can also be written asPV&nRT; the equation is thus of the same type as
Eq. (2.12).
Thevalidityof the quantitative relations given is illustrated in Table
2.1 for the freezing point depression by some solutes, Eq. (2.16). It is seen
that the agreement is reasonable, especially if the equations are used in their
most rigid form. Nevertheless, deviations remain, which implies that at the
concentration considered the solutions are not ideally dilute anymore.
Comparison with Figure 2.3 shows that, for sucrose, deviations become
large for concentrations over 20%. Actually, Eqs. (2.14–2.17) are all based
on the assumption thatx 1 ¼a 1 , i.e.,¼the water activityaw. By inserting
aw—which can often be measured—forx 1 , a much better agreement will be
obtained. As derived in Eq. (8.3),lnawis a property colligative with the
others mentioned.
The colligative properties are of importance by themselves, but they
can also be used to determine the molar mass of a solute, since they all
depend on the molar concentration and since the mass concentration
generally is known. To this end, the determination of the freezing point
often is most convenient. Because of nonideality, determinations should be
made at several concentrations and the results extrapolated to zero. For
determination of the molar mass of macromolecules, osmotic pressure
measurement is to be preferred, since membranes exist that are not
permeable for macromolecules, while they are for small-molecule solutes,
and even small quantities of the latter have a relatively large effect on the
colligative properties. Actually, a difference in osmotic pressure is thus
determined, the difference being due to the macromolecules only.
TABLE2.1 Estimates and Determined ValuesðDTf,inK)of
Freezing Point Depression of Some 20%(w/w) Aqueous SolutionsSolute Mm1.86m 103 x 2 103 lnx 1 DTf
Ethanol 46.1 4.20 7.8 9.2 9.6 10.9
Glycerol 92 2.27 4.22 4.80 4.92 5.46
d-Glucose 180 1.20 2.23 2.51 2.54 2.64
Sucrose 342 0.63 1.18 1.34 1.35 1.46
NaCl* 58.4 7.85 14.6 15.4 16.7 16.5
M¼molar massðDaÞ,m¼molarity,x 1 ¼mole fraction of water,x 2 ¼mole
fraction of solute.
* Assuming effective molarity to be twice the nominal value.