6. Transport of Dissolved Substances

Diffusion

Assume particles with a density (concentration) c in an environment (such as a liquid solvent). The current of amount of substance due to diffusion of these particles is equal to the product of cross section A, concentration c, and drift speed v_d (Fig. 1). The drift speed is modeled as the result of a chemical driving force delta_mu = mu(x2) – mu(x1). If we take the resulting speed to be proportional to the driving force per unit distance (with a factor called the diffusion constant D), we get

Delta_mu/Delta_x is called the gradient of the chemical potential. The minus sign indicates that the flow is positive in the direction of decreasing chemical potential of the diffusing substance.

If the chemical potential of the substance depends logarithmically upon the concentration c, small differences of mu are equal to Delta_c/c. Using the current density j_n = I_n/A, diffusive currents are given by

The term dc/dx is the concentration gradient. It can be replaced by Delta_c/Delta_x for intervals where c changes linearly with position. Equ. 4.29 is analogous to Ohm’s law (Chapter 2) or Fourier’s law (Chapter 3).

Transport between different environments

Substances flow from one environment into another as long as there is a difference of the chemical potentials of the substance in these environments. If the substance is dissolved, its chemical potential depends logarithmically upon its concentration. Therefore

The current of amount of substance is taken to be proportional to the difference of the chemical potentials and to the concentration of the substance in the originating environment A. Therefore, we have, near equilibrium,

K is the equilibrium constant similar to the one defined in Equ. 4.27. k is a transport coefficient similar to what is used in transports of entropy (Chapter 3).

Dynamical models

Dynamical models are obtained by introducing the flows of a species into its law of balance. Take the example of flow of a species from environment A to B: