# Answers to problems in "Electrode Potentials"

## Question 7

The reader is recommended to read pp. 63-66 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 | H2(g) (P= 1atm) | HCl(aq) | AgCl(s) | Ag(s)

Potential determining Equilibria

 RHE: AgCl(s) + e- ⇌ Ag(s) + Cl-(aq) LHE: H+(aq) + e- ⇌ ½H2(g)

Cell Reaction:

 AgCl(s) + ½H2(g) ⇌ Ag(s) + H+(aq) + Cl-(aq)

Nernst Equation: Ecell = E0cell - {RT/F}ln{aH+ aCl- a Ag / aAgCl PH2 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: Ecell = E0cell - {RT/F}ln{aH+ aCl-}

Now, aH+ = f+mH+ = f+mHCl, and aCl- = f-mCl- = f-mHCl

But in the electrolyte solution, the H+ ions and Cl- ions co-exist, 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:

Ecell = E0cell - 2.303{RT/F}log mHCl2 - 2.303{RT/F}log {f±2}

 E0cell = E0Ag|AgCl - E0H+|H2 = E0Ag|AgCl (by definition of E0H+|H2 = 0V)

Thus, Ecell = E0Ag|AgCl - 2.303{RT/F}log mHCl2 - 2.303{RT/F}log {f±2}

The Debye-Hückel Limiting Law (DHL) states that logf± = -A|z+||z-|I½, 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"; Prentice-Hall, New Jersey (1987).]

Hence, we can calculate a value for log f± .

 m / mol kg-1 I / mol kg-1 log f± 4 x 10-4 4 x 10-4 -0.01 9 x 10-4 9 x 10-4 -0.015 1.6 x 10-3 1.6 x 10-3 -0.02 2.5 x 10-3 2.5 x 10-3 -0.025

We can now determine a value for the Standard Electrode Potential for the silver/silver chloride couple.

Ecell = E0Ag|AgCl - 2.303(2){RT/F}ln m - 2.303(2){RT/F}log {f±}

or, Ecell + 4.606{RT/F}log {f±} = E0Ag|AgCl - 4.606{RT/F}log m

We now plot a graph of Ecell + 4.606{RT/F}log {f±} against log m. This gives an intercept of EAg|AgCl.

 E / V I / mol kg-1 Ecell + 4.606{RT/F}log {f±} log m 0.62565 4 x 10-4 0.62447 -3.398 0.58460 9 x 10-4 0.58283 -3.046 0.55565 1.6 x 10-3 0.55329 -2.796 0.53312 2.5 x 10-3 0.53020 -2.602

From the graph, E0Ag|AgCl = +0.222V