Your browser does not support JavaScript! (M)
Water site headerMasthead Island, Great Barrier Reef Print-me keygo to Water Visitor Book contributions
Go to my page Water Structure and Science

 

Osmotic flow countered by the osmotic pressure

 

Schematic showing the effect of osmotic flow countered by the osmotic pressure (Pi)

Osmotic pressure

Osmotic pressure is the tendency of solvent molecules to move in the direction of lower solvent activity. It plays a central role in biology.

 

V Mole fraction : molarity, molality, % w/w and % w/v

V Overview of colligative properties
V Diffusion

V Brownian motion

V Self-generation of osmotic pressure at interfaces
V Reverse osmosis

V Osmotic stress

V Osmotic pressure
V Osmotic flux

V Relation of osmotic pressure to vapor pressure lowering
V The osmotic pressure of polymers
V Osmotic potential

 

"...the cause of osmosis is the difference that exists between the free energy or activity

of the solvent in the pure state and in solution..."

Alexander Finlay 'Osmotic Pressure' 1919  

Osmotic pressure

When pure liquid water is separated by a membrane, permeable to water but not solute, from a solution containing a non-volatile solute, water will pass from the pure water side until sufficient extra pressure (Π) is caused or applied to the solution side [1872]. Water moves from high to low water activity due to osmosis and, if allowed, would equalize the water activity on both sides of the membrane. The rate of this osmotically-linked flow has been shown to equals the rate if a similar but hydrostatic pressure (in the opposite direction) was imposed. The pressure needed to stop the osmotic flow is the osmotic pressure (see below right). At this equilibrium, the vapor pressures of the solution and the pure water are identical (see below). Note that osmotic pressure is an equilibrium thermodynamic property a and gives no information concerning the rate of passage of the water, which depends primarily on the properties of the membrane [1873]. The osmotic pressure is the most biologically-important colligative property and correlates to other colligative properties [2749]. Solutes dissolved in a solvent lower both the chemical potential and the activity of the solvent in the solution relative to the chemical potential or activity of the pure solvent. Osmotic pressure, vapor pressure lowering, boiling point elevation, and freezing point depression, are all measures of the same chemical potential of the solvent. Either freezing point depression or osmotic pressure can be used to calculate the other.

 

Net movement of water towards solutions with lower

water activity, generating osmotic pressure

 

Net movement of water towards lower activity water solution, generating osmotic pressure

 

There is often confusion in the literature between the thermodynamic rationale and the kinetic process of osmosis. j The thermodynamics of the process (e.g., the colligative property) only describes the tendency for the process to occur but does not describe the mechanism or the rate of the flow of water in an osmotic process, apart from its (impending) direction. As such, all solutions possess the potential for osmotic pressure (but no actual osmotic pressure) if not adjacent to a semipermeable membrane. Also, there is no distension of the semipermeable membrane under osmotic pressure, but there may well be distension under hydrostatic pressure; the osmotic pressure only apparently acting around the pores of the semipermeable membrane whereas the hydrostatic pressure acts throughout the system. In this way, osmotic pressure m differs from hydrostatic pressure.

 

Osmotic pressure is determined at thermodynamic equilibrium in an osmometer similar to the diagram (see above right) with no net flow through the membrane. In line with the other colligative properties (boiling point raising, freezing point lowering) as determined, it is a pressure difference, not an absolute pressure. It cannot be determined by any kinetic process. Below right is shown the U-tube configuration whereby water flows from the pure water arm to the solution arm until the excess pressure on the solution in the right-hand arm (ρ ˣ g ˣ h) is balanced by the osmotic pressure.

 

As it takes more energy to remove an amount of water from a volume of low activity water than to remove the same amount of water from the same volume of high activity water, water molecules leaving the lower-water-activity compartment will require more energy as they cause the formation of a localized even lower activity water in their wake. Therefore, only higher energy water molecules can leave this compartment; necessarily a smaller amount than travel in the opposite direction. Note that the difference in activity may depend on the difference in the concentration of dissolved molecules and ions as discussed for other colligative properties, or may be due to other causes, such as surface interactions. The measured water flux is often much smaller than the theoretical water flux due to concentration polarization at the membrane surface and the membrane resistance, together with an allowance for leakage through the membrane (the osmotic reflection coefficient (σ) [3088]).

