452
10 Transport Processes
a model process such that an object repeatedly takes steps of fixed length in randomly
chosen directions. The displacement is proportional to the square root of the elapsed
time, not to the elapsed time, since as time passes there are more chances for the object
to reverse direction.
Exercise 10.5
Look up the integral and show that Eq. (10.2-19) is correct.
Although we obtained the root-mean-square distance traveled by a diffusing molecule by considering a special case in which all of the molecules started out at the same
value of z, the molecules diffuse in the same way for other initial conditions, and we
can use Eq. (10.2-19) for any kind of initial conditions and for the x and y directions
as well.
E X A M P L E 10.4
Find the root-mean-square distance in three dimensions diffused by glucose molecules in
30.0 minutes in water at 25 ◦ C.
Solution
The root-mean-square distance traveled in three dimensions is
r rms
x 2
+
y 2
+
z 2
1/2 3
z 2
1/2
6D 2 t
(10.2-20)
where we use the fact that all three directions are equivalent. From Table A.17 of Appendix
A, the value of the diffusion coefficient of glucose at 25 ◦ C is 0.673 × 10 −9 m 2 s −1 .
r rms 6D 2 t 1/2
6
0.673 × 10 −9 m 2 s −1
(30.0 min)(60 s min −1 )
2.70 × 10 −3 m 0.270 cm 2.70 mm
Newton’s Law of Viscous Flow
Consider the arrangement that is shown in Figure 10.1. There is a force per unit area
denoted by P zy that is exerted on the upper plate to keep it moving at a constant speed.
The first subscript on P zy indicates that the plate is perpendicular to the z direction,
and the second subscript indicates that the force is in the y direction. The y component of the velocity depends on z as indicated in Figure 10.1, corresponding to
shearing flow.
Newton’s law of viscous flow is named
for Sir Isaac Newton, 1642–1727, the
great British mathematician and
physicist who is famous for Newton’s
laws of motion and for being one of the
inventors of calculus.
Newton’s law of viscous flow is the linear law for viscous flow:
P zy η
∂u y
∂z
(Newton’s law of viscous flow)
(10.2-21)
The coefficient η is called the viscosity coefficient or the viscosity. It depends on the
temperature and the identity of the substance but does not depend on the rate of shear
if Newton’s law is obeyed. Table A.18 of Appendix A gives values for viscosity coefficients for a few liquids and gases. Equation (10.2-21) holds not only for the force
10 Transport Processes
a model process such that an object repeatedly takes steps of fixed length in randomly
chosen directions. The displacement is proportional to the square root of the elapsed
time, not to the elapsed time, since as time passes there are more chances for the object
to reverse direction.
Exercise 10.5
Look up the integral and show that Eq. (10.2-19) is correct.
Although we obtained the root-mean-square distance traveled by a diffusing molecule by considering a special case in which all of the molecules started out at the same
value of z, the molecules diffuse in the same way for other initial conditions, and we
can use Eq. (10.2-19) for any kind of initial conditions and for the x and y directions
as well.
E X A M P L E 10.4
Find the root-mean-square distance in three dimensions diffused by glucose molecules in
30.0 minutes in water at 25 ◦ C.
Solution
The root-mean-square distance traveled in three dimensions is
r rms
x 2
+
y 2
+
z 2
1/2 3
z 2
1/2
6D 2 t
(10.2-20)
where we use the fact that all three directions are equivalent. From Table A.17 of Appendix
A, the value of the diffusion coefficient of glucose at 25 ◦ C is 0.673 × 10 −9 m 2 s −1 .
r rms 6D 2 t 1/2
6
0.673 × 10 −9 m 2 s −1
(30.0 min)(60 s min −1 )
2.70 × 10 −3 m 0.270 cm 2.70 mm
Newton’s Law of Viscous Flow
Consider the arrangement that is shown in Figure 10.1. There is a force per unit area
denoted by P zy that is exerted on the upper plate to keep it moving at a constant speed.
The first subscript on P zy indicates that the plate is perpendicular to the z direction,
and the second subscript indicates that the force is in the y direction. The y component of the velocity depends on z as indicated in Figure 10.1, corresponding to
shearing flow.
Newton’s law of viscous flow is named
for Sir Isaac Newton, 1642–1727, the
great British mathematician and
physicist who is famous for Newton’s
laws of motion and for being one of the
inventors of calculus.
Newton’s law of viscous flow is the linear law for viscous flow:
P zy η
∂u y
∂z
(Newton’s law of viscous flow)
(10.2-21)
The coefficient η is called the viscosity coefficient or the viscosity. It depends on the
temperature and the identity of the substance but does not depend on the rate of shear
if Newton’s law is obeyed. Table A.18 of Appendix A gives values for viscosity coefficients for a few liquids and gases. Equation (10.2-21) holds not only for the force