Nernst Equation and Ionic Equilibrium

We're going to talk about the Nernst equation. First of all, what is the Nernst equation? Well, in physiology, put simply, the Nernst equation can be used to calculate an ion's equilibrium potential. That is, the voltage that is necessary to offset the movement of an ion down an ion's concentration gradient. For this particular situation, let's imagine we have an ionic concentration gradient such that we have a higher concentration of ions inside the cell and a lower concentration of ions outside the cell. It's worth noting that for each compartment, we have an equal concentration of cations and anions. This is in accordance to the law of electrical neutrality, which basically means that the amount of positive charge must equal the amount of negative charge. Another way of thinking about it, although it is a little simplified, is you can imagine that we have an equal concentration of cations and anions for each compartment. For the purpose of the Nernst equation, let's simplify this a little bit and just focus on the cations and ignore the presence of the anions. So the Nernst equation arises from the behavior of ions in solution, specifically on what determines the net movement of ions in an aqueous solution. There's really two things. First of all, ions tend to move down their concentration gradients. That is, they tend to move from regions of high concentration to regions of low concentration. And this can be symbolized here as a vector. The direction of the vector indicates the direction of net ionic movement from high to low concentrations. And then the size of our vector here indicates the magnitude or the strength of the tendency for that movement to occur. The greater the concentration gradient, the greater the strength of the tendency for the ion to move down its concentration gradient. In a situation where the concentration gradient is reduced, We could still expect ions to move from regions of high concentration to regions of low concentration, but the strength of that tendency is also reduced, which ultimately means that the rate of ionic movement will also be reduced. And you could imagine a situation where if we were to allow this concentration gradient to dissipate such that now the concentration inside and outside are equal, then now we would have no net movement of our ion. That is, the ion would be at equilibrium. The other factor that determines how ions move in a solution is the presence of an electrical field or an electrical gradient. Ions tend to move down their electrical gradients. In this situation, let's assume that we have an equal concentration of our cations both inside and outside of our cell, just so we can ignore the effects of a concentration gradient. But now we hooked up a battery such that the... Negative terminal is oriented towards the inside of the cell and the positive terminal is oriented towards the outside of the cell. This creates an electrical field that's going to cause the ions to tend to move inside the cell in this situation. This is because the ions are going to be attracted to the negative terminal on the inside of the cell and equally repelled from the positive terminal which is placed on the outside of our cell here. Again, the strength of this tendency for the ion to move down its electrical gradient can also be represented as a vector, in this case pointed inward, where once again, the magnitude of our vector represents the strength or the tendency for the ion to move down this electrical gradient. We can imagine that if the battery were a much smaller voltage, creating a weaker electrical field, that the strength of this tendency for the ion to move down. this electrical gradient would equally be reduced. So what about this situation here? So let's imagine a situation that combines our first two conditions where, first of all, we have an ionic concentration gradient, such that we have a greater concentration of cations inside the cell compared to outside, but then we also have this battery connected that creates an electrical field such that the negative terminal is oriented inside the cell, the positive outside the cell. of the cell. On the one hand, we have a concentration gradient which would tend to favor outward movement, but on the other hand, we have an electrical gradient which would favor inward movement. So the question is, which way is the ion going to move? If you think about it for a moment, there isn't an intuitive or obvious answer to this question. It's because we're comparing an ionic concentration gradient to the presence of an electrical gradient. These seem like two very different things. How do you compare the size of a concentration gradient to the size of an electrical gradient. Well, that's where the Nernst equation comes in. The Nernst equation does just that. So, first of all, here is the Nernst equation. already derived, the Nernst equation computes that equilibrium potential. It's going to tell us what that voltage is that can offset that concentration gradient. Equilibrium potential is equal to RT over Fz times the natural log of the concentration of our ion outside over the concentration of the ion inside. And this will compute an equilibrium potential in volts. So real quick, R for the gas constant. T for temperature in degrees kelvin. F is the Faraday constant, which relates charge to mass. And then Z is the ion valence. That is, whether it's a monovalent cation, such as sodium, or a monovalent anion, such as chloride, or a divalent cation, such as calcium. Now, I want to take a moment right here and point out that the appearance of the Nernst equation, or the form of the Nernst equation, can look a little different depending upon which textbook you might be looking at. For instance, certain physiology textbooks such as Boron and earlier editions of Guyton, they express the Nernst equation like this, the negative RT over FC times the natural log of the concentration inside versus the concentration outside. And it might seem like an error, but if you remember your log rules, remember that the log of A over B is equal to the negative log of B over A. These two forms are actually equivalent. For the purpose of this video, let's just focus on the simpler version where we have the concentration outside versus the concentration inside. In physiological context, we can often simplify this equation by computing the constants, our Faraday constant and our gas constant. We can assume a standard physiological temperature of 37 degrees C or 310.2 degrees Kelvin. We can convert the natural logarithm to the log base 10, and then also convert from volts to millivolts, because membrane potentials are often in the range of millivolts, to this. The equilibrium potential of an ion is equal to 61.5 over Z, which is that ionic valence, times the log base 10 of the concentration outside over the concentration inside in millivolts. Again, depending upon which textbook you're looking at, the Nernst equation is often simplified to 50. over Z times one. This is just assuming a room temperature of 25 degrees C. So let's go back to our original question about which way will the ion move. Well to address this, let's first specify some concentrations for both the inside and outside compartments. Let us assume that the inside concentration is 100 millimolar and the outside concentration is 5 millimolar. Let's also assume that we're dealing with a monovalent cation, so Z equals plus one. So we can simply insert these terms into our simplified expression of the Nernst equation. So it's simply this. So 61.5 over plus 1 times the log base 10 of the concentration outside, 5 millimolar, over the concentration inside, 100 millimolar. So we can first take care of our two ratios here. This is 61.5 times the log base 10 of 0.05. We compute the log of 0.05. It's equal to negative... 1.3 and simply take the product and there it is. Our equilibrium potential is equal to negative 80 millivolts under these conditions. So what does this mean in this case? Well, it means that given this particular concentration gradient, this battery would have to be precisely 80 millivolts to perfectly offset the movement of this ion down its concentration gradient. Therefore, if this battery is less than 80 millivolts, say it's a... a 50 millivolt battery such that we have negative 50 millivolts inside compared to outside, then the size of our voltage gradient would actually be weaker than our concentration gradient and we could expect ions to travel down the concentration gradient. So the concentration gradient would win in that case. Alternatively, let's imagine that this is a 100 millivolt battery such that we have a voltage of negative 100 inside of our cell. Well now the voltage is stronger than the equilibrium potential of this ion. And the battery would win. Therefore, we could actually expect ions to travel up the concentration gradient from the region of low concentration to the region of high concentration. So that's how the Nertz equation is applied in this case. We have a concentration gradient. The question is, how big does that electrical gradient have to be so that it perfectly offsets, perfectly opposes the tendency of that ion to travel down its concentration gradient? based on these ionic concentration gradients. That voltage is precisely negative 80 millivolts, or negative 80 millivolts inside the cell compared to outside the cell. So again, based on typical ion concentration gradients, keep in mind that those can vary depending upon conditions and species and locations. Here are some common equilibrium potentials for those, for the main ions that are often described in physiological contexts. So the last question you might have is, well, why do we care about these values? Why do we care about the equilibrium potential of ions? Put simply, the equilibrium potential of an ion describes its influence on the membrane potential. How that works exactly is beyond the scope of this video. Please look for a link in the video description for how the equilibrium potential of an ion can influence the membrane potential of a cell.

