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OSMOSIS (A self-instructional package)
Howard Kutchai
Department of Molecular Physiology & Biological Physics
University of Virginia
School of Medicine
Copyright 1980, 2001, 2003 by Howard Kutchai
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OBJECTIVES
1. Define osmosis, osmotic pressure, and ideal semipermeable membrane.
2. Write van't Hoff's Law (π = iRTc) and use this equation to estimate the osmotic
pressure of a solution. State that van't Hoff's Law applies exactly only to very dilute
solutions.
3. State that the osmotic coefficient (Φ) is used to correct for deviations of real
solutions from the predictions of van't Hoff's Law (ideal solution theory). Given values
for Φ, use π = ΦiRTc to calculate the osmotic pressure of a solution.
4. State that the osmotic pressure of biological solutions is usually estimated from their
freezing point depression. State that the effective osmotic concentration (ΦiC) in
osmoles/liter is computed as ∆Tf/1.86, where ∆Tf is the freezing point depression in oC.
5. State that 154 mM NaCl is isotonic to human erythrocytes (red blood cells) and that
this means that in 154 mM NaCl red blood cells will have the same volume they have in
plasma. State that red blood cells will swell in hypotonic NaCl (solutions more dilute
than 154 mM) and that red cells will shrink in hypertonic NaCl (solutions more
concentrated than 154 mM). Compute the steady-state volumes achieved by red blood
cells in NaCl solutions of different concentrations.
6. State that the volume flow (Jv) through a membrane caused by an osmotic pressure
difference (∆π) due to a concentration difference of a completely impermeable solute is
Jv = Lp∆π, where Lp is the hydraulic conductivity of the membrane. Use this equation
appropriately to calculate Jv.
7. State that solutes that are somewhat permeable to a membrane induce smaller osmotic
volume flows than impermeable solutes. State that for permeable solutes Jv =
σLp∆π.,where σ (sigma) is called the reflection coefficient of the membrane for the
solute in question. State that the reflection coefficient is equal to 1 for impermeable
solutes and approaches zero as solutes become more and more permeable. Use this
equation under appropriate conditions to calculate Jv.
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PRACTICE CYCLE 1
INPUT 1
Consider the situation diagrammed below. A 1M solution of sucrose is separated
from a chamber containing pure water by a membrane which is permeable to water, but
impermeable to sucrose. Water will flow across the membrane from chamber B to
chamber A.
This process is known as osmosis. The solute (sucrose) has the effect of lowering the
effective concentration of water molecules, or more properly lowering the chemical
activity of water, and this may be thought of as the driving force for osmosis. In this
example osmosis will continue until chamber B is empty.
Had we begun with a less concentrated sucrose solution in chamber B, say 0.5M,
instead of pure water, water would still flow by osmosis from chamber B to chamber A.
As this occurred, the sucrose in A would be diluted, while that in B became more
concentrated. When the sucrose concentrations in A and B are exactly equal, osmosis
will cease. Water will be in equilibrium between A and B.
Since the membrane is permeable to water, but impermeable to sucrose, the membrane
is called semipermeable. An ideal semipermeable membrane is permeable to water, but
completely impermeable to all solutes. (No such membrane actually exists.)
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PRACTICE 1
Define (a) Osmosis
(b) Ideal semipermeable membrane
FEEDBACK 1
Were your definitions something like this?
(a) Osmosis: The movement of water from a solution in which solute is less
concentrated to a solution in which solute is more concentrated through a membrane that
is permeable to water but impermeable to solute.
(b) Ideal semipermeable membrane: A membrane that is pereable to water, but
completely impermeable to all solutes.
Fine!
PRACTICE CYCLE 2
INPUT 2
It is useful to have a quantitative measure of the tendency of water to move from one
place to another. Such a purpose is served by the concept of osmotic pressure. Consider
the situation diagrammed below. The walls of chamber A and the membrane are
assumed to be rigid
As in the examples above water will flow by osmosis from chamber B to chamber A.
If we apply pressure to the solution in A by pressing on the piston, the net rate of osmotic
water flow will be smaller. If we press hard enough on the piston, the net rate of water
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flow will be zero. If we press harder than that, water will flow from A to B. The
pressure in chamber A needed to just prevent water entry is called the osmotic pressure
of the solution in chamber A. Thus osmotic pressure of a solution is defined as the
pressure which must be applied to the solution to just prevent pure water from entering
the solution across an ideal semi-permeable membrane.
