# 6.021/Notes/2006-09-20

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## Osmosis

• pressure given by van't Hoff law (prev lecture)
• π(x,t) = RTCΣ(x,t)
• semipermeable membrane reflects solutes $\Delta\pi \rightarrow \Delta p$
• osmotic pressure exactly the same as hydraulic pressure except opposite in sign
• only have to consider p − π
• ΦV = LV((pi − πi) − (po − πo))
• Note that the volume flux of Φ is different from the normal flux φ in its units. Φ has units of m/s whereas φ has units of mol/(m^2 s). As we are considering a volume of incompressible fluid (water), we can convert one to the other using a conversion factor (e.g. 55 mol/L for water).

## Osmosis in cells

• only water crosses membrane
• pi = po
• $\frac{dV^i}{dt} = -A(t)\Phi_V = -A(t)RTL_V(C^o_\Sigma(t)-C^i_\Sigma(t))$
• equilibrium: $\frac{dV^i}{dt} = 0 \rightarrow (C^o_\Sigma(\infty)=C^i_\Sigma(\infty))$
• solution is $v_c(\infty) = v_c' + \frac{N^i_\Sigma}{C^o_\Sigma}$ (perfect osmometer)
• non-linear relationship between $C^o_\Sigma$ and volume of cell
• Experimental data for many types of cells agrees with this equation