Heat of Diluting the Soil Solution Versus Soil Heat of Wetting
Lyle Prunty†
ABSTRACT
Adding water to dry soil generally results in release
of heat. Thermodynamically, the dilution of an aqueous
solution is an equivalent process in which, depending on the
solute, heat may be released or absorbed. For solutions the
applicable terms include 'heat (enthalpy) of solution' and
'partial molar enthalpy.' For soil the terms 'heat of wetting
(immersion)' and 'differential heat of wetting' apply.
Reference data available for solutions uses additional
nomenclature. Heat evolved or absorbed during either of
these processes is dependent on the beginning and ending
states but is not uniquely determined by them. Water
potential change and reversible heat flux are, however,
uniquely determined and their sum is the differential heat of
wetting. The reversible heat flux is tied to the rate of change
of water potential with respect to temperature. For strictly
capillary effects the change is positive, according to current
literature, while for solutions it is negative and generally
about 1/3 to 1/6 the magnitude of the capillary effect. The
heats of wetting and solution are determined by water
potential change and the temperature derivative of that
change.
1. Introduction
Mixing dry soil and water at constant temperature
generally results in release of heat. The amount of heat
released is known as the heat of wetting. When pure
water is added to a solution at constant temperature, heat
may be either released or captured. Thermodynamically,
these processes in soils and solutions are much the same.
Examining the addition of water to solutions can lead to
better understanding of soil heat of wetting.
During infiltration of pure, free water into oven
dry soil, temperatures may rise on the order of 10°C
[Prunty, 2002]. Yet, heat of wetting is not included in
most coupled heat and water transport models. Water
added to solutions may result in heating, as when mixing
sulfuric acid and water. Other solutions undergo cooling
when water is added.
The objective of this paper is to present a
thermodynamic analysis of the dilution process for a few
specific solutions with a range of properties.
Understanding the thermodynamics of diluting these
solutions enables understanding of soil heat of wetting.
KEY WORDS: heat of solution, heat of immersion, heat
of dilution, thermodynamic equilibrium
Enthalpy of Solution
Heats of formation of substances are found by
experiments which are symbolized in the format
A+ B → C
∆H = 1.23
J
(1)
where A and B are reactants producing product C at the
same temperature and pressure as the reactants. The
reaction heat of formation ∆H is positive if heat is added
during the reaction (endothermic reaction) and negative if
heat is released during the reaction (exothermic reaction).
If A is one mole of a water-soluble substance, B is n
moles of water, and C is the resulting solution then ∆H is
the integral heat of solution [Moore, 1962].
When the integral heat of solution, ∆H, is plotted
versus moles of water, n, as abscissa then the slope of the
curve is the differential heat of solution of water. Partial
molar enthalpy of water is another name for the same
quantity in this context.
Heat of Wetting of Soil
Edlefsen and Anderson [1943] defined q(w) as the
heat of wetting when completely dry soil is mixed with
free water, resulting in soil with gravimetric water
content, w. Differential heat of wetting was defined as
⎛ ∂q ⎞
where the subscript indicates a constant
⎜
⎟
⎝ ∂w ⎠T
temperature process. Heat of wetting is further described
as, "the heat developed when 1 gram of water is added
and distributed uniformly throughout the large mass of
soil." It is apparent that differential heat of solution and
differential heat of wetting are equivalent concepts with
different substances involved. Anderson [1986] makes
the same point, stating, "This concept is identical with
that of partial molar and partial specific quantities
developed in chemical thermodynamics. Therefore, the
differential heat of immersion may also be called the
partial molar or partial specific heat of immersion;…"
† Professor, Dept. of Soil Science, North Dakota State
University, P.O. Box 5638, Fargo, ND 58105
Email: [email protected]
© August 2004 Lyle Prunty
Diluting Soil Solution
Typical Solutions
Properties of many solutions have been tabulated.
Sulfuric acid and sodium chloride solutions are of interest
because they are such common substances. Additionally,
sulfuric acid and sodium chloride solutions have
contrasting thermodynamic properties in that when
diluted the resulting enthalpy changes are predominantly
of opposite sign.
