need is Fick's First Law because we know that we can say that the mass will simply equal the flux, which is mass per unit area per unit time. If I know the total area of the membrane and I know the time then that is all I need.

Fick's First Law is very thorough in describing a system at steady state. Last thing about this concentration profile, the operative definition of steady state is that J does not change with time. In other words, if I look at this at some later time, as long as I maintain constant end compositions, I will have this same profile.

But suppose I had a situation like this where the diffusion coefficient is not independent of concentration. I will give you an example. The diffusion coefficient of carbon in iron, this is steel, as a function of the concentration of carbon in iron.

It turns out that as more and more carbon goes into iron, the diffusion coefficient changes. You would expect that because now some of the sites are occupied. Which is it? Do you think that the diffusion coefficient, as you get more and more carbon into iron, starts to fall like this, or do you think the diffusion coefficient rises like this so that at higher concentrations it becomes more difficult to diffuse? Well, this might sound appealing because the sites are getting blocked.

It make sense. There are fewer empty sites. But, in point of fact, the diffusion coefficient actually rises. Why? Well, I have told you that the carbon atom is larger than the interstitial site. When the carbons go into the interstitial sites they act as wedges, and they actually wedge open the system and make it easier for the succeeding atoms to diffuse through.

In point of fact, this is what happens. But it could be either. Let's imagine, suppose we have case number one, where the diffusion coefficient falls as concentration rises. And now I have a steady state concentration profile.

I have c1 constant on the left, c2 constant on the right, and here is the equation we have to look at. Minus D, dc by dx. But this is now a function of concentration. If I have case number one, when I have the highest concentration up here, if this is going to remain constant, if the concentration dependence of the diffusion coefficient says D is low when c is high then that suggests to keep a constant flux, dc/dx must be high when c is high.

That means instead of a straight line, we are going to see something that looks like this. This is case number one. How about case number two where the concentration dependence of the diffusivity is indicated with a rise? Well, that means at high concentration my diffusion coefficient is going to be higher than it is at low concentration.

If this number is higher at high concentration then this number has to be lower so that the product will remain constant. This is case number two. These are all variants of steady state. J is not a function of time.

J is independent of time. Now I want to ask what happens in that initial moment when we first pressurize the system. Let's say we start off and there is nothing inside the membrane. Just to keep things simple, I am going to make P equal zero on the right-hand side.

Left side is P1. And at time zero, we initially pressurize so we have a concentration c1 instantly at the surface. What is the set of events that occurs leading up to the establishment of the steady state profile? I would expect, after a short time, we would see concentration profile shown like this.

And then after a longer period of time like this. And then after a longer period of time like this. And then finally, of course, we establish. This is time is increasing from low to high. This could be t1, t2, t3.

And so we can then say let's park ourselves at some value x1 on zero to L. And look at what is happening. Initially, if I take the slope it is a low number. If I take the slope at a later time it is a higher number.

If I take the slope at a later time it is an even higher number. And, ultimately, it is going to be as high as the steady state value. I could take that data set and plot the flux at x equals x1. I am sitting here watching the profile evolve with time.

Initially, it is zero because there is nothing in the membrane. And then it slowly rises and reaches the steady state value. This is J at steady state here, and this could be t1, t2 and t3. You can see that coming off the slope.

Clearly, when we are moving through this time period, J is very much a function of time. Because, if you close your eyes and wait, you will see that the profile has changed. And, if the profile has changed, it means that the instant value of flux has changed.

Flux is a function of time, so this is not steady state. Instead it is transient. We are in the transient regime. And so, as is always the case, we can draw these curves. They start at a high value, they attenuate to zero.

What is the magical question? What is the shape? That is the only reason I need math, to tell me the shape of those curves. Is this an exponential decay? Is this half of a sinusoid? What is the shape of the curves? I want to know c versus x at all times because it is changing as a function of time.

That means I need to know c is a function of x and t. I have got to solve this. And Fick gave us the solution to this as well. The solution to this is the solution to an equation known as Fick's Second Law.

Let's look at Fick's Second Law. I am going to write it in the special case when D is independent of composition. D has to be independent of composition for me to do what I am going to do next. And it is a pretty good assumption in many industrial settings.