PRACTICE 2
Define the osmotic pressure of a solution.
FEEDBACK 2
Did you say something like: the osmotic pressure of a solution is that pressure which
must be applied to the solution to just prevent pure water from entering the solution
across an ideal semipermeable membrane? Good!
If we compare two solutions, the one with the greater tendency to draw water (the
more concentrated solution) has the greater osmotic pressure. If these two solutions are
separated by a semipermeable membrane, water will flow from the one with the smaller
osmotic pressure to the one with the higher osmotic pressure. We say that in this case the
driving force for water movement is the osmotic pressure difference between the two
solutions. This is calculated as the difference between the higher and lower osmotic
pressures.
PRACTICE CYCLE 3
INPUT 3
As described above, the more concentrated a solution, the greater its osmotic
pressure. That is to say the greater the amount of pressure that is needed to keep water
from entering the solution across an ideal semipermeable membrane. The osmotic
pressure of a solution depends on the total number of ions or molecules in solution. Thus
a 0.5M NaCl solution has approximately the same osmotic pressure as a 1M glucose
solution. This is because NaCl completely dissociates in a Na+ and C1- giving 1M ions
(Na+ + C1-) in solution.
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A frequently used relationship between osmotic pressure (π) and concentration (c) is
π = iRTc
where i is the number of ions or molecules into which the dissolved species dissociates, R
is the ideal gas constant, T is the absolute temperature, and c is the solute concentration in
moles/1iter. This relationship is known as van't Hoff's Law after the great Dutch
physical chemist who first formulated it. van't Hoff's Law is strictly true only for very
dilute solutions.
PRACTICE 3
(a) Write van't Hoff's Law
(b) What is the osmotic pressure of a 5 mM solution of NaCl at 0oC? (at 0oC RT =
22.4 1iter atm/mole)
(c) Under what conditions does van't Hoff's Law apply exactly?
FEEDBACK 3
Did you say something like
(a) van't Hoff's Law is π = iRTc
(b) for 5 mM (0.005 M) NaCl at 0oC,
π = iRTc = 2 x 22.4 1iter atm/mole x 0.005 mole/liter = 0.224 atm
(c) van't Hoff's Law applies exactly only to very dilute solutions.
For a solution as dilute as 0.005 M NaCl, van't Hoff's Law can be expected to apply fairly
exactly. In the next section we describe one way to correct van't Hoff's Law predictions
for more concentrated solutions.
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PRACTICE CYCLE 4
INPUT 4
van't Hoff's Law is often used to estimate the osmotic pressures of solutions that are
too concentrated for van't Hoff's Law to apply exactly. One way of correcting van't
Hoff's Law is by means of correction factors called osmotic coefficients. With the
osmotic coefficient (Φ), van't Hoff's Law becomes
π = ΦiRTc
A table of osmotic coefficients for a number of solutes of physiological importance is
shown below
i
Substance
Osmotic
Coeff (Φ)
NaCl
2
0.93
KCl
2
0.92
HCl
2
0.95
NH4Cl
2
0.92
NaHCO3
2
0.96
CaCl2
3
0.86
MgCl2
3
0.89
Na2SO4
3
0.74
MgSO4
2
0.58
glucose
1
1.01
sucrose
1
1.02
Note that certain nonelectrolytes have osmotic coefficients slightly greater than 1.0, while
electrolytes have Φ values less than 1.0. Note also that the greater the valances of the
ions into which an electrolyte dissociates, the smaller the Φvalue. Keep in mind that for
all solutes Φ approaches zero as the solution becomes infinitely dilute.
PRACTICE 4
(a) What are osmotic coefficients used for?
(b) What is the osmotic pressure of 0.1 M NaCl at 0oC?
(c) What is the osmotic pressure of a solution of 0.1 M NaCl, 0.05 M CaCl2, 0.05 M
glucose at 0oC?
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FEEDBACK 4
Did you say that
(a) Osmotic coefficients are used to correct van't Hoff's Law for solutions of
appreciable concentration.
