VARIATION
OF SOLUBILITY
WITH
A THERMODYNAMIC
DEPTH IN THE
ANALYSIS
OCEAN:
Irving M. Klotx
Dcpartmcnt
of Chemistry,
Northwcstcrn
University’
ABSTRACT
A thermodynamic
equation can be obtained which gives the dcpendencc of solubility
on depth in terms of the partial specific volume of the solute. From published values of
these volumes it is shown that the ( equilibrium ) solubility
of nitrogen should decrease by
54% per 1,000 m depth whereas that of oxygen should change by less than 0.1%.
dF=O.
INTRODUCTION
In the general situation, in the absence of
equilibrium, r;’ would depend on x, the position of the solute in the gravitational field,
on P, the pressure on the solute at a given
depth, and on m, or preferably In m, where
m is the amount of solute dissolved in a
specified quantity of solvent at a particular
depth. Since
Questions of gas solubility in the ocean
have relevance to a number of biological
problems ( e.g., the accumulation of gases
in the swimbladder in deep-sea fishes ) . In
many situations the actual amount of dissolved gas is the important
parameter.
However, for some problems the critical
question is: what is the saturation or equilibrium solubility,
at great depths? This
may not be a quantity easily measured by
direct experiment. Nevertheless, since it is
an equilibrium
quantity, it is related thermodynamically
to other parameters which
are more accessible to experiment or already known from other investigations.
It
is the objective of this paper, therefore, to
point out these relationships and to make
some typical calculations based upon them,
THERMODYNAMIC
Fa=Fp.
Consequently,
a=f(x,P,lnm)
+
(
&
>
dlnm.
p, m
(4)
For an equilibrium
situation we may
place the restriction of equation (2) upon
(4) and, after rearrangement, obtain
dx -($) ’
dP
m, In m
* (5)
Values for the partial derivatives on the
right-hand side of equation (5) are generally known (Lewis and Randall 1923) :
(1) ’
we may also write:
l This paper was written during
the Marine Biological
Laboratory,
Massachusetts.
(3)
we may write
RELATIONSHIPS
Let us consider a solute (e.g., OZ or N,)
dissolved in the ocean and truly at cquilibrium at all depths, x (x to be measured positively in the downward direction).
In principle, the solubility at a deep level p may be
very different from that at the surface level
a. Nevertheless, if equilibrium
exists, then
the partial molal free energy, T;i, must be
the same for the solute at all levels (Lewis
and Randall 1923; Klotz 1950) :
(2)
a rcsidcnce at
Woods Hole,
149
=-Mg.
(6)
150
IRVING
M.
In these equations M represents the molecular weight of the solute, V, the partial
molal volume and g the acceleration due to
gravity.
To obtain ( d F//a In m)13, 5 we must know
something about the solubility behavior of
the solute at any fixed depth and pressure,
as a function of concentration.
For dissolved gases, Henry’s law is an adequate
description; in this case the relation between F’ and molarity is given by the equation:
F=RTlnm+B
(8)
does
where B is a constant if temperature
not vary. Consequently,
=RT.
If the relations of equations (6), (7),
and (9) are inserted into (5) we obtain
RTdlnm=Mgdx-VdP.
The pressure, being hydrostatic,
to the depth by the equation
dP = P g dx
where p is the density
that we may write
(10)
is related
(11)
of the solution,
RTdlnm=Mgdx-Vpgdx.
so
(12)
Furthermore, the partial molal volume is related to the partial specific volume, U, by
the equation
v=Mii.
Consequently,
verted into
equation
(13)
(12)
can be con-
KLOTZ
RESULTS
For some illustrative
computations, let
us take T=300"K (27"C), R=8.3~ lo7
ergs/deg. and g = 980 cm/seG. Then equation ( 15) can be rearranged into
logm -
mdepth
log,, -
dx.
mdepth
=Mg (l-up)
m
(14)
d.
