The
measurement of pH is one of the most often used in chemistry. The correct
expressions for pH are “acidity index” or sometimes “hydrogen index”.The formal
definition of pH is given by the equation:

рН=
-lg aH+, (1)

where aH+ is the thermodynamic activity or the effective concentration of the
hydrogen ions in the given solution. The thermodynamic activity is the
concentration of the hydrogen ions, corrected for the non-ideality of the
solution.

The correcting multiplier, which transforms the
concentration into activity is called the activity coefficient.

aH+ = γ[H+], (2)

where [H+] is the molar concentration of hydrogen ions
in the solution. The multiplier γ in many cases can be calculated theoretically. For
diluted solutions the multiplier γ is close to unity and in many
simple cases is neglected. In the present note we omit it, because it is not
connected with the primciple of the operating of a pH meter.

pHuses a logarithmic, rather than linear scale
because of two main reasons. First, the concentration of hydrogen ions is
varied in practical systems, say, while acid-base titrations, in a very wide
range 12 to 14 orders of magnitude (that is 10 to the 12^{th} or 1o to
the 14^{th} power times). The second reason is that the instrumental
measurements produce the output voltage that is varies linearly versus the logarithm
of the concentration, rather than versus the concentration itself. So the
application of the logarithmic scale, which allows to avoid very big numbers
and is compatible with the instrumental methods finds a wide range of
applications in chemistry and neighboring branches of science and technology.

How is pH measured
instrumentally. The primciple of operating of the system as a whole unit.

At present
the main means of the pH measurements isthe use of a special instrument, the pH meter.

The principle of the operating of such an instrument
is potentiometric one, i.e. the measured value is the voltage between two
electrodes, immersed into the tested solution.Normally the voltage between such
electrodfes is changed by tens and hundreds of millivolts.

Thus pH
met5er is actually a specialized voltmeter. The peculiarity of this voltmeter
is a very low input current? So that it can measure the voltage from very different
electrodes, including glass electrodes, which would not let big current flow.
In addition, the pH meter canprocess
the voltage thus measured and transform it into pH (and in more sophisticated
models into some other values, say the Turner degrees for the acidity of milk.)

Consider
the functions of the pH meters in moredetails. Two electrodes are immersed into the solution, that are the
indicator (work) electrode and the reference electrode. The potential of the
indicator electrode versus the solution is linearly dependent versus pH of the
solution (it is explained below.)

Uind= а + b*pH (3)

Here a and b are the constants, depending only versus
the temperature of the solution.

The potential of a perfect pH-electrode does not
depend upon the concentration of other ions. It is called the selectivity of
the indicator electrode to hydrogen cations. Thus the pH-electrode can be
classified as an ion-selective electrode.

The analytical response (the pH–dependent potential
drop) origins from the interface of the membrane of the indicator electrode and
the tested solution. It is transmitted to a metal wire in the construction of
the indicator electrode and then to the input of the pH-meter, which is
actually a special voltmeter. However the voltmeter cannot measure the
potential of a single electrode. So one needs also a reference electrode, which
is attached to the second input of the pH-meter.

The role of the reference electrode is to provide the
constant reference potential versus the solution. It establishes a constant
potential difference between the point in the solution, where it is inserted in
and its output. The potential of the refernce electrode does not depend upon
the potential of other ions. It is a whole device, using a series of scientific
and technological solutions.

If we
insert the reference electrode into the solution? A constant potential difference Uref is
generated between the solution and the output of the reference electrode. This
potential is applied to the second input of the pH-meter (reference input)/ The
potential drop in the solution equals zero, because no current flows (and if
flows, it interferes the pH measurement).

Thus, the potential between the inputs of the
pH-meter, that is the measured potential, is given by the expression:

DU = Uind – Uref = a+b*pH = c+ b*pH (4)

Thus there exists a linear relationship between the pH
or the tested solution and the measured voltage.

The constants c and b
for the expression (4) are determined with the help of standard solutions with
the known pH value. These constants are saved in the memory of the processor.
Since then the instrument can automatically transform the measured value of the
measured potential DU into the actual pH of the tested solution. The
measurement can be continuous, for example while monitoring the tap water. It
can also regulate the supply of the acid or alkali into the solution in order
to establish a wanted value of pH (for example for a pH-stat)

How the indicator electrode
works

Let us
consider in more detail the work of the pH-sensitive electrodes and trace the
origin of eq. (3)

The
pH-electrode is performed in the shape of a glass tube with a thin-walled glass
ball at one end.. The wall of this ball is a pH-sensitive membrane. A silver
wire, covered with a dense layer of silver chloride is place at the other end
of the tube and serves to be the output of the electrode. The tube is filled with 0.1 molar solution of
hydrochloric acid, so that acid completely, without bubbles fills the ball and
the wire is deeply inserted into the solution. The bubbles in the solution
interfere the normal operating of the system because they effect the contact
between the wire and the glass ball, which is conducted by the acid. So
sometimes the “broken” glass electrode is readily repaired by simple shaking.
Thus the electrochemical circuit within the pH electrode can be presented as
follows:

Denote the potential
difference between Ag and AgCl DU1, between AgCl and HCl (0,1 M) DU2, between HCl (0,1 M) and outer bulk
solution DU3.

