The Nernst Equation

So the nurse equation reveals the quantitative connection between the concentrations of compounds and sell potential. So the nurse equation is utilized when our concentrations of our compounds differ from one. Moeller were going to say that the nurse equation equals are self potential here. And this self potential represents some potential under nonstandard conditions, meaning that our concentration is unequal. Toe one Moeller. Our temperature wouldn't be 25 degrees Celsius are pH were not equal. Seven are pressure wouldn't be one atmosphere. All of those values represent standard conditions. So one atmosphere ph of seven, a temperature of 25 degrees Celsius as well as a concentration of one Moeller. When we have all of these conditions met, that means we're dealing with self self potential under standard conditions. So that's E zero cell. So this represents our self potential under standard conditions minus rt. Divided by N times F times Ellen A divided by a So here again we said that this represented our standard self potential are here is our gas constant. It is equal to 8.314 jewels over moles. Times K. Also remember here that when we talk about Jules, we're talking about energy. Ah, Julhas Equal toe vaults Times cool apps. Remember, a vault is equal to Jules over cool apps, so substitute in jewels over columns, times columns. And that's how it equals jewels. So we can say this or 8. Vault's times cool ums over moles times K and equals the number of electrons transferred within our Redox Reaction f equals fair days Constant, which is 9.649 times 10 to the four columns over moles of electrons. A represents our activities, so activity here would just be or activity coefficient times the concentration of the ion if necessary. Uh, many times we may not be given some type of coefficient, so you could just say activity. You could substitute and concentration for that value. Now we're going to say that this expression here represents products, overreact INTs in terms of our Redox reaction, and it's equal to cube our reaction quotient. Now, if we were to take a look at the portion, that's our times t divided by F here at 25 degrees Celsius, Remember, we have our our constant. We have our temperature of 25 degrees Celsius we add to 73.15 to this, so that gives me to 98.15. Kelvin F. Is our Faraday's constant here. We would see that from this, moles would cancel out. Kelvin's would cancel out what we have left. At the end is Jules over cool ums, which is equal to volts So our T over f reduces all the way down 2.257 volts. So that means that our nurse equation becomes now self potential under non static conditions equals self potential under standard conditions minus 0.257 volts divided by and the number of electrons transferred Times Ellen of Q. Remember, Q Is just your equilibrium expression Now here. If we multiply Ellen by 2.303 we can attain the log function. Now when we multiply this portion here, okay, by 2.303 we get a new value of 0.5916 volts divided by end Now log of Q. This is true because we say that log of X equals L N of X, divided by Ellen of 10. Ellen of 10 equals 2.303 So when I multiply both sides by 2. okay, we can see that multiplying by two points 303 helped us to establish this new relationship here of this value. Now this is important. We're going to say the self potential calculated from the nurse equation is the maximum potential at the instant the circuit. The self circuit is connected. So that's the moment that the current or the flow of electrons moves from the ano to the cathode. So we're talking about basically the transferring of electrons from one electrode to another electrode, we're going to say, As the cell discharges and current flows, the electrolyte concentrations will change. What's gonna happen here is that Q will begin to increase, and as a result, our self potential over time will decrease until it gets to a point where the self potential off my electrochemical cell equals zero. That's when we have a dead battery, because at that moment the battery has reached equilibrium, so over time the reaction will reach equilibrium. And then Q, which is our reaction question will equal K, which is our equilibrium constant, the self potential. Like I said, would equal zero. We'd have a dead battery at that moment. Now, as a result of this, once we've reached equilibrium, weaken substitute in K instead of cute. So now our new equation can become s l equals sl under standard conditions minus are TNF times lnk. So here, if we were to work this out, we could have zero equals e cell minus. Remember, this value here would be 0.257 Volz divided by n now times Ellen of K. So we can do here is we subtract sell potential from both sides so that be negative. Standard self potential equals negative 0.25 seven volts divided by n times Ellen F K Multiply both sides by n then divide both sides now by negative 0 to volts. So we get at this point, Alan of K equals end times your self potential under standard conditions divided by 0.257 volts. To get rid of this Ellen, we take the inverse of a natural law. So that means that K equals e to the end times yourself potential under standard conditions divided by 0. volts And that's when we're dealing with Elena. If we're to substitute and log instead. So if we're dealing with log function, then it would be 0.5916 volts divided by n times log of K. And this is instance, if we did the same exact, um, mathematical conversions in this case because we're dealing with long, we'd find a K at the end equals 10 to the n times yourself potential under standard conditions divided by 0.5916 volts. So this is how we connect. Are equilibrium constant to our standards. Self potential with these two formulas one when we're dealing with Ellen and one more 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 Delta G, under nonstandard conditions, equals Delta G under standard conditions minus r t l n k. When we have reached equilibrium, this Delta G equals zero. So that means zero equals Delta G zero minus r t l N k. Subtract Delta G zero from both sides so negative. Delta G zero equals negative. Rt Alan Kay divide both sides now by negative rt So Alan Kay equals Delta G zero, divided by rt. And so K equals E to the Delta G zero divided by rt. So we get this value at the end in terms off 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 self potential to K and then from K two Delta G. And we know that from earlier we could also connect to sell potential from Delta G as well. When we have Delta G equals negative end times. Faraday's constant times self potential. Later on, we'll talk about this connection between the three variables. But for now, just realize we deal with the nursed equation when we have concentrations that are not equal toe one Mueller and remember, nursed equation could be written in two different ways. We can write it in this version more dealing with Ln or but we could have the nurse equation in this version when we're dealing with lock to go from Alan toe log, you just use to you just multiply by 2. to go from your Alan function to your log function. So keep in mind the Inter Kid dynamics involved in the nurse equation and how it connects yourself potential in standard conditions to non standard conditions.