Editing Osmotic pressure (section)

== 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.