Standard data is available for heat of formation
(∆Hf°) of sulfuric acid solutions at 25°C in Rossini et al.
[1952] (Table 14-7) and also in Wagman et al. [1982].
Integral heat of solution (∆Hs) was calculated from the
formation data by subtracting the ∆Hf° of pure sulfuric
acid from ∆Hf° of the solution. These plotted data (Fig.
1) indicate two regions in which the relationship is
approximately logarithmic. The average differential heat
of solution (Ls) in kJ kg-1 on n1<n<n2 may be calculated
directly from the ∆Hs data using
Ls =
∆H s 2 − ∆H s1
1
0.01802
n 2 − n1
(2)
where n1 and n2 represent moles of water at successive
data points of Fig. 1 and the numerical factor expresses
the molar mass of water (0.018 kg mol-1). A better
estimate can be made, however, by using logarithmic
interpolation to estimate Ls at
Ls =
1
0.01802
1
n1 n 2
n1 n 2 using
∆H s 2 − ∆H s1
ln(n 2 / n1 )
The resulting Ls values, based on data of Fig. 1, are
plotted in Fig. 2.
0
H2SO4
H2SO4
-0.001
-0.01
-0.1
-1
-10
-100
-1000
-10000
0.1
1
10
100
1000
10000 1 x 105 1 x 106
n (moles H2O)
Fig. 2. Differential heat of solution for one mole H2SO4 in H2O.
Sodium chloride (NaCl) exhibits solution heats which are
for the most part of opposite sign to those of sulfuric acid.
That is, adding water to NaCl solutions, except quite
weak ones, results in an endothermic situation in which
heat must be absorbed in order to maintain constant
temperature. Detailed data for uni-univalent electrolyte
solution thermodynamics, including NaCl, are available
[Parker, 1965]. Additional nomenclature must be
introduced to discuss the electrolyte (NaCl) solutions and
compare their properties to those of sulfuric acid. We
begin with the molar enthalpy of solution at infinite
dilution (∆solH°), defined as the enthalpy change when 1
mol is dissolved in an infinite amount of water. The data
of Rossini et al. plotted in Fig. 1 yield for sulfuric
acid ∆ sol H ° = −96.19 kJ mol-1. According to Parker
[1965], for NaCl ∆ sol H ° = ∆H ∞o = 3.88 kJ mol-1 where
∆H ∞o is Parker's notation.
The function ΦL is defined [Parker, 1965] as
relative apparent molal enthalpy. In terms of quantities
already introduced above, it is defined by
Rossini et al., 1952
Wagman et al., 1982
-20
∆Hs (kJ)
(3)
-0.0001
Ls (kJ/kg)
2
-40
∆H − Φ = ∆ H o
s
L
sol
(4)
indicating that ∆Hs and ΦL differ only by the constant
∆solH°. Thus, the differential heat of solution is
-60
Ls =
-80
∂∆H s
∂Φ L
1
1
=
0.01802 ∂n
0.01802 ∂n
(5)
-100
0.1
1
10
100
1000
10000 1 x 105 1 x 106
n (moles H2O)
Fig. 1. Integral heat of solution for 1 mole of H2SO4 mixed
with n moles of H2O.
where n is moles of solvent, water. Since ΦL is
experimentally determined as a function of m, molality,
the relationship of m to n is needed. Because the amount
of solute is fixed for purposes of finding the partial
derivative, we consider, for simplicity, one mole of
solute, resulting in
Diluting Soil Solution
1
0.01802n
m=
(6)
and thus
∂m
−1
=
= −0.01802m 2
∂n 0.01802n 2
(7)
Graphical presentations of Φ L values (as for example in
Parker [1965]) use
m as the abscissa, so slopes
represent derivatives with respect to m . Thus, it is
useful to write
∂Φ L
1
=
0.01802 ∂n
∂Φ L ∂m1 / 2 ∂m
1
=
0.01802 ∂m1 / 2 ∂m ∂n
∂Φ L 1 / 2
1
(−0.01802m 2 )
0.01802 ∂m1 / 2 m1 / 2
Ls =
(8)
or
Ls = −
1 3 / 2 ∂Φ L
m
2
∂m1 / 2
(9)
and Ls is then easily computed when
∂Φ L
is tabulated
∂m 1 / 2
with the data, as in Parker [1965], Table X). Values of Ls
as computed from Parker’s Table X values and the
equation above are shown in Fig. 3.
