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4. Electrochemical cells

4.2. Electrochemical cell potential at equilibrium

This chapter shows the derivation of cell potential for the cell in the previous chapter at virtual equilibrium. At equilibrium, the cell cannot be called a galvanic or electrolysis cell. Equilibrium is an important starting point for the examination of cell potentials, as the function of an electrochemical cell can be considered to approach thermodynamic equilibrium when the current flowing through the cell is close to zero. Potentiometric measurements are in practice very close to this situation. The cell potential (or cell voltage) is derived in two ways, first with Gibbs free energy change and equation (1.6) and second by considering the different equilibria at the phase boundaries through the concept of electrochemical potential.

The Gibbs free energy change for reaction (4.3) is:

\( \displaystyle \Delta G=\Delta G^0+RT \ln \left[\frac{a_{\text{Fe}^{2+}}a_{\text{H}^{+}}}{a_{\text{Fe}^{3+}}(a_{\text{H}_2})^{1/2}}\right] \) (4.5) 

and taking into account the relation between potential and Gibbs free energy change in equation (1.6) (n = 1, as seen from reaction equation) we get:

\( \displaystyle E=E^0+\frac{RT}{F} \ln \left[\frac{a_{\text{Fe}^{3+}}(a_{\text{H}_2})^{1/2}}{a_{\text{Fe}^{2+}}a_{\text{H}^{+}}}\right] \)  (4.6)

In a general case, the Nernst equation for reaction

aA + bB + cC.... \(\ce{ <=> }\) dD + eE + f F...

is written as

\( \displaystyle E=E^0-\frac{RT}{nF}\ln\left(\frac{a^d_Da^{e}_Ea^f_F...}{a^{a}_Aa^b_Ba^c_C...}\right) \)

 E0 can be obtained from tabulated values (Chapter 4.3.1) for half-reactions and n is the number of electrons exchanged in the total reaction.

This is the Nernst equation for the potential of the cell in Figure 4.2. As seen from equation (4.6), the cell potential E corresponds to the standard cell potential E0, when the conditions are such that the activities inside the logarithm give the value 1. 

 

Let us derive the potential for cell in Figure 4.2 via an equilibrium treatment. At equilibrium, all the phases in an electrochemical cell that are in direct contact with each other are in thermic, mechanical and chemical equilibrium. An equilibrium between two phases that are in contact does not require the presence of all components and species in all phases, or the chemical potential of each component to be the same in all of the phases. For example, the equilibrium between a metallic gold electrode and Fe3+ and Fe2+  ions in the solution does not contain gold in the solution phase or iron ions in the gold phase. The equilibrium takes place through electrons that are able to move freely across the phase boundaries.

It should be noted that the concept of an equilibrium in the treatment of electrochemical cells corresponds in a way to an impeded cell reaction. This equilibrium in an electrochemical cell can be achieved only if the external circuit is open or the impedance in the external circuit is so large that an electrochemical equilibrium is maintained at the phase boundaries despite a small current flowing through the system. It is also assumed that the system does not have irreversible phenomena such as corrosion, or the system does not exhibit temperature, pressure or concentration gradients.

The phases of the example cell are depicted as Greek symbols in the cell diagram below:

 

Cu    |  Pt   | H+, H2   ||   Fe3+, Fe2+  |  Au   |Cu’              

 \( \alpha \)        \( \beta \)           \( \sigma \)                 \( \varepsilon \)           \(\gamma \)       \( \alpha \) .’

 

At equilibrium, the cell exhibits equilibria across  the following phase boundaries: phase \( \alpha \) | phase  \( \beta \), phase  \( \beta \) | phase  \( \sigma \), phase  \( \sigma \) | phase  \( \varepsilon \), phase  \( \varepsilon \) | phase  \( \gamma \) and phase  \( \gamma \) | phase  \( \alpha \)’. The chemical potential of an ion is an ill-defined quantity and hence the derivation uses the equality of the electrochemical potential (equation (2.7)) as the condition for equilibrium. The equilibria in the cell are as follows:

 

Phase \( \alpha \) | phase \( \beta \) i.e. the contact between copper wire and platinum electrode:

\( \tilde\mu^{\beta}_{\text{e}^-}=\tilde\mu^{\alpha}_{\text{e}^-} \) (4.7)

Phase \( \beta \) | phase \( \sigma \) i.e. the equilibrium at platinum electrode (reaction equation 4.2):

\( \tilde\mu^{\sigma}_{\text{H}^+}+\tilde\mu^{\beta}_{\text{e}^-}=\frac{1}{2}\tilde\mu^{\sigma}_{\text{H}_2} \) (4.8)

Phase \( \sigma \) | phase \( \varepsilon \), the interface between two liquid phases, in practice often a salt bridge. In this simple derivation, we will assume that the inner potential of the solution is the same in both sides of the interface. Some cases where this assumption is not done will be covered later on (junction potentials, chapter 4.4). 

