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uence on the temperature of maximum density and the
freezing point, T
f
. The former can be obtained from the equation of state (Eq.
2.4
) (in
principle, but
Salinity and pressure also in
fl
fits are therefore
employed), while the latter is an independent formula given in UNESCO (1981). The
the mathematics is cumbersome and lower order
t
for T
m
by Caldwell (1978) and the UNESCO freezing point formula are:
T
m
½
¼3
:
982
0
:
2229
S
0
:
02004
p
ð
1
þ
0
:
00376
S
Þð
1
þ
0
:
000402
p
Þ
ð
2
:
7a
Þ
½
¼
0
:
0575
S
þ
1
:
710523
10
3
S
3
=
2
2
:
154996
10
4
S
2
7
:
53
10
3
p
ð
T
f
2
:
7b
Þ
where the salinity is taken in parts per thousand (
‰
) and the pressure in bars. These
T
m
¼
4
:
93
C
at the zero gauge pressure. It is seen that both the freezing point and the temperature of
maximum density decrease with increasing salinity, the latter one even more rapidly. For
p = 0, the curves meet at T =
‰
T
f
¼
2
:
21
C
equations are valid for the salinity up to 40
, where
and
. This salinity is normally taken as the
boundary between brackish and saline waters; then in brackish water
−
1.28
°
C, S = 24.6
‰
T
m
T
f
, and in
saline water the opposite is true. Even though brackish water has a maximum density
above the freezing point, the maximum is weaker the larger is the salinity (Fig.
2.4
b). In
freshwater lakes, the freezing point is between
−
0.03 and 0
°
C, and the temperature of
maximum density is between 3.87 and 3.98
C.
Linear approximations of Eq. (
2.7a
,
b
) are:
°
T
m
¼ 3
:
98
0
:
223
S
0
:
0207
p
ð
2
:
8a
Þ
T
f
¼
0
:
0549
S
7
:
53
10
3
p
ð
2
:
8b
Þ
fit to the outcome from the equation of state,
exact at p = 0 and 100 bar, showing that the temperature of maximum density decreases
with increasing pressure by about 0.1
The pressure term in Eq. (
2.8a
) is a linear
°
C/5 bar. The freezing point is exact at p =0,S =0
‰
and S=35
.
Example 2.3.
Lake Baikal is a deep, freezing fresh water lake in Siberia. At 1-km depth,
the temperature of maximum density is 1.91
C, and at this temperature and depth the
density of water is 1004.9 kg m
−
3
. The freezing point is
°
C at this depth. The sub-
glacial Lake Vostok is beneath the Antarctic ice sheet. The pressure is of the order of
350 bar (Siegert et al. 2001), and, consequently, if the water is fresh, then Tf
f
=
0.75
°
-
2.64
°
C
-
and T
m
=
C.
In hypersaline waters, the freezing point depends more strongly on the chemical
composition of the dissolved substances. At high salinities, the freezing point depression
is no more linear, and different salts crystallize at their individual eutectic temperatures
3.95
°
-
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