6000
NaCl
Ls (J/kg)
4500
3000
1500
0
-1500
1
10
100
1000
10000
1 x 105 1 x 106 1 x 107
n (moles H2O)
Fig. 3. Differential heat of solution for one mole NaCl in H2O.
Clearly, differential heat of solution may be either
positive or negative and the range of magnitudes is large,
3
as implied for NaCl and H2SO4 by Fig. 2 and Fig. 3.
Differential heats of wetting for soil are generally
positive. We now turn our attention to heat generated by
transfer of water to, from, and between solutions and
soils.
Equilibrium of Soils and Solutions
Thermodynamic equilibrium with respect to
transport of matter between phases of a closed system
exists when the phases have equal pressure, temperature,
and chemical potentials, µi, where subscript i indicates the
system components. These conditions are summarized
for phases α and β by Tα=Tβ, Pα=Pβ, and µ iα = µ iβ
[Moore, 1962]. Since relative humidity is directly related
to the chemical potential of water, equilibrium with
respect to soil water content and solution strength is
established if at equal temperatures the same air pressure
and relative humidity coexist over soils and solutions of a
system.
Consider NaCl and H2SO4 solutions and moist soil
in equilibrium with air at atmospheric pressure and 25°C.
The presence of air in such systems disturbs the water
vapor equilibrium pressure by less than 0.1% [Moran and
Shapiro, 1992, p. 684]. We also assume in what follows
that the solutes and soil have negligible vapor pressures.
Values will be calculated for the soil and solutions at a
common water potential, ψ, of -10 kJ kg-1 (-100 bar). If
the soil is Fargo silty clay (fine smectitic frigid Typic
Epiaquert) the water content w corresponding to this
water potential is 0.125 g g-1. The NaCl solution strength
is about 2.04 m, based on interpolation of tabular values
presented by Lang [1967]. The sulfuric acid solution
strength is about 1.67 m, based on Harned and Owen
[1958, Table 13-10-1], which indicates water activities of
0.939 and 0.914 with solutions of 1.5 and 2.0 m,
respectively. Water potential is calculated from water
activity, a, as RT ln(a) .
If a small amount of pure, free (zero water
potential) water is added to the -10 kJ kg-1 water potential
soil or solution at constant temperature, heat will be
absorbed or evolved. For NaCl the heat absorbed (Ls)
will be 2500 J kg-1 (from Fig. 3; m = 2.04 occurs at n =
27.8) while for H2SO4 heat evolved will be 2330 J kg-1
(from Fig. 2). For other solutes the heat may be within or
outside the range defined by NaCl and H2SO4. The range
of heats for different solutions is very diverse, as may be
appreciated by examination of Parker [1965], Rossini et
al. [1982], and similar references.
The same process as described in the preceding
paragraph may be expressed also in the form of chemical
reaction equations. The equations for adding 1 kg mass
of water to an infinite amount of our NaCl and H2SO4
solutions at constant temperature, resulting in one
additional kg of solution and flow of heat ∆H are
4
Diluting Soil Solution
H 2 O(1) + 2.04 m NaCl (∞) →
2.04 m NaCl (∞ + 1) + ∆H (2.50 kJ )
H 2 O(1) + 1.67 m H 2 SO 4 (∞) →
1.67 m H 2 SO4 (∞ + 1) + ∆H (−2.33 kJ )
(10)
(11)
where we now used ∆H rather than Ls to represent the
differential heat of solution. In Eqs. (10) and (11), and
equations to follow, quantities in parentheses associated
with chemical species are the associated amounts in kg
for the example case. Now, adding the reverse of Eq.
(10) and Eq. (11) gives
2.04 m NaCl (∞ + 1) +
1.67 m H 2 SO 4 (∞) → 2.04 m NaCl (∞) +
(12)
1.67 m H 2 SO 4 (∞ + 1) + ∆H (−4.83 kJ )
where H2O(1) has been removed from both sides.
