tion, these reactions are generally chosen such that the reacting solute will be
completely consumed, and a concentration results. Generally, spectroscopic
methods give concentrations as well. Equilibrium methods, on the other
hand, yield activities. A good example is measurement of an electric
potential by means of an ion-selective electrode, as in pH measurement.
Also a partition equilibrium between two phases yields activity.
This provides an easy way of determining the activity of a substance if
it is volatile. It will then have the same activity in the gas phase as in the
solution, and at ambient conditions a gas generally shows ideal behavior.
The latter is true as long as the so-called ideal gas law,
pV¼nRT ð 2 : 12 Þ*holds, wherenis the number of moles in the system. The prime example is
determination of the water activity of a solution. Because of the ideality in
the gas phase, i.e.,a 1 ¼x 1 , thea 1 in the solution, mostly designatedaw,is
equal to the relative humidity of the air with which the solution is in
equilibrium, which can readily be measured. If the solute is also volatile, it is
often possible to determine its activity in the gas phase, hence in the
solution.
For a solution of one (nonvolatile) solute in water, whose water
activity is known over a concentration range, the activity of the solute can
be derived from theGibbs–Duhem relation, which can for this case be written
as
x 1 ?dlna 1 þx 2 ?dlna 2 ¼ 0 ð 2 : 13 ÞBy (numerical or graphical) integration,a 2 can now be derived. Figure 2.3
gives as an example the activities of sucrose solutions. It is seen that the
activities greatly deviate from the mole fractions at higher concentration.
For example, at x 2 ¼ 0 :1, the activity coefficient of water
& 0 : 85 = 0 : 90 ¼ 0 :94, that of sucrose & 0 : 26 = 0 : 1 ¼ 2 :6. For mixtures of
more than two components, the activities cannot be derived in this way.
2.2.4 Colligative Properties
The lowering of the chemical potential of a solvent by the presence of a
solute causes changes in a number of physical properties: vapor pressure,
boiling point, freezing point, osmotic pressure, etc. In an ideally dilute
solution the magnitudes of these changes all are proportional to the mole
fraction of solute; they follow from the same cause and are thus called
colligative solution properties. In Section 2.3, electrolyte solutions will be
discussed, but it is convenient to recall here that solutes that largely