What is the relationship between the number of particles in a solution and osmotic pressure?

Large quantities of water molecules constantly move across cell membranes by simple diffusion, often facilitated by movement through membrane proteins, including aquaporins. In general, net movement of water into or out of cells is negligible. For example, it has been estimated that an amount of water equivalent to roughly 100 times the volume of the cell diffuses across the red blood cell membrane every second; the cell doesn't lose or gain water because equal amounts go in and out.

There are, however, many cases in which net flow of water occurs across cell membranes and sheets of cells. An example of great importance to you is the secretion of and absorption of water in your small intestine. In such situations, water still moves across membranes by simple diffusion, but the process is important enough to warrant a distinct name - osmosis.

Osmosis and Net Movement of Water

Osmosis is the net movement of water across a selectively permeable membrane driven by a difference in solute concentrations on the two sides of the membrane. A selectively permiable membrane is one that allows unrestricted passage of water, but not solute molecules or ions.

Different concentrations of solute molecules leads to different concentrations of free water molecules on either side of the membrane. On the side of the membrane with higher free water concentration (i.e. a lower concentration of solute), more water molecules will strike the pores in the membrane in a give interval of time. More strikes equates to more molecules passing through the pores, which in turn results in net diffusion of water from the compartment with high concentration of free water to that with low concentration of free water.

The key to remember about osmosis is that water flows from the solution with the lower solute concentration into the solution with higher solute concentration. This means that water flows in response to differences in molarity across a membrane. The size of the solute particles does not influence osmosis. Equilibrium is reached once sufficient water has moved to equalize the solute concentration on both sides of the membrane, and at that point, net flow of water ceases. Here is a simple example to illustrate these principles:

Two containers of equal volume are separated by a membrane that allows free passage of water, but totally restricts passage of solute molecules. Solution A has 3 molecules of the protein albumin (molecular weight 66,000) and Solution B contains 15 molecules of glucose (molecular weight 180). Into which compartment will water flow, or will there be no net movement of water? [ answer ]

Additional examples are provided on how to determine which direction water will flow in different circumstances.

Tonicity

When thinking about osmosis, we are always comparing solute concentrations between two solutions, and some standard terminology is commonly used to describe these differences:

• Isotonic: The solutions being compared have equal concentration of solutes.
• Hypertonic: The solution with the higher concentration of solutes.
• Hypotonic: The solution with the lower concentration of solutes.

In the examples above, Solutions A and B are isotonic (with each other), Solutions A and B are both hypertonic compared to Solution C, and Solution C is hypotonic relative to Solutions A and B.

Diffusion of water across a membrane generates a pressure called osmotic pressure. If the pressure in the compartment into which water is flowing is raised to the equivalent of the osmotic pressure, movement of water will stop. This pressure is often called hydrostatic ('water-stopping') pressure. The term osmolarity is used to describe the number of solute particles in a volume of fluid. Osmoles are used to describe the concentration in terms of number of particles - a 1 osmolar solution contains 1 mole of osmotically-active particles (molecules and ions) per liter.

The classic demonstration of osmosis and osmotic pressure is to immerse red blood cells in solutions of varying osmolarity and watch what happens. Blood serum is isotonic with respect to the cytoplasm, and red cells in that solution assume the shape of a biconcave disk. To prepare the images shown below, red cells from your intrepid author were suspended in three types of solutions:

• Isotonic - the cells were diluted in serum: Note the beautiful biconcave shape of the cells as they circulate in blood.
• Hypotonic - the cells in serum were diluted in water: At 200 milliosmols (mOs), the cells are visibly swollen and have lost their biconcave shape, and at 100 mOs, most have swollen so much that they have ruptured, leaving what are called red blood cell ghosts. In a hypotonic solution, water rushes into cells.
• Hypertonic - A concentrated solution of NaCl was mixed with the cells and serum to increase osmolarity: At 400 mOs and especially at 500 mOs, water has flowed out of the cells, causing them to collapse and assume the spiky appearance you see.

Predict what would happen if you mixed sufficient water with the 500 mOs sample shown above to reduce its osmolarity to about 300 mOs.

