Geoscience Reference

In-Depth Information

3.6.3.3

Short-Range Potentials

The consideration remains simple for the weak potentials such that ˇU.R/
1.In

this case the function F.r/at large r
a behaves as in the case of free condensation

with the particle radius replaced by a, where the modifier depends on the details

of the potential and

r
1

2
a
2

R
2
:

F.R/
D

(3.178)

Let us introduce
z
.a/, the free molecule enhancement factor

˛
fm
.a/

a
2
v
T

z
.a/
D

:

(3.179)

We obtain,

z
.a/
v
T
a
2

2DR
2

2
a
2

2R
3
q
1

D

(3.180)

2
a
2

R
2

On solving Eq.
3.180
gives

s
2D
2

z
.a/
v
T

2

R
D

C
2
a
2
:

(3.181)

and

2a
2
v
T
z
.a/

˛.a/
D

r
1
C

(3.182)

a
v
T
z
.a/

2D
2
2

1
C

It is possible to find .a/. To this end we simplify F.R/by ignoring U.R/ and

replacing the lower limit of integration by 0. After these simplifications we find

dx
e
x
r
x
.x
C
ˇ
j
U.r
/
j
/
r
2

Z

2

p

R
2

F.R/
D

0

dx
e
x
p
x
1

1
C

Z

r
2

R
2

2

p

1

2

ˇ
j
U.r
/
j

x

0