 

At equilibrium, the chemical potential (μ) of the water must be the same on both sides of the semipermeable membrane. c

 

(1)

chemical potential of pure water at pressure P = chemical potential of a solution at pressure P+osmotic pressure (Pi)                                 

The effect of the solute may be included as for Raoult's law

 

see below d (2)

chemical potential of a solution at pressure P+osmotic pressure (Pi) = chemical potential of pure water at pressure P+osmotic pressure (Pi)+ RTLn(xw)

where xw is the mole fraction of the ‘free’ water (0 < xw < 1). b xw = 1 - xs

 

The change in chemical potential with pressure may be included.

 

(3)

chemical potential of pure water at pressure P+osmotic pressure (Pi) = chemical potential of pure water at pressure P+ integral from pressure P to P+osmotic pressure of (the molar volume of water) dT     

Combining equations 1, 2 and 3 and then assuming the molar volume of water (Vm, m3 mol-1 ) varies little within the likely pressure range.

chemical potential of a solution at pressure P+osmotic pressure (Pi) = chemical potential of pure water at pressure P+ integral from pressure P to P+osmotic pressure of (the molar volume of water) dT+ RTLn(xw)

Using equation 1

 

(4 ) l

-RTLn(xw) = integral from pressure P to P+osmotic pressure of (the molar volume of water) dT = Π Vm        

[250] (5)

 

Π Vm = -RTLn(1-xs)

The piezoscopic constant (kp = R/Vm).

When xs is low, the following approximation is valid k

(6)

Ln(1 - xs) = - xs           

Hence,

 

(7)

ΠVm = xsRT

 

(8)

At low xs,

       xs = ns/nw                  

where ns is the number of moles of solute and nw is the number of moles of water. 

           

Given the volume of water (V),

V = nw ˣ Vm                                                             

 

The molar concentration of the solute (Ms mol ˣ m-3) is

Ms = ns/V = xsnw/nwVm  = xs/Vm

 

the van't Hoff equation (9) f' g

                                     Π = MsRT

 

Note that Ms is the molarity and not the molality that is used to describe the other colligative properties.

 

Given the water activity is different on each side of the membrane (aA < aB), from equation 5

Π = (RT/Vm) Ln(aB/aA)

 

The osmotic behavior of glucose solutions

The osmotic behavior of glucose solutions

Opposite shows the osmotic behavior of glucose solutions. e The lower blue line follows the 'ideal' colligative equation, Π = MsRT. The thick red line follows the osmotic pressure as calculated from the experimental freezing point data [70]. The middle green line shows the osmotic pressure calculated from the 'ideal' colligative equation, Π = MsRT, but correcting the concentration errors made by assuming Ln(1 - xs) = - xs and xs = ns/nw (but not the error in the molar volume of water). This is also the line given by Morse and Frazer's equation [2749]

Π = (n/V')RT

that avoids the approximation; where V’ is the volume of pure solvent in which n moles of solute are dissolved. The dotted black line is the best fit to the equation,

 

          Π = RT(Ms+AMs2)               (10)

 

that is a commonly used form of approximation [1104], where the experimentally-determined empirical virial coefficient is A = 0.2 M-1 in this case, with most of the 'A' parameter due to the approximations used and not due to non-ideality. Further empirical virial coefficients may be used at higher concentrations;Π = RT(Ms+A1Ms2+A2Ms3+A3Ms4). They are determined empirically by fitting them to the experimental osmotic pressure data and do not have any physical meaning. The thin red line is described below. Note that at low osmotic pressures, up to about five atmospheres pressure (0.5 MPa) the ideal colligative equation holds well.

 

Osmotic coefficients of bromides and acetates

Osmotic coefficients of bromides and acetates

 

Experimental osmotic pressures (Π ) may be compared with the 'ideally expected' osmotic pressure (Π 0; as from Raoult's law) by the osmotic coefficient (ϕ); ϕ =Π /Π0. The osmotic coefficients of salts vary in a complex ion-specific manner (as ion activities, [2225], see Hofmeister series). For example, at a given concentration, the osmotic coefficients of bromide and iodide salts decreases in the sequence: Li+ > Na+ > K+ > Rb+ > Cs+ (see left), whereas the osmotic coefficients of acetate or fluoride salts decreases in the opposite sequence: Cs+ > Rb+ > K+ > Na+ > Li+ [2224].

