Physical Chemistry Third Edition

8.2 Electrochemical Cells 357

0 

(

∂G

∂ξ

)

dξ

 2 μ(HCl)+ 2 μ(Ag)−μ(H 2 )− 2 μ(AgCl)+ 2 μ(e−(L))− 2 μ(e−(R))dξ

(8.2-5)

which is equivalent to

0 

(

∂Gchem

∂ξ

)

T,P

dξ+ 2 μchem(e−(L))−μchem(e−(R))+Fφ(R)−Fφ(L)dξ

(8.2-6)

whereGchemincludes the chemical potentials of substances other than electrons. The

partial derivative is equal to

(

∂Gchem

∂ξ

)

T,P

 2 μ(HCl)+ 2 μ(Ag)−μ(H 2 )− 2 μ(AgCl) (8.2-7)

All of these substances are electrically neutral, so only the chemical parts of these

chemical potentials are included. The electric potential occurs only in the chemical

potential expression of the electrons. If the reaction equation of Eq. (8.2-3) were used,

the same equation would result since the electric potential terms of the H+and the Cl−

would cancel.

Exercise 8.2

Write the equation for

(

∂Gchem

∂ξ

)

T,P

based on Eq. (8.2-3) and show that it is identical to

Eq. (8.2-6).

Both terminals are made of platinum and are at the same temperature and pressure,

so the chemical parts of the chemical potential of the electrons are the same at the two

terminals, and we can write from Eq. (8.1-5)

0 

(

∂Gchem

∂ξ

)

T,P

+ 2 FE (8.2-8)

whereEis thecell voltage, defined by

Eφ(R)−φ(L) (8.2-9)

This corresponds to our second convention:The cell voltage is defined as the electric

potential of the right electrode minus that of the left electrode.

Since electrons are negative, a positive value ofEmeans that the chemical potential

of the electron is larger in the left electrode. If a passive circuit (one that does not exert

its own voltage) is connected between the terminals, electrons will move spontaneously

from the left terminal to the right terminal through this circuit. The cell reaction proceeds

spontaneously in the direction in which we wrote it. IfEis negative, the reverse of the

cell reaction proceeds spontaneously and electrons move spontaneously from right to

left in the external circuit.

Equation (6.3-6) relates the chemical potential of each uncharged substance to its

activity:

μiμ◦i+RTln(ai) (uncharged substance) (8.2-10)