And so I am just going to write it. It is a partial differential equation. What it is telling you is that the partial, with respect to time, is equal to the product of the diffusion coefficient times the partial of the gradient in concentration.

And so, clearly, this is a partial differential equation. And that is bad. You cannot solve this one, not yet. But the good news is it is a linear partial differential equation. That is good because I can show you a solution that applies for a given setting, and then you can play with the boundary conditions and use the same solution.

That is the good news, it is linear, but the solution of this is both in x and t. And the solution looks like this. Let's say we have an initial concentration c naught. C is a function of x. I have an initial nonzero concentration c naught and I have a surface concentration fixed at cs, and this is the system that we are trying to model.

And the general solution to Fick's Second Law is c of x and t is some constant A plus Aaconstant B times a function, which I am going to define for you in a second, the Gaussian error function of x divided by square roots of Dt These are both constants to be determined.

This is the Gaussian error function. X is position, t is time, and d is the diffusion coefficient. And this is nothing but a transcendental function that varies from zero to one. And I could have said sine.

If I had said it is sine x over two roots of Dt -- You know the definition of sine. It has some weird integral and blah, blah, blah. Nobody pays any attention to it. You know it goes from zero to one.

It is linear at low values of theta. So what. We are going to learn what the properties of this are, but I can tell you something right off the bat. What do you think the shape of the error function is? What is the shape of the error function? That is the shape.

That is what math is. Math is finding the shape. Otherwise, it doesn't make any sense. And what are these? These are scale factors. You have done this on a computer where you take some image and you pull it, stretch it and so on, and you can lock the aspect ratio.

Well, that is all you are doing here mathematically. We are going to pull this. Why are we going to pull it? Because at a later time it looks like this. But the relationship between all of these points is the same.

That is what math does. If it doesn't fire it. Math works for you. You don't work for math. Let's put this thing to work, lazy. What we are going to do is solve for A and B. You know that when you differentiate, you throw away information.

Every time you throw away information we need to add some. We need to have one time-based boundary condition, and we need to have two spatially based boundary conditions because we threw away three bits of information.

That is what we are going to do. And, if we go ahead and do that, here is what A and B will solve to. And this works for all cases, c minus the surface concentration divided by the initial concentration minus the surface concentration equals error function of x over two roots of Dt.

C is what you are looking for. This is c as a function of x and t, cs is surface concentration, c0 is the initial concentration. And let's take a look at what that function plots out to be. I am going to plot erf of z as a function of z.

It varies from zero to one. And it starts off, like so many functions, it is linear. For low values of z, erf of z equals z. And, in fact, up to about 0.6 you are good to within 1%. Erf of z equals z to within 1%.

And then once it gets up to 0.6 then it starts to veer off and then asymptotically approach erf z equals one. In fact, we can write here erf of 1 equals 0.84. You can see now it is not linear. And erf of 2 equals 0.995.

Once you get out here to two, you are practically at the asymptote. Here is the integral. And it is defined erf of z is the integral from zero to z of e to the minus u squared du. That is this, e to minus u squared as a function of u is the Gaussian error function.

That is the bell curve. This is the area under the bell curve. If we start at u equals zero, we are going to integrate as far as we need to out here. And the integral from zero to infinity of this thing is root pi over two.

And we want this to go from zero to one, so we will put the factor two over root pi out in front. That way, when we integrate this from zero to infinity, erf of infinity becomes one. Erf runs from zero to one.

And that is the area under this curve. That is why some people call this the Gaussian error function because the Gaussian curve is associated with random statistics in terms of errors. This now is the template.

If I have any reaction this is all I have to fit using these multiplication factors. Let's play "Mr. Dress Up" here. We are going to use this function to describe all of the typical industrial processes.

There are only two types of processes, those that are diffusing substance in and those that are pulling substance out. Let's look at the two cases. Let's look first of all at the one that is shown right up there.

This would be for something like out-gassing or drying. Many processes involve drying which is to get the water out of the content of a solid piece. And so, as you can imagine, you have some initial concentration c naught.

And you take the surface concentration, whatever this is, and obviously if you want to draw something out and move matter from right to left then the surface concentration, by necessity, must be less than the initial concentration.