Osmotic coefficient of NaCl, broken down

from [3432]

Osmotic coefficient of NaCl, broken down, from [3432]

 

The form of the curves involves a drop in activity due to direct contact ion-pairing reducing the apparent salt concentration counteracted (see the red-dashed line on the right, [3432]) by an increase in osmotic coefficients due to the binding of water by the salt and its consequential loss from the dissolving solution (see the blue-dotted line on the right), so increasing the apparent concentration. The order of the salts is governed by their tendency to form direct contact ion-pair as opposed to solvent separated ion-pairs.

 

Osmotic pressure and simple diffusion may be treated as distinct, unrelated phenomena [2100], one being thermodynamic and the other kinetic. However, osmotic pressure presents the driving force (osmosis) for the movement of water from a region of high 'free' water concentration to a region of lower 'free' water concentration (for example. one with higher salt content). Such a relationship cannot exist between a thermodynamic property (osmotic pressure) and a kinetic process (rate of diffusion), but osmosis and diffusion can both be treated using thermodynamics or kinetics. The thermodynamic argument connects osmosis with the direction of diffusion whereas the kinetic argument connects the pressure differences with the movement of molecules. A diffusion-based theory for osmotic pressure has been described [1315], which involves several assumptions and gives the equation

Pi=MsRTx(xwV^2/(1+xsV))

where xw and xS are the mole fractions of water and solute and V is the volume of the solution (in liters) containing one liter of water. This equation contains no empirical parameters (such as 'A' above in equation 10) and gives a good fit; it is shown as the top thin dashed red line for glucose above. It is noteworthy that this equation is valid for single solutes only and includes the experimentally determined volume (V) term that varies non-linearly with concentration and depends on the size of the solute (that is, it is not a colligative equation).

 

Proteins may remove much of the 'free' water from inside cells causing much intracellular water to be osmotically unresponsive. As an example, one molecule of bovine serum albumin may bind 5000-43000 molecules of water [1103]. The osmolality of blood serum is about 282 - 295 mOsm ˣ (kg water)-1, close to that of normal saline (0.90% w/v of NaCl, 308 mOsm ˣ L-1). Serum values greater than 380 mOsm ˣ (kg water)-1 or less than 240 mOsm ˣ kg water-1 may be fatal.

[Back to Top to top of page]

Osmotic flux

The flux of water is from the solution with the higher water activity (lower osmotic pressure) towards the solution with the lower water activity (higher osmotic pressure). The (net) rate of water flow depends on the relative rates of flow in the opposite directions. This, in turn, will depend on the ease with which the water molecules may leave the compartments on each side of the membrane, if present (dependent on the mole fraction of 'free' (bulk) water; the water activity), and the ease with which they can pass through such a membrane. Although it is not certain that the hydraulic conductivity is identical under hydrostatic pressure (P) and osmotic pressure (Π ), the volume flow (F, L ˣ m-2 ˣ s-1) is often said to depend on the hydraulic conductivity (H, L ˣ N-1ˣ s-1) of the membrane

 

F = H ˣ (ΔP - ΔΠ )

 

where ΔP is the difference in hydrostatic pressure on the two sides of the membrane and ΔΠ   is the difference in osmotic pressure between the two solutions. The net flow may be several times greater than that possible by diffusion alone, clearly indicating that it is not diffusion controlled. Where there is no difference in hydrostatic pressure but differences in the solutions' osmotic pressure, this equation reduces to,

F = H ˣ ΔΠ

 

The net flux will be lower than this if the membrane is leaky. Then, the following equation is sometimes used [3277],

 

F = (1/K) ˣ Ln(Π /Π b)

 

where K is related to the structural properties of the membrane (S) and solute diffusion coefficient (D),

 

K = S/D

[Back to Top to top of page]

