ð 4 = 3 Þpr^3 , but another molecule cannot come closer to it than a distance 2r
(taking the position of the molecules to be at their centers), which implies
that a volume of ð 4 = 3 Þpð 2 rÞ^3 is excluded for a second molecule. If the
volume fraction occupied by the molecules isj, this would imply that a
volume of 4jis not available as a solvent (not 8j, because we then would
count the excluded volume twice). The ‘‘effective’’ concentration of the
solute would be increased by a factor 1=ð 1 4 jÞ,ifr 2 (solute molecules)
r 1 (solvent molecules). Mostlyr 1 andr 2 differ less; the numerical factor
would then be <4, and it is zero if r 1 ¼r 2. This reasoning is an
oversimplification, because (a) only for smalljdoes the excluded volume
indeed equal 4j, whereas for largerjit becomes relatively less; (b) for
nonspherical molecules the excluded volume is less well defined, though
generally higher; and (c) interactions between solute and solvent mole-
cules—i.e., the solvent quality mentioned above—may modify the result.
Nevertheless volume exclusion is a very real and important source for
FIGURE2.5 Hypothetical examples of the dependence of osmotic pressureðPÞ
divided by concentration against molar concentrationðmÞ.