Equation 12 represents the process in which 1 kg of water
is extracted from an infinite amount of 2.04 m NaCl
solution and added to the infinite amount of 1.67 m
H2SO4 solution. Since the reversible work required to
extract the water from 2.04 m NaCl is equal to the
reversible work recoverable by adding the same water to
the 1.67 m H2SO4 the process requires no net work from
the environment. When the reversible work per kg of
water, ∆W=-10 kJ, is considered as part of ∆H in Eq.
(10), for instance, we have for 2.04 m NaCl
∆H = 2.50 kJ = −10.00 kJ + 12.50 kJ =
∆W + ∆L = L s
(13)
where ∆L = 12.5 kJ represents a latent heat of transfer
when water is added to the NaCl solution from a pool of
pure water in equilibrium with the solution. Note that the
difference between ∆L and Ls is that ∆L applies when
water is reversibly added to the solution while Ls applies
when free (zero water potential) water spontaneously
mixes with the solution, resulting in a loss of free energy.
Thus one may write, in place of Eq. (10)
H 2 O(1) + 2.04 m NaCl (∞) →
2.04 m NaCl (∞ + 1) + ∆W (−10 kJ ) +
Relationship to Temperature
Thus far, no variations of properties with
temperature have been considered. When phase changes
are involved, however, it is evident from various forms of
the Clausius-Clapeyron equation that enthalpy of phase
change for a pure substance is directly related to the rate
of change of equilibrium pressure with temperature. One
form of the Clausius-Clapeyron equation [Edlefsen and
Anderson, 1943] is
dP
∆h
=
dT T∆v
For the Fargo soil, the equivalent expression is
(15)
(16)
where ∆h and ∆v are differences in enthalpy and specific
volume occurring upon phase change.
A similar relationship important with respect to
soil solutions is the rate of change of water potential with
respect to temperature at constant composition. Let's
consider again the NaCl and H2SO4 solutions. From the
dψ (2.0 m NaCl )
data of Lang (1967) the value of
dT
ranges from -50 J kg-1 K-1 at 2.5°C to -38 J kg-1 K-1 at
37.5°C. For 1.5 m H2SO4, values of ψ are -8.157, -8.654,
and -8.920 kJ kg-1 at 0, 25, and 40°C, respectively. Thus,
for 1.5 m H2SO4, the average rates of change of ψ are
-19.9 J kg-1 K-1 from 0°C to 25°C and -17.7 J kg-1 K-1
from 25°C to 40°C.
A basic relationship from thermodynamics
(equation 3-9 of Pitzer, [1995]) is
(∂G / ∂T ) P = − S
(17)
where the relationship holds for a system at constant
pressure with S the entropy and G the Gibbs energy.
With the superscript degree symbol used to represent the
standard state pressure of 100 kPa then at any
temperature
G = G° + ∆G ; S = S ° + ∆S
(14)
∆L(+12.5 kJ )
H2O(1) + 0.125w Fargo(∞)→
0.125w Fargo(∞ + 1)+
∆W(-10 kJ) + ∆L(∆H + 10 kJ)
This partitioning of the differential heat of wetting into a
part due to the water potential, ∆W, and a part due to
latent heat, ∆L, is similar in form to that indicated by
Taylor and Ashcroft [1972, p. 92].
(18)
are used with the standard state being 100 kPa and a
constant temperature. In Eq. (18) ∆G and ∆S represent
differences due to departure of the actual state from the
standard pressure condition. Thus,
(∂∆G / ∂T ) P = −∆S
(19)
where the P subscript means that the external pressure on
Diluting Soil Solution
the system remains constant during the process which
causes ∆G. Equation (19), when expressed in terms of
specific quantities for soil water as expressed in relation
to free water, was given by Edlefsen and Anderson [1943,
p. 100] as
(d (∆f ) / dT ) P = −∆S P
(20)
where ∆f is the Gibbs specific energy or, equivalently, the
free energy, as it is called by Edlefsen and Anderson and
∆SP is specific entropy measured with respect to free
water.
