What is the potential of the electrode?

Question

The selectivity coefficient, 𝐾K+,H+, for a K+ ion-selective electrode is 0.00394. When the electrode is placed into a 2.23×10−5 M K+ solution at pH 6.71, the potential is −0.160 V vs. SCE. If the pH of the solution is lowered to 1.26 and the ionic strength is kept constant, what is the potential of the electrode? Assume 𝛽=1 for the electrode.

Answer 1

This is the Nikolsky-Eisenman equation. An ion-selective electrode does not respond only to its target ion - it also responds, weakly, to interferents, and the selectivity coefficient is how weakly. With beta = 1 and both ions singly charged:

E = constant + 0.05916 log( [K+] + K(K,H) [H+] )

The strategy is to use the first measurement to pin down the unknown constant, then reuse that same constant at the new pH. Nothing about the electrode changed - only the solution did.

Step 1 - the first condition, pH 6.71

[H+] = 10^-6.71 = 1.95 x 10^-7 M

interference term = (0.00394)(1.95 x 10^-7) = 7.7 x 10^-10 M

Compare that to [K+] = 2.23 x 10^-5 M. The interference is about 30000 times smaller, so at this pH the electrode is effectively seeing potassium alone. Sum = 2.230 x 10^-5.

-0.160 = constant + 0.05916 log(2.230 x 10^-5)
-0.160 = constant + 0.05916(-4.6517)
-0.160 = constant - 0.2752

constant = +0.1152 V

Step 2 - the second condition, pH 1.26

[H+] = 10^-1.26 = 0.0550 M

interference term = (0.00394)(0.0550) = 2.165 x 10^-4 M

Now the picture inverts. The hydrogen contribution is roughly ten times larger than the actual potassium signal, even though the electrode is 250 times more selective for K+ - because there is so much more acid present.

sum = 2.23 x 10^-5 + 2.165 x 10^-4 = 2.388 x 10^-4

E = 0.1152 + 0.05916 log(2.388 x 10^-4)
E = 0.1152 + 0.05916(-3.6219)
E = 0.1152 - 0.2143

E = -0.099 V vs. SCE

Check the size of the shift

The apparent activity went up by a factor of 2.388/0.2230 = 10.7, and a tenfold change should move a one-electron Nernstian response by about 59 mV. Indeed 0.05916 log(10.7) = 61 mV, matching the shift from -0.160 to -0.099 V.

The practical lesson: a small selectivity coefficient does not make an interferent safe. What matters is the product of the coefficient and the interferent concentration. This is exactly why real potassium measurements are run in a pH-buffered ionic strength adjuster rather than in whatever the sample happens to be.