Otherwise, why is something going to move? This I will call effusion, something is effusing. This is effusion. The flux is moving from right to left, and the determinant here is c naught is greater than cs.

The surface concentration is less than the initial concentration. I am going to pin the concentration at cs. I have c0 deep inside. And this is telling me what the shape of the curve is. I just write with impunity c minus cs over c naught minus cs equals error function x over two roots of Dt.

**C-Cs / (C0-Cs) = erf (x / (2 sqrt(Dt))** And that is the answer for all such problems. And because it is linear it doesn't matter. If I change the initial concentration to a new value, I can just add.

We can add and multiply because it is a linear equation, linear functionality. I think here we looked at those. This is the tabulation. This is in the supplemental text. There is a table of error function values so that if you happen to get into this regime beyond z equals 0.6 you will have the values that you need.

Let's look at another case. Let's look at driving material in. That, for example, is something like doping or nitriding, carburizing. If we want to case harden a material, we will raise the carbon content at the free surface, have something that is soft and ductile inside.

For that here is the scheme. We have a surface concentration cs. We have an initial concentration c naught. And in this case c naught is less than cs. Surface concentration is higher so material now wants to move from left to right.

And so this is infusion. And what is the curve going to look like? It looks like this, which is simply that one flipped upside down by subtracting it from one, which I will show you in a second. Again, this same general solution applies.

The surface concentration is higher or lower than the initial bulk concentration, it comes out in the math here. Again, we write c minus cs over c naught minus cs equals erf x over two roots of Dt. And this one comes up very often.

This is the way it works in doping of semiconductors and comes up so often that it is convenient to define another function where, in many instances, c naught equals zero. When we are doping pure silicon, the initial concentration of boron or phosphorus in pure silicon is going to be zero.

For many of those cases, you will end up with something that if you go through the math this is a zero, transpose, you will end up with this as the solution, c equals cs times the complimentary error function, erf compliment x over two roots of Dt where the complimentary error function, erf compliment of z is simply one minus erf of z.

If you see that in the literature, that is all they are doing. Now, the question is, I have told you that there are special conditions that validate the error function solutions. What are the conditions? When to use erf? Because that is not the only solution to Fick's second law.

When these are the conditions, c of x at time zero is a constant. This is boundary condition number one. We threw away some time data, so here is data at time equals zero. This is called the initial condition.

The initial condition must be a constant. And that is OK. There are many systems for which the initial value of the material that is being diffused is constant throughout the material, as opposed to bopping all over the place or maybe already having some kind of a gradient.

As long as you can make this assumption, you are on the road. Now, we need two more boundary conditions. We need boundary conditions two and three over here because we have a double derivative. You have some function plus constant.

When you do d by dx, constant goes into the trash bin. We need to recover that information somehow. Here is what I have. I have c for x equals zero at all time is a constant. The surface concentration is fixed as a constant.

And you say, well, what else could it be? Well, I could have a slowly rising surface concentration or I could have a surface concentration that varies sinusoidally or something. Those won't work. It must be fixed at a constant value.

That is one. This is boundary condition number one. This is boundary condition number two. And then I need a third one. The third one I am going to get is this one. It looks funny when I write it mathematically, but you will understand what I mean in a second.

At infinity the concentration does not change. You say, well, duh. This really has some meaning. We know that the size of our piece is not infinite, but here is what we are assuming. We are assuming that over the time scale of the process that, for all intents and purposes, the dimension of the silicon wafer, for example, is so large in comparison to the depth of the diffusion profile that assuming the wafer is semi-infinite, because we have it diffusing in from both sides, but let's just assume from the one side, that we can assume that it is infinite.

Ultimately, what happens? If we wait long enough, we start to have a tail over here because the system still thinks that the wafer exists. But we are saying no, it stops here. But this value is so small during the time scale of the process that we can neglect it.

And by allowing ourselves to assume that the material is semi-infinite it simplifies the math. If we have a finite-sized specimen, we cannot use this simplified solution and things get very, very messy.

But there is another way of expressing it. You see this relationship? It is always erf x over two times the square root of Dt. Something would be infinite would mean it is very large. Large dimension.

Let's say large length scale. That is a large physical dimension. That means the argument of this is very, very large. But there is another way to get the same result: short time scale. That means every diffusion experiment initially is operating in a material that, for all intents and purposes, is infinite.

