tentials of constituent | in the two phases
is then
wi - w= RT in
(4)
ai
To obtain a measure of this difference
of the chemical potentials, use is made
of the experimental fact that whena certain pressure difference is imposed across
the barrier there will be no net passage
of constituent | from phase A to phase
where x is the so-called osmotic pressure.
It is important to note that the osmotic
ptessure multiplied by the partial molal
volume of the constituent under consideration is equal to the difference between
chemical potentials that obtained when
the pressures on the two phases were
equal. Nowhere does “the pressure of
solute molecules” enter.
Thelimiting law, or van’t Hoff law, of
B. Experimentally, it is found that the
pressure on phase B, Pg, must be greater
than the pressure on phase A, P,. Equilibrium is established with respect to
osmotic pressure is derived from Eq. 13
chemical potential, u’,,*,, of constituent
1 in phase A at the pressure P, is equal
for Eq. 13
constituent | in this manner, and the
by the use of simplifying assumptions.
Theactivity, a, is the product of the
mole fraction, N, and the mole fraction
activity coefficient, {. We may then write
x= Rt,Mi
to the chemical potential, p’’, pg, of constituent | in phase B at the pressure Pp.
Wehavenowtherelationships
whe =ae,gtRT Ina:
(5)
and
B's, pp = Bs, py t RT Ina’s
(6)
But equilibrium obtains; hence,
Wie, SW". PR
.
(7)
and
woe, + RT Ina’, =
Ho,ppt+ RT Ina”,
Vi Nf"
The convention that the activity of constituent 1 in phase A is unity is now applied. Equation 14 reduces to:
tions of 4 with P. This variation is given
by
es
. =V
(10)
molal volume of the constituent under
consideration. For sufficiently small varia-
tions of P—that is, if P, and P, differ by-
a sufficiently small amount, and if the
compressibility of the constituents may
be neglected—we may write
H9. by ~ Weg = Vil Pa - Pa)
(11)
From Eq. 11 and Eqs. 4 and 9, there
follows immediately
wi- w= Vi(Pe- Pa) =RT In = (12)
whence
1 SEVTEMBER 1956
1 n Not”
af"
the present simplified system, we have
NN”, +N’, = 1. Hence, N’”",f’, may be
replaced by (1 — N’’,). We nowhave
RT
x=- In (1-N"s)
(13)
(16)
The logarithmic portion is expressed as
a series to give
w= FE enor
— AUN) |
(17)
Ic is now assumed that all terms in the
expansion of degree higher than 1 may
be neglected. Thus
RT
n= — - N",
B ut
me
Ngee
(18)
(19)
where n denotes number of moles. But
n’’, is small compared with n’,; there-
fore, we may write N”’,=n,/n”,. We
then have
RT
t= >
n 1 Vi
na”
(20)
But n’”’,V, is simply the volume occupied
by constituent | and may be written V,.
Then n”,/V, is the mnolal concentration
of constituent 2 and is denoted by C”,,.
We have finally
n= C0", RT
Pa~ Pyne dn 22
The relationship is analogous in form
to the simple gas law. However, this
analogy does not require that the pressure difference x, experimentally determined, be the “bombardment pressure of
the solute molecules against the barrier.”
Indeed,if this were the case, the pressure
“exerted” by the solute molecules against
the barrier would have to be a negative
bombardment pressure in order to accountfor the passage at equal hydrostatic
pressures of the solvent from the pure
solvent phase to the solution phase. It
maybe pointed out that there is no justi-
negative values.
Use of the term osmotic pressure inthe
where the subscripts T and n indicate
that the temperature and composition remain constant, and whereVis the partial
Conclusions
If constituent 2 is present in sufficiently
small concentrations, it may be assumed
that Nf’, approaches N”, in value. In
=- RT
=i
Vi,
where 2°, pg and »°, », are determined
by temperature and pressure. The system
is considered to be isothermal, and we
need be concerned only with the varia-
librium with respect to constituent 1.
15
(15)
= =In
x=
V, In ——
NF",
(8)
(9)
present system in orderto establish equi-
fication or basis for the concept of negative pressure: pressure approaches zero
From Eqs. 7 and 8 we have
Upp ~U%s.e,= RT In
(14)
does no more than indicate the approximaterelationship between the concentration of constituent 2 and the pressure
difference that must be imposed in the
(21)
This is van't Hoffs limiting haw of osmotic pressure. [Cis an approximation, It
as the molecular density approaches zero,
but pressure does not and cannot assume
sense of the “pressure of the solute against
a membrane permeable only to the salvent” cannot be justified either on theoretical or on experimental grounds see,
for example, 6, 8, 9).
On occasion, the conventent: operational designations, * Ssolvent and “sa.
lute,” have been used with the implications of fundamental ditferences an
properties. It will be noted that these
designations have been avoided in’ the
derivation givenhere. In a svstem suchas
is considered here, “solvent” would desig-
nate constituent | and “solute” constituent 2, But the meaning of such designations would be simply that the barner is
permeable to constituent | and not per-
meable to constituent 2. The designations
would be reversed if a barrier permeable
to constituent 2 but not to constituent |
were used. A system frequently considered in physiology is that formed by the
plasma, the capillary walls, and the in-
terstitial fluid. In a simplified form of the
system, water would be the solvent, the
plasmaproteins the solute, and the capillary walls the barrier. In such a simplified
system, the osmotic pressure of the
plasma as determined in the laboratory
provides a measure of the effect of the
proteins on the properties of water,
The term osmatic pressure is in itself
misleading. [t is probable thac this has
contributed to the confusion surrounding
the meaning of the terin. As we have
noted, osmotic pressure as experimentally
determined is a pressure difference and
not an absolute pressure. The experimental implications have been retainedin, for
example, the terms freezing- point de pres473