(b) For 0.1 M NaCl at 0oC:
π = ΦiRTc = 0.93 x 2 x 22.4 liter atm/mole x 0.1 mole/liter = 4.17 atm
(c) For solutions of more than one solute the osmotic pressures add algebraically, so
for 0.1 M NaCl, 0.05 M CaCl2, 0.05 M glucose
π = πNaCl + πCaCl2 + πglucose
= ΦNaCl 2 RT CNaCl + ΦCaCl2 3 RT CCaCl2 + Φglucose RT Cglucose
= RT (2ΦNaClCNaCl + 3ΦCaCl2CCaCl2 + ΦglucoseCglucose)
= 22.4 L-atm/mole [(2 x 0.93 x 0.1) + (3 x 0.86 x 0.05) + (1.01 x 0.05)]mole/L
= 22.4 [0.186 + 0.129 + 0.0505] = 22.4 [0.3655] = 8.19 atm
PRACTICE CYCLE 5
INPUT 5
Because the osmotic pressure depends on the number of molecules or ions per liter of
solution, osmotic pressure is known as colligative property. Other colligative properties
of solutions are vapor pressure lowering, boiling point elevation, and freezing point
depression. Experimentally it is usually easier and more convenient to measure freezing
point depression than to measure boiling point elevation or osmotic pressure. The
freezing point depression (∆Tf) has been determined to be ∆Tf = 1.86Φic
If we consider the effective osmotic concentration to be Φic, then Φic = ∆Tf /1.86
The effective osmotic concentration is often expressed in osmoles/1iter or osmolar. For
example, the effective osmolar concentration (Φic) of 0.1 M NaCl is calculated as
ΦNaC liNaC lCNaCl = 0.93 x 2 x 0.1M = 0.186 osmolar
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PRACTICE 5
(a) If a particular solution depresses the freezing point of water by 0.5oC, what is its
effective osmotic concentration in osmoles/liter?
(b) What is the effective osmotic concentration in osmoles/1iter of a solution of
0.15 M NaCl?
(c) What is the effective osmotic concentration in osmoles/1iter of a solution of 0.1 M
NaCl. 0.05 M CaCl2?
(d) How much would the solution in part (c) depress the freezing point of water?
(e) How is the effective osmotic concentration of biological solutions usually
measured?
FEEDBACK 5
(a) ∆Tf = 1.86Φic, so
Φic = ∆Tf /1.86 = 0.5/1.86 = 0.269 osmole/1iter
(b) Φic = 0.93 x 2 x 0.15 M = 0.279 osmoles/1iter
(c) Effective osmotic concentration = (ΦNaCl iNaCl CNaCl) + (ΦCaCl2 iCaCl2 CCaCl2)
= (0.93 x 2 x 0.1) + (0.86 x 3 x 0.05) = 0.315 osmoles/1iter
(d) ∆Tf = 1.86Φic = 1.86 x 0.315 = 0.586oC
(e) Freezing point depression is usually used to estimate the effective osmotic
concentration of biological solutions.
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PRACTICE CYCLE 6
INPUT 6
We have briefly dealt with the osmotic properties of solutions. Let us now consider
the osmotic behavior of typical animal cells. For this purpose we will use a very simple
cell, the human erythrocyte, as a model. For our purposes we will consider the red cell to
be a solution of hemoglobin, salts, and metabolites surrounded by a membrane that is
permeable to water and some solutes but impermeable to many other solutes.
If a red cell is placed in a solution of an impermeant solute with a higher osmotic
pressure than the effective osmotic pressure of the cell interior, water will leave the cell
by osmosis. The water loss will continue concentrating intracellular solutes until
the osmotic pressure of the cell interior is equal to that of the suspending solution. If a
red cell is placed in a solution of an impermeant solute that has less osmotic pressure than
the cell interior, water will move by osmosis into the cell. The cell will swell and the
cell solutes will be diluted. If the volume increase is more than about 40%, a fraction of
the cells will lyse (burst). The cell will strive toward an equilibrium in which internal
and external osmotic pressures are equal.
Red cells are fairly impermeable to most cations. Thus, for our purposes, NaCl can be
considered an impermeant solute. It is found that if a human red cell is placed in a
0.154 M NaCl solution, the cell neither swells nor shrinks. 0.154 M NaCl is thus said to
be an isotonic solution for the red cell. In solutions of NaCl more concentrated than
0.154 M the red cell will shrink; such solutions are called hypertonic to the cell.
Solutions of NaCl less concentrated than 0.154 M will cause the red cell to swell; such
solutions are called hypotonic to the red cell.
