6.021/Notes/2006-09-18

Water Transport

 * Cells specific for water transport
 * ~15 pounds of water secreted and reabsorbed daily
 * hydraulic pressure: blood
 * osmosis: cells

Osmosis

 * Transport of solvent due to differences in solute concentration
 * described by Dutrochet (early 1800s)
 * developed 1st osmometer
 * Wilhelm Pfeffer
 * osmosis can be stopped by hydraulic pressure
 * pressure proportional to concentration of solute
 * isotonic: concentration at which cells don't change in size
 * osmosis is colligative
 * depends on molar concentration not chemical properties of solute

Van't Hoff

 * Found relationship identical to ideal gas law
 * $$\pi(x,t)= RTC_\Sigma(x,t)$$
 * Also works for salts if count ions of salt
 * $$C_\Sigma(x,t)$$ is called osmolarity in units of (osmol/volume)
 * 1 osmol is the same as 1 mol
 * $$\pi$$: units of pressure (Pa)
 * ocean about 1000 osmol/m^3, $$\pi\approx 25\cdot 10^5{\rm Pa}\approx 25{\rm atm}$$

Model

 * No one really understands osmosis
 * requires semipermeable membrane
 * solute collides and bounces off membrane
 * membrane exerts force due to changing momentum of solute
 * solute transfers momentum to solvent
 * change in solvent momentum is equivalent to hydraulic pressure
 * change in hydraulic pressure is change in osmotic pressure
 * momentum of solvent increase away from membrane due to solute bouncing back off membrane

Darcy's Law

 * flow through porous medium
 * $$\Phi_V(x,t)= -\kappa\frac{\partial p}{\partial x}$$
 * solvent flux is proportional to hydraulic pressure gradient
 * continuity: $$-\frac{\partial}{\partial x}(\rho_m \Phi_V) = \frac{\partial \rho_m}{\partial t}$$
 * water is incompressible so $$\rho_m$$ is constant
 * Therefore, flux gradient is zero so flux is constant and $$p(x,t)$$ is a linear function of space