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Chemistry: The Nernst Equation

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  • Type: Video Tutorial
  • Length: 12:04
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  • Size: 130 MB
  • Posted: 07/14/2009

This lesson is part of the following series:

Chemistry: Full Course (303 lessons, $198.00)
Chemistry: Electrochemistry (12 lessons, $19.80)
Chemistry: Galvanic Cells (6 lessons, $11.88)

This lesson was selected from a broader, comprehensive course, Chemistry, taught by Professor Harman, Professor Yee, and Professor Sammakia. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/chemistry. The full course covers atoms, molecules and ions, stoichiometry, reactions in aqueous solutions, gases, thermochemistry, Modern Atomic Theory, electron configurations, periodicity, chemical bonding, molecular geometry, bonding theory, oxidation-reduction reactions, condensed phases, solution properties, kinetics, acids and bases, organic reactions, thermodynamics, nuclear chemistry, metals, nonmetals, biochemistry, organic chemistry, and more.

Dean Harman is a professor of chemistry at the University of Virginia, where he has been honored with several teaching awards. He heads Harman Research Group, which specializes in the novel organic transformations made possible by electron-rich metal centers such as Os(II), RE(I), AND W(0). He holds a Ph.D. from Stanford University.

Gordon Yee is an associate professor of chemistry at Virginia Tech in Blacksburg, VA. He received his Ph.D. from Stanford University and completed postdoctoral work at DuPont. A widely published author, Professor Yee studies molecule-based magnetism.

Tarek Sammakia is a Professor of Chemistry at the University of Colorado at Boulder where he teaches organic chemistry to undergraduate and graduate students. He received his Ph.D. from Yale University and carried out postdoctoral research at Harvard University. He has received several national awards for his work in synthetic and mechanistic organic chemistry.