\( \phi^{\sigma}= \phi^{\varepsilon} \) (4.9)

Phase \( \varepsilon \) | phase \( \gamma \) i.e. the equilibrium at the gold electrode (reaction equation 4.1)

\( \tilde\mu^{\varepsilon}_{\text{Fe}^{3+}}+\tilde\mu^{\gamma}_{\text{e}^-}=\tilde\mu^{\varepsilon}_{\text{Fe}^{2+}} \) (4.10)

Phase \( \gamma \) | phase \( \alpha \)’ i.e. the contact between copper wire and gold electrode:  

\( \tilde\mu^{\gamma}_{\text{e}^-}=\tilde\mu^{\alpha}_{\text{e}^-} \) (4.11)

By using equation (2.7), these equilibrium conditions can be presented as:

\( F\left(\phi^{\beta}-\phi^{\alpha}\right)=\mu_{\text{e}^-}^{\beta}-\mu_{\text{e}^-}^{\alpha} \)

 (4.12)
\( \displaystyle F\left(\phi^{\sigma}-\phi^{\beta}\right)=\left(\frac{1}{2}\mu^0_{\text{H}_2}-\mu_{\text{H}^+}^0-\mu_{\text{e}^-}^{\beta}\right)+RT\ln\left(\frac{(a_{\text{H}_2})^{1/2}}{a_{\text{H}^+}}\right) \)

 (4.13)
\( \displaystyle F\left(\phi^{\gamma}-\phi^{\varepsilon}\right)=\left(\mu^0_{\text{Fe}^{3+}}-\mu_{\text{Fe}^{2+}}^0+\mu_{\text{e}^-}^{\gamma}\right)+RT\ln\left(\frac{a_{\text{Fe}^{3+}}}{a_{\text{Fe}^{2+}}}\right) \)

 (4.14)
\( F\left(\phi^{\alpha '}-\phi^{\alpha\gamma}\right)=\mu_{\text{e}^-}^{\alpha'}-\mu_{\text{e}^-}^{\gamma} \) (4.15)

Combining equations (4.12)-(4.15) and taking into account (4.9) the cell potential i.e. the galvanic potential difference between the two copper wires is obtained as (note that the chemical potentials of the wires are equal because the phases are identical):

 \( F\left(\phi^{\alpha}-\phi^{\alpha'}\right)=\left(\mu^0_{\text{Fe}^{3+}}-\mu_{\text{Fe}^{2+}}^0+ \frac{1}{2}\mu^0_{\text{H}_2}-\mu_{\text{H}^+}^0\right)+RT\ln\left(\frac{a_{\text{Fe}^{3+}}(a_{\text{H}_2})^{1/2}}{a_{\text{Fe}^{2+}}a_{\text{H}^{+}}}\right) \)

  (4.16)
\( \displaystyle E_{\text{cell}}=E^0+\frac{RT}{F} \ln \left[\frac{a_{\text{Fe}^{3+}}(a_{\text{H}_2})^{1/2}}{a_{\text{Fe}^{2+}}a_{\text{H}^{+}}}\right] \) (4.17)

which naturally corresponds to equation (4.5) that was obtained earlier. Let us consider equation (4.17) further, keeping in mind the relationship between the Gibbs free energy change and the potential (equation (1.6)). If the virtual total cell reaction (4.3) of our cell is exergonic (\( \Delta \)G < 0), hydrogen gas can, as described in the reaction equation, reduce Fe3+. This reaction produces work, and if the potentiometer in Figure 4.2 is replaced by a load, the work of the reaction can be utilized. This work corresponds to the transfer of electrons from the gold electrode to the platinum electrode (in solution), and in an ideal case the value of this work is –FEcell.



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