Chapter 1
Physical Properties of the Body Fluids
1.2 The Thermodynamics of Osmosis
We will consider the equilibrium state of liquid mixtures in two regions separated by a
membrane that is permeable to some of the species present and impermeable to others. This
situation is illustrated in Figure 1.2-1 where a semi-permeable membrane separates regions A
that contains a nondiffusing solute and region B that contains only water.
Water and
non-diffusable
solute
A
h
water
B
PA
PB
Figure 1.2-1 Osmotic pressure  = PA  PB  Agh
Water will diffuse from region B into region A until the chemical potential or fugacity of
water on each side of the membrane is the same. This phenomenon is called osmosis and the
pressure difference between regions A and B at equilibrium is the osmotic pressure of region
A. We now want an expression that gives us the solution osmotic pressure as a function of
the solute concentration. At equilibrium
f WA (T, PA, xs) = f WB (T, PB)
(1.2-1)
where f WB (T, PB) is the fugacity of water as a pure component and f WA (T, PA, xs) is the
fugacity of water as it exists in solution with solute at mole fraction xs. A similar equation is
not written for the solute since it cannot diffuse through the membrane. Equation (1.2-1) can
be expressed in terms of the pure water fugacity using the activity coefficient  WA
 WA xWA f WA (T, PA) = f WB (T, PB)
(1.2-2)
The fugacity is a thermodynamic function defined by
 G(T , P )  G IG (T , P ) 
 1
f(T, P) = Pexp 
 = Pexp 
RT
 RT



P
0
RT  

V 
dP
P  

where G(T, P) is the molar Gibbs free energy and GIG(T, P) is the molar Gibbs free energy as
the fluid approached ideal gas state. The water fugacities at states (T, PA) and (T, PB) are then
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1
f (T , P A )
ln
=
A
RT
P

PA

PB

PB
0
RT 

V 
dP
P 

RT 

V 
dP
P 

ln
1
f (T , P B )
=
B
RT
P
ln
1
f (T , P A )
=
A
RT
P
ln
1
f (T , P A )
f (T , P B )
=
ln
+
A
B
RT
P
P
0
Since PA > PB
0
1
RT 

V 
dP +
RT
P 


PA
P

PA
P
V dP 
B
B
RT 

V 
dP
P 


PA
P
B
dP
P
V is the molar volume of water, an incompressible liquid
ln
f (T , P A )
PA
f (T , P B ) V ( P A  P B )
=
ln
+

ln
PA
PB
RT
PB
ln
f (T , P A )
f (T , P B ) V ( P A  P B )
=
ln
+
PA
PA
RT
V ( P A  P B ) 
f (T , P A )
f (T , P B )
=
exp


PA
PA
RT


From the equality of fugacity, equation (1.2-2)  WA xWA f WA (T, PA) = f WB (T, PB), we have
V ( P A  P B ) 
B
B
 WA xWA f WA (T, PB) exp  W
 = f W (T, P )
RT


L
Since f WA (T, PB) = f WB (T, PB) = pure water fugacity
  V L (P A  PB ) 
 WA xWA = exp  W

RT


The osmotic pressure is then
 = PA  PB = 
RT
ln  WA xWA 
L
VW
(1.2-3)
For an ideal aqueous solution at 298oK with xW = 0.98, W = 1, the osmotic pressure is
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 = PA  PB = 
RT
ln(xW)
L
VW

bar  m 3 
 8.314  10 5
( 298o K )
o
mol

K

=  
ln(0.98) = 27.8 bar
6
3
18  10 m / mol
For ideal solution and small solute concentration, xWA  1, and ln( xWA )   (1  xWA )
Hence
=
x SA =
RT
RT
RT
ln( xWA )  L (1  xWA ) = L x SA
L
VW
VW
VW
Moles
x SA
=
L
VW
(1.2-4)
Moles solute
Moles solute

solute  Moles solvent
Moles solvent
Moles solute
solute
=
= CS
 Volume solvent  Volume solvent

solvent )
 Moles solvent 
Moles
( Moles
The ideal dilute solution osmotic pressure, described by equation (1.2-4), is known as van’t
Hoff’s law. This equation can also be written in terms of the mass concentration S
=

RT A
x S = RT CS = RT S
L
Mw S
VW
(1.2-5)
Mass solute
and MwS = molecular weight of solute. Equation (1.2-3) can be
Volume solvent
used to determine solvent activity coefficient in a solvent-solute system provided a semipermeable membrane can be found.
where S =
 = PA  PB = 
RT
ln  WA xWA 
L
VW
Osmotic pressure measurements are more commonly used to determine the molecular
weights of proteins and other macromolecules using an osmometer shown in Figure 1.2-2. At
equilibrium the osmotic pressure  is equal to gh, where  is the solution density and h is
the difference in liquid heights. Equation (1.2-5) is then solved for the molecular weight of
the solute.
MwS = RT
S