Because when those first phosphorus atoms start diffusing into the silicon, they don't know the silicon is only a millimeter thick. As far as they are concerned it is infinitely thick. This solution is valid everywhere.

As long as you have concentration initial fixed, concentration surface fixed at short times, this solution is valid. Valid over a short time. That means it has great utility. Now I want to show you something that takes everything we have learned and crashes it into a simple rule of thumb.

And these are the kinds of rules I hope you will retain long after. What was it Franklin said? Education is what remains when you have forgotten all of your schooling. This is what you will remember.

You have c minus cs over c naught minus cs, and that is equal to over here erf x over two roots of Dt. That is the accurate solution. If I wanted to capture the whole diffusion process in one tiny little sentence, what would be an average value? This is a normalized ratio, isn't it? It goes from zero to one.

What is an average value between zero and one? I don't know. How about a half? Suppose I just choose this as an average value for the whole diffusion process, everything that can ever happen in diffusion, well, a half is less than 0.6.

Since a half is less than 0.6, I can approximate erf x over two roots of Dt as simply x over two roots of Dt. Now I have this equals this. I will cancel out the twos and cross-multiply. That gives me x goes as two roots of Dt as an overarching equation for everything that happens in short-term diffusion.

**x = 2 sqrt (Dt)** I cannot tell you how many times I have been in situations where I am in some conference room and a bunch of people are sitting around trying to figure out what to do and somebody is getting ready to do some finite element calculation to go mathematically model this whole process, and I sit back and go, gee, well, I know the length scales.

Well, why don't we do one? Let's do one. Have you ever been told this thing about cathedral windows, why the glass at the bottom of a cathedral window is thicker than it is at the top? It has to do with the fact that glass is really a viscous liquid and over 500 years it trickles and it gets wider.

That is nuts. And I can show you like this. It's very simple. You already know that the average -- What is the average temperature in a cathedral in Northern Europe? It is zero in the winter and 20 degrees in the summer.

Let's pick a number, 10 degrees C. What do you think the diffusion coefficient of silicate glass at 10 degrees C would have to be for this thing to go anywhere? Here is what I did. I said let's use this.

Here is my simple model of the cathedral window. This is the cross-section. And I figure I need a length scale. I chose a length scale, if something is going to diffuse, on the order of ten centimeters.

You can quibble. If you want to make it one centimeter, I don't care. I am going to blow this thing so far out of the water it won't matter. This is about ten centimeters, and it is going to get thicker on the bottom.

I figure I am going to give this a really long run time, so I am going to make it 500 years. I want to show you how to put this problem to the test. What I am going to do is derive a quantity that I can say yes or no to.

What I am going to try to do is pull out a value of the diffusion coefficient, so I know that x squared equals Dt, if I just square both sides of that thing. That means that D, the diffusion coefficient should be x squared divided by t.

I am going to do this in centimeters. This is ten squared on the top and time is 500 years, so that is 500. And how many seconds in a year? Pi times ten to the seventh. Pi times ten million is good to within about 1%.

That is it. Who cares? This is a diffusion calculation. You don't need anymore. You are not going to go 365 days and how many leap years were there? Come on. Get with it. 6.4 times ten to the minus nine centimeters squared per second.

I would need a diffusion coefficient of a silicate glass of ten to the minus nine. That is the diffusion coefficient of a close-packed metal at its melting point. I went and looked up some data, and here is what I found for the diffusion coefficient of sodium in a soda lime glass such as the type that would be used in those windows.

And, using those data, at 10 degrees C, the diffusion coefficient would be ten to the minus nineteen centimeters squared per second. It is off by a mere ten orders of magnitude. I don't care if you want to make this one centimeter or if you want to make it one millimeter.

There is no way that is the explanation. We know what the reason is. This proves that it cannot be. What I want to show you is that you can come at it from both sides. The other thing is to figure out how they make the class.

You know how they made the glass? They pull it out of a layer and put it on a spinning table. That is how they could flatten it. They didn't have the float glass process where you pour silicate on top of liquid tin.

