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The Nernst-Einstein equation

It is possible to determine the equilibrium potential for an ion across a delimiting membrane using the Nernst-Einstein equation. Let us see how this equation is derived.

We wish to determine the quantity of a material which (in this case an ion in aqueous solution) will move across the cell membrane under given circumstances of ionic concentration inside the cell and electrical potential across that cell membrane. From the thermodynamic point of view this depends on the sum of the free energy of the material inside the cell (Gin) and outside the cell (Gout) at equilibrium. The equilibrium potential is the electrical potential at which no such movement occurs when Gin — Gout =0.

For a charged particle (an ion) this energy is represented by the sum of the diffusional and electrical forces acting on the particle.

Let us consider the electrical forces

According to Faraday the charge on a mole of material is 96483 z Coulombs where z is the charge of each atom or ion (the valency of an ion). This is the Faraday constant or F.

Thus the electrical energy in a mole of an ion may be expressed as

Equation #1 Gelect = zFE

Where z =valency of ion

E = electrical potential across the delimiting membrane.

Let us consider the diffusional forces

Given a mass of gas in thermal equilibrium we may measure its pressure (p) temperature (T) and volume (V). Boyle demonstrated that pV/T is a constant Volume occupied is proportional to the mass of gas, we can write the above constant as ÁR where Á is the mass in moles and R is a constant. R = 8.134 joule/mole K

Thus
Equation #2
pV=ÁRT

Now the work done by an expanding gas can be calculated as follows:

Equation #3

Where the gase expands from an initial volume Vi to a final volume Vf. Using equation #2, we can see that the work done per mole is

Equation #4

At a constant temperature this may be written as

Equation #5

We can use this result to describe the free energy of diffusion of a particular ion in our cell or outside of the cell as follows:

Equation #6 Gdiff = RTln(c)

Where c = concentration (M)

The free energy of ions inside or outside of the cell is the sum of these

So the free energy (G) inside or outside of the cell may be expressed as

Equation #7 G = Gdiff + Gelect = RTln(c) + zFE (from #1 and #2)

The difference between free energy inside the cell and outside defines the free energy driving movement of the ion across the cell membrane which may be expressed as follows:

Equation #8 DG= (RTln(cout) +zFEout) —(RTln(cin) + zFEin)

Which simplifies to:

Equation #9) DG = R.T.ln(cout/cin) + z.F.DE

By definition at equilibrium DG = 0 thus the equilibrium potential for any given ion is given by:

Equation #10 DE = RT/ZF ln(cin/cout) This is the Nernst-Einstein equation.

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