J . Phys. Chem. 1989, 93, 2079-2082
2079
Single-Ion Heat of Transport in Electrolyte Solutions. A Hydrodynamic Theory J. N. Agar, Department of Physical Chemistry, Cambridge University, Cambridge, England
C. Y. Mou, Department of Chemistry, National Taiwan University, Taipei, Taiwan 107, The Republic of China
and Jeong-long Lin* Department of Chemistry, Boston College, Chestnut Hill, Massachusetts 02167 (Received: April 25, 1988; In Final Form: August 26, 1988)
The standard single-ion heat of transport Q i * O in electrolyte solutions is derived based on a hydrodynamic approach. It is shown that Qi*" may be related to the structure and dynamics of the solution through the velocity field induced by the ionic motion and the entropy density of the solvent around the ion. The hydrodynamic approach suggests analogies between the heat of transport and dielectric properties of the solutions. The concepts of the thermal dipole moment and thermal polarization are found to be useful in the discussion of the heat of transport. The hydrodynamic theory permits an analysis of the relationship between the entropy of transport and the entropy of hydration and shows that the entropy of hydration may be obtained from the entropy of transport at the thermodynamic limit. The hydrodynamic expression of Q i * O also made it possible to carry on a systematic investigation of the Soret effect using the structural and dynamical models of electrolyte solutions. This is illustrated by an example using the Born and Stokes-Einstein models.
I. Introduction In a nonisothermal solution in which the temperature is steady but not uniform, transports of matters and heat (thermal diffusion) may be commonly observed. This effect in condensed phases is known as the Soret effect. The transport of matter is in general coupled with the transport of heat, and according to Eastman'v2 when a solute particle is transferred between regions of differential temperature difference in a stationary solvent, a quantity of heat Q* is absorbed from the heat reservoir behind and given out ahead of the moving particle. This Q* is the heat of transport of the solute and may be referred to in the Hittorf frame of reference from the concentration gradient established at the (Soret) steady state. Eastman obtained where p and m are respectively the chemical potential and molality of the solute species. The subscript s denotes the steady state. The result obtained in eq 1.1 also arises from the Onsager nonequilibrium thermodynamic^.^ Q* is proportional to the Onsager cross phenomenological coefficient for flow of matter and heat. Furthermore, because of the reciprocity relation, Q* may be shown to be identical with the Dufour heat, reflecting the isothermal diffusion thermoeffect. Thus Q* may be regarded as the amount of heat which must be supplied behind or evolved ahead of the diffusing particle in order to keep the temperature constant. Q* is a useful property of solutions. This may become even more obvious when the discussion is carried out in terms of the corresponding entropy of transport S*, which may be obtained from the equation S* = Q * / T . S* is the entropy absorbed behind (or evolved ahead) of a moving particle. However, the entropy (or heat) of transport has remained somewhat an unfamiliar quantity since it is not exactly clear how the Soret data may be related to other properties (such as the structure and dynamics) of the solutions. In order to provide a theoretical interpretation of Q* Agar4,5 has proposed a hydrodynamic theory for the single-ion heat of transport in electrolyte solution. In the course of the discussion Agar pointed out that there are similarities between the heat of transport and the dielectric properties and the heat of transport
* To whom correspondence regarding this article should be addressed. 0022-3654/89/2093-2079$01 SO10
may be viewed in terms of thermal polarization which may be computed from thermal dipole moments. Agar, however, provided only a greatly abbreviated hydrodynamic derivation of the heat of transport. The purpose of this paper is to present a detailed hydrodynamic derivation of the standard single-ion heat of transport in electrolyte solutions, Qi*". We shall identify the thermal dipole moment and show that Qi*O is related to the integral of the thermal dipole moment over the volume, the thermal polarization. Our derivation refers to the (isothermal) diffusion thermoeffect which is not the same as in the previous work of Agar who considered both transports of heat (entropy) by diffusion and conduction. Since Eastman's work depicts an essential basic physical picture, we begin by giving a summary of his calculation in the next section. Eastman's experiment is then discussed briefly in terms of Onsager nonequilibrium thermodynamics to provide the necessary basis for the hydrodynamic approach presented in section 111. Finally, in section IV we discuss our treatment. Here we show how the Soret effect may be interpreted in terms of the structure and dynamics of electrolyte solutions. To illustrate this, structure-making and -breaking effects of the ion are discussed, using values of Qi*O derived form the Born and Stokes-Einstein models. 11. Thermodynamics
Consider the transfer of a mole of solute (component A) in a stationary solvent between regions of differential (infinitesimal) temperature difference in an infinitely large tube maintained at constant pressure P. The reference plane where the temperature changes from T to T + dT is denoted as BB shown in Figure 1. Following Eastman, a quantity of heat is absorbed from the reservoir at T and given out at T + dT. The changes in the partial Eastman, E. D. J. Am. Chem. SOC.1926,48, 1482. Eastman, E. D. J. Am. Chem. SOC.1928,50, 283, 292. Fitts, D. D. Nonequilibrium Thermodynamics; McGraw-Hill: New York, 1962. (4) Agar, J. N. Thermal Diffusion in Electrolyte Solutions. In The Structure of Electrolyte Solutions; Hamer, W. J., Ed.; Wiley: New York, 1959; p 200. ( 5 ) Agar, J. N. Thermogalvanic Cells. Advances in Electrochemistry and Electrochemical Engineering, Interscience: New York, 1963; Vol. 3, Chapter 2.