And liquid tin has a perfectly flat surface, the silicate glass is less dense and is insoluble, and so it floats on the liquid tin surface and is perfectly flat. They did not do that back in Medieval times.

They spun cast it. It was like a spin coating technique. This is top view. You have some table spinning, and you plop some glass on it and it goes whoosh by centrifugal force. Now the cross-section of the glass, what does it look like? Let's start from the center and go out.

Guess what? Not to scale. It is ever so slightly thicker on the outside than the inside. And people would actually be very careful about choosing the pieces. And then the glazers said when you mount the windows, let's put all the glass in with the thick side down.

Wouldn't that make sense? And that is why the glass is thicker, because it was made that way. Simple answer. But you can do the calculation. I don't know how many times, but I get asked this at least once a year.

Someone says is it true that the cathedrals, blah, blah, blah. And I go does this person know Fick's Law or not? Don't know, so we have to give a different explanation. Now I want to finish the story of the automobile exhaust catalyst and show how we can bring a lot of what we have been learning in the last several days together.

Just to review. This is an old sketch because it was taken from an early General Motors publication from the ë80s. Basically, what you have is - we have seen the catalytic converter. And it is taking the exhaust gases and trying to convert some of the toxins into something less offensive.

And I had reported to you that we need to control in order to achieve both oxidation and reduction. And how do we control? We control with the aid of an oxygen sensor. Could we switch to the document camera, please, for a second? Take that input.

What I am showing you is an oxygen sensor. What you are looking at is this is the metal sleeve on top of the oxygen sensor, and you can see veins here. This is inserted just past the engine, before the catalytic converter.

And what I want to do is to describe the solid state chemistry that is going on inside there. May we go back to the computer, please? What I just showed you is sitting right here just as the exhaust gases leave the engine.

And what they are doing is measuring the oxygen pressure in the exhaust gases so that we can control the air-to-fuel ratio. Because it turns out that to convert the CO and the hydrocarbons to the desired products, in other words, carbon dioxide, water vapor and carbon dioxide, you need to oxidize.

And the catalyst is pretty good at very, very high air ratios. And to reduce NOX to nitrogen you need to have rich fuel ratios. And mercifully nature was kind. And there is this tiny window at about 14.6 to one where you get fairly high yields on both catalytic reactions.

This is where you want to park yourself. Now you need to have a sensor, and that sensor looks like this. It is zirconia. It is an ionic oxide, and it is doped with calcium oxide to create oxygen vacancies.

We are going to get defects involved and we are going to talk about diffusion. When we put calcium oxide into zirconia, we have two plus sitting on a four plus site. And so we need to make an oxygen vacancy to compensate.

As we add calcium oxide, we make more oxygen vacancies which means that the diffusion coefficient goes up. And that shortens the response time. It is no good having a sensor that responds on a time scale of minutes when you're driving conditions are changing on a time scale of seconds.

And so we dope with a subvalent oxide in order to improve the performance. And here is a cartoon that shows this is what is underneath the vein. This blue that I have drawn is a closed one end zirconia tube.

There are electrodes on the inside. This is the exhaust. And on the outside it is open to the air. And this measures a voltage across the zirconia sensor. And that voltage goes to the CPU. Every car has several computers on it, but the first cars to have computers had them in order to control air to fuel ratio to make the catalytic converter effective.

And then that goes to control either the carburetor or nowadays the fuel injectors, which then keeps the fuel-to-air ratio so that this keeps measuring 14.6 to one. And if you change altitude, if you get stuck in the Callahan Tunnel and the ambient conditions are changing, this can respond to changes in temperature.

What do you think the partial pressure of oxygen is? About 20% in air. But how much oxygen per unit volume do you have at 100 degrees Fahrenheit versus zero degrees Fahrenheit? This thing can figure all that out and, thereby, run the system at the optimum fuel-to-air ratio to maximally get the output from the catalytic converter.

And so in 3.091 we have learned about the doping of silicon to make what goes into here. We have learned about the ceramics that go into the catalytic converter and the catalyst that goes on it. And now we have learned about diffusion and everything that goes into the oxygen sensor.

That is why I tell you chemistry is the central science to energy efficiency and environmental conservancy. Learn your chemistry. It is the most important subject. And with that I will say have a nice day and we will see on Wednesday.

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