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

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