Calculating Osmotic and Hydrostatic Pressure

The flow of water across a membrane in response to differing concentrations of solutes on either side - osmosis - generates a pressure across the membrane called osmotic pressure. Osmotic pressure is defined as the hydrostatic pressure required to stop the flow of water, and thus, osmotic and hydrostatic pressures are, for all intents and purposes, equivalent. The membrane being referred to here can be an artifical lipid bilayer, a plasma membrane or a layer of cells.

The osmotic pressure P of a dilute solution is approximated by the following:

where R is the gas constant (0.082 liter-atmosphere/degree-mole), T is the absolute temperature, and C1 ... Cn are the molar concentrations of all solutes (ions and molecules).

Similarly, the osmotic pressure across of membrane separating two solutions is:

where ΔC is the difference in solute concentration between the two solutions. Thus, if the membrane is permeable to water and not solutes, osmotic pressure is proportional to the difference in solute concentration across the membrane (the proportionality factor is RT).

Advanced and Supplemental Topics

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Measure of the tendency of a solution to take in pure solvent by osmosis

Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane.[1] It is also defined as the measure of the tendency of a solution to take in a pure solvent by osmosis. Potential osmotic pressure is the maximum osmotic pressure that could develop in a solution if it were separated from its pure solvent by a semipermeable membrane.

Osmosis occurs when two solutions containing different concentrations of solute are separated by a selectively permeable membrane. Solvent molecules pass preferentially through the membrane from the low-concentration solution to the solution with higher solute concentration. The transfer of solvent molecules will continue until equilibrium is attained.[1][2]

Theory and measurement

Jacobus van 't Hoff found a quantitative relationship between osmotic pressure and solute concentration, expressed in the following equation:

Π = i c R T {\displaystyle \Pi =icRT}

where Π {\displaystyle \Pi }

P = n V R T = c gas R T {\textstyle P={\frac {n}{V}}RT=c_{\text{gas}}RT}

For more concentrated solutions the van 't Hoff equation can be extended as a power series in solute concentration, c. To a first approximation,

Π = Π 0 + A c 2 {\displaystyle \Pi =\Pi _{0}+Ac^{2}}

where Π 0 {\displaystyle \Pi _{0}}

The Pfeffer cell was developed for the measurement of osmotic pressure.

Applications

Osmotic pressure measurement may be used for the determination of molecular weights.

Osmotic pressure is an important factor affecting biological cells.[4] Osmoregulation is the homeostasis mechanism of an organism to reach balance in osmotic pressure.

• Hypertonicity is the presence of a solution that causes cells to shrink.
• Hypotonicity is the presence of a solution that causes cells to swell.
• Isotonicity is the presence of a solution that produces no change in cell volume.

When a biological cell is in a hypotonic environment, the cell interior accumulates water, water flows across the cell membrane into the cell, causing it to expand. In plant cells, the cell wall restricts the expansion, resulting in pressure on the cell wall from within called turgor pressure. Turgor pressure allows herbaceous plants to stand upright. It is also the determining factor for how plants regulate the aperture of their stomata. In animal cells excessive osmotic pressure can result in cytolysis.

Osmotic pressure is the basis of filtering ("reverse osmosis"), a process commonly used in water purification. The water to be purified is placed in a chamber and put under an amount of pressure greater than the osmotic pressure exerted by the water and the solutes dissolved in it. Part of the chamber opens to a differentially permeable membrane that lets water molecules through, but not the solute particles. The osmotic pressure of ocean water is approximately 27 atm. Reverse osmosis desalinates fresh water from ocean salt water.

Derivation of the van 't Hoff formula

Consider the system at the point when it has reached equilibrium. The condition for this is that the chemical potential of the solvent (since only it is free to flow toward equilibrium) on both sides of the membrane is equal. The compartment containing the pure solvent has a chemical potential of μ 0 ( p ) {\displaystyle \mu ^{0}(p)}

p {\displaystyle p}

0 < x v < 1 {\displaystyle 0<x_{v}<1}

p ′ {\displaystyle p'}

μ v ( x v , p ′ ) {\displaystyle \mu _{v}(x_{v},p')}

p ′ = p + Π {\displaystyle p'=p+\Pi }

μ v 0 ( p ) = μ v ( x v , p + Π ) . {\displaystyle \mu _{v}^{0}(p)=\mu _{v}(x_{v},p+\Pi ).}

Here, the difference in pressure of the two compartments Π ≡ p ′ − p {\displaystyle \Pi \equiv p'-p}

In order to find Π {\displaystyle \Pi } , the osmotic pressure, we consider equilibrium between a solution containing solute and pure water.

