heat of diluting the soil solution versus soil heat of wetting · heat of diluting the soil...

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 CBA →+ 23.1=∆H 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

Twq⎟⎠⎞

⎜⎝⎛

∂∂ where the subscript indicates a constant

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

2 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

12

12

01802.01

nnHH

L sss −

∆−∆= (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 21nn using

)/ln(

101802.0

1

12

12

21 nnHH

nnL ss

s∆−∆

= (3)

The resulting Ls values, based on data of Fig. 1, are plotted in Fig. 2.

n (moles H O)

∆H

(kJ)

0.1 1 10 100 1000 10000 1 x 10 1 x 10

0

-20

-40

-60

-80

-100

2

s

Rossini et al., 1952

Wagman et al., 1982

5 6

H SO 2 4

Fig. 1. Integral heat of solution for 1 mole of H2SO4 mixed with n moles of H2O.

0.1 1 10 100 1000 10000

-0.0001

-0.001

-0.1

-100

-1000

-10000

n (moles H O)2

L (

kJ/k

g)

s

H SO2 4

-10

-1

-0.01

61 x 10 51 x 10

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 19.96−=°∆ Hsol kJ mol-1. According to Parker

[1965], for NaCl 88.3=∆=°∆ ∞o

sol HH kJ mol-1 where oH ∞∆ 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

oHsolLsH ∆=Φ−∆ (4)

indicating that ∆Hs and ΦL differ only by the constant ∆solH°. Thus, the differential heat of solution is

nn

HL Ls

s ∂Φ∂

=∂∆∂

=01802.0

101802.0

1 (5)

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 3

n

m01802.0

1= (6)

and thus

22

01802.001802.0

1 mnn

m −=−=∂∂ (7)

Graphical presentations of LΦ 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

)01802.0(2/101802.0

101802.0

101802.0

1

22/12/1

2/1

2/1

mmm

nm

mm

m

nL

L

L

Ls

−∂

Φ∂

=∂∂

∂∂

∂Φ∂

=∂Φ∂

=

(8)

or

2/1

2/3

21

mmL L

s∂

Φ∂−= (9)

and Ls is then easily computed when 2/1m

L

∂Φ∂

is tabulated

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.

n (moles H O)

L (

J/kg

)

1 10 100 1000 10000 1 x 10 1 x 10 1 x 10

6000

4500

3000

1500

-1500

2

0

s

NaCl

5 6 7

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,

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 βα µµ ii = [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 )ln(aRT . 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


)50.2()1(04.2

)(04.2)1(2

kJHNaClmNaClmOH

∆++∞→∞+

(10)

)33.2()1(67.1

)(67.1)1(

42

422

kJHSOHmSOHmOH

−∆++∞→∞+

(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

)83.4()1(67.1)(04.2)(67.1

)1(04.2

42

42

kJHSOHmNaClmSOHm

NaClm

−∆++∞+∞→∞

++∞ (12)

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

sLLW

kJkJkJH=∆+∆

=+−==∆ 50.1200.1050.2 (13)

where kJL 5.12=∆ 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)

)5.12(

)10()1(04.2)(04.2)1(2

kJLkJWNaClm

NaClmOH

+∆+−∆++∞

→∞+ (14)

For the Fargo soil, the equivalent expression is H2O(1) + 0.125w Fargo(∞)→ 0.125w Fargo(∞ + 1)+ (15) ∆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]. 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

vT

hdTdP

∆∆= (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

data of Lang (1967) the value of dT

NaClmd )0.2(ψ

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 STG P −=∂∂ )/( (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 GGG ∆+°= ; SSS ∆+°= (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, STG P ∆−=∂∆∂ )/( (19) where the P subscript means that the external pressure on

Diluting Soil Solution 5

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 PP SdTfd ∆−=∆ )/)(( (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 TdqdS rev /= (21) 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

WLL S ∆−=∆ . Also in Fig. 4 are some points representing H2SO4 and KCl solutions. Figure 4 verifies Eq. (20) within the accuracy of the data available.

-d∆f

/dT=

-d∆W

/dT

(J k

g K

)

0 10 20 30 40 50

50

40

30

20

10

0

KClH SONaCl

2 4

-1-1

∆L/T=(L -∆W)/T (J kg K )-1 -1S

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 Pvf ∆=∆ [Edlefsen and Anderson, 1943 - their Eq. 156]. When ∆P is produced by capillary pressure of soil water which is free of solutes then

dT

pdv

dTPdv

dTfd c )(−

=∆=∆

(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

LW ∆+∆ . Considering the definitions of these quantities results in the identifications fW ∆=∆ and revqL ∆=∆ . From Eq. (21) we have STqrev ∆=∆ . Then, through Eq. (20)

dT

fdTfLW

)(∆−∆=∆+∆ (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 thermo-dynamically 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


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 Anderson, D. M., Heat of immersion, In A. Klute, (ed.)

Methods of Soil Analysis, Part 1. 2nd Ed., ASA,CSA,SSSA, Madison, WI, no. 9, pp. 969-984, 1986.

de Vries, D. A., Simultaneous transfer of heat and

moisture in porous media, Eos Trans. AGU 39, 909-916, 1958.

Edlefsen, N. E., and A. B. C. Anderson, Thermodynamics

of soil moisture, Hilgardia, 15, 31-288, 1943. Grant, S. A., Extension of a temperature effects model for

capillary pressure saturation relations, Water Resour. Res., 39(1), 1003, doi: 10.1029/2000WR000193, 2003.

Harned, H. S., and B. B. Owen., The physical chemistry

of electrolytic solutions, Reinhold Publishing Corp., New York, 1958.

Lang, A. R. G., Osmotic coefficients and water potentials of sodium chloride solutions from 0 to 40°C., Aust. J. Chem. 20, 2017-2023, 1967.

Moore, W. J., Physical Chemistry, 3rd ed., Prentice-Hall,

Englewood Cliffs, N.J., 1962. Moran, M. J., and H. N. Shapiro, Fundamentals of

engineering thermodynamics, 2nd Ed., John Wiley & Sons Inc, New York, 1962.

Parker, V. B., Thermal properties of aqueous uni-

univalent electrolytes, National Standard Reference Data Series National Bureau of Standards 2, Washington, DC, 1965.

Prunty, L., Spatial distribution of heat of wetting in

porous media, Paper 023119, American Society of Agricultural Engineers, St. Joseph, MI., 2002.

Pitzer, K. S., Thermodynamics, 3rd ed., McGraw-Hill,

New York, 1995. Rossini, F. D., D. D. Wageman, W. H. Evans, S. Levine,

and I. Jaffe, Selected values of chemical thermodynamic properties, Circular of the National Bureau of Standards 500, Superintendent of Documents, Washington, D.C., 1952.

Taylor, S. A., and G. L. Ashcroft, Physical edaphology,

W. H. Freeman and Company, San Francisco, 1972. Wagman, D. D., W. H. Evans, V. B. Parker, R. H.

Schumm, I. Halow, S. M. Bailey, K. L. Churney, and R. L. Nutall, The NBS tables of chemical thermodynamic properties: Selected values for inorganic and C1 and C2 organic substances in SI units, J. of Physical and Chemical Reference Data 11 (Suppl. 2), 2-58,59, 1982.