## Theory

The Donnan equilibrium relationship is derived from thermodynamics. Under conditions of equilibrium the chemical potential ^ of any dissolved species i is the same in every phase present:

It should be noted that the \$ terms cancel because the same standard state exists in both liquid phases. But the co-ions are not free to move through the membrane that separates the two liquid phases, so there is no opportunity for their concentrations to change. Whenever salt concentrations on opposite sides of the membrane differ, there will be a potential difference across the membrane caused by the concentration difference. This potential difference, called the 'Donnan potential', EDon, is described by rearrangement of eqn [3]:

l/zi

Since the Donnan potential acts on all mobile ionic species, the value of (ail/ai2)1/zi is the same for all of the counterions in the system. In other words, the concentration difference of the co-ions causes an electrical potential that acts on the counterions.

As Donnan pointed out in his seminal description of the theory, a precise treatment of the equilibria would require the use of activities rather than concentrations of ions in the solutions. But the use of molar concentrations greatly simplifies the presentation of the theory, so that is the approach taken here. For the experiment described by Donnan where zi =+ 1 for both Na + and K + ions, the equilibrium written with concentrations would be:

[Na + ]i/[Na + L = [K + ]i/[K + ^ or [Na + ]i/[K + ]i

Here, ¡if is the chemical potential of species i in the standard state, R is the gas law constant, T is the absolute temperature, and ai is the activity of the particular chemical species i being considered. However, electrical potentials must also be considered when the chemical species are ionic, so the electrochemical potential q must be used to describe the equilibrium:

where zi is the ionic charge, F is Faraday's constant, and 6 is the electrical potential. When the two liquids, phase 1 and phase 2, are at equilibrium with the membrane, there is also equilibrium between the two liquid phases, and the electrochemical potential of any mobile species i in the two phases can be equated.

% = ^i2, or RT ln ai1 + ziF61 = RT ln ai2 + zF62 [3]

Figure 1 illustrates the flow of ions in the approach to Donnan equilibrium. Two dilute salt solutions NaCl and KCl are separated a cation-exchange membrane, labelled C, which is permeable to the cations Na + and K + but impermeable to the common anion Cl~. The concentration difference of Na + ions across the membrane provides a driving force for their diffusion through the membrane. There is no net flow of electric current through the membrane, so any net transfer of Na + to the right must be balanced by transfer of an equivalent amount of K+ to the left. Those diffusive processes will occur until an equilibrium is established.

The equilibrium concentrations can be expressed in terms of the initial molar concentrations c1 of NaCl on the left and c2 of KCl on the right, x moles transported through the membrane (still the same for both cations) and the volumes V1 and V2 of the

Figure 1 Donnan potential forces K+ ions to higher concentration.

solutions, expressed in litres:

Solving this equation for x yields:

Donnan dialysis is particularly effective for recovery or removal of multivalent ions. The Donnan equilibrium for a divalent Ca2 + ion and a univalent K + is described by the equation:

For maintenance of electroneutrality in the system, the transport of x moles of Ca2# ions through the membrane requires the transport of 2x moles of K# ions in the opposite direction. Thus the equilibrium is described by:

For Vi = i0, V2 = i and initial concentrations of ci = 0.0i and c2 = i, the value of x = 0.095 is calculated by eqn [9], which means that more than 95% of the calcium would be driven through the membrane. The effect of valence is even more dramatic when the c1/c2 ratio is much lower than the value used in this example.

## Solar Panel Basics

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