It's worth noting that for each compartment, we have an equal concentration of cations and anions. This is in accordance to the law of electrical neutrality, which basically means that the amount of positive charge must equal the amount of negative charge. Another way of thinking about it, although it is a little simplified, is you can imagine that we have an equal concentration of cations and anions for each compartment.

For the purpose of the Nernst equation, let's simplify this a little bit and just focus on the cations and ignore the presence of the anions. So the Nernst equation arises from the behavior of ions in solution, specifically on what determines the net movement of ions in an aqueous solution. There's really two things.

First of all, ions tend to move down their concentration gradients. That is, they tend to move from regions of high concentration to regions of low concentration. And this can be symbolized here as a vector.

The direction of the vector indicates the direction of net ionic movement from high to low concentrations. And then the size of our vector here indicates the magnitude or the strength of the tendency for that movement to occur. The greater the concentration gradient, the greater the strength of the tendency for the ion to move down its concentration gradient. In a situation where the concentration gradient is reduced, We could still expect ions to move from regions of high concentration to regions of low concentration, but the strength of that tendency is also reduced, which ultimately means that the rate of ionic movement will also be reduced.

And you could imagine a situation where if we were to allow this concentration gradient to dissipate such that now the concentration inside and outside are equal, then now we would have no net movement of our ion. That is, the ion would be at equilibrium. The other factor that determines how ions move in a solution is the presence of an electrical field or an electrical gradient.

Ions tend to move down their electrical gradients. In this situation, let's assume that we have an equal concentration of our cations both inside and outside of our cell, just so we can ignore the effects of a concentration gradient. But now we hooked up a battery such that the...

Negative terminal is oriented towards the inside of the cell and the positive terminal is oriented towards the outside of the cell. This creates an electrical field that's going to cause the ions to tend to move inside the cell in this situation. This is because the ions are going to be attracted to the negative terminal on the inside of the cell and equally repelled from the positive terminal which is placed on the outside of our cell here. Again, the strength of this tendency for the ion to move down its electrical gradient can also be represented as a vector, in this case pointed inward, where once again, the magnitude of our vector represents the strength or the tendency for the ion to move down this electrical gradient. We can imagine that if the battery were a much smaller voltage, creating a weaker electrical field, that the strength of this tendency for the ion to move down.

this electrical gradient would equally be reduced. So what about this situation here? So let's imagine a situation that combines our first two conditions where, first of all, we have an ionic concentration gradient, such that we have a greater concentration of cations inside the cell compared to outside, but then we also have this battery connected that creates an electrical field such that the negative terminal is oriented inside the cell, the positive outside the cell.

of the cell. On the one hand, we have a concentration gradient which would tend to favor outward movement, but on the other hand, we have an electrical gradient which would favor inward movement. So the question is, which way is the ion going to move? If you think about it for a moment, there isn't an intuitive or obvious answer to this question. It's because we're comparing an ionic concentration gradient to the presence of an electrical gradient.