Relation of osmotic pressure to vapor pressure lowering

An alternative definition of the osmotic pressure of a solution of given uniform concentration at a given temperature is the liquid pressure under which the internal vapor pressure of such solution is equal to the vapor pressure of the pure solvent [2750]. The interface between liquid water and the gas phase is a form of semipermeable membrane, with water molecules able to pass through (by evaporation and condensation) but non-volatile solutes held within the liquid phase. It is unsurprising, therefore, that the osmotic pressure should be related to the lowering of vapor pressure. The relationship is

 

V ˣ dP = v ˣ dp

 

where V is the volume of the vapor at vapor pressure P and v is the volume of the solvent in the solution at liquid pressure p [2786].

 

The vapor pressure lowering is given by                                      Psolute = xwPpure liquid

Therefore,                                                                  Ppure liquid - Psolute = Ppure liquid ˣ (1 - xw)

From the gas laws ( PV = RT ) and  ( xs =  1 - xw ),    Ppure liquid - Psolute = xsRT/Vg               

where Vg is the molar volume of the gaseous solvent.

Relation of osmotic pressure to vapor pressure

 

Relation of osmotic pressure to vapor pressure

 

Compare this equation with equation 7 above for dilute solutions

Π = xsRT/Vm

where Vm is the molar volume of the liquid solvent. For water at STP, the ratio Vg/Vm = 1232.

Therefore,                   Ppure liquid - Psolute =Π /1232              at STP

 

For example from an experiment, the vapor pressure lowering of a 1-M solution of NaCl at 100 °C, 101,325 Pa is 3.360 kPa. From the van't Hoff equation, the osmotic pressure of this solution is

2000 ˣ 8.314 ˣ 373.15 = 6.205 MPa

At 100 °C, Vg/Vm = 1613 comparing well to the ratio of the pressures of 6205/3.36 = 1846 considering the approximations, experimental data error, and non-ideality involved.

 

Increasing solutes lower the water activity and the vapor pressure of a solution. The vapor pressure of a solution is increased by increasing the hydrostatic pressure on it. The osmotic pressure is the liquid pressure under which the vapor pressure of the solution is equal to the vapor pressure of the pure solvent. Both the osmotic pressure and the vapor pressure are intensive properties of solutions (do not depend on the amount of substance present in the system). As at constant temperature and composition,

 

ΔG = V ˣ Δp

 

the pressure change is related to the free energy change; that is, the amount of work that the system can perform.

[Back to Top to top of page]

The osmotic pressure of polymers

At low concentrations in suitable solvents, linear polymers obey the van't Hoff equation but not at higher concentrations. For all solvents at higher concentrations, this gives rise to complex osmotic behavior with far greater than expected (from van't Hoff equation) osmotic pressures that depend on the degree of polymerization and the interaction between the chains. Polymer chains have random configurations sweeping through space and occupying much larger volumes than the molecular partial volume. The empirical equation

Π = RT(Ms+AMs2

 

is often used, where A is the second virial coefficient that takes into account the interaction between two chains. Note that equation (4)

Π Vm = -RTLn(xw)        

 

still holds throughout this section but the approximation

 

Ln(xw) = Ln(1 - xs) = - xs

 

presents greater error and should not be used. The polymers effectively remove (screen) some of the solvent from the solution by reducing the 'free' volume of solvent available.

 

The osmotic pressure of pullulan from [2709]

osmotic pressure of pullulan from [2709]

As the concentration of the neutral polymers increases, an overlap between the chains increases with increasing contacts between the monomers and the solution is no longer homogenous. Under these conditions, the des Cloiseaux law

Π ∝ Ms9/4

 

holds[2710] (see right [2709]). At low concentrations, the van't Hoff equation osmotic pressure (πvH ) is given by,

 

osmotic pressure equation 1

where C is polymer mass concentration, Mp is polymer molecular weight, Mm is monomer molecular weight, and N is the number of monomers in the polymer chain. As the concentration increases (often called the semidilute regime), the osmotic pressure (π) varies as [2710],

osmotic pressure equation 2

where C* is a characteristic N-dependent polymer concentration associated with the crossover between the dilute and semidilute regimes (see figure above right) and k is a structurally-specific constant. C* has been related to the partial specific volume (V bar) [2710],

osmotic pressure equation 3

giving                                        osmotic pressure equation 5                and                osmotic pressure equation 5

with log-log plots of the osmotic pressure against mass concentration changing from unit gradient to a gradient of 9/4 around C* (polyelectrolytes change from 9/8 gradient to a gradient of 9/4 [2710c]).