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 5, 1989
2080
Agar et al.
B
B
+
+
x
,QA
QA
1
A
JA
X
1
'\
\
T
\\J
T+dT
B
B Figure 1. Transport of solute A in a stationary solvent between regions of differential temperature difference. BB is the reference plane where the temperature changes from T to T + dT. QA* is evolved to the reservoir at T + dT to the right of BB and absorbed to the left of BB from the reservoir at T .
molal enthalpy and entropy respectively are denoted by dHAand dSA. The total entropy change of the reservoirs (6SR)is then ~ S =R-dHA/T- QA*/T + QA*/(T dT)
+
= - r ' [ ( a H A / a T ) P , m , d T + (aHA/amA)P,T dmAl -
QA* d T 7 (2.1)
The total entropy change of the system 6SS =
dSA
=
[(aSA/aT)P.mA
(ass)is
d T + (aSA/amA)P,T
dmAl
(2.2)
where mA is the molality of the solution. Eastman solved the heat of transport QA* in eq 2.1 by equating 6SR+ 6Ss = 0 at the steady state. Eastman thus obtained QA* = -(apA/amA)P,T(dmA/d In
(2.3)
T)S
+
The relationship 6SR 6Ss = 0 is an assumption for the steady state. However, this assumption may be verified by rederiving the result obtained in (2.3) using Onsager nonequilibrium thermodynamics. The Hittorf flux for A, JA, and the heat flux, JQ, are given by JA
= LAXA+ LAQXQ
(2.4)
(2.5) = LQAXA+ L ~ X Q where L's are the phenomenological coefficients and X A and XQ are driving forces. By taking x as the direction of transport JQ
XA = -dpAT/dX
(2.6)
XQ = -d In T/dx
(2.7)
where, in Eastman's experiment, dMAT
= dC(A - ( a p A / a T ) P , m A d T = (apA/amA)P,T
dmA
(2.8)
Setting J A = 0 for the steady state, and from eq 2.4-2.8, one finds LAQ/LA= -(XA/XQ) = - ( a ~ ~ / a m ~ ) ~ , T ( d In m ~T), / d= QA* (2.9) as in eq 2.3. Equation 2.9 indicates that QA* is proportional to according the heat-matter cross coefficient LAO. Since LA, = bA to the reciprocity relation QA* = LAQ/LA= LQA/LA= (JQ/JA)x,-o or JQ
= JAQA*
(2.10)
at the limiting isothermal state. Equation 2.10 shows that QA* is identical with the Dufour heat. It establishes a basis for an isothermal calculation of QA*. 111. Hydrodynamic Theory of Q i * O
We now proceed to formulate a hydrodynamic theory of the heat of transport in electrolyte solutions. We shall first present a derivation following the method sketched in Agar.s Then a simpler alternative derivation which enables us to identify thermal
Figure 2. Hydrodynamic calculation of the single-ion heat of transport. e,*" is the total amount of heat evolved to the right of BB when the ion is transported across BB from X = --m to X = m ,
dipole moment and thermal polarization is given. For the calculation of the standard single-ion heat of transport Qi*" we take a spherical ion immersed in its solvent and transport the ion across the reference plane BB shown in Figure 2. The heat of transport Qi*O is the amount of heat evolved to the right of this reference plane. While there is an evolution of heat ahead of the ion, the steady state is maintained by a reverse process, an absorption of heat behind the ion. That is, the same amount of heat is absorbed to the left of the reference plane BB. Our calculation is based on the consideration of the diffusion thermoeffect and thus the system is maintained at a constant temperature T . The ion is assumed to move in the x direction with a constant velocity V under a force F. As the ion migrates, solvent molecules move in and out of the ionic field, giving rise to the absorption or evolution of heat (in order to keep the temperature constant). To calculate Qi*O we first identify the heat source surrounding the moving ion and then calculate the total heat evolved to the right of the reference plane BB when the ion is transported from x = --m to x = -m. The entropy density of the solvent at a distance r from the ion is denoted as S ( r ) and the velocity of solvent relative to the ion is U. For a stationary-state problem where S(r) is not an explicit function of time, dS/dt = as/& + Usgrad S = Usgrad S
