POST-LECTURE!
4-4 NERNST EQUILIBRIUM
When the membrane is permeable to only one kind of ion and the steady potential difference is reached, the permeant ions are said to be in Nernst equilibrium. However the ions which can't get through the membrane are not in equilibrium. If they could get into the membrane the field would push them through to the low concentration side.
Nernst equation
The equation for the Nernst potential is usually written using natural logarithms rather than base- 10 logs. ∆V = kTze ln ËÁ Ê ¯ ˜ ˆ C1 C2 . .... (4.1)
Here C1 and C2 are the concentrations on opposite sides of the membrane, k is the Boltzmann constant , T is the absolute temperature, z is the degree of ionisation of the permeant ion (usually 1 or 2) and e is the electronic charge. This is the only form of the equation that you need to know.
The equation can also be written in terms of number densities, which are proportional to the concentrations: ∆V = kTze ln ËÁ Ê ¯ ˜ ˆ n1 n2 . ... (4.1a) Note
It is not worth trying to remember how to order concentrations and apply the sign of z in equation 4.1. The sense of the electro-diffusive potential is more easily determined by considering which sign of ion diffuses through the membrane.
4-5 MEASUREMENT OF pH
One convenient and elegant way of determining the pH of a solution is to measure the Nernst potential across a membrane which is permeable only to hydrogen ions. The voltage across this membrane is then proportional to the logarithm of the ratio of the hydrogen ion concentrations on either side of this membrane.
Since ∆V µ log ËÁ Ê ¯ ˜ ˆ C1 C2 and pH µ log ËÁ Ê ¯ ˜ ˆ CH mol.L-1 then ∆V = const ¥ (pH1 - pH2) .
Thus if the pH of a standard solution on one side of the membrane is known, measurement of the membrane potential (the Nernst potential for H+ ) gives a direct measure of the pH of the unknown solution on the other side of the membrane.
Thus pH can be directly measured with a high-resistance (ideal) voltmeter. This is the basis of operation of the pH meter shown in the video lecture. In this instrument and most other commercial pH meters, the ‘membrane’ is a specially compounded glass which is permeable only to H+ ions. (See the box on page 50.)
E4: Ion Diffusion 49
A Theory of the Nernst equilibrium
In this theory we regard Nernst equilibrium as the balance between electrostatic force on a layer of charge, and the difference in the forces exerted by the ions in collisions with the surfaces of that layer. We assume that the motion of the ions is similar to that of the molecules of a gas. This is a valid assumption provided that the concentration of the solute is not too great. (This presentation is slightly different from that given in the video lecture. The main difference is the replacement of the symbol c
(concentration) by n (number destiny) in the early part of the derivation. The final result is identical.) Suppose that the Nernst equilibrium has
already occurred. Consider a thin layer of positive ions that are within the membrane, moving through. All those ions will feel an electrostatic force to the right from the electric field within the membrane. However this layer does not move, as the pressure of the ions on the right hand side of the layer is greater than that on the left. This pressure difference arises simply because there is a greater concentration of ions at the right of the layer. In a Nernst equilibrium the electric force is balanced by the force resulting from the pressure difference on the layer and there is no net flow of ions into, or out of, the layer.
For the layer, this equilibrium condition can be expressed as the equation:
(Magnitude of electric field) ¥ (total charge) = force from pressure difference ;
E. nze.A dx = A dP
Hence, by cancelling the area A, we get E.nze.dx = dP .... (1) where n is number density of ions, z is the ionisation state of the ion, and e is the electronic charge.
Assuming that the ions obey the ideal gas law (detailed investigation shows this to be true for dilute solutions), P = nkT or P = CRT ... (2) where k is the Boltzmann constant (R/N0), T is the temperature, R is the gas constant and N0 is the Avogadro constant.
From equation 2 we get dP = kT.dn or dP = RTdC .
This, together with the relations n = N0C ; k = R/N0
allows us to rewrite equation 1 as E.Cze.dx = kT.dC .... (3)
or E dx = dV = kT
ze
dC C
Integration through the thickness of the membrane gives the Nernst equation: ∆V = kT ze ln ËÁ Ê ¯ ˜ ˆ C 1 C2 where ∆V = ıÛ x1 x 2
!
E! dx is the potential difference across the membrane, and C1 and C2 are the
concentrations on opposite sides of the membrane.
Expressed with different constants and base-10 logarithms the equation takes the form ∆V = 2.303 R T zF log1 0 ËÁ Ê ¯ ˜ ˆ C1 C 2 ...(4) where F is the Faraday constant (see chapter E3).
You should satisfy yourself that the constant factor is indeed about 60 mV for a singly ionised ion at room temperature, as found experimentally.
E4: Ion Diffusion 50
Demonstration - pH meter
A pH meter is used to check the pH of the blood of a patient recovering from anaesthesia. The pH meter shown is specifically designed for a small sample of blood. The glass membrane is a capillary tube sealed into a glass tube filled with a liquid of known, standard pH.
The head is separated from its stand to draw a blood sample up the plastic capillary to fill the glass capillary, which is permeable to H+ ions. (The stand and head are shown separated in this diagram.) The standard reference solution is in the glass tube surrounding the glass capillary.
When the head is returned to the stand the blood in the plastic capillary makes electrical contact with the saturated KCl solution which is part of the calomel reference electrode.
The reference electrodes used in this instrument are designed so as to minimise the spurious potentials which arise at the various liquid-liquid interfaces. (For a discussion of bioelectrodes such as the calomel electrode see chapter E5.) AgCl electrode Standard solution Glass capillary (H+ ion permeable 'membrane') Plastic capillary Well for capillary filled with saturated KCl solution
Calomel electrode