(15)
RlWfilCC
2 We shall assume that U and p are independent
of depth. Where this is not a valid assumption we
must know the compressibilitics
of the dissolved
solute and of the solution and a more general solution of equation ( 14 ) can be obtained.
md
= 2.7 x 10-O d
msul.fnce
If we integrate* this equation from the surface of the water to any depth, cl, we obtain
RTln-
&! ( 16)
where d is expressed in centimeters. Thus
the direction that ma takes in comparison
with msurfnccis determined by the factor
( 1 - c p ) . If 2j p > 1, then solubility will decrease with increasing depth in the ocean.
Contrariwise, if up < 1, solubility will be
greater at greater depths.
For sea water, we may use a value
(Shanklin 1954) of 1.03 for p. The partial
specific
volume depends, of course, on the
substance. Values for some gases have been
computed from data in - the literature
(Kritchevsky and Iliinskaya 1945) and are
assembled in Table 1,
Qualitatively, we see immediately that in
regard
to air the solubility of nitrogen
- should
decrease with depth in the ocean (Up =
1.47 > 1) whereas that of oxygen should
remain practically constant (Up = 1.00).
Changes in salinity or temperature will affect (1 - Up) markedly for oxygen, since the
density p would be changed thereby. Thus
if p drops from 1.035 to 1.025, (1 - up) will
change from -0.005 to +0.005. In neither
case, however, will the solubility of oxygen
change very much with depth. Thus for
water of density 1.025, equation (16) reduces to
RTdlnm=Mgdx-Mupgdx
=Mg(l-up)
= 1.71 x 1O-8M(l-vp)
m surface
TABLE
1.
Partial
specific
(17)
uol24mes of gases
cZissoluedin water at 25°C
--
-.
GM
112
co
N2
02
CO2
--
--Molecular
weight
~~.
2
28
28
32
44
u
( cln:~/g)
~_____.
~__
13.0
1.29
1.43
0.97
0.75
VARIATION
OF SOLUBILITY
or if D is the depth in meters
ml
= 2.7 x 1O-7D .
log10 msllrfacc
(18)
Thus even at a depth of 1,000 m,
md
= 2.7 x lo-”
log*, msurface
and the solubility of oxygen increases by
< 0.1%. Could the density of the water be
increased to 1.035, the solubility of oxygen
would decrease by somewhat less than 0.1%
at 1,000 m below the surface.
On the other hand, the situation with
nitrogen is markedly different. Here variations in salinity and temperature will havs
little effect on the factor ( 1 - Up) since the w
of 1.43 is relatively so large. Thus for nitrogen, equation (16) reduces to
md
= -2.3 x lo-” D .
log10 m surfnce
log -EL
1 -0.023
m surface
which corresponds to a decrease in solubility of nitrogen of 5-6%. At lesser depths,
the solubility
would be decreased by a
151
DEPTII
smaller amount, which can be calculated
from equation ( 19 ) . The important distinction with nitrogen, as contrasted to oxygen,
is that the solubility should decrease progressively with depth.
For substances other than gases, the
analysis would bc substantially the same.
For electrolytes in water, equation
(8)
would not be valid but a modified form
containing mn, whcrc n is an integer, would
be more nearly correct. The right-hand side
of equation (10) would still hold, however.
Consequently, the right-hand side of equation ( 15) would still be of the same general
form. Clearly then, the crucial criterion
with regard to change of solubility
with
depth remains, what is the sign of the factor
(1 - u p)? For most salts u is appreciably less
than unity. Consequently, in general, the
solubility of salts will increase with increasing depth.
(19)
At a depth of 1,000 m
WIT11
HEFERENCES
I. M. 1950. Chemical thermodynamics.
Prentice-Hall, Inc., New York, pp. 224-228.
KHITCEIEVSKY,
I., AND A. ILIINSKAYA.
1945. Acta
Physicochim. URSS, 20: 327-348.
LEWIS, G. N., AND M. RANDALL.
1923. Thermodynamics and the free energy of chemical
substances. McGraw -Hill Book Co., New
York, pp. 242-244.
SHANKLIN,
D. R. 1954. Nature, 173: 82.
KLOTZ,