# What is N in the Nernst equation

## Nernst-Equation / Gold man-Equation

The equation established by Walther Nernst originally comes from electrochemistry; it can be used to calculate the concentration-dependent voltage that exists between two galvanic half-cells. People who are familiar with chemistry, especially electrochemistry, are welcome to skip the following excursus and jump straight to the calculation of the membrane potential.

### Digression into electrochemistry

A half cell consists of an electrode, for example zinc, and an electrolyte, i.e. a saline, electrically conductive solution that must match the electrode. A solution of zinc chloride would be suitable for the zinc electrode, for example, a solution of copper sulfate for a copper electrode.

A standard half-cell is a half-cell in which the concentration of the electrolyte is exactly 1 mol / l.

A galvanic element is obtained when two half-cells are connected to one another. For example, if you connect a zinc half-cell with a copper half-cell, you can measure an electrical voltage. The level of this voltage is calculated from the standard redox potentials of the half-cells. These redox potentials can be looked up in tables. For the standard zinc half-cell the redox potential is -0.76 volts, for the standard copper half-cell +0.35 volts. A galvanic element, which consists of two such standard half-cells, then supplies a voltage of 1.11 volts. That is exactly the difference between the two standard redox potentials of zinc and copper.

You can find more information about half-cells and galvanic elements on my chemistry pages under "The voltage series of metals".

However, it is often the case that a battery or an accumulator or other electrochemical processes do not have standard half-cells with 1-molar salt solutions. Usually the electrolytes have other concentrations, for example C (ZnSO4) = 0.35 mol / l and c (CuSO4) = 1.22 mol / l. How can you now calculate the voltage that exists between these two half-cells? The following equation is used for this, which was named after Walther Nernst.

$ U_ {H} = U_ {H} ^ {\ 0} + \ frac {0.059V} {z} * lg \ frac {c (Me ^ {z +})} {mol / l} $$ U_ {H} ^ {\ 0} $ is the standard redox potential of the half-cell, 0.059 V is a constant, z is the number of electrons that can be absorbed by the metal ions (for copper or zinc, z = 2, for silver 1, for aluminum 3 and so on), and $ c (Me ^ {z +}) $ is the concentration of the electrolyte.

Let's play that for the copper half-cell with c (CuSO_{4}) = 1.22 mol through. The standard redox potential for the copper cell is +0.35 volts.

$ U_ {H} = 0.35 V + \ frac {0.059V} {2} * lg \ frac {1.22 \ mol / l} {mol / l} $

$ U_ {H} = 0.35 V + \ frac {0.059V} {2} * 0.086 $

$ U_ {H} = 0.35 V + 0.0025 V = 0.3525 V $

The redox potential of this copper half-cell is therefore somewhat greater (more positive) than that of the standard copper half-cell. This is because in the copper-zinc element, the copper plays the role of the electron acceptor, i.e. it accepts electrons:

$ Cu ^ {2 +} _ {(aq)} + 2 e ^ {-} \ to Cu _ {(s)} $

The higher the concentration of the copper sulfate solution, the greater the probability that the excess electrons of the copper electrode can be absorbed by copper ions.

This ends our little chemical digression, and we now come to the actual topic of this article, namely the Nerst equation, as it is important for calculating the membrane potential of a nerve cell.

### Calculation of the membrane potential

The following section assumes that you have read and understood the basics of the development of the resting potential. If not, please do it very quickly, otherwise you will not understand anything about the following statements!

##### The K^{+}-Equilibrium potential

Assume that only the potassium ions play a role in creating the membrane potential, so we can ignore the influence of the sodium, chloride and organic anions. Then as long as K^{+}-Ions flow outwards until the electrical potential that forms is exactly as large as the slowly decreasing diffusion pressure or the chemical K^{+}-Potential. An electrochemical equilibrium has then been established between chemical and electrical potential, exactly as many K diffuse per unit of time^{+}-Ions outwards as well as inwards. The electrical potential that can then be measured on the membrane is called K^{+}- equilibrium potential.

With the Nernst equation this K^{+}-Calculate equilibrium potential now:

R is the so-called gas constant, T the absolute temperature, measured in Kelvin and F the Faraday constant. $ K ^ {+} _ {outside} $ and $ K ^ {+} _ {inside} $ are the concentrations of potassium ions in the external medium and inside the nerve cell.

At a temperature of 37 ºC, such as that found in human cells, the expression is simplified to

$ E_ {K} = -61mV \ * \ log (\ frac {K ^ {+} _ {inside}} {K ^ {+} _ {outside}}) $

Attention: Outside and inside are now swapped, for this the $ \ frac {R * T} {F} $ - term has been given a negative sign.

Let’s now take a look at the concentration ratios in a human nerve cell. In the cytoplasm of the nerve cell there is a concentration of approx. 155 mmol / l, in the external medium a concentration of 4 mmol / l^{ [1]}. If we plug this value into the simplified Nernst equation, we get

$ E_ {K} = -61mV \ * \ log (\ frac {155 \ mol / l} {4 \ mol / l}) $

The result is then -96.88 mV. That is the equilibrium potassium potential. And this value would also be the resting potential of the human nerve cell if the resting potential were solely dependent on the potassium diffusion.

