6.021/Notes/2006-09-20

Osmosis

 * pressure given by van't Hoff law (prev lecture)
 * $$\pi(x,t)= RTC_\Sigma(x,t)$$
 * 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 $$p-\pi$$
 * $$\Phi_V = L_V((p^i-\pi^i)-(p^o-\pi^o))$$
 * Note that the volume flux of $$\Phi$$ is different from the normal flux $$\phi$$ in its units. $$\Phi$$ has units of m/s whereas $$\phi$$ 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
 * $$p^i=p^o$$
 * $$\frac{dV^i}{dt} = -A(t)\Phi_V = -A(t)RTL_V(C^o_\Sigma(t)-C^i_\Sigma(t))$$
 * 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