Transcript Document
Osmotic Pressure Jim Parker Osmotic Pressure = Pressure to stop inward flow of solvent into volume • Pressure proportional to concentration • P=ckT van’t Hoff relation where c = N / V Same form as Ideal Gas Law • Osmotic Pressure – Derive from Free-Energy F= - k T ln Z – Similar to Ideal gas derivation • Same constraints -- particles do not interact (internal energies - affect chemical potentials not discussed) • At high enough concentrations these relations break down – Partition function formed from all combinations of r and p times the Boltzmann factor. • In continuous media, sums -> integrals •These are either gaussian integrals, constants or just L • Z(L) = constant * V where V = L3 –Fixed Volume •F = - k T N ln V + constant •P = - d F/ dV = k T N / V (ideal gas law) – Need to find change in Free Energy / volume to get dimensions of pressure (force / area) – Fixed Pressure - find average L displacement of piston Kinetic Energy of Piston Potential Energy of gas – Average value of thermodynamic property is <L> = integral L x P and integral of P = 1 • In P, only 1 factor depends upon L -- exp( -fl/ kT) • All other terms will cancel lots of terms to yield eq 7.5 • Make substitutions f = p A and L = V / A Biological import • Importance of osmotic pressure to cells – Concentration of protein in cell computed from volume fraction – V(protein) / V(total) = N/V(total) * V(protein)/N = c * V(1 protein molecule) – Surface tension comparison to osmotic pressure • Osmotic pressure = 300 Pa (from 0.12 mM protein conc) R Surface Tension must balance osmotic pressure Therefore for Energy balance p dV - dA = 0 – Lysis (cell burst) when unable to balance osmotic pressure • For protein conc 0.12 mM, near rupture threshold • Salts at 1 M conc, 10,000 x stronger pressure burst without compensating mechanism • Depletion force – Push larger molecules together by smaller – Volume 1R around large molecule excludes small molecule – Small molecule could use that volume for more states (higher entropy) – Thus when large molecules get close, they tend to get closer Depletion Force Experiment • 0.24 micron and 0.04 micron spheres • Circular boundary • b & c histogram plots -- white=many observations Osmotic Flow – Transfer of momentum from solute to fluid – Low Reynolds numbers of solute particles • Negligible inertial forces • Reynolds = v R – Viscous drag balanced by force on solute – Action/reaction - viscous drag net force on solvent – Barrier acts in only 1 direction Osmotic Flow B) Pressure for solute C) Osmotic pressure at equilibrium (no pressure on left) D) Solid Line - Pressure on left = right - pressure gradient in membrane --> flow to right, - put in osmotic pressure drop, - connect pressures with steady gradient (quasi-steady state) D) Reverse osmosis, push solvent out of “dirty solution”