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h
Solvent
solute mixture
Solvent
Semipermeable
membrane
Figure 1.2-2 A graphical depiction of a simple osmometer.
Example 1.2-1. ---------------------------------------------------------------------------------(Sandler, Chemical and Engineering Thermodynamics, Wiley, 1999, p.605)
The polymer polyvinyl chloride (PVC) is soluble in solvent cyclohexanone. At 25oC it is
found that if a 2 g of a specific batch of PVC per liter of solvent is placed in an osmometer,
the height h to which the pure cyclohexanone rises is 0.85 cm. Use this information to
estimate the molecular weight of the PVC polymer. Density of cyclohexanone is 0.98 g/cm3.
Solution ----------------------------------------------------------------------------------------- = gh = 980
MwS = RT
S

MwS = 8.314
kg
m
9.81 2 8.510-3 m = 81.72 Pa
3
m
s
g
Pa  m 3
298.15 K2,000 3 /81.72 Pa = 60,670 g/mol
m
mol  K
--------------------------------------------------------------------------------------------------If the dilute solution contains N ideal solutes then
N
 = RT  C S ,i
i 1
The term osmole is defined as one mole of a nondiffusing and nondissociating substance.
One mole of a dissociating substance such as NaCl is equivalent to two osmoles. The number
of osmoles per liter of solution is called osmolarity. For physiological solutions, it is
convenient to work in terms of milliosmoles (mOsm) or milliosmolar (mOsM). The number
of particles formed by a given solute determines osmotic pressure. Each nondiffusing particle
in the solution contributes the same amount to the osmotic pressure regardless of the size of
the particle.
The osmotic pressure difference between the interstitial and plasma fluids is due to the
plasma proteins since the proteins do not readily pass through the capillary wall. The osmotic
pressure created by the proteins is given the special name of colloid osmotic pressure or
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oncotic pressure. For human plasma, the colloid osmotic pressure is about 28 mmHg; 19
mmHg caused by the plasma proteins and 9 mmHg caused by the cations within the plasma
that are retained through electrostatic interaction with the negative surface charges of the
proteins.
Figure 1.2-3 Osmosis of water through red blood cell.
If a cell such as red blood cell is placed in a hypotonic solution that has a lower concentration
of solutes or osmolarity, then the establishment of osmotic equilibrium requires the osmosis
of water into the cell resulting in swelling of the cell. If the cell is placed in a hypertonic
solution with a higher concentration of solutes or osmolarity, then osmotic equilibrium
requires osmosis of water out of the cell resulting in shrinkage of the cell. An isotonic
solution has the same osmolarity of the cell and will not cause any osmosis of water as
shown in Figure 1.2-3. A 0.9 weight percent solution of sodium chloride or a 5 weight
percent solution of glucose is just about isotonic with respect to a cell.
1.3 Flow of Fluid Across the Capillary Wall
Membrane
Pore
Pfeed-Ppermeate
feed-permeate)
Low pressure
(permeate) side
High pressure
(feed) side
Figure 1.3-1 Forces acting to cause a flow of fluid across a porous membrane.
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The flow of fluid across a porous membrane is driven by a difference in hydrostatic pressure
across the membrane and a difference in osmotic pressure between the fluids separated by the
membrane. The volumetric fluid transport (J) across the membrane is given by
J = LpS[(Pfeed  Ppermeate)  (feed  permeate)]
(1.3-1)
where Lp is the hydraulic conductance, S is the surface area of the membrane, (Pfeed 
Ppermeate) is the hydrostatic pressure difference, and (feed  permeate) is the osmotic pressure
difference. When Pfeed > Ppermeate the flow is from the feed to the permeate side, and when
feed > permeate the flow is from the permeate to the feed side. Equation (1.3-1) can be
rearranged to
J = LpS[(Pfeed  feed)  (Ppermeate  permeate)] = P
where P is the effective pressure difference. When (Pfeed  feed) > (Ppermeate  permeate) the
flow is from the feed to the permeate side.
The transport equation (1.3-1) can be applied for the transport of plasma between the
capillary with subscript C and the surrounding interstitial fluid with subscript if.
J = LpS[(Pc  Pif)  (P  if)]
Capillary wall
-(C-if)
PC
C
Pif
if
PC-Pif
Interstitial fluid
Figure 1.2-4 Forces acting to cause a flow of fluid across the capillary wall.
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