10.1: Single-ion Quantities
Consider a solution of an electrolyte solute that dissociates completely into a cation species and an anion species. Subscripts \(+\) and \(-\) will be used to denote the cation and anion, respectively. The solute molality \(m\B\) is defined as the amount of solute formula unit divided by the mass of solvent.
We first need to investigate the relation between the chemical potential of an ion species and the electric potential of the solution phase.
The electric potential \(\phi\) in the interior of a phase is called the inner electric potential , or Galvani potential . It is defined as the work needed to reversibly move an infinitesimal test charge into the phase from a position infinitely far from other charges, divided by the value of the test charge. The electrical potential energy of a charge in the phase is the product of \(\phi\) and the charge.
Consider a hypothetical process in which an infinitesimal amount \(\dif n_+\) of the cation is transferred into a solution phase at constant \(T\) and \(p\). The quantity of charge transferred is \(\delta Q=z_+F\dif n_+\), where \(z_+\) is the charge number (\(+1\), \(+2\), etc.) of the cation, and \(F\) is the Faraday constant. (The Faraday constant is the charge per amount of protons.) If the phase is at zero electric potential, the process causes no change in its electrical potential energy. However, if the phase has a finite electric potential \(\phi\), the transfer process changes its electrical potential energy by \(\phi \delta Q=z_+F\phi\dif n_+\). Consequently, the internal energy change depends on \(\phi\) according to \begin{equation} \dif U(\phi) = \dif U(0) + z_+F\phi\dif n_+ \tag{10.1.1} \end{equation} where the electric potential is indicated in parentheses. The change in the Gibbs energy of the phase is given by \(\dif G = \dif(U-TS+pV)\), where \(T\), \(S\), \(p\), and \(V\) are unaffected by the value of \(\phi\). The dependence of \(\dif G\) on \(\phi\) is therefore \begin{equation} \dif G(\phi) = \dif G(0) + z_+F\phi\dif n_+ \tag{10.1.2} \end{equation}
The Gibbs fundamental equation for an open system, \(\dif G=-S\dif T+V\difp+\sum_i\mu_i\dif n_i\) (Eq. 9.2.34), assumes the electric potential is zero. From this equation and Eq. 10.1.2, the Gibbs energy change during the transfer process at constant \(T\) and \(p\) is found to depend on \(\phi\) according to \begin{equation} \dif G(\phi) = \left[ \mu_+(0) + z_+F\phi \right] \dif n_+ \tag{10.1.3} \end{equation} The chemical potential of the cation in a phase of electric potential \(\phi\), defined by the partial molar Gibbs energy \(\bpd{G(\phi)}{n_+}{T,p}\), is therefore given by \begin{equation} \mu_+(\phi)=\mu_+(0)+z_+F\phi \tag{10.1.4} \end{equation} The corresponding relation for an anion is \begin{equation} \mu_-(\phi)=\mu_-(0)+z_-F\phi \tag{10.1.5} \end{equation} where \(z_-\) is the charge number of the anion (\(-1\), \(-2\), etc.). For a charged species in general, we have \begin{equation} \mu_i(\phi)=\mu_i(0)+z_iF\phi \tag{10.1.6} \end{equation}
We define the standard state of an ion on a molality basis in the same way as for a nonelectrolyte solute, with the additional stipulation that the ion is in a phase of zero electric potential. Thus, the standard state is a hypothetical state in which the ion is at molality \(m\st\) with behavior extrapolated from infinite dilution on a molality basis, in a phase of pressure \(p=p\st\) and electric potential \(\phi{=}0\).
The standard chemical potential \(\mu_+\st\) or \(\mu_-\st\) of a cation or anion is the chemical potential of the ion in its standard state. Single-ion activities \(a_+\) and \(a_-\) in a phase of zero electric potential are defined by relations having the form of Eq. 9.7.8: \begin{equation} \mu_+(0)=\mu_+\st+RT\ln a_+ \qquad \mu_-(0)=\mu_-\st+RT\ln a_- \tag{10.1.7} \end{equation} As explained in Sec. 9.7, \(a_+\) and \(a_-\) should depend on the temperature, pressure, and composition of the phase, and not on the value of \(\phi\).
From Eqs. 10.1.4, 10.1.5, and 10.1.7, the relations between the chemical potential of a cation or anion, its activity, and the electric potential of its phase, are found to be \begin{equation} \mu_+=\mu_+\st + RT\ln a_+ + z_+ F\phi \qquad \mu_-=\mu_-\st + RT\ln a_- + z_i F\phi \tag{10.1.8} \end{equation} These relations are definitions of single-ion activities in a phase of electric potential \(\phi\).
