# 6.021/Notes/2006-09-20

## Osmosis

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