So the Nernst equation reveals the quantitative connection between the concentrations of compounds and cell potential. The Nernst equation is utilized when the concentrations of our compounds differ from 1 molar. We're going to say that the Nernst equation equals our cell potential here. This cell potential represents cell potential under non-standard conditions. Meaning that our concentration isn't equal to 1 molar. Our temperature wouldn't be 25 degrees Celsius. Our pH would not equal 7. Our pressure wouldn't be 1 atmosphere. All of those values represent standard conditions. So 1 atmosphere, a pH of 7, a temperature of 25 degrees Celsius, as well as a concentration of 1 molar. When we have all of these conditions met, that means we're dealing with cell potential under standard conditions. So that's E_{0} cell. This represents our cell potential under standard conditions. Minus RT divided by N times F times LN(A) divided by A. Here again, we said that this represented our standard cell potential. R here is our gas constant. It is equal to 8.314 joules over moles times k. Also, remember here that when we talk about joules we're talking about energy. A joule is equal to volts times coulombs. Remember a volt is equal to joules over coulombs. So substitute in joules over coulombs times coulombs and that's how it equals joules. So we can say this or 8.314 volts times coulombs over moles times k. N equals the number of electrons transferred within our redox reaction. F equals Faraday's constant which is 96490 coulombs over moles of electrons. A represents our activities so activity here which is to be our activity coefficient times the concentration of the ion if necessary. Many times we may not be given some type of coefficient so you can just say activity, you could substitute in concentration for that value. Now, we're going to say that this expression here represents products over reactants in terms of our redox reaction and it's equal to q, our reaction quotient.

Now if we were to take a look at the portion that's RT divided by F, here at 25 degrees Celsius, remember we have our R constant, we have our temperature of 25 degrees Celsius. We add 273.15 to this. So that gives me 298.15 Kelvin. F is our Faraday's constant. Here we would see that from this moles would cancel out, Kelvins would cancel out. What we'd have left at the end is joules over coulombs which is equal to volts. So RT over F reduces all the way down to 0.0257 volts. So that means that our Nernst equation becomes now cell potential under non-standard conditions equals cell potential under standard conditions minus 0.0257 volts divided by n, the number of electrons transferred times ln of q. Remember, q is just your equilibrium expression. Now, here if we multiply ln by 2.303 we can attain the log function. Now when we multiply, this portion here by 2.303, we get a new value of 0.05916 volts divided by n, now log of q. This is true because we say that log of x equals ln of x divided by Ln of 10. Ln of 10 equals 2.303. So when I multiply both sides by 2.303, we can see that multiplying by 2.303 helped us to establish this new relationship here of this value.

This is important. We're going to say the cell potential calculated from the Nernst equation is the maximum potential at the instant the circuit the cell circuit is connected. That's the moment that the current or the flow of electrons moves from the anode to the cathode. What's going to happen is the electrolyte concentrations will change. Over time, the reaction will reach equilibrium and then q, which is our reaction quotient, will equal k, which is our equilibrium constant. The cell potential, like I said, would equal 0. We'd have a dead battery at that moment. Now as a result of this, once we've reached equilibrium we can substitute in k instead of q. So now our new equation can become e_{cell} equals e_{cell} under standard conditions minus RTNF times ln(k). Here, if we were to work this out we could have 0 equals e_{cell} minus remember this value here would be 0.0257 volts divided by n, now times ln of k. So what we can do here is we'd subtract cell potential from both sides. That would be negative standard cell potential equals negative 0.0257 volts divided by n times ln of k. Multiply both sides by n. Then divide both sides now by negative 0.0257 volts. So we get at this point, ln of k equals n times your cell potential under standard conditions divided by 0.0257 volts. To get rid of this ln, we take the inverse of the natural 0257 volts. And that's when we're dealing with ln. If we were to substitute in log instead, so if we're dealing with the log function, then it would be 0.05916 volts divided by N times log of k. In this instance, if we did the same exact mathematical conversions. In this case, because we're dealing with log, we'd find that k at the end equals 10 raised to the power of (n times your cell potential under standard conditions divided by 0.05916 volts). So this is how we connect our equilibrium constant to our standard cell potential with these two formulas. One when we're dealing with ln and one where we're dealing with log. Now we could also say that when you're at equilibrium, we can talk about the connection to Gibbs free energy and your equilibrium constant k. So here we'd say that ΔG under non-standard conditions equals ΔG under standard conditions minus RT ln k. When we've reached equilibrium, this ΔG equals 0. So that means 0 equals ΔG_{0} minus RT ln k. Subtract ΔG_{0} from both sides. So -ΔG_{0} equals -RT ln k. Divide both sides now by -RT. So ln k equals ΔG_{0} divided by RT. And so, K equals e raised to the power of (ΔG_{0} divided by RT). So we get this value at the end in terms of our connection between Gibbs Free Energy and your equilibrium constant K. So, just keep in mind some of the connections that we've seen here in terms of how things are connected to one another. These two equations are just a way of us connecting cell potential to k, and then from k to ΔG. And we know that from earlier, we could also connect to cell potential from ΔG as well, when we have ΔG equals negative n times Faraday's constant times cell potential. Later on, we'll talk about this connection between the three variables. But for now, just realize we deal with the Nernst equation when we have concentrations that are not equal to 1 molar. And remember, the Nernst equation can be written in 2 different ways. We can write it in this version when we're dealing with ln or we can have the Nernst equation in this version when we're dealing with log functions. So keep in mind the intricate dynamics involved in the Nernst equation and how it connects your cell potential in standard conditions to nonstandard conditions.