pH Meter Calibration Equations

I am sometimes asked whether,  given that a pH electrode is just a feeble battery whose voltage depends on the pH,  one couldn't just hook up a pH electrode to a voltmeter and do the math necessary to calculate pH from voltage. The answer is "Yes, provided…". The provisions are that you have a good electrode and that the voltmeter has very, very high input impedance - much higher than even most FET based multimeters. This derives from the fact that the current produced by the meter is so weak (it is, after all, flowing through a glass membrane and glass is an insulator). So if you want to use your Fluke you will have to design (or buy) an instrumentation amplifier to connect between the electrode and the meter.  The pH electrode produces about 58 mV for each unit away from pH 7 so that if you are measuring beer at pH 4 the voltage would be about +232 mV and thus 1 or 10 might be good choices for the amplifier's gain. But even with a complete pH meter system there are advantages to putting it in mV mode (this mode is not usually found on inexpensive meters) in which case it acts just like the combination instrumentation amplifier/volt meter we have just described. 

There are a couple of reasons for doing this. The most significant is that many modern meters will detect that your electrode is aging by looking at it's 'slope' (see below) which they calculate during calibration and refulse to complete calibration if it  falls below some value (typically 95%) programmed into the instrument. This means you must replace the electrode. As, clearly, the manufacturer will sell more electrodes with a threshold of 95% than he will with a threshold of 90% the writer's cynical nature causes him to suspect that perhaps thresholds are set higher than they really need to be. By processing millivolt readings yourself you can continue to use the meter with any reasonable value of electrode slope. Less than 75% would probably not be reasonable.

A second reason is that modern pH electrodes have isoelectric pH (pHi - see below) very close to 7 and there is no way to insert another value into the meter's algorithm. In the unlikely event you own an electrode with pHi not close to seven (and the writer does) doing the math yourself allows you to continue to use this electrode and still obtain accurate ATC (Automatic Temperature Correction).

A third reason is, of course, that you will learn something about the workings of pH meters.

The calculations are not difficult. The electrode produces a voltage: 

E = Eo - s*(R*T/F)(pH(T) - pHi)

E is the electrode voltage in volts

Eo is the electrodes offset in volts - you get this number from calibration

s is the "slope" of the electrode. It is a number close to 1 (a value of 1 is often displayed by meters as 100%). If the electrode is ideal it will be very close to 1 and decline gradually as the electrode ages. s is the other number you get from calibration.

R and F are the gas constant and the Faraday constant. They can be found on the web or in textbooks

T is the temperature of the sample in Kelvins (add 273.15 to centigrade).

RT/F = 58.167 mV/pH at 20 °C (293.15 K)

pH is the thing you are trying to find and I wrote it pH(T) to indicate that it is a function of temperature (both in the case of buffers and samples being measured).

pHi is the "isoelectric pH" of the electrode i.e. the pH at which the electrode is insensitive to temperature. In a modern electrode pHi is close to, and the meter's algorithms always assume that it is equal to,  7.00

RT/F = 58.167 mV/pH at 20 °C (293.15 K) so you can write the electrode voltage as

E = Eo - s*(58.167*(T + 273.15)/293.15)*(pH(T) - 7)

If your pHi is not equal to 7 and you know what it is insert it's value in place of 7 in this and all the formulas which follow. It is rather difficult to obtain the actual value of pHi but the math given here suggests how it might be done by making observations of buffers at several temperatures. pHi isn't very 'observable' so that multiple observations are required and a system of multiple, non consistent equations will have to be solved to get the solution. This is well beyond the scope of this note but the terminology will suggest a path to those familiar with these techniques.

To get s we measure 2 buffers at pH1(T1) and pH2(T2). The pH values for buffers vary with temperature and pH values for various temperatures are found on the packages they come in or can be calculated from simple formulas. For NIST pH 7 and pH 4 technical buffers these pH values are:

pH 7 Buffer:   pH(T) = 1911.4/K -5.5538 + 0.022635*K - 6.8146e-6*K2

pH 4 Buffer:   pH(T) = 1617.3/K -9.2852 + 0.033311*K - 2.3211e-5*K2

K is the temperature of the buffer in Kelvins: K = °C + 273.15 

The voltages produced by the two buffers are:

E1 = Eo - s*(58.167*(T1 + 273.15)/293.15)*(pH1(T1) - 7)
E2 = Eo - s*(58.167*(T2 + 273.15)/293.15)*(pH2(T2) - 7)

Now define

X1 = (58.167*(T1 + 273.15)/293.15)*(pH1(T1) - 7)


X2 = (58.167*(T2 + 273.15)/293.15)*(pH2(T2) - 7)

They give 2 equations in 2 unknowns

E1 = Eo - s*X1
E2 = Eo - s*X2

Subtracting gives

E1 - E2 = s*(X2 -X1) so s = (E1 - E2)/(X2 - X1)

and then

Eo = E1 + s*X1

Now to calculate pHs of a sample at Ts from the voltage, Es, the electrode produces when immersed in it

pHs = (Eo-Es)/(s*(58.167*(Tas + 273.15)/293.15)) + 7

Using this algorithm (easily set up in a spreadsheet including calculating pH1(T) and pH2(T)) you can calibrate with the buffers at any temperature and measure pH at any temperature as long as pHi is approximately 7. In other words, this equation for pH includes ATC. Modern electrodes are so good that modern meters don't allow you to set pHi. The reason I mention this is because I have an electrode with pHi = 8.6. That's not a problem for this algorithm unless the "ATC" is forced to 'work hard'  i.e. if the 2 buffers and sample are not at temperatures within a few degrees of one another.© A.J. deLange 2013