474 10 Transport ProcessesFit the natural logarithm of the viscosity against 1/Tand
find the activation energy for the viscosity. Compare it
with that of carbon tetrachloride from Example 10.17,
and compare it with the enthalpy change of vaporization
of benzene, 34.1 kJ mol−^1.10.36The diffusion coefficient of bovine serum albumin in
water at 20.0◦C equals 7× 10 −^11 m^2 s−^1.
a.Assuming the molecule to be spherical, estimate its
radius.
b. If the density of the protein molecule equals
1.25 g cm−^3 , estimate the molecular mass and the
molar mass.
c.Estimate the sedimentation coefficient of the
protein.
10.37The diffusion coefficient of horse heart myoglobin in
water at 20◦C is equal to 1. 13 × 10 −^10 m^2 s−^1 , and its
sedimentation coefficient is equal to 2.04 svedberg.
Assume that its density is equal to that of hemoglobin,
1.335 g cm−^3 , and find its molar mass.
10.38Consider the buret described in Example 10.6. If the
distance from the 50.00 mL mark to the constriction is
4.50 cm, find the time required for the meniscus to drain
down to the 50.00 mL mark.
10.39Many substances with fairly small molecules have
liquid-state diffusion coefficients near 1× 10 −^9 m^2 s−^1.
What effective molecular radius corresponds to this
value if water is the solvent and if the temperature is
25 ◦C?
10.40The value of the diffusion coefficient of hemoglobin in
water is 6. 9 × 10 −^11 m^2 s−^1 at 20◦C. The viscosity of
water at 20◦Cis1. 002 × 10 −^3 kg m−^1 s−^1. Find the time
required forxrmsto equal 1.00 cm.
10.41The Brownian motion of colloidal particles is observed
for a length of time such that the root-mean-square
displacement of the particles is 0.100 cm. How must the
observation time be changed so that the root-mean-square
displacement of the same set of particles is 0.200 cm at
the same temperature?
10.42Assume that a hemoglobin molecule is spherical with a
radius of 27 Å 2. 7 × 10 −^9 m.
a.Assuming that the Brownian-motion result and
Stokes’ law can be applied, estimate the diffusion
coefficient of hemoglobin in water at 298.15 K.
b. Estimate the time required for a hemoglobin molecule
to diffuse 1.00 mm in one direction. (Use the
root-mean-square displacement in one direction.)
10.43Assume that a ship is towing an underwater SONAR
antenna encased in a spherical shell 0.280 m in radius.
Assume that the sea-water temperature is 15◦C and that
its viscosity is the same as that of pure water at this
temperature, 0.001139 kg m−^1 s−^1.a.Assuming that the Reynolds number can be applied
with the diameter of the sphere used instead of the
diameter of a tube, estimate the maximum speed at
which the sphere can be towed and still have laminar
flow around the sphere.
b.Using the maximum speed you calculated in part a,
find the frictional force on the sphere.
10.44Calculate the rms distance diffused in one direction
in 30.0 minutes by hemoglobin molecules in water
at 20◦C. The value of the diffusion coefficient is
6. 9 × 10 −^11 m^2 s−^1.
10.45A certain bowling ball has a diameter equal to 7.0 inches
and a density equal to 2.5 g mL−^1. If the bowling ball is
falling in a lake at 25◦C and has achieved a steady speed,
estimate its speed, assuming laminar flow. Do you think
the flow is laminar?
10.46Solve Newton’s second law for a sphere initially moving
upward in a fluid with velocity component given byvz(0)
and initially atz0. Draw a sketch of its trajectory as a
function of time. The equation of motion is an
inhomogeneous equation.^9
10.47The molar mass of hemoglobin is 68 kg mol−^1 , and its
average density is 1.335 g cm−^3.a.Estimate the radius of the molecule, assuming it to be
spherical.
b.Estimate the diffusion coefficient of hemoglobin in
water at 25◦C.
c.Estimate the root-mean-square distance diffused in
one direction by hemoglobin molecules in 30.00
seconds in water at 25◦C.
d.Assume that hemoglobin molecules are sedimenting
in water at 25◦C in an ultracentrifuge rotor spinning at
100000 revolutions per minute (10472 radians per
second) and with the sample at a distancer 5 .00 cm
from the axis of rotation. Estimate the sedimentation
speed of the hemoglobin molecules.(^9) See Robert G. Mortimer,Mathematics for Physical Chemistry, 3rd ed.,
Academic Press, San Diego, CA, 2005, pp. 247–249, or any textbook of
differential equations.