Editing Osmotic pressure

{{Short description|Measure of the tendency of a solution to take in pure solvent by osmosis}}
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[[File:Osmosis diagram.svg|thumb|250px|alt=Progression: (1) a U-tube is filled with water and has a membrane ....in the middle (2) sugar is added to the left part (3) water crosses the membrane and fills the left side more than the right.|Osmosis in a U-shaped tube]]

'''Osmotic pressure''' is the minimum [[pressure]] which needs to be applied to a [[Solution (chemistry)|solution]] to prevent the inward flow of its pure [[solvent]] across a [[semipermeable membrane]].<ref name=voet>{{Cite book| edition = Rev.| publisher = Wiley| isbn = 978-0-471-41759-0| vauthors = Voet D, Aadil J, Pratt CW | title = Fundamentals of Biochemistry| location = New York| year = 2001| page= 30}}</ref>
It is also defined as the measure of the tendency of a solution to take in its pure solvent by [[osmosis]].{{citation needed lead|date=March 2025}} '''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 ''osmotic equilibrium'' is attained.<ref name=voet /><ref>{{cite book | vauthors = Atkins PW, de Paula J  |title=Physical Chemistry|edition=9th |year=2010 |publisher=[[Oxford University Press]] |isbn=978-0-19-954337-3|chapter= Section 5.5 (e) }}</ref>
<!-- (Hidden until reworked/removed) In order to visualize this effect, imagine a U-shaped tube with equal amounts of water on both sides of a membrane which is permeable to water and which is impermeable to sugar molecules. [[Dialysis tubing]] is an example of a material which can act as a semi-permeable membrane. When sugar is added to the water in one arm, the height of the liquid column on that side will then rise and that on the other side will drop. This process will continue until the osmotic pressures of the water and sugar solution are equal. Osmotic pressure can then be obtained from a measurement of the difference in height between the liquids in the two arms. -->

==Theory and measurement==
[[File:Pfeffer Osmotische Untersuchungen-1-3.jpg|thumb|right|200 px|A [[Pfeffer cell]] used for early measurements of osmotic pressure]]

[[Jacobus Henricus van 't Hoff|Jacobus van&nbsp;'t Hoff]] found a quantitative relationship between osmotic pressure and solute concentration, expressed in the following equation:
:<math>\Pi = icRT</math>
where <math>\Pi</math> is osmotic pressure, ''i'' is the dimensionless [[van 't Hoff factor|van&nbsp;'t Hoff index]], ''c'' is the [[molar concentration]] of solute, ''R'' is the [[ideal gas constant]], and ''T'' is the [[absolute temperature]] (usually in [[kelvins]]). This formula applies when the solute concentration is sufficiently low that the solution can be treated as an [[ideal solution]]. The proportionality to concentration means that osmotic pressure is a [[colligative property]]. Note the similarity of this formula to the [[ideal gas law]] in the form <math display="inline">P = \frac{n}{V} RT = c_\text{gas} RT</math> where {{mvar|n}} is the total number of moles of gas molecules in the volume ''V'', and ''n''/''V'' is the molar concentration of gas molecules. [[Harmon Northrop Morse]] and Frazer showed that the equation applied to more concentrated solutions if the unit of concentration was [[molal]] rather than [[molar (unit)|molar]];<ref name=":0">{{Cite journal|  vauthors = Lewis GN | date=1908-05-01| title=The Osmotic Pressure of Concentrated Solutions and the Laws of the Perfect Solution.| journal=Journal of the American Chemical Society| volume=30| issue=5| pages=668–683| doi=10.1021/ja01947a002| issn=0002-7863| url=https://zenodo.org/record/1428858| access-date=2019-07-04| archive-date=2022-06-18| archive-url=https://web.archive.org/web/20220618111258/https://zenodo.org/record/1428858| url-status=live}}</ref> so when the molality is used this equation has been called the '''Morse equation'''.

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

:<math> \Pi = \Pi_0 + A c^2 </math>

where <math>\Pi_0 </math> is the ideal pressure and ''A'' is an empirical parameter. The value of the parameter ''A'' (and of parameters from higher-order approximations) can be used to calculate [[Pitzer parameter]]s. Empirical parameters are used to quantify the behavior of solutions of ionic and non-ionic solutes which are not [[ideal solution]]s in the thermodynamic sense.

The [[Wilhelm Pfeffer|Pfeffer cell]] was developed for the measurement of osmotic pressure.

== Applications ==
[[Image:Osmotic pressure on blood cells diagram.svg|thumb|upright=1.15|Osmotic pressure on red blood cells]]
Osmotic pressure measurement may be used for the determination of [[molecular weight]]s.

