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### Course: AP®︎/College Chemistry > Unit 14

Lesson 5: Cell potentials under nonstandard conditions# Nernst equation

Deriving a few different forms of the Nernst equation, the relationship between Gibbs free energy and reaction quotient Q. Created by Jay.

## Want to join the conversation?

- As far as the MCAT goes, are we going to be expected to have all of these equations memorized? Does anyone know? And are there any tips or tricks for remembering different equations?(1 vote)
- The Nernst equation will be given to you on the MCAT if necessary.(8 votes)

- Why is the gas constant used?(4 votes)
- Constants are used is most linear equations. They are used to describe the magntiude of an effect of increase/decreasing on variable on the other.(2 votes)

- But why is cell potential changes with concentrations. In another video, Jay says that voltage is intensive property. Then why does it depends on concentration. Can anyone explain me where I misunderstood?(3 votes)
- Voltage/cell potential isn't an intensive property (this is Ecell). The intensive property is
**standard cell**potential.(this is E0cell)

The two quantities are related by the Nernst equation.

Ecell = E0cell -RTlnQ/nF

https://www.khanacademy.org/test-prep/mcat/physical-processes/intro-electrochemistry-mcat/v/nerst-equation-edited

Standard cell potential is called so because special standard conditions are maintained.(3 votes)

- Hi, in all of my books nerst equation is E = E0 + ( 0.0592/2 ) + logQ; Why are you substract 0.0592/2 from E0 instead of add it, does it depend of E0? I am little bit confused.(2 votes)
- If i'm right, in Gibb's free energy equation, we use Q as Conc. of products/Conc. of Reagents, so in this kind of equation Conc. of cathode ions (which are reducted)/ Conc. of anode ions (which are oxidated), so [Cu2+]/[Zn2+], obtaining an higher value of k as the reaction flows.

In nerst equation, the concentrations are swapped, so [Conc of anode]/[Conc. of cathode], changing the sign from - to +.

Hope i'm right!(3 votes)

- If I have a lithium ion battery (for arguments sake let it be lithium-cobalt-oxide), does the Nernst equation apply? I'm trying to find mathematically what changing the temperature does to the performance of the battery. Thanks(2 votes)
- Why is n a constant in both equations?(1 vote)
- n is the number of moles of electrons transferred during the reaction.

Both equations refer to the same reaction, so n is constant in each equation.(2 votes)

- In the Nernst equation..I am confused about the n variable. Some sources have this variable listed as "z", or the valence/charge of the ion. Here it is considered the # of moles of electrons transferred in the reaction. Is this the same? Am I missing something?(1 vote)
- The Nernst equation is used to determine the potential of a cell.

n or z (from Zahl for "number") is the number of moles of electrons transferred in the cell reaction.(2 votes)

- Just curious, why convert Q to base 10 instead of just leaving it at lnQ?(1 vote)
- It is just traditional in chemistry to use certain equations with base 10 and others with the natural log. There is not a good reason except in the case of pH (that is base 10 so that one pH unit amounts a 10 fold change in concentration).(2 votes)

- if E0 is E under standard state condition, then wouldn't E0 just be equal to logK*0.592V/n? and by that, wouldn't E be zero?(1 vote)
- In the electrolysis of dilute sulphuric acid, how many faradays, F required to produce 224 cm3 of gas at the anode measured at STP?(1 vote)

## Video transcript

- [Instructor] We've
already seen that the change in free energy, delta G, can be related to the cell
potential E, by this equation. Under standard state conditions, this would be the standard
change in free energy, so delta G zero which is related to the standard cell potential E zero, right, by the same equation. This equation down here
comes from thermodynamics, and we're going to plug in
for delta G and delta G zero, so we're going to plug
this in for delta G, and we're going to plug
this in for delta G zero. So that gives us negative nFE is equal to, negative nFE zero, plus RT, natural log of Q, where Q is the reaction quotient. Let's divide everything by negative nF, so we're going to divide everything in here by negative nF, and let's see what cancels out. So all of this would cancel out, all of that would cancel out, and we get the Nernst equation. So let me go ahead and write it, the cell potential E, is equal to the standard
cell potential, E zero, minus RT over nF, times the natural log of Q, where Q is the reaction quotient. So this is the Nernst equation, alright, we'll talk about
why the Nernst equation is so important, we'll
talk more about that at the end of the video. Right now, let's go ahead and derive another form of the Nernst equation, I should say, the form when you're talking about a certain temperature. So at 25 degrees C, right, most of our reactions take
place at 25 degrees C, well, temperature, right, temperature in here, this in
the temperature in Kelvin, so we need to convert to Kelvin. So if you add 273.15, alright, you get your temperature in Kelvin, so that would be 298.15, so what we're going to do
is solve for RT over F, to write a different form
of the Nernst equation. And so the temperature is 298.15 Kelvin, so let's write that in, so 298.15 Kelvin. R is the gas constant, so remember, the gas
constant is 8.314, 8.314, joules over mole Kelvin. And F is Faraday's constant, remember Faraday's constant
from an earlier video, right, that's 96,500 coulombs per mole. So Faraday's constant is the charge of one mole of electrons. Alright, so let's solve for what all this is equal to, so let's get some more space, and let's get out the calculator. So we've already done this calculation in an earlier video, but I'm going to go ahead and do it again, for this Nernst equation video. So 8.314 times 289.15, we need to divide that by
Faraday's constant, 96,500. So that gives us 0.0257, alright, so this is equal to, this is equal to 0.0257, and for units, Kelvin would cancel out, moles would cancel out, that gives us joules over coulombs, which is equal to volts, alright. So this is equal to volts, and we can write another
form of the Nernst equation. So if your reaction is at 25 degrees C, you can write the Nernst
equation this way, you could say that the cell potential E, is equal to the standard
cell potential E zero, minus, so all that RT over F is
equal to this, 0.0257, right. So 0.0257 volts, we still have n, remember n is the number of moles I'm going to use green for that, n is the number of moles of electrons that are transferred in your redox reaction, right. So we're going to put n in here, and we still have the natural log of Q, the reaction quotient, right. So here is another form
of the Nernst equation, alright, so at 25 degrees,
you can use this form, and then we can also write this into base 10 logarithms
that have natural logs. So we can do that conversion, right. So we also did that in an earlier video, but if you're trying
to convert this 0.0257, you need to multiply by
the natural log of 10. So when we do that, so we have 0.0257 times
the natural log of 10, that gives us 0.0592, so this is equal to 0.0592. So we can write the Nernst
equation once again, alright, so E, or the cell potential, is equal to the standard
cell potential E zero, minus 0.0592 over n. And we essentially just change this from natural logarithm
to base 10 logarithm, so this would be log of Q,
log of the reaction quotient. So here is just another
form of the Nernst equation. So why is the Nernst equation important and why is it useful? It's useful, because it
allows us to calculate a cell potential under
non-standard state conditions, so think about this cell potential as being the instantaneous cell potential, and you can relate that to the progress of the reaction, right. So as you change concentrations, the reaction quotient Q changes, right, and that means that the instantaneous cell potential changes, alright, so if you change
if the concentrations, if you change the concentrations, you're changing the cell potential. And so we'll see more about
that in the next video, so I think that the Nernst
equation makes much more sense when you do some problems with it, and then you can understand
better what it means.