These seem like two very different things. How do you compare the size of a concentration gradient to the size of an electrical gradient. Well, that's where the Nernst equation comes in. The Nernst equation does just that.

So, first of all, here is the Nernst equation. already derived, the Nernst equation computes that equilibrium potential. It's going to tell us what that voltage is that can offset that concentration gradient. Equilibrium potential is equal to RT over Fz times the natural log of the concentration of our ion outside over the concentration of the ion inside. And this will compute an equilibrium potential in volts.

So real quick, R for the gas constant. T for temperature in degrees kelvin. F is the Faraday constant, which relates charge to mass. And then Z is the ion valence.

That is, whether it's a monovalent cation, such as sodium, or a monovalent anion, such as chloride, or a divalent cation, such as calcium. Now, I want to take a moment right here and point out that the appearance of the Nernst equation, or the form of the Nernst equation, can look a little different depending upon which textbook you might be looking at. For instance, certain physiology textbooks such as Boron and earlier editions of Guyton, they express the Nernst equation like this, the negative RT over FC times the natural log of the concentration inside versus the concentration outside. And it might seem like an error, but if you remember your log rules, remember that the log of A over B is equal to the negative log of B over A. These two forms are actually equivalent.

For the purpose of this video, let's just focus on the simpler version where we have the concentration outside versus the concentration inside. In physiological context, we can often simplify this equation by computing the constants, our Faraday constant and our gas constant. We can assume a standard physiological temperature of 37 degrees C or 310.2 degrees Kelvin.

We can convert the natural logarithm to the log base 10, and then also convert from volts to millivolts, because membrane potentials are often in the range of millivolts, to this. The equilibrium potential of an ion is equal to 61.5 over Z, which is that ionic valence, times the log base 10 of the concentration outside over the concentration inside in millivolts. Again, depending upon which textbook you're looking at, the Nernst equation is often simplified to 50. over Z times one. This is just assuming a room temperature of 25 degrees C.

So let's go back to our original question about which way will the ion move. Well to address this, let's first specify some concentrations for both the inside and outside compartments. Let us assume that the inside concentration is 100 millimolar and the outside concentration is 5 millimolar.

Let's also assume that we're dealing with a monovalent cation, so Z equals plus one. So we can simply insert these terms into our simplified expression of the Nernst equation. So it's simply this.

So 61.5 over plus 1 times the log base 10 of the concentration outside, 5 millimolar, over the concentration inside, 100 millimolar. So we can first take care of our two ratios here. This is 61.5 times the log base 10 of 0.05. We compute the log of 0.05.

It's equal to negative... 1.3 and simply take the product and there it is. Our equilibrium potential is equal to negative 80 millivolts under these conditions. So what does this mean in this case? Well, it means that given this particular concentration gradient, this battery would have to be precisely 80 millivolts to perfectly offset the movement of this ion down its concentration gradient.

Therefore, if this battery is less than 80 millivolts, say it's a... a 50 millivolt battery such that we have negative 50 millivolts inside compared to outside, then the size of our voltage gradient would actually be weaker than our concentration gradient and we could expect ions to travel down the concentration gradient. So the concentration gradient would win in that case.

Alternatively, let's imagine that this is a 100 millivolt battery such that we have a voltage of negative 100 inside of our cell. Well now the voltage is stronger than the equilibrium potential of this ion. And the battery would win. Therefore, we could actually expect ions to travel up the concentration gradient from the region of low concentration to the region of high concentration. So that's how the Nertz equation is applied in this case.

We have a concentration gradient. The question is, how big does that electrical gradient have to be so that it perfectly offsets, perfectly opposes the tendency of that ion to travel down its concentration gradient? based on these ionic concentration gradients. That voltage is precisely negative 80 millivolts, or negative 80 millivolts inside the cell compared to outside the cell.

So again, based on typical ion concentration gradients, keep in mind that those can vary depending upon conditions and species and locations. Here are some common equilibrium potentials for those, for the main ions that are often described in physiological contexts. So the last question you might have is, well, why do we care about these values? Why do we care about the equilibrium potential of ions?

Put simply, the equilibrium potential of an ion describes its influence on the membrane potential. How that works exactly is beyond the scope of this video. Please look for a link in the video description for how the equilibrium potential of an ion can influence the membrane potential of a cell.

Transcript for:Nernst Equation and Ionic Equilibrium

Transcript for:
Nernst Equation and Ionic Equilibrium