The potential difference between Ag and AgCl is determined by the fast exchange
of silver cations at this interface and is constant and stable, because the
composition of the both Ag and AgCl is
constant. Thus DU1 is constant

The potential difference between AgCl and HCl (0.1 m) is determined by the
equilibrium, caused by the fast exchange of chloride ions at this interface and
is constant and stable, because the composition of the both AgCl and HCl is
constant and also because of the constant solubility product of AgCL.

The potential-determing
reaction can be written as follows:

Cl- in AgCl↔ Cl- in HCl(6)

The potential drop in this case can be expressed by a
famous Nernst equation:

DU2 = DU20 + (RT/F)
ln ([Cl-] in AgCl/[ Cl-] in HCl(7)

Here the notations are standard for Nernst equation.

Thus DU2 is constant

Only one term namely DU3 remains of the successively
attached elements of the electrochemical chain.

It is the potential drop on the glass membrane. The
type of the glass is selected so, that only hydrogen ions penetrate through the
membrane. The other ions would not penetrate through this membrane.

The multiple experimental research have shown, that
this potential difference is determined by the expression:

DU3 = (RT/F) ln ([H+]out /[ H+] in HCl (8)

No perfect theory to explain this fact is available so
far, though several good explanations exist.

The logarithm of the ratio equals to the difference of
the logarithms:

The second
term in eq.(9) does not depend upon the concentration of hydrogen ions, so we
can consider it to be a constant.

Thus, the
total potential of the pH electrode versus the solution can be expressed as
follows:

Uind = DU1 +DU2+DU3=DU1 +DU2 - (RT/F) ln [ H+]in HCl +
(RT/F) ln ([H+] out) = = a +(RT/F) *2,303*lg([H+] out) = a -(RT/F) *2,303* рН out = a +
b рНout (10)

So we have derived eq.(3).

The following notatiopns are used:

a= DU1 +DU2 - (RT/F) ln [ H+] in HCl

and

b =-(RT/F)
*2,303.

We also used the following expression to turn from
natural logarithms to decimal ones:

Lnx= 2.303 lgx

How the reference
electrode works?

Consider
now in more detail the work of the reference electrode. The most widely used in
the laboratory is so-called saturated silver-chloride electrode.

The
reference electrode is performed in the shape of a glass tube with a filtering
element of porous ceramics, porous glass asbestos or other fibers at one end..
This element is called an electrochemical bridge. It is needed to provide the
electric contact of the inner solution with the outer tested solution but with
the lrate of the leak of the inner solution to be very slow. . A silver wire,
covered with a dense layer of silver chloride is place at the other end of the
tube and serves to be the output of the electrode.The tube is filled with the saturate (or
concentrated with the concentration fixed) solution of potassium chloride, so
that the salt solution completely, without bubbles fills the tbe and the wire
is deeply inserted into the solution. The bubbles in the solution interfere the
normal operating of the system because they effect the contact between the wire
and the electrochemical bridge, which is conducted by the salt solution. So
sometimes the “broken” reference electrode is readily repaired by simple
shaking. Thus the electrochemical circuit within the reference electrode can be
presented as follows:

Denote the potential
difference between Ag and AgCl DU1, between AgCl and KCl (sat) DU4,between KCl (sat) and
outer bulk solution DU5

As follows from the considerations equivalent to ones
for the pH-electrode, DU1 is a constant, being the same as for the pH-electrode
with the same temperature dependence. DU4 is also a constant, which is close to
DU2, but not equal to it because the silver/silver chloride wire is inserted
into hydrochloric acid in the pH-electrode and into poitassium chloride in the
reference electrode.

However in the both cases the equilibrium is
determined by the exchange between the crystal and the solution with chloride
ions. Chloride is redistributed till the electric potential would balance the
difference in the energies of chemical nature. After that the potential
difference would not change DU4= const.

Only one term namely DU5 remains of the successively
attached elements of the electrochemical chain.

It is the potential drop on the filtering element. There
exists a theoretical equation of Albert-Sergeant for the potential drop on the
tube, where the concentrated salt diffuses into the diluted solution (it is the
case):

DU5 =
(RT/F) ln (Λ+/Λ-) (12)

Here Λ+ and Λ-) are the equivalent electric conductivities of the
cation of the salt (that is potassium in our case) and thew anion (chloride in
our case).

The peculiarity of potassium and chloride is that for
such a pair Λ+
= Λ-
,and the expression under the logarithm turns into unity, and the value of the
logarithm turns into zero, because the logarithm of unity is zero. Thus DU5=0.

Thus the overall potential of the reference electrode
versus the solution can be calculated as:

U ref.= DU1 +DU4 = const (13)

Such a stability of the
potentialo of the output contact of the reference electrode versusthe pointin the solution, where its electrochemical bridge is inserted is the
basis to use this electrode as a reference point in potential measurements.