(3.1)
Since the ion is assumed spherical and the effect under consideration is linear in the velocity, to the first order in U we may ignore the asymmetry of the entropy density due to ionic motion and write -T(dS/dt) = -dQ/dt = q = -TU,(dS/dr)
(3.2)
where T dS = dQ and q is the local rate of heat evolution per unit volume. In eq 3.2 the usual assumption for linear processes, namely, the principle of local equilibrium is implied in the first equality. The radial velocity of the solvent with respect to the ion U, may be expressed in terms of the value at r = by an equation of the form U, = UJ(r) cos 0 (3.3) where 0 is the angle between V and r. By lettingf(r) = 1 at r = -m, U , = -Vand q is given by q = Tvf(r)(as/ar) COS 0
(3.4)
The quantity q gives the local rate of heat evolution per unit volume to the first order in the velocity. The amount of heat liberated by the solvent lying inside the spherical shell between r and r + dr during a time interval dt is, therefore, given by 2 r r 2 d r ( L ' q sin 0 d0) dt = 2 ~ T V f ( r ) ( a S / a r ) r ~dr !I2sin2 0 dt (3.5) Notice that, when 0 = K, that is, when the spherical shell is lying wholly to the right of the plane (or of course, wholly to the left), no contribution is made to the heat of transport. Thus by placing r = 0 at x = 0, the time integral is reduced to an integration between t = - r / V to t = r/V. Noting that x = Vr and sin2 0 = (r2 - x2)/r2,the total contribution from the spherical shell to the
Single-Ion Heat of Transport in Electrolytes
The Journal of Physical Chemistry, Vol. 93, No. 5, 1989 2081
heat of transport may be computed. This is
Ri may not be necessarily identical with the ionic radius ri of the ion. For the present discussion, however, the distinction is not important and we can effectively consider that (as/&) = 0 for r C Ri.Equation 3.7 therefore may be written as
dQi*O = 2 r T V j ( r ) ( a S / d r ) $ d r $(sin2 8 / 2 ) dt = ( 4 r / 3 ) T f l r ) ( B / a r ) r 3d r
(3.6) Qi*" = (4r/3)TJm(&!7/ar)f(r)$ Ri dr
It follows therefore that Qi*O = T ( 4 r / 3 ) $ j ( r ) ( a S / a r ) r 3 d r
(3.7)
Equation 3.7 is the basic hydrodynamic expression for the single-ion heat of transport at infinite dilution. In order to show that the right-hand side may be interpreted in terms of thermal polarization we shall rederive eq 3.7 by an alternative method. The steady state is maintained by the heat flux JQ, given by q = -div JQ
(3.8)
where (see eq 2.10)
JQ = JjQj*O = (V/R)Qj*"
(3.9)
Here R is the volume of the system and the flux Jion= V/fl. By operating eq 3.8 with r and an integration over the volume, one arrives a t $19 dR = - $r div JQdfl =
$JQ dR =$(V/R)Qi*O
dQ = VQi*O (3.10)
Combining eq 3.10 and 3.4 VQi*O =
$ $rTVf(r)(dS/ar)
cos 0 2 d d r sin 0 d0
Taking a dot product with ev, the unit vector in the direction of V and taking into account that e y r = r cos 0 VQi*O = 2rVT$flr)(8S/ar)r3 d r
Jr
From arguments based on experimental data, Chakraborty and Lin' concluded that for small ions the heat and entropy of transport measure predominantly the second-shell effects, that is, effects not in the immediate vicinity of the ion but rather those occurring at a distance (in the second hydration shell, for example) from the ion. Their explanation is consistent with the expression given in eq 4.4. This is because in the region where r i= Ri both aS/dr and f ( r ) are small so that the contribution to the integral from this region is not significant. On the other hand with a moderate increase in r, and especially going from one hydration cosphere to another where the hydration structure undergoes significant changes, aS/ar is expected to be large, resulting in a substantial