Derivation of Osmotic Pressure
In order to find, the osmotic pressure, we consider equilibrium between a solution containing solute and pure water.
We can write the left hand side as:
where is the activity coefficient of the solvent. The product is also known as the acitivity of the solvent, which for water is the water activity . The addition to the pressure is expressed through the expression for the energy of expansion:
where 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:
If the liquid is incompressible the molar volume is constant and the integral becomes . Hence we can get an equation with high accuracy
For pure substances the activity coefficient can be found as a function of concentration and temperature, but in the case of mixtures we are often forced to assume it is 1.0, so
For aqueous solutions, when determining the mole fraction of water, it is necessary to take into account the ionisation of salts. For example 1 mole of NaCl ionises to 2 moles of ions, and the mole fraction of water reduces accordingly.
Historically chemists have found it time consuming to calculate natural logs, so they used molal concentrations for dilute solutions as shown below. With spreadsheets the equations above are easy to use and offer much greater accuracy over a wider range of concentrations.
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