 

The osmotic behavior of branched polymers is complicated by the degree and order of the branching with the osmotic pressure dependent on the 'free' volume of solvent available.

 

When the solvent is water, there are two ways that water becomes bound (i.e., it is not 'free' water) due to the number of monomers; (a) water hidden from the bulk due to the above overlapping of the polymer chains and (b) water held firmly to the monomers by electrostatic effects and hydrogen bonds. As there is likely to be overlap between these effects they may well be only partially or insignificantly additive. As above, the osmotic behavior of polyelectrolytes in water depends on the mass concentration relative to that concentration at which the molecules overlap in solution [1493]. Polyelectrolyte gels are may swell to a much greater extent (e.g., hydrogels) than neutral gels due to the presence of counter-ions increasing the osmotic pressure. The elasticity of the gel counteracts this by exerting pressure on the contained water.

 

Shows the area that polymers cannot penetrate

 

cartoon showing the area that polymers cannot penetrate
A different effect occurs in suspensions of inert particles in polymer solutions. It is found that such particles do not exert any osmotic pressure. However, the solution between them (see right) exerts a negative pressure that drives the particle together and destabilizes their suspension [2786]. This negative pressure causes an outward flow of water from the high activity polymer free volume between the inert particles, towards the lower activity polymer bulk solution, and consequently an attraction between these inert particles. The magnitude of this force depends on the product of the osmotic pressure and the cross-sectional area of the excluded volume. Effectively, the small access to biomolecules between the inert particles acts as a semipermeable membrane.

It should be noted that the osmotic drive does not require a material membrane to separate the two solutions [1744] provided that there are two distinct volumes (e.g., [1669]).

 

The same phenomenon applies to inert surfaces with narrow clefts and pores, such as the ZIF-8 material h that has great energy storage capacity (up to 30 MPa negative osmotic pressure in 2M NaCl) due to its sub-nanometric pores [2791].

 

Interfaces can also generate osmotic pressure without the requirement for dissolved solutes [1772, 2057].

[Back to Top to top of page]

Osmotic potential

The osmotic potential is a measure of the tendency of a solution to withdraw water from pure water by osmosis across a differentially permeable membrane. Net diffusion of water occurs from regions of less negative potential to ones of more negative (or lower) potential and continues until the potentials become equal. The osmotic potential (Ψπ) of pure water is defined to be zero (= 0 Pa), with all solutions having negative values, in pressure units. Thus at low concentrations, Ψπ is given by the van't Hoff equation,

 

Ψπ = - MsRT

 

If a living cell is surrounded by a more concentrated solution (with more negative osmotic potential), the cell (hypotonic) will tend to lose water to the surrounding environment. If the living cell is surrounded by a less concentrated solution (less negative osmotic potential), the cell (hypertonic) will tend to gain water from the surrounding environment. Water flows from higher osmotic potential to lower osmotic potential. In osmosis, water molecules move down the water potential gradient, from a high water potential to a lower water potential. Water moves from the ground surrounding trees to their upper branches in response to differences in the osmotic potential between the roots and the leaves; where water evaporates (transpiration [2434]) to increase solute concentration and decrease water potential. The maximum height that a tree (122-130 m) can grow depends on the maximum water potential (≈ 2 MPa) that can be generated in the leaves compatible with retaining the water within the leaf cells [2246].