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3012 - The Nernst Equation
Okay, so now you're an expert on understanding reduction potentials and how to use them. And one of the consequences from our discussion was that we could talk about how spontaneous a reaction was - a chemical reaction. So whenever we say that word, we know that that's got a lot to do with free energy. And all that's left for us to do now is make the connection, formally, between electrochemical potential and free energy. And once we do that, we know that free energy is connected to equilibrium and so we can kind of tie this whole circle together.
I remind you that we can find out about equilibrium constants by looking directly at the ratio of reactants and products. And we can find out about free energy by looking at calorimetric measurements - measuring the heat that comes out of a reaction - and looking up, or calculating, entropy. And so, with that as our raw data, along with the electrochemical data, which we can measure directly, that tells us the information we need. And once we know one, we can find the other, between free energy and equilibrium. So this kind of completes this package.
So let's do this. Let's make this connection here. Okay, so free energy, for an electrochemical reaction, is essentially the maximum work done by your electrochemical cell. And that's going to be charge times potential. Now the charge, what is that? The charge, in this case, is going to be the charge of all those electrons that you're pushing through the circuit. Okay, well that's going to be a faraday - now remember, that is the charge due to one mole of electrons, so that's 96,480 coulombs, times the number of moles of electrons that you're sending through that circuit. So this is going to be 1 or 2 or 3 or what have you, depending on the stoichiometry of your reaction. Multiply that by the potential, which is our electromotive force of the cell.
Now, note that minus sign. That tells us that when we have a positive potential, that this term, overall, is negative, or the free energy at standard state is negative. That tells you it's a spontaneous reaction in the forward direction. Okay, so when we have a positive voltage, we note that the forward reaction is spontaneous. Now, again, this is all talking standard state so far.
All right, now, one other connection, then, going back to equilibrium - the idea that G° at standard state is related to an equilibrium constant - remember, we've seen that. So we can tie, then, electrochemical information to an equilibrium constant. And FE is going to RT natural log of K. So, if we know something about the cell, we can talk about what the ultimate equilibrium constant is going to be for that reaction. So, we've got just about all the pieces in place.
Now that we know something about free energy, let's talk about not being at standard state. What's the big deal? Well, being at standard state is a big deal. Everything has to be exactly one mole per liter and one atmosphere. Most of the time, you're not going to be at standard state. We need to be able to deal with that real-world situation. So, when we need to be away from standard state or when we're talking about being away from standard state, we talk about free energy. We know how to deal with that already, so let's take the relationship - go from electrochemical information back to G° now. So at standard state, we know that it's related to G°. Well, also, when we're not at standard state, that same relationship's going to apply and so I can write G° - in fact, I already have. If you recall, we talked about G° not at standard state. That's equal to G° at standard state plus this term, RT natural log of Q.
Okay, so this is stuff we've seen before. G°, we've got an expression for, in terms of electrochemical information. G° at standard state we've got information for. So I'm just going to substitute these equations in for the G°s here. And then we still have this RT natural log of Q term. Okay, last step - I'm going to divide everything by minus NF, and that gives me what I want, and that is that the potential at any starting point, whether I'm at standard state or not, is the potential at standard state minus . This is the Nernst equation, one of the most powerful tools of electrochemistry. It tells us how the potential of a cell changes when we're away from standard state. And again, a lot of the time, we're not going to be as standard state, so we need to know information about being at concentrations where we're not at one mole per liter. And so, this is going to give us that information. You may note that this looks a lot, in structure, like the Henderson-Hasselbalch equation: pH = Pka. And then there was a log term that modified it. This is the same idea. The electrochemical potential is the electrochemical potential at the standard, but modified by whatever your ratio is. Okay, if your ratio happens to be 1, that term drops out, just like it did in the Henderson-Hasselbalch equation.
Now, let's actually put this to use and see what value there is in knowing the Nernst equation. Let's suppose that we're interested in measuring the pH of solution using an electrical means rather than indicator. Well, one thing that I could do would be to set up an electrochemical cell and look at how the potential - what I measure - changes as I change concentration of acid. And that would give me a direct connection, then, between concentration of acid and voltage and I could calibrate that somehow to let me know what my pH was. So let's see how this would work.
Okay, to start with, let's think about a zinc cell at standard state - and that will just be a reference cell for us. And then we're going to have a hydrogen cell that's not at standard state. So this is not a standardized cell anymore, now. This is just a cell where I've gotten my H plus concentration at 10^-3 molar. And I want to know what my potential will be for this electrochemical cell. And I'm also going to go ahead and keep my hydrogen pressure at an atmosphere, so I don't have to worry about that changing. So I'll leave that at standard state. The only thing not at standard state is my H^+ concentration, which is what I'm interested, ultimately, in probing for.
Now, before we go any further, let me remind you that if we wanted to know the potential where everything was in standard state - I'm talking then about this equation - and I just take the standard reduction potentials for the half-reactions here. Except, of course, I'd have to reverse the zinc one, because I'm going from zinc metal to zinc 2 plus. So this becomes positive .76 volts, rather than minus. And that, in fact, is the potential for this reaction. So, I want you to just remember that number. That is the potential I would measure if I had one mole per liter of everything up here. Because, again, I want to know what happens as I change H^+ concentration - what happens to this number.
All right, well as soon as I go away from standard state, I need to know Q. Q, in this case, is going to be concentration of products over reactants. Concentration of products, in this case, would be zinc 2 plus times the partial pressure of hydrogen, because this is a gas, divided by concentration of H^+ squared, because of that coefficient. And so that reduces in this particular problem to . Why? Because partial pressure of hydrogen is 1 and zinc concentration has been set at 1. You know, I've stacked the deck here. I've simplified things as much as I could, so I'm only looking at what happens as I change H^+ concentration.
So I have a value for Q, then: . And so now I'm ready to use the Nernst equation. And so here is our Nernst equation again, that we saw a moment ago. And I want to point out to you that lots of times, what people will do is take the term, R and T, where T is 298° Kelvin, and then F - those are constants - and then pull those out along with a conversion term that takes us from natural log to log - base 10 - and just group them all together so we just don't have to mess with it. That value turns out to be - at room temperature - .0592. Certainly don't memorize this number, but it's a number that you'll often see listed in books. That is just combining all these terms together. I still have N here, because N varies on what my reaction is. In this case, N is 2, the number of moles of electrons transferred, in this case. Again, going from zinc to zinc 2 plus requires 2 moles of electrons per mole of zinc. And then, log base 10 now of Q. Okay, so plugging in everything - it's just a bunch of numbers now. So stick in numbers and we end up ultimately solving for the cell potential - the electromotive force of our cell - and it's got a value of .58 volts. Well, look what happened compared to the original potential. .76 volts has now become .58 volts as a result of lowering concentration of H^+.
Well, what does that mean? First of all, does it make sense? Well, think about your chemical intuition here now. You're lowering concentration of reactants. So there's a driving force to go in this direction, but it's not as great as if you had a mole concentration of H^+. You've got less stuff over here that's going to go this way, so your driving force goes down. Your potential decreases from .76 to .58. So again, that matches, at least, our sense of what should happen.
Now notice what we've done. What we've done is, by changing concentration of H^+, we've seen a direct change in voltage. And that's something you can measure. You can measure that with a little voltmeter that you buy at a store for about ten bucks. That's a trivial thing to get a hold of that just measures voltage, and we've seen a big voltage change with concentration of H plus change. And that, again, effectively is a pH meter - a device that allows us to measure the concentration of H^+ - or other ions, if we're interested in measuring those - that gives us, as a response, just a voltage change. So the Nernst equation describes how that type of a device works. So a really, really useful relationship to have, in terms of electrochemistry. These are used for sensors for all kinds of different things - sensing different levels of ions in your blood to solutions in a lab to all kinds of different ways. And the nice thing about this is it's just simply measuring a voltage. And so it's a really powerful tool, in that sense.
Okay. We've completed this connection now between electrochemical information and free energy and we know free energy is related to equilibrium. That's the step that we haven't said anything about yet. So the last stop on our tour, as far as the basics of electrochemistry, is how do we go from electrochemical information to information about equilibrium constants. And that's what's coming up next.
Electrochemistry
Galvanic Cells
The Nernst EquationPage [3 of 3]

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