Biomedical Engineering Reference
In-Depth Information
that is, a pressure difference is necessary to prevent a net flow of water from traveling to
the left side. The pressure difference that causes
Q
¼
0 is called the osmotic pressure, with
P
osmotic
¼
(
p
1
p
2
)
¼
RTc
1
. The traditional symbol used to denote osmotic pressure is p,
where p
¼
RTc
1
. If there is no pressure difference, then water is driven from the right side
to the left side at a rate
Q
¼
RTc
1
.
R
m
If there is solute on both sides of the membrane, then the osmotic pressure equals
p
¼
RT c
1
c
2
ð
Þ
ð
7
:
16
Þ
where
c
2
are the solute concentration on either side of the membrane. Equation (7.16)
is called the van't Hoff law. If there are a number of different impermeable particles on
either side of the membrane, then
p
1
p
2
c
1
and
Þ
and the total osmotic pressure equals the sum of the concentration differences for each
particle (ion or molecule) on either side of the membrane
p
ð
Þ
RT c
A
1
c
A
2
ð
ð
Þ þ
c
B
1
c
B
2
ð
Þþþ
c
Z
1
c
Z
2
ð
Þ
Þ ¼
R
m
Q
ð
7
:
17
¼
RT c
A
1
c
A
2
ð
ð
Þ þ
c
B
1
c
B
2
ð
Þþþ
c
Z
1
c
Z
2
ð
Þ
Þ
EXAMPLE PROBLEM 7.1
Find the initial osmotic pressure at room temperature for a cell if the only ions present are
NaCl
Na
þ
and
Cl
from Table 7.2 and
on either side of the membrane. Assume the concentrations for
that the ions cannot cross the membrane, and the cell volume is
V
i
¼
2 nL.
Solution
Note first that the cell cannot withstand a pressure gradient across the membrane, so
p
i
¼
p
o
.
Therefore, we use Eq. (7.17) to find the initial osmotic pressure at room temperature as
¼
RT Na
þ
i
Na
þ
o
þ
Cl
i
Cl
p
½
½
½
½
o
Cl
] are the concentrations of sodium and chlorine on the inside and outside
of the membrane.
8
From Table 7.2, we substitute the concentrations into this equation, giving
Na
þ
] and [
where [
p ¼
62
:
3637
310 10
ð
ð
142
Þ þ
ð
4
103
Þ
Þ ¼
4466 mmHg
The osmotic pressure initially drives water out of the cell to equalize water concentration,
which reduces the cell volume. This continues until the inside water concentration equals the
outside water concentration. To compute the final cell volume size, note that the number of moles
of
Na
þ
and
Cl
inside the cell remains constant, since it cannot pass through the membrane, and is
V
i
¼
q
i
¼
Na
þ
i
þ
Cl
10
9
10
9
M
½
½
i
ð
10
þ
4
Þ
2
¼
28
To have a zero osmotic pressure at steady state, we require that the inside steady state concen-
tration,
c
i
ss
, equal the outside concentration,
c
o
ss
,
Continued
8
We use the symbol [
þ
] and [
].
I
] to denote the concentration of ion “I.” In this case, [
Na
Cl