Osmotic pressure is an important factor affecting biological cells.<ref>{{cite journal | vauthors = Esteki MH, Malandrino A, Alemrajabi AA, Sheridan GK, Charras G, Moeendarbary E | title = Poroelastic osmoregulation of living cell volume | language = English | journal = iScience | volume = 24 | issue = 12 | pages = 103482 | date = December 2021 | pmid = 34927026 | pmc = 8649806 | doi = 10.1016/j.isci.2021.103482 | bibcode = 2021iSci...24j3482E }}</ref> [[Osmoregulation]] is the [[homeostasis]] mechanism of an organism to reach balance in osmotic pressure.

*[[Tonicity#Hypertonic solution|Hypertonicity]] is the presence of a solution that causes cells to shrink.
*[[Tonicity#Hypotonicity|Hypotonicity]] is the presence of a solution that causes cells to swell.
*[[Tonicity#Isotonicity|Isotonicity]] is the presence of a solution that produces no change in cell volume.

When a [[biological tissue|biological]] [[cell (biology)|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 cell]]s, the [[cell wall]] restricts the expansion, resulting in pressure on the cell wall from within called [[turgor pressure]]. Turgor pressure allows [[herbaceous plant]]s 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]] due to the absence of a cell wall.

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 [[Atmosphere (unit)|atm]]. Reverse osmosis [[desalination|desalinates]] fresh water from [[seawater|ocean salt water]] and is applied globally on a very large scale.

== Derivation of the van&nbsp;'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 <math>\mu^0(p)</math>, where <math>p</math> is the pressure. On the other side, in the compartment containing the solute, the chemical potential of the solvent depends on the [[mole fraction]] of the solvent, <math>0 < x_v < 1</math>. Besides, this compartment can assume a different pressure, <math>p'</math>. We can therefore write the chemical potential of the solvent as <math>\mu_v(x_v, p')</math>. If we write <math>p' = p + \Pi</math>, the balance of the chemical potential is therefore:

:<math>\mu_v^0(p)=\mu_v(x_v,p+\Pi).</math>

Here, the difference in pressure of the two compartments <math>\Pi \equiv p' - p</math> is defined as the osmotic pressure exerted by the solutes. Holding the pressure, the addition of solute decreases the chemical potential (an [[entropy|entropic effect]]). Thus, the pressure of the solution has to be increased in an effort to compensate the loss of the chemical potential.

In order to find <math>\Pi</math>, the osmotic pressure, we consider equilibrium between a solution containing solute and pure water.
:<math>\mu_v(x_v,p+\Pi) = \mu_v^0(p).</math>

We can write the left hand side as:
:<math>\mu_v(x_v,p+\Pi)=\mu_v^0(p+\Pi)+RT\ln(\gamma_v x_v)</math>,
where <math>\gamma_v</math> is the [[activity coefficient]] of the solvent. The product <math>\gamma_v x_v</math> is also known as the activity of the solvent, which for water is the water activity <math>a_w</math>. The addition to the pressure is expressed through the expression for the energy of expansion:
:<math>\mu_v^o(p+\Pi)=\mu_v^0(p)+\int_p^{p+\Pi}\! V_m(p') \, dp',</math>
where <math>V_m</math> is the molar volume (m³/mol). Inserting the expression presented above into the chemical potential equation for the entire system and rearranging will arrive at:
:<math>-RT\ln(\gamma_v x_v)=\int_p^{p+\Pi}\! V_m(p') \, dp'.</math>

If the liquid is incompressible the molar volume is constant, <math>V_m(p') \equiv V_m</math>, and the integral becomes <math>\Pi V_m</math>. Thus, we get
:<math>\Pi = -(RT/V_m) \ln(\gamma_v x_v) .</math>

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
:<math>\Pi = -(RT/V_m) \ln(x_v) .</math>

The mole fraction of solute, <math>x_s</math>, is <math>1-x_v</math>, so <math>\ln(x_v)</math> can be replaced with <math>\ln(1 - x_s)</math>, which, when <math>x_s</math> is small, can be approximated by <math>-x_s</math>.

:<math>\Pi=(RT/V_m)x_s.</math>

The mole fraction <math>x_s</math> is <math>n_s/(n_s+n_v)</math>. When <math>x_s</math> is small, it may be approximated by <math>x_s = n_s/n_v</math>. Also, the molar volume <math>V_m</math> may be written as volume per mole, <math>V_m = V/n_v</math>. Combining these gives the following.

:<math>\Pi = cRT.</math>

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 ==
{{Reflist}}

== External links ==
* [https://arxiv.org/ftp/physics/papers/0305/0305011.pdf What is Osmosis? Explanation and Understanding of a Physical Phenomenon]

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[[Category:Amount of substance]]
[[Category:Cell biology]]
[[Category:Membrane biology]]
[[Category:Solutions]]
[[Category:Jacobus Henricus van&nbsp;'t Hoff]]
[[Category:Pressure]]