Osmotic pressure

Applications

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

Osmotic pressure is an important factor affecting biological 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 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 atm. Reverse osmosis desalinates fresh water from ocean salt water and is applied globally on a very large scale.

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, ${\displaystyle 0<x_v<1}$. Besides, this compartment can assume a different pressure, ${\displaystyle p^{\prime }}$. We can therefore write the chemical potential of the solvent as ${\displaystyle {\mu }_v(x_v,p^{\prime })}$. If we write ${\displaystyle p^{\prime }=p+\mathrm{\Pi }}$, the balance of the chemical potential is therefore:

${\displaystyle {\mu }_v^0(p)={\mu }_v(x_v,p+\mathrm{\Pi }).}$

Here, the difference in pressure of the two compartments ${\displaystyle \mathrm{\Pi }\equiv p^{\prime }-p}$ 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 ${\displaystyle \mathrm{\Pi }}$, the osmotic pressure, we consider equilibrium between a solution containing solute and pure water.

${\displaystyle {\mu }_v(x_v,p+\mathrm{\Pi })={\mu }_v^0(p).}$

We can write the left hand side as:

${\displaystyle {\mu }_v(x_v,p+\mathrm{\Pi })={\mu }_v^0(p+\mathrm{\Pi })+RT\ln ({\gamma }_v{x}_v)}$,

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:

${\displaystyle {\mu }_v^o(p+\mathrm{\Pi })={\mu }_v^0(p)+{\int }_p^{p+\mathrm{\Pi }}{V}_m(p^{\prime })d{p}^{\prime },}$

where ${\displaystyle V_m}$ 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:

${\displaystyle -RT\ln ({\gamma }_v{x}_v)={\int }_p^{p+\mathrm{\Pi }}{V}_m(p^{\prime })d{p}^{\prime }.}$

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

${\displaystyle \mathrm{\Pi }=-(RT/V_m)\ln ({\gamma }_v{x}_v).}$

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

${\displaystyle \mathrm{\Pi }=-(RT/V_m)\ln (x_v).}$

The mole fraction of solute, ${\displaystyle x_s}$, is ${\displaystyle 1-x_v}$, so ${\displaystyle \ln (x_v)}$ can be replaced with ${\displaystyle \ln (1-x_s)}$, which, when ${\displaystyle x_s}$ is small, can be approximated by ${\displaystyle -x_s}$.

${\displaystyle \mathrm{\Pi }=(RT/V_m)x_s.}$

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

${\displaystyle \mathrm{\Pi }=cRT.}$

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