## Analysis of Isotherms to Provide Thermodynamic Data

For a fully reversible isotherm a mass action quotient (Km) can be used to defined the process, as with any other reversible chemical process, namely:

Km = AzzBmZA/BZAmZAB [8]

From this the thermodynamic constant (Ka) can be determined using eqn [6]:

where:

yA and yB are the single ion activity coefficients of A A and B B, respectively, in solution, and fA,B are the activity coefficients of the same ions in the solid phase.

Ka can be determined by graphical integration of a plot of ln Kmr against AZ (or by an analytical integration of the polynomial that gives the computed best fit to the experimental data). The quantity Kmr can be described as:

where Kc is the Kielland coefficient related to Ka by the simplified Gaines and Thomas equation:

Values for yA,B cannot be determined, but r is available from the mean stoichiometric activity coefficients in mixed salt solutions via eqn [10]:

r = yZA/yZB = ([y(±AXX]ZA(ZB + ZX>/[y(±XX]ZB(ZA-ZX))1/ZX

In eqn [13], ZX is the charge on the common anion y(±XX, and y±AX can be calculated from y±BX and y( AX using the method of Glueckauf. fA,B values are available from the Gibbs-Duhem equation.

Having obtained Ka, a value of AG6 can be gained from:

where R and T have their usual meanings, and AG9 is the standard free energy per equivalent of charge.

The standard states of the exchanger relate to the respective homoionic forms of the exchanger immersed in an infinitely dilute solution of the corresponding ion. This implies that the water activity in the solid phase in each standard state is equal to the water activity in the ideal solution, and that the standard states in the solution phase are defined as the hypothetical ideal, molar (mol dm~3) solutions or the pure salts according to the Henry Law definition of an ideal solution.

At this point it should be commented that this approach is based on a simplified Gaines and Thomas treatment. In the complete version of eqn [12] the LHS should be ln Ka — A, where A is a water activity term. For most selectivity studies the AG6 values measured using the simplified treatment are adequate.

To obtain a selectivity series, isotherms should be constructed for a homoionic exchanger initially in, say, sodium form in contact with solutions of ingoing ions (for instance Li, K, Rb, Cs). This yields AG6 values that, when arranged in order of decreasing negativity, provide an assessment of the affinity the exchanger has for the alkali metals.

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