Environmental Engineering Reference
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of (1+
x
)
2/3
≈ 1+2
x
/3 when
x
is small enough (e.g.
x
< 0.1), Eq. (5.6-b) can be rewritten as
γ
lv
(
T
) ≈ (
m
′
H
v
-
TS
)[1+2
u
(
T
-
T
m
)/3]/(λ
N
a
1/3
V
2/3
). Because (
m
′
H
v
-
TS
)[1+2
u
(
T
-
T
m
)/3] =
v
+(
T
-
T
m
){-
S
+(2
uv
/3)[1-
S
(
T
-
T
m
)/
v
]} with
v
=
m
′
H
v
-
T
m
S
, there is
γ
lv
(
T
) ≈ {
v
+(
T
-
T
m
){-
S
+(2
uv
/3)[1-
S
(
T
-
T
m
)/
v
]}}/(λ
N
a
1/3
V
2/3
).
(5.11)
In terms of
m
′,
H
v
,
T
m
and
S
values listed in table 13,
S
(
T
-
T
m
)
/v
< 0.1 for the most metals
at
T
-
T
m
< 800 K and thus negligible as a first order approximation. Thus, Eq. (5.11) can be
simplified as γ
lv
(
T
) ≈ [
v
+(-
S
+2
uv
/3)(
T
-
T
m
)]/(λ
N
a
1/3
V
2/3
). Since
v
/(λ
N
a
1/3
V
2/3
) = γ
lv0
(
T
m
) and (-
S
+2
uv
/3)/(λ
N
a
1/3
V
2/3
) = γ′
lv0
(
T
m
), the agreement between Eq. (5.3) and Eq. (5.6-b) shown in
figure 10a is not only understandable, but also inevitable.
1.0
Na
0.8
0.6
Eq. (5.15): Current model
Eq. (5.13): Tolman model
Simulated [Ref. 212]
0.4
0.2
0.0
0
10
20
30
D
/
δ
Figure 11. Comparison of the
D
/
δ
(
δ
=
h
) dependence of
γ
lv
(
D
)/
γ
lv0
described by various models and
computational results for Na [212].
Since γ
lv0
(
T
) is related to the bond strength of atoms and its derivative corresponds to the
bond strength change or electron orbitals change as temperature varies, thus, as long as the
substance remains liquid and
T
has a temperature distance from
T
c
, electron orbitals of liquids
change linearly with temperature. Deviation of Eq. (5.11) with respect to
T
leads to [181],
γ′
lv0
(
T
) ≈ γ′
lv0
(
T
m
)+
e
(
T
-
T
m
)
(5.12)
with
e
= -4
uS
/(3λ
N
a
1/3
V
m
2/3
) being the temperature coefficient of γ′
lv0
(
T
).
Eq. (5.12) indicates that γ′
lv0
(
T
) is positively proportional to
T
since
e
> 0 induced by
u
<
0. This tendency against
T
is in reverse to that of γ
lv0
(
T
) because the energetic difference
between liquid and gas drops as
T
increases. Although Eq. (5.12) ultimately gives positive
temperature dependence, it should be indicated that
e
has only a secondary effect and can be
neglected as a first order approximation. For instance,
e
≈ 3.0×10
-5
and 1.6×10
-5
mJ/m
2
-K
2
for
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