- The big picture[[Student version, January 17, 2003]] 247
The big picture
Returning to the Focus Question, we have seen how the concentration of asolutecan cause a flux
ofwater across a membrane, with potentially fatal consequences. Chapter 11 will pick up this
thread, showing how eukaryotic cells have dealt with the osmotic threat and even turned it to
their advantage. Starting with osmotic pressure, we generalized our approach to include partially
entropic forces, like the electrostatic and hydrophobic interactions responsible in part for the crucial
specificity of intermolecular recognition.
Taking a broader view, entropic forces are ubiquitous in the cellular world. To take just one
example, each of your red blood cells has a meshwork of polymer strands attached to its plasma
membrane. The remarkable ability to red cells to spring back to their disklike shape after squeezing
through capillaries many times comes down to the elastic properties of this polymer mesh—and
Chapter 9 will show that the elastic resistance of polymers to deformation is another example of
an entropic force.
Key formulas
- Osmotic: Asemipermeable membrane is a thin, passive partition through which solvent, but
not solute, can pass. The pressure jump across a semipermeable membrane needed tostop
osmotic flow of solvent equalsckBTfor a dilute solution with number densitycon one side
and zero on the other (Equation 7.7).
The actual pressure jump ∆pmay differ from this value. In that case there is flow in the
direction of the net thermodynamic force, ∆p−(∆c)kBT.Ifthat force is small enough then
the volume flux of solvent will bejv=−Lp(∆p−(∆c)kBT), where thefiltration coefficient
Lpis a property of the membrane (Equation 7.15). Appendix B gives a few numerical values
forLp. - Depletion Interaction: When large particles are mixed with smaller ones of radiusa(for
example, globular proteins mixed with small polymers), the smaller ones can push the larger
ones together, to maximize their own entropy. If the two surfaces match precisely, the corre-
sponding reduction of free energy per contact area is ∆F/A=ckBT· 2 a(Equation 7.10). - Gauss: Suppose there is a plane of charge density−σqatx=0and no electric field at
x<0. Then the Gauss Law gives the electric field in the +ˆxdirection, just above the surface:
E|surface=−εσq(Equation 7.18). - Poisson: The potential obeys Poisson’s equation, d
(^2) V
dx^2 =−ρq/ε,whereρq(r)isthe charge
density atrandεis the permittivity of the medium, for example water or air.
• Bjerrum Length: (^) B=e^2 /(4πεkBT)(Equation 7.21). This length describes how closely two
like-charged ions can be brought together withkBT of energy available. In water at room
temperature, (^) B≈ 0. 71 nm.
- Debye: The screening length for a monovalent salt solution, for example NaCl at concentra-
tionc∞,isλD=(8π Bc∞)−^1 /^2 (Equation 7.34). At room temperature, it’s 0.31nm/
√
[NaCl]
(for a 1:1 salt like NaCl), or 0.18nm/√
[CaCl 2 ](2:1 salt), or 0.15nm/√
[MgSO 4 ](2:2 salt),
where [NaCl] is the concentration measured in moles per liter.