# 6.021/Notes/2006-09-18

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## 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