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### Activity Coefficient

A knowledge of the variation with temperature and composition of the activity of component k, ak, or the activity coefficient of k, γk, is of primary importance in solution thermodynamics. This is required for the determination of the partial molar Gibbs energy of mixing of the component; which again is required for the determination of equilibrium state of any chemical reaction involving component k in the solid solution. Generally, the variation of ak or γk with temperature and composition must be determined experimentally.

The activity coefficient is a unitless thermodynamic function. If γk = 1 (an ideal solution), the activity of component k is equal to its mole fraction and the behavior of k, from the point of view of its chemical potential, is completely determined by its composition. If γk > 1 (a nonideal solution where the component k is said to exhibit a positive deviation from Raoult's law), then ak > xk and in the evaluation of its chemical potential, component k "acts as if" the solution contains more of k than the mole fraction suggests. Similarly, if γk < 1 (a nonideal solution where the component k exhibits a negative deviation from Raoult's law) so that ak < xk, the component "acts as if" there is less of it present than the composition suggests.

In general, the higher the temperature of a nonideal solution, the smaller the deviation of its components from ideal behavior. In other words, if γk > 1 then an increase in temperature causes γk to decrease toward 1, and if γk < 1 an increase in temperature causes γk to increase toward 1.

The relationship between the activity coefficient of k, temperature, and the partial molar heat of solution of k, ΔHkM, is given as

∂(R·ln γk) / ∂(1/T) = ΔHkM

Thus, if γk > 1 then ΔHkM is a positive quantity, indicating that the mixing process is endothermic. Conversely, if γk < 1 then ΔHkM is a negative quantity, indicating that the mixing process is exothermic.

In an A-B binary system, endothermic mixing indicates a tendency toward phase separation or clustering in the solution. The A-A and B-B attractions are greater than A-B attractions, i.e., the A atoms attempt to be coordinated only by A atoms, and the B atoms attempt to be coordinated only by B atoms. Conversely, exothermic mixing indicates a tendency toward compound formation or ordering between the two components. The A-B attractions are greater than either A-A or B-B attractions, i.e., the A atoms attempt to have only B atoms as nearest neighbors, and the B atoms attempt to have only A atoms as nearest neighbors.

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