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male fraction salt, xs Fig. 8.16 The NaCl/HiO phase diagram

On the top of the diagram is a large single-phase region entitled liquid (sol'n); the common name is brine. On the left-hand ordinate is pure ice and the right hand ordinate (x = 1, not shown) is pure salt. The shaded zone in Fig. 8.16 is the only single-phase solid region in the diagram. It is a compound with the water-to-salt mole ratio denoted by n. It is shown as the region is labeled NaCl(H2O)x, with x varying from the lower phase boundary at a salt mole fraction of xHL = 0.078 to the upper phase boundary where the salt mole fraction is xH = 0.092.

The freezing point of the salt-water solution decreases with salt concentration along the curve ab, which is given by:

As the temperature of the salt solution in the concentration range a-b is reduced, pure solid water (ice) appears when the temperature reaches the value given by Eq (8.34). The maximum depression occurs at point b, where the salt concentration is xL = 0.078 and

T = -18 oC, or 0 oF. Point a represents the freezing point of pure water, which is 32oF. These two reference temperatures formed the basis of the Farenheit temperature scale, named after its eponymous author.

An approximation to Eq (8.34) is obtained by equating the chemical potentials of the two phases in equilibrium along the curve ab:

where g I and g w are the molar free energies of pure ice and pure liquid water, respectively. xw = 1 - xS is the mole fraction of water and yw is the activity coefficient of water in the salt solution. The difference in the free energies of the liquid and ice phases of water is given by Eq (3.27b):

Here AhM = 6008 J/mole is the enthalpy of melting of ice and TM = 273 K is the freezing point of pure water.

Example: What is the activity coefficient of water in an aqueous solution containing 120 g NaCl in a liter of water?

Converting 120 g to 2.05 moles of NaCl and 1000 g water to 55.6 moles gives xS = 0.036. For this composition, Eq (8.34) gives T = -6.7oC, or 266 K. Solving Eq (8.36) yields yw = 0.968.

Determination of the activity coefficient of salt in the brine would require use of Eq (7.32), with B = S and A = w. This involves calculating yw as illustrated above for a sufficient number of salt concentrations to permit the integral on the right side of Eq (7.32) to be accurately evaluated.

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