The potential of an electrochemical cell consisting of two Ag|AgCl electrodes

In our routine work, it is sometimes necessary to test Ag/AgCl reference electrodes. This document presents the calculation of the theoretical potential difference for a cell consisting of a saturated KCl Ag/AgCl reference electrode and a 3 M KCl Ag/AgCl test electrode. Both electrodes were immersed in a 3M KCl electrolyte.

Potential Difference Between Ag/AgCl Electrodes Containing Saturated KCl and 3 M KCl at 298 K

Consider the electrochemical cell Ag|AgCl|KCl(sat.) || KCl(3,M)|AgCl|Ag at T=298 K.

1. Electrode reaction

For each silver–silver chloride electrode, the reduction half-reaction is

$$\mathrm{AgCl}(\mathrm{s})+{\mathrm{e}}^{-}\rightleftharpoons \mathrm{Ag}(\mathrm{s})+\mathrm{C}{\mathrm{l}}^{-}(\mathrm{aq})$$

The Nernst equation is

$$E=E^{\circ }-\frac{RT}{F}\ln a_{\mathrm{C}{\mathrm{l}}^{-}},$$

because the activities of the pure solids Ag and AgCl are equal to unity.

At 298 K,

$$E=E^{\circ }-0.05916\log a_{\mathrm{C}{\mathrm{l}}^{-}}.$$

Here, ($E^{\circ }$) is the thermodynamic standard potential of the AgCl/Ag electrode reaction.

2. Electrode containing saturated KCl

For the Ag/AgCl electrode containing saturated KCl,

$$E_{\mathrm{sat}}=E^{\circ }-\frac{RT}{F}\ln a_{\mathrm{C}{\mathrm{l}}^{-}}^{\mathrm{sat}}.$$

Because saturated KCl has a relatively high chloride-ion activity, this electrode has the lower reduction potential.

At 298 K, its commonly tabulated potential relative to the standard hydrogen electrode is approximately

$$E_{\mathrm{sat}}\approx +0.197\mathrm{V}vs.\mathrm{SHE}.$$

3. Electrode containing 3M KCl

For the Ag/AgCl electrode containing 3 M KCl,

$$\ln a_{\mathrm{C}{\mathrm{l}}^{-}}^{3\mathrm{M}}.$$

The chloride-ion activity in 3 M KCl is lower than that in saturated KCl. Therefore,

$$E_{3\mathrm{M}}>E_{\mathrm{sat}}.$$

At 298 K, the commonly tabulated value is approximately

$$E_{3\mathrm{M}}\approx +0.210\mathrm{V}vs.\mathrm{SHE}.$$

4. Potential difference of the cell

For the cell written as

$$Ag|AgCl|KCl(sat.)||KCl(3M)|AgCl|Ag,$$

the cell potential is defined as

Thus,

Using the tabulated electrode potentials,

Therefore,

$$\boxed{E_{\mathrm{cell}}\approx +0.013\mathrm{V}\approx +13\mathrm{mV}}$$

The 3 M KCl electrode is the positive electrode, and the saturated-KCl electrode is the negative electrode.

5. Equivalent derivation using chloride-ion activities

Starting from the two Nernst equations,

$$\ln a_{\mathrm{C}{\mathrm{l}}^{-}}^{3\mathrm{M}},$$

and

$$\ln a_{\mathrm{C}{\mathrm{l}}^{-}}^{\mathrm{sat}},$$

their difference is

Therefore,

$$\frac{RT}{F}\ln \left(\frac{a_{\mathrm{C}{\mathrm{l}}^{-}}^{\mathrm{sat}}}{a_{\mathrm{C}{\mathrm{l}}^{-}}^{3\mathrm{M}}}\right).$$

At 298K,

\log \left( \frac{a_{\mathrm{Cl^-}}^{\mathrm{sat}}} {a_{\mathrm{Cl^-}}^{3\mathrm{M}}} \right) }.

The activities cannot be replaced reliably by the numerical concentrations 4M and 3M, because concentrated KCl solutions are strongly non-ideal. The activity coefficients are significantly different from unity. Using tabulated Ag/AgCl reference-electrode potentials automatically accounts for this non-ideal behavior.

6. Liquid-junction potential

The notation

$$\mathrm{KCl}(sat.)||KCl(3M)$$

also implies a liquid junction between two KCl solutions of different concentrations.

The experimentally measured voltage is therefore

$$E_{\mathrm{sat}}+E_{\mathrm{junction}}.$$

Because the ionic mobilities of (${\mathrm{K}}^{+}$) and ($\mathrm{C}{\mathrm{l}}^{-}$) are similar, the liquid-junction potential is usually relatively small, but it is not strictly zero.

Thus, neglecting the junction potential,

$$\boxed{E_{\mathrm{cell}}\approx +13\mathrm{mV}\text{at }298\mathrm{K}}.$$

Appendix: Using Other Ag/AgCl Reference Electrodes

The derivation presented above is general and can be applied to any Ag/AgCl reference electrode containing KCl of a known concentration.

At 25 °C (298 K), the commonly accepted electrode potentials versus the Standard Hydrogen Electrode (SHE) are approximately:

Filling solution	Electrode potential (V vs. SHE)
Saturated KCl	+0.197 V
4 M KCl	+0.199 V
3 M KCl	+0.210 V
1 M KCl	+0.235 V

If a different Ag/AgCl reference electrode is used, no changes to the derivation are required. The only modification is to replace the corresponding electrode potential in the final calculation.

Thus, for a cell consisting of a reference electrode and a test electrode,

$$E_{\mathrm{cell}}=E_{\mathrm{test}}-E_{\mathrm{reference}},$$

where ($E_{\mathrm{test}}$) and ($E_{\mathrm{reference}}$) are the tabulated potentials for the corresponding KCl filling solutions.

For example:

• Saturated KCl vs. 4 M KCl: $$E_{\mathrm{cell}}=0.199-0.197=2\mathrm{mV}$$
• Saturated KCl vs. 1 M KCl: $$E_{\mathrm{cell}}=0.235-0.197=38\mathrm{mV}$$

The same procedure applies to any other KCl concentration, provided that the corresponding Ag/AgCl electrode potential is known.