Chemistry Reference
In-Depth Information
in the curve, and so the energy per molecule is
Energy
=
mol
¼
sr
3
2
This energy assists the motion to the right in Figure 3.9 and would oppose it
in returning to the left. Using V
m
as the minimum volume occupied by
Avogadro's number of molecules enables us to write the jump frequency as
exp
E
sV
m
N
¼
k
B
T
h
ð
Þ=
2
ð
3
:
39
Þ
RT
The shear rate is the distance moved in each second over the distance separating
the layers:
g
¼
rN
N
r
0
E
where r
0
is the perpendicular distance between moving layers.
Recalling that sinh x
¼
e
x
e
x
2
, we can write eqn (3.39) as:
2 sinh
exp
E
RT
g
¼
k
B
T
h
ð
sV
m
Þ
ð
3
:
40
Þ
2RT
Also, recalling that sinh x
-
xasx
-
0 gives us the zero-shear viscosity as:
E
RT
Z
ð
0
Þ¼
hN
A
V
m
exp
ð
3
:
41
Þ
3.4 SUPERFLUIDS
Helium provides an interesting variant of normal liquid behaviour. When
4
He
is cooled at atmospheric pressure, the gas liquefies to He I at 4.2K. On further
cooling a second-order transition to He II, another liquid form, occurs at 2.2K.
This form persists down to as close to 0K as can be attained. To form solid
helium, a pressure of 25 atmospheres is required at 2.2K. The mass of the
4
He
atom is low and the intermolecular interactions are very weak. This means that
the motion of the helium atoms is unusually large, too large in fact for a solid-
like structure to persist except when high pressure forces closer distances and
hence restricts the motion.
The density of He I at the boiling point at 1 atm is 125 kgm
3
and the viscosity
3
10
6
Pa s. As we would anticipate, cooling increases the viscosity until
He II is formed. Cooling this form reduces the viscosity so that close to 0K a
liquid with zero viscosity is produced. The vibrational motion of the helium
atoms is about the same as or a little larger than the mean interatomic spacing
and the flow properties cannot be considered in classical terms. Only a quantum-
mechanical description can prove satisfactory. We can consider this condition to
give the limit of D
e
-
is
B
0 as we have difficulty in defining a relaxation when we
have the positional uncertainty for the structural components.
