256 Chapter 7. Entropic forces at work[[Student version, January 17, 2003]]
measured to be about 60gL−^1. Use these data to estimate the average molar massMing/mole
for these plasma proteins, assuming the validity of the dilute limit.
b. The filtration coefficient of capillary membranes is sometimes quoted asLp=7· 10 −^6 cm s−^1 atm−^1.
If we put pure water onbothsides of a membrane with a pressure drop of ∆p,the resulting volume
flux of water isLp∆p.Assume that a normal person has rough osmotic balance across his capillaries,
but that in a particular individual the blood plasma proteins have been depleted by 10%, due to
anutritional deficiency. What would be the total accumulation of fluid in interstitial space (liters
perday), given that the total area of open capillaries is about 250m^2 ?Whydo you think starving
children have swollen bellies?
7.5Depletion estimates
Section 7.2.1 said that a typical globular protein is a sphere of radius 10nm. Cells have a high
concentration of such proteins; for illustration suppose that they occupy about 30% of the interior
volume.
a. Imagine two large, flat objects inside the cell (representing two big macromolecular complexes
with complementary surfaces). When they approach each other closer than a certain separation,
they’ll feel an effective depletion force driving them still closer, caused by the surrounding suspension
of smaller proteins. Draw a picture, assuming the surfaces are parallel as they approach each other.
Estimate the separation at which the force begins.
b. If the contact area is 10μm^2 ,estimate the total free energy reduction when the surfaces stick.
Youmayneglect any other possible interactions between the surfaces, and as always assume that
wecan still use the van ’t Hoff (dilute-suspension) formula for osmotic pressure. Is it significant
compared tokBTr?
7.6 T 2 Weak-charge limit
Section 7.4.3 considered an ionizable surface immersed in pure water. Thus the surface dissociated
into a negative plane and a cloud of positive counterions. Real cells, however, are bathed in a
solution of salt, among other things; there is an external reservoir ofbothcounterions and negative
coions. Section 7.4.3′on page 250 gave a solution for this case, but the math was complicated; here
is a simpler, approximate treatment.
Instead of solving Equation 7.33 exactly, consider the case where the surface’s charge density is
small. Then the potentialV(0) at the surface will not be very different from the value at infinity,
which we took to be zero. (More precisely, the dimensionless combinationVis everywhere much
smaller than 1.) Approximate the right-hand side of Equation 7.33 by the first two terms of its
series expansion in powers ofV.The resulting approximate equation is easy to solve. Solve it, and
give an interpretation to the quantityλDdefined in Equation 7.34.
7.7Effect of H-bonds on water
According to Section 7.5.1, the average number of H-bonds between a molecule of liquid water and
its neighbors is about 3.5. Assume that these bonds are the major interaction holding liquid water
together, and that each H-bond lowers the energy by about 9kBTr. Find a numerical estimate
for the heat of vaporization of water based on these ideas (see Problem 1.6), then compare your
prediction to the actual value.
7.8 T 2 Diffusion increases entropy
Suppose we prepare a solution at timet=0with a nonuniform concentrationc(r)ofsolute. (For
example, we could add a drop of ink to a glass of water without mixing it.) This initial state is