# Osmotic pressure

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 pure solvent (which belongs to the solution under discussion) 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 selectively permeable membrane. The phenomenon of osmosis arises from the propensity of a pure solvent to move through a semipermeable membrane and into its solution containing a solute to which the membrane is impermeable. This process is of vital importance in biology as the cell's membrane is semipermeable.

## Background

Osmosis in a U-shaped tube.
A Pfeffer cell used for early measurements of osmotic pressure

In order to visualize this effect, imagine a U-shaped tube with equal amounts of water on each side, separated by a water-permeable membrane made from dialysis tubing at its base that is impermeable to sugar molecules. Sugar has been added to the water on one side. The height of the liquid column on that side will then rise (and that on the other side will drop) proportional to the pressure of the two solutions due to movement of the pure water from the compartment without sugar into the compartment containing the sugar water. This process will stop once the pressures of the water and sugar water on both sides of the membrane become equal.[2]

Jacobus van 't Hoff first proposed a "law" relating osmotic pressure to solute concentration

${\displaystyle \Pi =iM_{\text{solute}}RT}$

where ${\displaystyle \Pi }$ is osmotic pressure, i is the dimensionless van 't Hoff index that addresses solute that dissociates an frequently non-idealities, Msolutes is the molar concentration, R is the ideal gas constant, and T is the temperature 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 ${\displaystyle p={n \over V}RT=c_{\text{gas}}RT}$ where n is the total number of moles of gas molecules in the volume V, and n/V is the molar concentration of gas molecules.

van 't Hoff was the first recipient of the Nobel prize for chemistry for his work on osmotic pressure and chemical equilibrium. [1]

A simple extension of this equation was proposed by Harmon Northrop Morse and Frazer.[3][4]

${\displaystyle \Pi =im_{\text{solutes}}RT}$,

where molal, m,concentration units are used in place of molarity. This formula extends the range of applicability osmotic pressure calculations to more concentrated solutions. However, it is still limited.

There are no cases in which the law of van 't Hoff or the modified form of this law proposed by Morse and Frazer have been shown to hold at concentrations higher than normal. (In a normal solution in water the fraction of the solute is about 0.02.)

Indeed, at very high concentrations van 't Hoff's law cannot hold, for the osmotic pressure of a solution approaches infinity as the percentage of solvent approaches zero, while the osmotic pressure calculated from the van 't Hoff equation never exceeds a few hundred atmospheres even when we approach the condition of pure solute. On the other hand, it will be shown presently that the law proposed by Morse and Frazer ordinarily gives, at higher concentrations, osmotic pressures far higher than those which actually exist. But often the law of Raoult (and the modified law of Henry) has been shown to hold at all concentrations from 0 per cent. to 100 per cent. of solute, and while in many other cases this law does not hold, the greatest deviations are always found in those cases in which we have reason to believe that the solvent and the solute form complex compounds either with themselves or with each other.[4]

The Raoult-Lewis derivation of osmotic pressure further extends concentration range and accuracy of osmotic pressure calculations is derived below and conventional written as,

${\displaystyle \Pi =(-ln(a_{\text{solvent}})/V_{\circ })RT}$,

in modern text where the molal or molarity concentration units are replaced with the negative natural log of the solvent activity, asolvent, divided by the molar volume, V° , of the solvent.

Historically, van 't Hoff's law and osmotic pressure measurement have been used for the determination molecular weights. The Pfeffer cell was developed for the measurement of osmotic pressure in biology.

## Applications

Osmotic pressure on red blood cells

Osmotic pressure is an important factor affecting cells. 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 about 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 ${\displaystyle \mu ^{0}(p)}$, where ${\displaystyle p}$ 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, ${\displaystyle 0. Besides, this compartment can assume a different pressure, ${\displaystyle p'}$. We can therefore write the chemical potential of the solvent as ${\displaystyle \mu _{v}(x_{v},p')}$. If we write ${\displaystyle p'=p+\Pi }$, the balance of the chemical potential is therefore:

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

Here, the difference in pressure of the two compartments ${\displaystyle \Pi \equiv p'-p}$ is defined as the osmotic pressure exerted by the solutes. Holding the pressure, the addition of solute decreases the chemical potential (an 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 ${\displaystyle \Pi }$, the osmotic pressure, we consider equilibrium between a solution containing solute and pure water.

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

We can write the left hand side as:

${\displaystyle \mu _{v}(x_{v},p+\Pi )=\mu _{v}^{0}(p+\Pi )+RT\ln(\gamma _{v}x_{v})}$,

where ${\displaystyle \gamma _{v}}$ is the activity coefficient of the solvent. The product ${\displaystyle \gamma _{v}x_{v}}$ is also known as the activity of the solvent, which for water is the water activity ${\displaystyle a_{w}}$. The addition to the pressure is expressed through the expression for the energy of expansion:

${\displaystyle \mu _{v}^{o}(p+\Pi )=\mu _{v}^{0}(p)+\int _{p}^{p+\Pi }\!V(p')\,\mathrm {d} p'}$,

where ${\displaystyle V}$ 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:

${\displaystyle -RT\ln(\gamma _{v}x_{v})=\int _{p}^{p+\Pi }\!V(p')\,\mathrm {d} p'}$.

If the liquid is incompressible the molar volume is constant, ${\displaystyle V(p')\equiv V}$, and the integral becomes ${\displaystyle \Pi V}$. Thus, we get

${\displaystyle \Pi =-(RT/V)\ln(\gamma _{v}x_{v})}$.

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

${\displaystyle \Pi =-(RT/V)\ln(x_{v})}$.

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

## Non-ideal solutions

A general extension of the van 't Hoff equation uses a virial expansion. For an ideal gas this has the form

${\displaystyle pV=RT+Bp+Cp^{2}...}$

were B, C ... are virial coefficients. A similar expression applies to osmotic pressure with ${\displaystyle \Pi }$ in place of pressure, p. Determination of virial coefficients from osmotic coefficients is the main basis for the determination of Pitzer parameters which are used to quantify non-ideal behaviour of solutions of ionic and non-ionic solutes.