This experiment shows one method of determining the activity of water in a solution. The solute is HCl, and a range of concentrations is studied. Since the solvent is in equilibrium with the solid, the Gibbs energies of the two phases are equal:
G 1 (solid) = G 1 (solution) = Go 1 (liquid) + RT ln a1
where the G's represent the Gibbs energies of their respective phases, and a1 is the activity of the solvent. Division by T and differentiation of this equation with respect to T gives:
H 1 (solid) = H 1 (liquid) - RT^2(d ln a1/dT)P
where the H's are the enthalpies of their respective phases. Since the difference in the enthalpies is just the molar heat of fusion,
(d ln a1/dT)P = DHfusion/RT^2
The heat of fusion is temperature dependent, so the equation becomes
(d ln a1/dT)P = DHm/RT^2 - DCP (Tm-T)/RT^2
where DHm is the heat of fusion at the melting point, DCP is the difference in heat capacity between the liquid and solid phases, and Tm is the melting point. This equation may be integrated to give
ln a1 = -DHm/RT + DCP/RT + (DCP ln T)/R + C
where C is the constant of integration. Since a1 is equal to unity when T=Tm, we may rearrange this equation :
ln a1 = -((DHm - DCP Tm)/RTm )(Tm/T -1) - (DCP ln(Tm/ T))/R
For water, DHm is 6.008 kJ mol-1, Tm is 273.16 K, and DCP is 38.1 J K-1 mol-1. One can therefore calculate the activity, and hence the activity coefficient, for a solution from its freezing point.
Prepare 250 mL each of solutions 1, 1.5, 2, 2.5, and 3 M HCl. Once prepared, these should be stored on ice. Place 125 mL of ice into the Dewar flask, and just cover the ice with distilled water (also prechilled at 0oC). Stir the ice-water mixture until a constant temperature reading is observed. Record this temperature. Remember, the Beckman thermometer reads temperature differences, rather than actual temperatures. Since most of the measurements should be below 0oC, the ice-water reading should be at the top of the scale. Drain off the water, and cover the ice with the 3 M HCl. Stir and record the new temperature. Withdraw 25 mL of the solution, and set it aside, allowing it to warm to room temperature. Dilute a 10 mL aliquot (the remainder of the room temperature solution is used in the density determination) of this solution to 100 mL and titrate a 10 mL aliquot of that with 0.1 M NaOH using phenolphthalein. Repeat this procedure with each of the HCl solutions, adjusting aliquot sizes to allow sensible amounts of NaOH. Determine the density of the room temperature solutions using the pycnometer.
Calculate the activity of water for each of the five solutions. From the titrations and density measurements, calculate the mole fraction of water in each solution, then use this to calculate the activity coefficients.