6.021/Notes/2006-09-20

Osmosis

• pressure given by van't Hoff law ([[../2006-09-18/|prev lecture]])
• [math]\displaystyle{ \pi(x,t)= RTC_\Sigma(x,t) }[/math]
• semipermeable membrane reflects solutes [math]\displaystyle{ \Delta\pi \rightarrow \Delta p }[/math]
• osmotic pressure exactly the same as hydraulic pressure except opposite in sign
• only have to consider [math]\displaystyle{ p-\pi }[/math]
• [math]\displaystyle{ \Phi_V = L_V((p^i-\pi^i)-(p^o-\pi^o)) }[/math]
• Note that the volume flux of [math]\displaystyle{ \Phi }[/math] is different from the normal flux [math]\displaystyle{ \phi }[/math] in its units. [math]\displaystyle{ \Phi }[/math] has units of m/s whereas [math]\displaystyle{ \phi }[/math] 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
• [math]\displaystyle{ p^i=p^o }[/math]
• [math]\displaystyle{ \frac{dV^i}{dt} = -A(t)\Phi_V = -A(t)RTL_V(C^o_\Sigma(t)-C^i_\Sigma(t)) }[/math]
• equilibrium: [math]\displaystyle{ \frac{dV^i}{dt} = 0 \rightarrow (C^o_\Sigma(\infty)=C^i_\Sigma(\infty)) }[/math]
  • solution is [math]\displaystyle{ v_c(\infty) = v_c' + \frac{N^i_\Sigma}{C^o_\Sigma} }[/math] (perfect osmometer)
  • non-linear relationship between [math]\displaystyle{ C^o_\Sigma }[/math] and volume of cell
• Experimental data for many types of cells agrees with this equation