Answers to problems in "Electrode Potentials"
Question 7
The reader is recommended to read pp. 6366 of the book, to see how to solve this particular problem. Readers looking for innovative and novel ways to solve it are invited to read on!
Harned Cell:
Pt  H_{2}(g) (P= 1atm)  HCl(aq)  AgCl(s)  Ag(s) 
Potential determining Equilibria
RHE:  AgCl(s) + e^{} ⇌
Ag(s) + Cl^{}(aq)

LHE:  H^{+}(aq) + e^{} ⇌ ½H_{2}(g) 
Cell Reaction:
AgCl(s) + ½H_{2}(g) ⇌ Ag(s) + H^{+}(aq) + Cl^{}(aq) 
Nernst Equation: E_{cell} = E^{0}_{cell}  {RT/F}ln{a_{H+} a_{Cl} a _{Ag} / a_{AgCl} P_{H2}^{ ½}}
Notice that since hydrogen is a gas at the temperature of the experiment (298K), we use its partial pressure. ( Strictly, we should talk about the fugacity of hydrogen. This is similar to the concept of activity. However, we can consider hydrogen [to an excellent approximation] as being an ideal gas, with a fugacity coefficient equal to unity.)
Also, since silver and silver chloride are pure solids, their activities can be equated to unity.
Therefore, the Nernst Equation simplifies to: E_{cell } = E^{0}_{cell}  {RT/F}ln{a_{H+} a_{Cl}}
Now, a_{H+} = f_{+}m_{H+} = f_{+}m_{HCl}, and a_{Cl} = f_{}m_{Cl} = f_{}m_{HCl }
But in the electrolyte solution, the H^{+} ions and Cl^{} ions coexist, and hence, the activity coefficients of the individual ions have no physical meaning. We therefore use the mean ionic activity coefficient f_{±}, which is defined on p.48, for each ionic species.
Therefore, since we are dealing with a 1:1 electrolyte,
f_{±}= (f_{+} f_{})^{½}
Hence, substituting these results into the Nernst Equation, and changing from natural logarthims to decadic logarthims gives:
E_{cell} = E^{0}_{cell}
 2.303{RT/F}log m_{HCl}^{2}  2.303{RT/F}log {f_{±}^{2}}
E^{0}_{cell} = E^{0}_{AgAgCl}  E^{0}_{H+H2} = E^{0}_{AgAgCl}  (by definition of E^{0}_{H+H2}
= 0V) 
Thus, E_{cell} = E^{0}_{AgAgCl}  2.303{RT/F}log m_{HCl}^{2}  2.303{RT/F}log {f_{±}^{2}}
The DebyeHückel Limiting Law (DHL) states that logf_{±} = Az_{+}z_{}I^{½}, where z_{+}, z_{} are the charges on the ions, A is a temperature and solvent dependent parameter (= 0.509 in water at 298K) and I is the ionic strength of the solution, as defined on p.45. In this case, I = m. [The committed reader will find a derivation of the DHL in chapter 2, Rieger, P H, "Electrochemistry"; PrenticeHall, New Jersey (1987).]
Hence, we can calculate a value for log f_{±} .
m / mol kg^{1}  I / mol kg^{1}  log f_{±} 
4 x 104 
4 x 104  0.01 
9 x 104 
9 x 104  0.015 
1.6 x 103 
1.6 x 103  0.02 
2.5 x 103 
2.5 x 103  0.025 
We can now determine a value for the Standard Electrode Potential for the silver/silver chloride couple.
E_{cell } = E^{0}_{AgAgCl
} 2.303(2){RT/F}ln m  2.303(2){RT/F}log {f_{±}}
or, E_{cell} + 4.606{RT/F}log {f_{±}} = E^{0}_{AgAgCl}  4.606{RT/F}log m
We now plot a graph of E_{cell} + 4.606{RT/F}log {f_{±}} against log m. This gives an intercept of E_{AgAgCl}.
E / V  I / mol kg^{1}  E_{cell} + 4.606{RT/F}log {f_{±}}  log m 
0.62565 
4 x 104  0.62447  3.398 
0.58460 
9 x 104  0.58283  3.046 
0.55565 
1.6 x 103  0.55329  2.796 
0.53312 
2.5 x 103  0.53020  2.602 
From the graph, E^{0}_{AgAgCl} = +0.222V