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
• [math]\displaystyle{ \pi(x,t)= RTC_\Sigma(x,t) }[/math]
• Also works for salts if count ions of salt
• [math]\displaystyle{ C_\Sigma(x,t) }[/math] is called osmolarity in units of (osmol/volume)
• 1 osmol is the same as 1 mol
• [math]\displaystyle{ \pi }[/math]: units of pressure (Pa)
• ocean about 1000 osmol/m^3, [math]\displaystyle{ \pi\approx 25\cdot 10^5{\rm Pa}\approx 25{\rm atm} }[/math]

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
• [math]\displaystyle{ \Phi_V(x,t)= -\kappa\frac{\partial p}{\partial x} }[/math]
• solvent flux is proportional to hydraulic pressure gradient
• continuity: [math]\displaystyle{ -\frac{\partial}{\partial x}(\rho_m \Phi_V) = \frac{\partial \rho_m}{\partial t} }[/math]
• water is incompressible so [math]\displaystyle{ \rho_m }[/math] is constant
• Therefore, flux gradient is zero so flux is constant and [math]\displaystyle{ p(x,t) }[/math] is a linear function of space