# Pitzer-Debye-Hückel long-range electrostatic correction¶

Systems with charged species often demonstrate behavior that is not easily captured with standard thermodynamic models meant for neutral molecules. A major reason for this that the chemical potential requires an additional contribution due to the electrostatic potential. The electrostatic potential decays with $$r^{-1}$$, where $$r$$ is the distance between two charges. This inverse dependence on $$r$$ makes the electrostatic potential a long-range term, especially as compared to other intermolecular energetic contributions, which are typically only relevant at short distances. One successful model for adding necessary electrostatic corrections for systems with charged species is the Pitzer-Debye-Hückel (PDH) 1 model. In the COSMO-RS package, we use a modification of the PDH term that allows for mixed solvents.

## Mixing rules and required property inputs for the PDH term¶

In the original publication, the PDH model is used only for pure solvents. To generalize to mixed solvent systems, we have included mixing rules to estimate required parameters for the PDH model. These are discussed below in addition to required parameters.

### Molecular weight¶

No additional input is needed for this property as this is directly determined by the atomic composition of a molecule. The average molecular weight for the entire system, $$M$$, is given as a function of all the individual molecular weights $$m_i$$ :

$M = \frac{\sum\limits_i n_i m_i}{\sum\limits_i n_i}$

where $$n_i$$ is the number of moles of species $$i$$.

### Density¶

The density of each compound is also required for the calculation of the PDH term. For ionic species, one may simply use the density of the combined salt. The average density for the entire system, $$D$$, is given as a function of all the individual densities $$d_i$$ :

$D = \frac{\sum\limits_i n_i}{\sum\limits_i \frac{n_i}{d_i}}$

### Dielectric constant¶

Additionally, the dielectric constant of each compound is required for the PDH term. These terms do not need to be experimentally accurate numbers, but should be approximate. The dielectric constant of a combined salt can be used for the individual ionic species. The average dielectric constant for the entire system, $$\mathcal{E}$$, is given as a function of all the individual dielectric constants $$\epsilon_i$$ :

$\mathcal{E} = \frac{\sum\limits_i n_i \frac{m_i \epsilon_i}{d_i}}{\sum\limits_i n_i \frac{m_i}{d_i}}$

## Derivation of the PDH term for general mixtures¶

In 1 , the Debye-Hückel model is adapted to account for higher concentrations of ions (the Debye-Hückel model works well for low concentrations). The authors choose from a variety of possible models, ultimately deciding on one that is deemed most accurate empirically. The expression for the PDH excess energy (long range electrostatic correction) is given below.

$\frac{G^{PDH}}{RT} = -4 \left({\frac{1000}{M}}\right)^{1/2} \frac{N A_x I_x}{\rho} \ln( 1+\rho I_x^{1/2} )$

where $$I_x$$ is the ionic strength of the system, which is a function of the charge $$z_i$$ of each species:

$I_x = \frac{1}{2} \frac{\sum\limits_i n_i z_i^2}{\sum\limits_i n_i}$

and $$A_x$$ is the Debye-Hückel parameter, defined as:

$A_x = \frac{1}{3} \frac{(2\pi N_A D e^6 )^{1/2}}{(4 \pi \epsilon_0 \mathcal{E} kT)^{3/2}}$

where $$e$$ is the charge of the electron, $$N_A$$ is Avogadro’s number, $$k$$ is the Boltzmann constant, and $$\epsilon_0$$ is the permittivity of free space.

Finally, $$\rho$$ is the closest approach parameter. The specifics of this parameter are still under development, and for now the following simple formula from 1 is used.

$\rho = 2150 \left( \frac{D}{ \mathcal{E} T } \right)^{1/2}$

Finally, we can calculate the PDH contribution to the activity coefficient as follows:

$\begin{split}\ln(\gamma_i^{PDH}) =& \frac{\delta}{\delta n_i} \left( \frac{G^{PDH}}{RT} \right)_{T,P} \\ =& -4 \left({\frac{1000}{M}}\right)^{1/2} \frac{A_x I_x}{\rho} \Bigg[ \ln( 1+\rho I_x^{1/2}) \bigg( \frac{M-m_i}{2M} + \frac{z_i^2}{2I} \\ & + \frac{1}{2}\left(1-\frac{D}{d_i}\right) +\frac{3N}{2} \frac{m_i}{d_i \sum\limits_j n_j \frac{m_j}{d_j}} \left(1-\frac{\epsilon_i}{\mathcal{E}}\right) -\frac{N}{\rho} \frac{\delta\rho}{\delta n_i} \bigg) \\ & + \frac{1}{1+\rho I_x^{1/2}} \left( \frac{\rho I_x^{1/2}}{2N} \left( \frac{z_i^2}{2I}-1 \right) + I_x^{1/2} \frac{\delta\rho}{\delta n_i} \right) \Bigg]\end{split}$

Note that in the above, the analytical form of the term $$\frac{\delta\rho}{\delta n_i}$$ is not used to simplify the expression.

## Tutorial on using the PDH correction¶

There is a tutorial on using the PDH correction that demonstrates the use of this correction with the GUI.

References

1(1,2,3)

Pitzer, Kenneth S., and John M. Simonson. Thermodynamics of multicomponent, miscible, ionic systems: theory and equations. The Journal of Physical Chemistry 90 (1986): 3005-3009.