10.2 Transport Processes 459
PROBLEMS
Section 10.2: Transport Processes
10.3 One end of a bar of aluminum is maintained at 100.0◦C
and the other is maintained at 20.0◦C. The length of the
bar is 0.500 m, and its cross-sectional area is 1.00 cm^2
0 .000100 m^2. The thermal conductivity of aluminum is
equal to 237 J s−^1 m−^1 K−^1 237 watt m−^1 K−^1. How
much heat is transferred through the bar in 1.00
minute?
10.4 The thermal conductivity of copper at 300 K is equal to
398 J s−^1 m−^1 K−^1. A bar of copper with cross-sectional
area 1.00 cm^2 and length 10.00 cm is maintained with one
end at 305.0 K and the other end at 295.0 K. Find the rate
at which heat is conducted through the bar.
10.5 a.Show that the concentration expression in Problem
10.2 can satisfy Fick’s second law of diffusion,
Eq. (10.2-13).
b.Find the expression for the constantb 2 in terms ofa 2
andD 2 , the diffusion coefficient, assuming Fick’s law
to be valid. What is the physical interpretation of the
constanta 2?
c.IfD 2 1. 00 × 10 −^9 m^2 s−^1 and ifa 2 10 .0m−^1 ,
find the value ofb 2.
10.6At 25.4◦C, the diffusion coefficient of methane in
glycerol equals 9. 5 × 10 −^10 m^2 s−^1.
a.Find the root-mean-square displacement in one
direction of a methane molecule in 60 minutes and in
120 minutes.
b.Find the total root-mean-square distance traveled in 90
minutes by a methane molecule in glycerol at 25.4◦C.
10.7 a.A uniform tube with cross-sectional area
1. 00 × 10 −^5 m^2 and length 0.100 m connects two
large vessels in which effective stirring is maintained.
Each contains a solution of 1,1,1-trichloroethane
(methyl chloroform) in carbon tetrachloride, and a
uniform temperature of 25.00◦C is maintained in the
entire system. If one vessel contains a solution of
concentration 0.500 mol L−^1 and the other contains a
solution of concentration 0.100 mol L−^1 , and if a
steady state has been achieved, find the rate in mol s−^1
at which 1,1,1-trichloroethane is being transported
through the tube. The diffusion coefficient at 25.0◦Cis
equal to 1. 4 × 10 −^9 m^2 s−^1.
b.Find the root-mean-square distance in one direction
traveled by a 1,1,1-trichloroethane molecule in30.0 minutes. How long is required for an average
molecule to traverse the length of the tube?
10.8Two large containers are connected by a tube of length
10.0 cm and inside diameter 0.60 cm. Chamber A is filled
with an aqueous solution of glucose with molar
concentration 0.500 mol L−^1 and container B is filled
with a solution of glucose with molar concentration
0.100 mol L−^1. The two containers have effective stirrers
operating in them, but we can assume that the liquid in
the connecting tube is not stirred. Calculate the amount of
glucose that diffuses through the tube in 10.0 minutes.
The diffusion coefficient of glucose in water at this
temperature is equal to 6. 73 × 10 −^10 m^2 s−^1.
10.9Estimate the time required for molecules of a
neurotransmitterto diffuse across asynapse(the gap
between two nerve cells) by calculating the time required
for〈
x^2〉 1 / 2
to equal 50 nm, a typical synapse spacing, if
D 5 × 10 −^10 m^2 s−^1.
10.10The diffusion coefficient of 1,1,1-trichloroethane (TCE)
in carbon tetrachloride at 25◦C is equal to
1. 36 × 10 −^9 m^2 s−^1. Find the time required for the
root-mean-square displacement of TCE molecules in one
dimension to equal 1.00 cm at 25◦C.
10.11Liquid water at 20◦C is flowing through a tube of radius
5.00 mm with a speed at the center of the tube (atr0)
equal to 4.55 cm s−^1.a.Find the speed atr 2 .00 mm.
b.FindP 2 −P 1 if the length of the tube is 0.500 m.
c.FinddV /dt, the volume rate of flow through the tube.
d.Find the value of the Reynolds number. Is the flow
laminar?
10.12Find the root-mean-square distance in thezdirection
traveled by glucose molecules in 1.00 hour in a dilute
aqueous solution at 25◦C.
10.13A lead sphere of radius 0.500 cm is falling at a steady
speed in water at 25.0◦C. The density of lead is
11.35 g mL−^1. Find its speed. Comment on whether you
think your value is reasonable. If not, what could be the
reason that it is not?
10.14A glass marble is falling at a steady speed in a
swimming pool at 20.0◦C. If the density of the marble is
2. 2 × 103 kg m−^3 and the radius of the marble is
0.0075 m, find its speed, assuming laminar flow. If the