Physical Chemistry Third Edition

374 8 The Thermodynamics of Electrochemical Systems

ions to establish chemical equilibrium across the membrane. Since the chemical part

of the chemical potential of the hydrogen ions is not the same on the two sides of the

membrane, an electrical potential difference is established across the glass membrane

when the system equilibrates.

Although the voltage of the cell depends on the exact nature of the glass membrane

and on the hydrogen ion activity inside the bulb, the dependence of the voltage on

the pH of the unknown solution is the same as in the cell of Figure 8.10, so that

Eq. (8.4-7) can be used if a reference solution of known pH is available. A typical pH

meter consists of a glass electrode, a reference electrode (usually a calomel electrode),

a voltage-measuring device, an analogue or digital display, and a circuit that gives the

pH directly without requiring the operator to substitute numbers into Eq. (8.4-7). Since

the temperature occurs in Eq. (8.4-7), some pH meters have a control knob with which

one can set the temperature, and others automatically adjust for different temperatures.

The first commonly available pH meter
was made possible in the 1930s when
Arnold Beckman, then a chemistry
professor at the California Institute of
Technology, invented an amplifier that
allowed the cell voltage to be read
easily. Professor Beckman left Caltech
in 1940 and founded a company that
sold the pH meters and the famous
Beckman DU spectrophotometer, which
used the same amplifier. Beckman
became a wealthy man and a great
benefactor of Caltech. He died in 2004
at age 104.

Exercise 8.12

Calculate the difference between the cell voltages that occur for a pH reading of 7.00 and one

of 12.50 at 298.15 K.

PROBLEMS

Section 8.4: The Determination of Activities and Activity
Coefficients of Electrolytes

8.18 In a cell as pictured in Figure 8.2, a solution of HCl with
molality 1.000 mol kg−^1 and a hydrogen pressure of
0.986 atm produces a cell voltage of 0.2332 V at 298.15 K.
Find the mean molar activity coefficient of HCl in the
solution. Assume the hydrogen gas to be ideal.

8.19 From the data in Problem 8.8, find the value of the mean
molar activity coefficient of HCl at 0.0500 mol kg−^1 and
298.15 K.

8.20 Following are data on the activity of water in a calcium
chloride solution at 25◦C^4 :

m(CaCl 2 )/ a(H 2 O) m(CaCl 2 )/ a(H 2 O)

mol kg−^1 mol kg−^1

0.1 0.99540 0.6 0.96998

0.2 0.99073 0.7 0.96423

0.3 0.98590 0.8 0.95818

0.4 0.98086 0.9 0.95174

0.5 0.97552 1.0 0.94504

Find the activity coefficient of calcium chloride at

1.000 mol kg−^1 , using a Gibbs–Duhem integration.

8.21 If a solution of NaOH with a molality of 0.100 mol kg−^1 is

placed in a cell with a standard hydrogen electrode and a

normal calomel electrode at 298.15 K, find the cell voltage

and the pH of the solution. Neglect the liquid junction

potential. State any other assumptions.

8.5 Thermodynamic Information from Electrochemistry

Thermodynamic information about many chemical reactions that occur outside of elec-

trochemical cells can be obtained from electrochemical data. For a general reaction

written in the form of Eq. (8.2-18), the analogue of Eq. (8.2-8) is

−nFE

(

∂Gchem

∂ξ

)

T,P

(8.5-1)

(^4) Robinson and Stokes,op. cit., p. 478.