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2. Thermodynamics of electrolyte solutions

2.2. Chemical potential

We already introduced the chemical potential \( \mu_i \) of component i in the Gibbs-Duhem equation. It is defined as

\( \displaystyle\mu_i= \left( \frac{\partial G}{\partial n_i} \right) _{P,T,n_{j \neq i }}=\mu_i^0+RT\ln a_i \),

(2.6)

where G is the Gibbs free energy of the solution. For ions, an electrochemical potential can be defined as

\( \displaystyle\tilde{\mu_i}= \mu_i+z_iF \phi= \mu_i^0+RT\ln a_i+z_iF \phi \)

(2.7)

where F is the Faraday constant, zi the charge number, \( \mu_i^0 \) is the standard chemical potential and ai is the activity of ion i, respectively. Thus, for an uncharged species \( \tilde{\mu_i}=\mu_i \). Definition (2.6) is not, however, relevant for an ion because it is not possible to change the concentration of a single ion  while keeping the concentrations of the other ions constant.


The fundamental Gibbs equation expresses the change of the system’s internal energy dU as


dU = dq + dw = TdS - PdV    only PV work

(2.8)


where q is heat evolved from the system or exchanged with its environment, and dw is the work done against the system. In addition to the PV work it can contain, for example work against surface tension, \( \gamma dA \), or electrical work, \( \phi \)dqe. (\( \gamma \) is surface tension, qe electric charge and \( \phi \) electric potential.)

The other thermodynamic state functions, H (enthalpy), F (Helmholzin free energy) and G are easily derived from Equation (2.8):

 \( H=U+PV \Rightarrow dH=dU+PdV+VdP=TdS+VdP \)

(2.9)

\( F=U-TS \Rightarrow dF=dU-TdS-SdT=-SdT-PdV \)

(2.10)

\( G=H-TS \Rightarrow dG=dH-TdS-SdT=-SdT+VdP \)

(2.11)

In all these functions, the term \( \sum\limits_{i}\mu_idn_i \) can be added that takes into account the amount of material in the system. Adding the term that takes into account ions,\( \sum\limits_{i}\tilde{\mu_i}dn_i \), into Equation (2.11) gives

\( dG=-SdT+VdP+\sum\limits_{i}(\mu_i+z_iF\phi)dn_i \)

\( =-SdT+VdP+\sum\limits_{i}\mu_idn+F\phi\sum\limits_{i}z_idn_i \)

\( =-SdT+VdP+\sum\limits_{i}\mu_idn_i+F\phi dq_e\)


(2.12)

where the definition of the electric charge \( q_e=\sum\limits_{i}z_in_i \) is used. In an electroneutral system \( q_e=0 \), but as will be seen later on, the concept of the electrochemical potential is most useful. The Galvani potential \( \phi \) also is conceptually a bit problematic; in metals it is related to the Fermi energy of electrons (see paragraph 1.3.3). Anyway, applying the equality of the electrochemical potential between the phases at equilibrium, Galvani potential differences between phases and the electromotive force (Chapter 4) can be derived. In the Gibbs-Duhem Equation (2.5) the chemical potential can thus be replaced by the electrochemical potential.


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