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PRACTICE 6
A human red cell is placed in a large volume of 0.2 M NaCl.
(a) Draw a graph showing the cell volume changes with time that occur.
(b) What will be the equilibrium volume?
FEEDBACK 6
Did you get something like this
(a) The shrinking of the red cell will occur quite rapidly because the red cell membrane
is very permeable to water. The shrinking will be complete in about 0.1 sec!
(b) This one is tougher! The final volume of the red cell will be (0.308/0.4)Vo You
may (or may not!) grasp that intuitively. Since the solutes inside the red cell can be
treated as impermeants, their mass will not change as the cell swells and shrinks. Since
mass = concentration (g/ml or mole/ml) x volume (ml), cV = constant. Thus if the
subscripts o and f denote the initial and final states of the red cell, coVo = cfVf
So that Vf =
co
Vo
cf
Since the red cell behaves like it was filled initially with 0.154 M NaCl or 0.308 M
(Na + Cl), co = 0.308. In the example above we know the red cell will shrink until the
internal concentration is osmotically equivalent to 0.2 M NaCl or 0.4 M (Na + Cl), so that
cf = 0.4. Thus we obtain Vf = (0.308/0.4)Vo
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PRACTICE CYCLE 7
INPUT 7
In the diagram below a semipermeable membrane separates water in chamber A from
water in chamber B. If pressure is applied to chamber A by pushing on the piston, water
will flow from A to B.
The rate of water flow (Jv) will be proportional to the pressure difference (∆P) across the
membrane: Jv = Lp(πA-πB) = Lp∆P. The coefficient of proportionality in this equation
(Lp) is called the hydraulic conductivity. Typical units for the hydraulic conductivity are
ml/(min-atmosphere) or ml/(min-mm Hg).
When an osmotic pressure difference across a membrane causes water flow (Jv) a
similar equation holds: Jv = Lp∆π, where ∆π is the osmotic pressure difference across
the membrane. For the same membrane, the value of Lp for water flow due to hydrostatic
pressure is the same as Lp for water flow caused by osmotic pressure.
Practice 7
In the diagram below a membrane permeable to water, but impermeable to Na+and Clseparates two NaCl solutions, both at 0oC.
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If the Lp for the membrane is 0.05 ml/(min-atm), what is the rate of osmotic water flow
from B to A?
FEEDBACK 7
Did you compute an osmotic flow of 0.1875 ml/min for B to A? Great!
If not, here's how it can be done. The relevant equation is Jv = Lp∆π
where the osmotic pressure difference can be computed from
∆π = ΦiRT∆C = ΦiRT(CA - CB)
= 0.93 x 2 x 22.4 1-atm/mole x 0.009 mole/1 = 0.375 atm
So Jv = Lp∆π = 0.5 ml/(min-atm) x 0.375 atm = 0.1875 ml/min
PRACTICE CYCLE 8
INPUT 8
In the previous practice cycle we considered the osmotic water flow induced by an
impermeant solute. In the case of a permeant solute the osmotic water flow induced is
transient and is less than the flow predicted by Jv = Lp∆π. The more permeable the
solute, the greater the discrepancy between the observed osmotic water flow and Lp∆π.
The table below shows the results of a set of experiments in which the osmotic water
flow induced across a porous dialysis membrane by different solutes was determined.
solute
osmotic flow solute radius
µl min-1M-1
Å
BSA
25
37
Inulin
19
12
Raffinose
11
6.1
Sucrose
9.2
5.3
Glucose
5.1
4.4
Urea
0.6
2.7
D2O
0.06
1.9
σ ≅ 1−
PS
PW
σ
1.0
0.76
0.44
0.37
0.21
0.02
0.002
where PS is the permeability of the membrane to solute
and PW is the permeability of the membrane to water
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The larger the molecular radius of the solute, the less its permeability, and the greater the
osmotic water flow it induces. The ratio of the observed osmotic flow to the theoretical
osmotic flow that would be induced were the solute impermeant (Lp∆π) is called the
reflection coefficient (σ). The reflection coefficient is equal to 1 for impermeable solutes
and approaches zero as the solute becomes more and more permeable.
