Ion Selective Electrode membrane potential (E_mem) based on a glass membrane

Many ion selective electrodes use a glass membrane for the ion-selective component. The potential across the membrane is similar to a liquid junction potential. As ion selective electrode measurements are a classic type of potentiometry measurement, we can begin to understand the membrane potential by recognizing the relationship of the electrochemical potentials of the ions on either side of the membrane, as well as the relationship. The glass membrane can be broken into three distinct regions: the inner hydrated membrane (m''), the outer hydrated membrane (m'), and the region between (m, dry glass). At the interface between each of these regions, along with the interface between the hydrated membranes and the solution phases (alpha- the outer, test solution, and beta- the inner solution), a phase potential difference is established.

We can say that:

Breaking this into all of the contributions from each of the interfaces in the above diagram:

Now, using electrochemical potentials and liquid junction potential relationships, we can generate expressions for each of the phase potential differences in the above E_mem equation. For example, at the interface between the solution phase (alpha) and the hydrated glass (m') an equilibrium is established between H⁺ ions in both phases (see the above figure). Consequently, we can say:

Now, solve for the phase potential difference:

An analogous expression can be derived for the interface between the inner solution (beta) and the inner hydrated membrane (m''):

The "interfaces" between the hydrated glass layers (m' and m'') and the dry glass (m) are similar to liquid junctions. Consequently, we use a variant of the Henderson equation to express the "Junction potentials" at each of these interfaces. The result is given below:

and

For these liquid junctions, we assumed that there were 2 charge carriers (H⁺ and Na⁺). This comes from the composition of the glass, and introduces the possibility of other ions contributing to the membrane potential. Now combining these four contributions to the membrane potential:

This equation can be simplified by combining terms, and by recognizing that:

1. The standard state chemical potential of a proton in any aqueous phase are equal to each other:

2. A similar statement can be made about the standard state chemical potential of a proton in the two hydrated layers:

Making these assumptions, and combining terms leaves:

Now combining the logarithm terms and rearranging leaves:

From this equation, we can see that the membrane potential is dependent on the concentrations of the protons in the solutions on either side of the glass membrane (the alpha and beta solutions).

At the membrane/solution interface, we can consider that an equilibrium reaction is occurring:

This equation says that there is an exchange of charge carriers (Na⁺ and H⁺) at the solution/membrane interface. From this equilibrium reaction, we can write an equilibrium constant expression:

Looking carefully, we see that the membrane potential equation (above) has a combination of activities that are very similar to this equilibrium constant expression. Rearranging the equilibrium constant expression gives:

Substituting into the membrane potential expression:

We can define a new term, the potentiometric selectivity coefficient from the above Membrane potential expression. The Potentiometric selectivity coefficient is simply a measure of how selective the membrane is for a particular ion relative to an interfering ion.

In the case above, the potentiometric selectivity coefficient expression is for the selectivity of the membrane for protons relative to an interfering sodium ion.

This expression for the membrane potential has separated the contributions from the test solution (alpha phase) from the inner solution (beta phase) in the numerator and denominator of the logarithm term. Ideally, we wish to have the membrane potential measure the concentration of only the test solution. This cannot be done, however, one can establish that the inner solution composition (the beta phase) is constant and does not change during the course of the measurement. In this case, the denominator of the logarithm term remains constant and can be separated from the contribution to the membrane potential by the test solution.

This expression has only a contribution from one interfering ion (Na⁺). If other interfering ions are present, they must also be accounted for. The membrane potential equation, therefore, can be generalized by: