Passive Mass Transport Processes in Cellular Membranes and their Biophysical Implications - Porous Media: Applications in Biological Systems and Biotechnology - page 300

The chemical potential µ i of the i th component can be expressed as

µ i = µ st + V i P + µ i (8.5)

i

where µ st i is the standard ch emical potential, V i P is the pressure component
of the chemical potential ( V i is the molar volume, and P is the pressure), and
µ i is the concentration component of the potential. In particular, for water
the chemical potential difference across the membrane equals

µ ∆ w = V w P 0 P ∆ x +( µ c w ) 0

( µ c w ) ∆ x

∆ µ w = µ w −

− −

= V w ∆ P +∆ µ c w = V w (∆ P

−

∆Π) (8.6)

where ∆ P=P 0 P ∆ x , the osmotic pressure difference ∆Π is given by the

van't Hoff formula ∆Π =RT ( c s −

−

c ∆ s ), R is the universal gas constant, T
is the temperature, and c s ,c ∆ s are the concentrations on both sides of the
membrane. For well-mixed solutions c s = c 1 and c ∆ s = c 2 . For the solute ( s )
∆ µ s = V s ∆ P +∆ µ s = V s ∆ P + ∆Π
c s

(8.7)

∆Π
∆ µ s . For diluted solutions we can use

where the mean concentration is c s =

an approximation:

c s = c s + c ∆ x

s

(8.8)

2

and the dissipation function can be written as
ψ = j w V w + j s V s ∆ P + j s V w j w ∆Π .

c s −

(8.9)

The term j w V w + j s V s is the volume flux:

J v = j w V w + j s V s = J vs + J vw (8.10)

The mean volume fraction of the solute and the solvent (water) in a solution
can be written as

y s = c s V s
y w = c w V w ≈

1 (8.11)

Then the second term in equation (8.9) can be written as

j s
c s −

V w j w = j s

j w
c w

c s −

(8.12)

The flux j i of the i th component of the solution is

A ∆ x ∆ t = ∆ n i

∆ n i ∆ x ∆ x

∆ t

j i = = c i v i (8.13)

∆ V