Let's examine the relationship given by Eq. (20) in
terms of experimental data for NaCl solutions. Entropy
changes in a closed system are by definition
dS = dq rev / T
(21)
-d∆f/dT=-d∆W/dT (J kg-1 K-1)
where dqrev indicates heat flow during a reversible
process. For the 2.04 m NaCl solution, for instance, this
is ∆L as expressed in Eq. (13). Using water potential data
[Lang, 1967] along with data of Fig. 3 allows calculation
of the left and right sides of Eq. (20) for NaCl solutions
and results in the comparison presented in Fig. 4, where
∆L = L S − ∆W . Also in Fig. 4 are some points
representing H2SO4 and KCl solutions. Figure 4 verifies
Eq. (20) within the accuracy of the data available.
50
KCl
H2SO4
NaCl
40
5
d (− p c )
d∆f
d∆P
=v
=v
dT
dT
dT
(22)
where v is the specific volume of liquid water and pc>0 is
capillary pressure. With respect to the capillary pressure,
it "appears to be universal" [Grant, 2003] that capillary
pressure at constant saturation is a linearly decreasing
(approaching zero) function of temperature. Thus, when
∆f is due to capillary pressure effects only, as is the case
in solute-free soil with the gas at standard atmospheric
pressure, Grant's observation means d∆f/dT>0.
The quantity d∆f/dT is greater than zero where
capillarity only is involved and Grant [2003, Fig. 1]
indicates a relative rate of change ((d∆f/dT)/∆f) of -0.012
K-1 for Elkmound sandy loam. Where solution effects
only are involved the previous (between Eq. (16) and Eq.
(17)) values of dψ/dT for 2.0 m NaCl may be used as an
example, since ψ=∆f. Using the example NaCl value just
mentioned and making similar calculations for other
solutions reveals that (d∆f/dT)/∆f has positive values of
roughly 0.005, 0.004, and 0.002 K-1 for NaCl, KCl, and
H2SO4 solutions, respectively, depending somewhat on
temperature. Limited data examined for other solutes
indicates that the range represented above by the three
solutions is representative. Based on Elkmound soil
[Grant, 2003] and the three solutions noted, the
magnitude of the relative rate of change of water potential
due to temperature change at constant water content is 2.5
to 6 times greater due to the capillarity effect than due to
solute effects.
Connection to Heat of Wetting.
Differential heat of wetting of soil of Eq. (15) is
∆W + ∆L . Considering the definitions of these quantities
results in the identifications ∆W = ∆f and ∆L = ∆q rev .
From Eq. (21) we have ∆q rev = T∆S . Then, through Eq.
(20)
30
20
10
∆W + ∆L = ∆f − T
0
0
10
20
30
40
50
∆L/T=(L S -∆W)/T (J kg-1 K-1)
Fig. 4. Verification of Eq. (20) for KCl, NaCl and H2SO4
solutions.
The derivative on the left of Eq. (20) has been of
interest in soil systems because ∆f is related to pressure
change in an incompressible liquid by ∆f = v∆P
[Edlefsen and Anderson, 1943 - their Eq. 156]. When ∆P
is produced by capillary pressure of soil water which is
free of solutes then
d (∆f )
dT
(23)
where the right side corresponds to the expression given
by deVries [1958] as his equation 12 for differential heat
of wetting. In his paper, deVries [1958] attributed the
expression on the right of Eq. (23) to Edlefsen and
Anderson [1943].
Conclusions
Dilution of an aqueous solution is thermodynamically a parallel process to adding water to soil or
another porous media. For a porous media with zero or
small capillary pressure the solution dilution effect would
6
Diluting Soil Solution
dominate, producing either absorption or evolution of
heat. For instance, a matrix of 2-mm diameter glass
beads half saturated with sulfuric acid would produce a
very large exothermic heat flux if water were added to
saturate it. On the other hand, the same bead media
mixed with solid NaCl would absorb heat upon saturation
with water. When strictly capillary effects are involved,
as with adsorption to clays, the evidence is [Grant, 2003]
that the reaction is always exothermic.
The sign of the derivative on the right side of Eq.
(23) seems to be dependent on the source of the water
potential. Capillary potential corresponds to a positive
derivative. Solute potential corresponds to a negative
derivative. Further investigation of the generality of this
conclusion is advised.
References
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