However, the actual resting potential of human nerve cells is -70 mV.

##### The Na^{+}-Equilibrium potential

The resting potential of a mammalian nerve cell is not quite as "large" as the calculated K^{+}-Equilibrium potential. The K^{+}- Equilibrium potential was caused by the leakage of positive charges. If the actual membrane potential is now a little weaker, this could be explained in two ways:

- Either other positive charges flow into the cell
- or negative charges also flow out of the cell.

We can rule out the second case. There are chloride ions in the cell, but their concentration is negligible, so that they shouldn't play a role. And then there are the large organic anions. But these are too big to be able to pass through the membrane. So we can forget about these negative charges too. Only the first option remains, namely the influx of positive charges. Calcium ions and sodium ions come into question here, both of which are present in very high concentrations in the external medium. But as has been found out, the calcium ions play Ca^{2+} also plays a rather subordinate role in membrane potential. So only the sodium ions Na remain^{+}.

The equilibrium sodium potential for T = 37 ºC is calculated similarly to the equilibrium potassium potential:

$$ E_ {K} = -61mV \ * \ log (\ frac {Na ^ {+} _ {inside}} {Na ^ {+} _ {outside}}) $$

If we put in the values from [1] again, we get

$ E_ {K} = -61mV \ * \ log (\ frac {12 \ mol / l} {145 \ mol / l}) $

The Na is then calculated from this^{+}-Equilibrium potential with +67.1 mV.

So a small proportion of the Na also flows into the actual resting potential of a nerve cell^{+}-Equilibrium potential.

##### Experimental evidence for the influence of Na^{+}-Ions

If you have the Na^{+}-Ions in the external medium are replaced by larger cations, which under no circumstances can fit through the cell membrane, then a resting potential arises that is roughly equal to the K^{+}- Equilibrium potential. Choline^{+} is for example such a cation^{ [2]}.

The choline cation

### The influence of membrane permeabilities

The permeabilities of the membrane for the various types of ions play an important influence on the membrane potential. If the permeability for potassium ions is set to the arbitrary value 1, chloride ions have a relative permeability of 0.45 and sodium ions have a relative permeability of 0.04. If you take these values into account, you arrive at the Goldmann equation, which can be used to calculate the membrane potential as a function of the three types of ions mentioned:

$ E_ {membrane} = \ frac {R \ cdot T} {F} \ cdot \ log (\ frac {P_ {K} \ cdot K ^ {+} _ {a} + P_ {Na} \ cdot Na ^ { +} _ {a} + P_ {Cl} \ cdot Cl ^ {+} _ {a}} {P_ {K} \ cdot K ^ {+} _ {i} + P_ {Na} \ cdot Na ^ {+ } _ {i} + P_ {Cl} \ cdotp Cl ^ {+} _ {i}}) $However, you have to be careful when inserting the values. In [2], values that deviate completely from [1] are used for the permeabilities, namely P_{N / A}= 1, P_{K}= 20 and P_{Cl} = 0.1. However, the value for the chloride permeability is quite unusual. Should an error have occurred in [2]?

$ E_ {membrane} = \ frac {R \ cdot T} {F} \ cdot \ log (\ frac {237} {3112}) = 61 mV \ cdot -0.39 = 68.2 mV $

This theoretical value agrees fairly well with the measured value of the mammalian membrane potential at rest.

##### Annotation:

The question of the actual membrane permeabilities is still open. Two important sources [1] and [2] differ considerably. A more detailed research in the specialist literature is sometimes necessary.

A search for the keyword "membrane permeability" on July 19, 2020

Great, the first thing that gets listed is my own pages on Google, so I seem like total capacity in terms of membrane permeability.

The same permeabilities can be found in the lexicon of biology published by Spektrum-Verlag as in [1]:

"The permeability constants of the 3 types of ions are roughly PK: PCl: PNa = 1: 0.45: 0.04." ^{[3]}

In a script from the University of Munich, the following values are given for the permeabilities at rest: P_{K} = 15, P_{N / A} = 1, P_{Cl} = 1. If you set the potassium permeability to 1, these values result: P_{K}K = 1, P_{N / A} = 0.07, P_{Cl} = 0,07 [4].

In Shepherd's book^{ [5]} the permeability value of Schmidt can be found again.

Swell:

- Schmidt: Outline of Neurophysiology, Berlin Heidelberg 1987
- Dudel, Menzel, Schmidt: Neuroscience, Heidelberg 2001.
- Lexicon of Biology, Spektrum-Verlag 1999, keyword "Membrane Potential".
- Script for the lecture biophysics (WS 03/04) at the University of Munich
- Shepherd: Neurobiology, Berlin Heidelberg 1993.

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