Steric Mass Action

The steric mass action model takes charges of the molecules into account [23] and is, thus, often used in ion-exchange chromatography. Each component has a characteristic charge \(\nu\) that determines the number of available binding sites \(\Lambda\) (ionic capacity) used up by a molecule. Due to the molecule’s shape, some additional binding sites (steric shielding factor \(\sigma\)) may be shielded from other molecules and are not available for binding.

\[\begin{aligned} \frac{\mathrm{d} q_i}{\mathrm{d} t} = k_{a,i} c_{p,i}\left( \frac{\bar{q}_0 }{q_{\text{ref}}} \right)^{\nu_i} - k_{d,i}\: q_i\: \left(\frac{c_{p,0}}{c_{\text{ref}}}\right)^{\nu_i} && i = 1, \dots, N_{\text{comp}} - 1, \end{aligned}\]

where \(c_{p,0}\) and \(q_0\) denote the salt concentrations in the liquid and solid phase of the beads, respectively. The number of free binding sites

\[\begin{aligned} \bar{q}_0 = \Lambda - \sum_{j=1}^{N_{\text{comp}} - 1} \left( \nu_j + \sigma_j \right) q_j = q_0 - \sum_{j=1}^{N_{\text{comp}} - 1} \sigma_j q_j \end{aligned}\]

is calculated from the number of bound counter ions \(q_0\) by taking steric shielding into account. In turn, the number of bound counter ions \(q_0\) (electro-neutrality condition) is given by

\[\begin{aligned} q_0 = \Lambda - \sum_{j=1}^{N_{\text{comp}} - 1} \nu_j q_j, \end{aligned}\]

which also compensates for the missing equation for \(\frac{\mathrm{d} q_0}{\mathrm{d}t}\).

The concept of reference concentrations (\(c_{\text{ref}}\) and \(q_{\text{ref}}\)) is explained in the respective paragraph in Section Reference concentrations.

For more information on model parameters required to define in CADET file format, see Steric Mass Action.