For a charged species in general, we can write \begin{equation} \mu_i=\mu_i\st + RT\ln a_i + z_i F\phi \tag{10.1.9} \end{equation} Note that we can also apply this equation to an uncharged species, because the charge number \(z_i\) is then zero and Eq. 10.1.9 becomes the same as Eq. 9.7.2.
Some thermodynamicists call the quantity \((\mu_i\st+RT\ln a_i)\), which depends only on \(T\), \(p\), and composition, the chemical potential of ion \(i\), and the quantity \((\mu_i\st+RT\ln a_i+z_iF\phi)\) the electrochemical potential with symbol \(\tilde{\mu}_i\).
Of course there is no experimental way to evaluate either \(\mu_+\) or \(\mu_-\) relative to a reference state or standard state, because it is impossible to add cations or anions by themselves to a solution. We can nevertheless write some theoretical relations involving \(\mu_+\) and \(\mu_-\).
For a given temperature and pressure, we can write the dependence of the chemical potentials of the ions on their molalities in the same form as that given by Eq. 9.5.18 for a nonelectrolyte solute: \begin{equation} \mu_+=\mu_+\rf + RT\ln\left(\g_+\frac{m_+}{m\st}\right) \qquad \mu_-=\mu_-\rf + RT\ln\left(\g_-\frac{m_-}{m\st}\right) \tag{10.1.10} \end{equation} Here \(\mu_+\rf\) and \(\mu_-\rf\) are the chemical potentials of the cation and anion in solute reference states. Each reference state is defined as a hypothetical solution with the same temperature, pressure, and electric potential as the solution under consideration; in this solution, the molality of the ion has the standard value \(m\st\), and the ion behaves according to Henry’s law based on molality. \(\g_+\) and \(\g_-\) are single-ion activity coefficients on a molality basis.
The single-ion activity coefficients approach unity in the limit of infinite dilution: \begin{gather} \s{ \g_+ \ra 1 \quad \tx{and} \quad \g_- \ra 1 \quad \tx{as} \quad m\B \ra 0} \tag{10.1.11} \cond{(constant \(T\), \(p\), and \(\phi\))} \end{gather} In other words, we assume that in an extremely dilute electrolyte solution each individual ion behaves like a nonelectrolyte solute species in an ideal-dilute solution. At a finite solute molality, the values of \(\g_+\) and \(\g_-\) are the ones that allow Eq. 10.1.10 to give the correct values of the quantities \((\mu_+-\mu_+\rf)\) and \((\mu_- -\mu_-\rf)\). We have no way to actually measure these quantities experimentally, so we cannot evaluate either \(\g_+\) or \(\g_-\).
We can define single-ion pressure factors \(\G_+\) and \(\G_-\) as follows: \begin{equation} \G_+\defn\exp\left(\frac{\mu_+\rf-\mu_+\st}{RT}\right) \approx \exp\left[ \frac{V_+^{\infty}(p-p\st)}{RT} \right] \tag{10.1.12} \end{equation} \begin{equation} \G_-\defn\exp\left(\frac{\mu_-\rf-\mu_-\st}{RT}\right) \approx \exp\left[ \frac{V_-^{\infty}(p-p\st)}{RT} \right] \tag{10.1.13} \end{equation} The approximations in these equations are like those in Table 9.6 for nonelectrolyte solutes; they are based on the assumption that the partial molar volumes \(V_+\) and \(V_-\) are independent of pressure.
From Eqs. 10.1.7, 10.1.10, 10.1.12, and 10.1.13, the single-ion activities are related to the solution composition by \begin{equation} a_+=\G_+\g_+\frac{m_+}{m\st} \qquad a_-=\G_-\g_-\frac{m_-}{m\st} \tag{10.1.14} \end{equation} Then, from Eq. 10.1.9, we have the following relations between the chemical potentials and molalities of the ions: \begin{equation} \mu_+=\mu_+\st + RT\ln(\G_+\g_+m_+/m\st) + z_+F\phi \tag{10.1.15} \end{equation} \begin{equation} \mu_-=\mu_-\st + RT\ln(\G_-\g_-m_-/m\st) + z_-F\phi \tag{10.1.16} \end{equation}
Like the values of \(\g_+\) and \(\g_-\), values of the single-ion quantities \(a_+\), \(a_-\), \(\G_+\), and \(\G_-\) cannot be determined by experiment.