In terms of the reflection coefficient (r) the osmotic volume flow induced by a
solute is Jv = σLp∆π. This expression holds for both permeant and impermeant
solutes.
PRACTICE 8
The average human red cell has a hydraulic conductivity (Lp) of 1.5 x 10-11 ml
H20/(sec-atm). Malonamide is a slow permeant and the red cell membrane has a
reflection coefficient for malonamide equal to 0.8. If an average red cell is put in a large
volume of isotonic saline containing enough malonamide (about 8 mM at 37C) to
contribute 0.2 atm to the osmotic pressure of the solution, what will
be the initial rate of red cell shrinking?
FEEDBACK 8
Did you calculate that the initial rate of red cell shrinking will be 2.4 x 10-12 ml/sec.
Good! Since the normal red cell volume is about 100 x 10-12 ml, this represents a change
of about 2.4% of the red cell volume per second.
Here's how I got that answer: The appropriate equation is: Jv = σLp∆π
so that
Jv = 0.8 x (1.5 x 10-11 ml H2O sec-1atm-1) x 0.2 atm
= 2.4 x 10-12 ml H2O/sec
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POST - TEST
1. Define (a) osmosis
(b) osmotic pressure
(c) ideal semi-permeable membrane
2. (a) Write van't Hoff's Law
(b) What is the approximate osmotic pressure (at 0oC) of a 0.01 M CaCl2 solution?
(c) Under what conditions does van't Hoff's Law apply?
3. (a) What is the osmotic coefficient (Φ) used for?
(b) If Φ for CaCl2 is 0.86, what is a better estimate of the osmotic pressure of the
solution in problem 2?
4. (a) How are the osmotic pressures of biological solutions usually determined in the
laboratory?
(b) What is meant by the "effective osmotic concentration" and what are its units?
(c) How is the effective osmotic concentration obtained by the method you described
in part (a)?
5. Draw graphs of red blood cell volume vs. time to show what happens when a human
red blood cell is placed in a large volume of a) 154 mM NaCl, b) 125 mM NaCl, and
c) 175 mM NaCl. Are these solutions isotonic, hypotonic, or hypertonic relative to the
red cell?
6. A true semipermeable membrane has a hydraulic conductivity of 5 µl/(min-atm) per
cm2 membrane area at 0oC. Water is present on one side of the membrane and 10 mM
CaCl2 on the other. What is the net rate of osmotic volume flow across each cm2 of
membrane?
7. A particular membrane has a reflection coefficient for sucrose of 0.6 and a hydraulic
conductivity of 0.2 µl/(min-atm) per cm2 of membrane area at 0oC. If water is present on
one side of the membrane and 100 mM sucrose on the other, what will be the initial rate
of osmotic volume flow across the membrane?
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ANSWERS TO POST-TEST
1. (a) Osmosis - The flow of water induced by a concentration difference of solute
across a membrane that is permeable to water but impermeable to solute.
(b) Osmotic pressure - The pressure which must be applied to a solution to just
prevent pure water from entering the solution across an ideal semi-permeable
membrane.
(c) Ideal semi-permeable membrane - a membrane that is permeable to water but
completely impermeable to all solutes.
2. (a) π = iRTc
(b) π = 0.672 atm
(c) van't Hoff's Law applies exactly only for very dilute solutions.
3. (a) The osmotic coefficient (Φ) is used to make van't Hoff's Law apply more exactly
to real solutions.
(b) π = 0.86 x 0.672 atm = 0.578 atm
4. (a) Osmotic pressures are usually estimated from measurements of the freezing point
of the solution.
(b) The effective osmotic concentration is equal to Φic. Its units are osmole/1iter or
osmolar.
(d) The effective osmotic concentration can be obtained from the freezing point
depression (∆Tf) by Φic = ∆Tf/1.86
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6. Jv = Lp∆π, ∆π = ΦiRT∆C = 0.86 x 3 x 22.4 1-atm/mole x 0.01 mole/1iter = 0.578 atm
Jv = 5 µl/(min-atm) x 0.578 atm = 2.89 µl per cm2 of membrane
7. Jv = σLp∆π,
∆π = ΦiRT∆C = 1.02 x 1 x 22.4 1-atm/mole x 0.1 mole/1 = 2.28 atm
Jv = 0.6 x 0.2 µl/(min-atm) x 2.28 atm = 0.274 µl/min per cm2 membrane