[Back to Top to top of page]


Footnotes

a Osmotic pressure is a pressure difference existing at equilibrium between two solutions separated by a semipermeable membrane. An osmotic pressure is a physical quantity dependent only on the concentration(s) and temperature of the solution. Water moves from a solution with low osmotic pressure to a solution with high osmotic pressure due to osmosis and, if allowed, would equalize the pressure on both sides of the membrane. Osmotic pressure can also apply to gases and supercritical fluids [2100]. [Back]

 

b Thus it is the mole fraction of water that is important, not the water concentration by itself. [Back]

 

c Similar derivations hold for the osmotic pressures in liquids other than water. [Back]

 

d The energy change occurring over a small change (dx) in the mole fraction of water (xw) is RTdx/xw. Therefore the total energy change going from pure water (xw= 1) to this new value (xw) is RT times the integral dx of 1/xw = RTLn(xw/1) = RTLn(xw). [Back]

 

e The osmotic properties of carbohydrate aqueous solutions from vapor pressure determinations have been described [2556]. [Back]

 

f This van't Hoff equation for the osmotic pressure (Π = MsRT) has the same algebraic form as the Ideal Gas Law (PV = nRT) as n/V = Ms. Thus the (ideal) osmotic pressure of a solute behaves as though it is alone as an (ideal) gas within the space occupied by the volume of the solution. If N2 gas is assumed ideal, and a glucose solution is assumed ideal then the pressure exerted by one mole of N2 gas in a one-liter container (2.437 MPa) is identical to the osmotic pressure exerted by a one molar aqueous solution of glucose (2.437 MPa). The average kinetic energy of molecules is the same in gases and liquids (3/2 RT ˣ mol-1). As first stated by van't Hoff in 1892 however, this conceptual picture may mislead as, although gas pressure is due to the bombardment pressure of the gas molecules against its container, the osmotic pressure is not the bombardment pressure of the solute molecules against the barrier. [Back]

 

g Jacobus Henricus van't Hoff, Jr. (30 August 1852 - 1 March 1911) derived his equation in 1885. He was a Dutch chemist and the first winner of the Nobel Prize in Chemistry (1901), partially for this work. [Back]

 

h Zeolitic imidazolate frameworks (ZIFs) are topologically isomorphic with zeolites, but with the tetrahedrally-coordinated transition metal ions connected by imidazolate linkers rather than silica. [Back]

 

j Osmosis is an essential topic in microbial, plant and animal physiology, physical chemistry and water purification. In the research literature and relevant textbooks, there are almost as many different descriptions and derivations of the process of 'osmosis' as there are 'osmosis' articles; some descriptions are mistaken or misleading whereas others are inadequate or opaque. Some are knowingly wrong while attempting to 'simplify' the subject. The result is one of the most muddled areas of modern science with little consistency or agreement and with most authors vehemently rejecting each other's views. As this is a field that has stretched many great minds, including Albert Einstein, it may be concluded that it is difficult to find clarity. This confusion is compounded as the word 'osmosis' has been defined by both thermodynamics (the tendency of water to pass through a semipermeable membrane) and kinetics (the diffusion of water through a semipermeable membrane). More recently, osmotic pressure models using molecular dynamics have appeared, but these may mislead if they do not behave in a colligative manner. [Back]

 

k The logarithmic term Ln(1 - x) may be expanded in terms of a Maclaurin series

 

Ln(1 - x) = -(x + x2/2 + x3/3 + x4/4 + x5/5 + x6/6 + x7/7 + . . . . . . . .)

 

This series converges for -1 ≤ x < 1 and quickly converges for values of x close to zero. [Back]

 

l This equation is also applicable to non-ideal solutions.

Π Vm = -RTLn(xw)

Π Vm = -RTLn(aw)

[Back]

 

m It is relevant that the units of osmotic pressure, (Pa, kg ˣ m-1 ˣ s-2) are identical to the units of energy per volume (J ˣ m-3, kg ˣ m-1ˣ s-2). Osmotic pressure is an intensive property of the solution rather than just associated with what happens at a membrane. It is also capable of doing work, like raising the level of a solution. Considering osmotic pressure as energy per volume allows easier comprehension of its links to the water potential, solution potential, and the other colligative properties. [Back]

 

[Back to Top to top of page]

 

Home | Site Index | Molality and molarity | Water activity | Colligative properties | Water: Introduction | The icosahedral water clusters | LSBU | Top

 

This page was established in 2006 and last updated by Martin Chaplin on 3 October, 2018


Creative Commons License
This work is licensed under a Creative Commons Attribution
-Noncommercial-No Derivative Works 2.0 UK: England & Wales License