The Debye–Hückel activity model for aqueous electrolyte solutions.

An instance of this class can be used to control how activity coefficients of ionic and neutral species, \(\gamma_i\) and \(\gamma_n\) respectively, as well as the activity of solvent water, \(a_\mathsf{H_2O(aq)}\), are calculated using the Debye–Hückel activity model.

The activity coefficients of ionics species are calculated using the following modified Debye–Hückel equation^[6]:

\[\log\gamma_{i}=-\dfrac{AZ_{i}^{2}\sqrt{I}}{1+B\mathring{a}_{i}\sqrt{I}}+b_{i}I,\]

while the activity coefficients of neutral species are calculated using:

\[\log\gamma_{n}=b_{n}I.\]

In these equations, \(Z_i\) is the electrical charge of the ionic species; \(\mathring{a}_i\) is the size or an effective diameter of the ionic species (in \(\mathrm{Å}\), where \(1 \mathrm{Å}=10^{-10}\text{m}\)); \(I\) is the ionic strength of the aqueous solution (in molality), calculated using:

\[I=\frac{1}{2}\sum_{j}m_{j}Z_{j}^{2},\]

with \(m_{j}\) denoting the molality of the \(j\)th ion. The constants \(A\) and \(B\) in the Debye–Hückel model are calculated using (see Anderson and Crerar (1993)^[1], page 439, and Langmuir (1997)^[6], page 128):

\[A=1.824829238\cdot10^{6}\rho_{\mathrm{H_{2}O}}^{1/2}(\epsilon_{\mathrm{H_{2}O}}T)^{-3/2}\]

and

\[B=50.29158649\rho_{\mathrm{H_{2}O}}^{1/2}(\epsilon_{\mathrm{H_{2}O}}T)^{-1/2},\]

with \(A\) in \(\mathrm{(mol/kg)^{-1/2}}\) and \(B\) in units of \(\mathrm{(mol/kg)^{-1/2}}/\mathrm{Å}\). In these equations, \(T\) is temperature (in K); \(\epsilon_{\mathrm{H_{2}O}}\) is the dielectric constant of pure water (dimensionless), calculated using the Johnson and Norton (1991) model (see waterElectroPropsJohnsonNorton); and \(\rho_{\mathrm{H_{2}O}}\) is the density of pure water (in \(\mathrm{g/cm^{3}}\)), calculated using either the equation of state of Haar–Gallagher–Kell (1984)^[3] or the equation of state of Wagner and Pruss (2002)^[12] (see waterThermoPropsHGK and waterThermoPropsWagnerPruss).

The activity of water is calculated using the following equation:

\[\log a_{w}=-\frac{1}{n_{w}^{\circ}}\left[\frac{m_{\Sigma}}{2.303}+\sum_{i}^{{\scriptscriptstyle \mathrm{ions}}}m_{i}\log\gamma_{i}+\frac{2}{3}AI^{\frac{3}{2}}\sum_{i}^{{\scriptscriptstyle \mathrm{ions}}}\sigma(\Lambda_{i})-I^{2}\sum_{i}^{{\scriptscriptstyle \mathrm{ions}}}\frac{b_{i}}{Z_{i}^{2}}\right],\]

which is thermodynamically consistent with the previous equations for the activity coefficients of both ionic and neutral species, since it was derived from the Gibbs–Duhem equation. In this equation, \(n_{w}^{\circ}=55.508472\) is the number of moles of water per kilogram; \(m_{\Sigma}\) is the sum of the molalities of all solutes (both ionic and neutral species); and \(\sigma(\Lambda_{i})\) is defined as:

\[\sigma(\Lambda_{i})=\frac{3}{(\Lambda_{i}-1)^{3}}\left[(\Lambda_{i}-1)(\Lambda_{i}-3)+2\ln \Lambda_{i}\right],\]

with \(\Lambda_{i}\) given by:

\[\Lambda_{i}=1+B\mathring{a}_{i}\sqrt{I}.\]