μ v ( x v , p + Π ) = μ v 0 ( p ) . {\displaystyle \mu _{v}(x_{v},p+\Pi )=\mu _{v}^{0}(p).}

We can write the left hand side as:

μ v ( x v , p + Π ) = μ v 0 ( p + Π ) + R T ln ⁡ ( γ v x v ) {\displaystyle \mu _{v}(x_{v},p+\Pi )=\mu _{v}^{0}(p+\Pi )+RT\ln(\gamma _{v}x_{v})}

where γ v {\displaystyle \gamma _{v}}

γ v x v {\displaystyle \gamma _{v}x_{v}}

a w {\displaystyle a_{w}}

μ v o ( p + Π ) = μ v 0 ( p ) + ∫ p p + Π V m ( p ′ ) d p ′ , {\displaystyle \mu _{v}^{o}(p+\Pi )=\mu _{v}^{0}(p)+\int _{p}^{p+\Pi }\!V_{m}(p')\,dp',}

where V m {\displaystyle V_{m}}

− R T ln ⁡ ( γ v x v ) = ∫ p p + Π V m ( p ′ ) d p ′ . {\displaystyle -RT\ln(\gamma _{v}x_{v})=\int _{p}^{p+\Pi }\!V_{m}(p')\,dp'.}

If the liquid is incompressible the molar volume is constant, V m ( p ′ ) ≡ V m {\displaystyle V_{m}(p')\equiv V_{m}}

Π V m {\displaystyle \Pi V_{m}}

Π = − ( R T / V m ) ln ⁡ ( γ v x v ) . {\displaystyle \Pi =-(RT/V_{m})\ln(\gamma _{v}x_{v}).}

The activity coefficient is a function of concentration and temperature, but in the case of dilute mixtures, it is often very close to 1.0, so

Π = − ( R T / V m ) ln ⁡ ( x v ) . {\displaystyle \Pi =-(RT/V_{m})\ln(x_{v}).}

The mole fraction of solute, x s {\displaystyle x_{s}}

1 − x v {\displaystyle 1-x_{v}}

ln ⁡ ( x v ) {\displaystyle \ln(x_{v})}

ln ⁡ ( 1 − x s ) {\displaystyle \ln(1-x_{s})}

− x s {\displaystyle -x_{s}}

Π = ( R T / V m ) x s . {\displaystyle \Pi =(RT/V_{m})x_{s}.}

The mole fraction x s {\displaystyle x_{s}} is n s / ( n s + n v ) {\displaystyle n_{s}/(n_{s}+n_{v})}

x s = n s / n v {\displaystyle x_{s}=n_{s}/n_{v}}

V m = V / n v {\displaystyle V_{m}=V/n_{v}}

Π = c R T . {\displaystyle \Pi =cRT.}

For aqueous solutions of salts, ionisation must be taken into account. For example, 1 mole of NaCl ionises to 2 moles of ions.

See also

• Gibbs–Donnan effect

References

1. ^ a b Voet, Donald; Judith Aadil; Charlotte W. Pratt (2001). Fundamentals of Biochemistry (Rev. ed.). New York: Wiley. p. 30. ISBN 978-0-471-41759-0.
2. ^ Atkins, Peter W.; de Paula, Julio (2010). "Section 5.5 (e)". Physical Chemistry (9th ed.). Oxford University Press. ISBN 978-0-19-954337-3.
3. ^ Lewis, Gilbert Newton (1908-05-01). "The Osmotic Pressure of Concentrated Solutions and the Laws of the Perfect Solution". Journal of the American Chemical Society. 30 (5): 668–683. doi:10.1021/ja01947a002. ISSN 0002-7863.
4. ^ Poroelastic osmoregulation of living cell volume, Iscience, 24(12), 103482 (2021) doi=10.1016/j.isci.2021.103482

External links

• What is Osmosis? Explanation and Understanding of a Physical Phenomenon

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