contribution to the integral. A qualitative illustration of the behavior of the integrand may be given by calculating the entropy according to the theory of Born hydration. In the Born model,6 (dS/dr) is due entirely to the ion-dipolar polarization and the entropy gradient decays uniformly as rw5. Accordingly, the integrand grows from 0 at r = Ri to a maximum and then vanishes to zero at large r. The maximum is located at r = 1.9Ri and 1.5Ri, respectively, depending on whether the boundary condition is stick or slip. For a typical ion where Ri = 2 A, the predominant contribution to the integral in eq 4.4 is thus mainly from the region r = 3 ~ 4 A . For numerical applications, it is convenient to write Qi*O in terms of S ( r ) rather than (aS/ar). This is accomplished by a partial integration giving
cos2 6 sin 8 d0
= V ( 4 r / 3 ) T $ f ( r ) ( d S / a r ) r 3d r
( 3 . 1 1)
a
Qi*' = - ( 4 r / 3 ) T l m [ S ( r )- S ( m ) ] - [ r 3 f l r ) ] d r (4.5) Ri ar At the limit where f ( r )
which leads to eq 3.7 when divided by V. rq in eq 3.10 may be regarded as the "thermal dipole moment" and hence Qi*O V may be identified as the "thermal polarization". This equation implies that the reason the heat of transport may be measured is because of the existence of polarizable structures surrounding the ion. When there are no distinguishable structures that can be polarized by the ion [that is, when (as/&) = 01, q(r) = 0 and Qi*O = 0, even though the ion may move with a finite velocity. IV. Discussion Sinceflr) determines the solvent velocity field induced by the ionic motion and ( a S / d r ) is the entropy density gradient surrounding the ion, eq 3.7 opens to a systematic investigation of the heat of transport in terms of dynamic and structural models of electrolyte solutions. When the solvent is considered as a simple hydrodynamic continuum with a uniform dielectric constant e, f ( r ) may be obtained from the solution of the linearized Navier-Stokes equation.6 Choosing Ri as the hydrodynamic radius of the ion such thatf(Ri) = 0, the function f ( r ) may be readily obtained. For a stick boundary condition fstick
= 1 - (3Ri/2r)
+ (Ri3/2$)
(4.1)
h i p
= 1 - (Ri/r)
(4.2)
Ri is related to the self-diffusion coefficient of the ion Di through the Stokes-Einstein equation
Di = k,T/6a7Ri
(stick)
= kBT/4rvRi (slip) (4.3) where k B is the Boltzmann constant and TJ the solvent viscosity. (6) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon: New
York, 1959.
Qi*"
-
1 , eq (4.5) is reduced to
= - T L r [ S ( r ) - S ( m ) ] 4 r $ dr
(4.6)
The integral in the right-hand side of (4.6)is the standard entropy of solvation. This is an expected result and may be regarded as the thermodynamic limit for the entropy of transport. This is 1 implies that the ion is transported because the limit f ( r ) without disturbing the equilibrium structure of the solvent. The amount of entropy absorbed ahead of the ion must, therefore, be the standard entropy of solvation at this limit. The deviation from the equalityflr) = 1 is a measure of the perturbation of the solvent equilibrium induced by the ionic motion. This gives rise to the measurability of the heat of transport. Equation 4.5 may be integrated if the Born theory is used for S(r) - S(m).Denoting Zi as the ionic valence and e the electronic charge, Born theory gives -+
S(r)- S(m) = (e2Zf/8rr4c2)(dt/aT) which leads to Qi*O
= -T(e2Zi2/c2)(8e/dT)/4Ri
= -T(e2Zf/t2)(a e / d T ) / 3 R i
and for a slip boundary condition
(4.4)
For water at 25 eq 4.7 gives
OC t
(stick) (slip)
(4.7)
= 78.54 and d In t/dT = -4.579 X 10-3k-',
Qi*O = 6 . 0 4 Z i 2 / R ikJ mol-' = 8.05Zi2/RikJ mol-'
(stick) (slip)
(4.8a) (4.8b)
Although the Born model may not be realistic, the model describes a useful reference state for the discussion of structural (7) Chakraborty, B. P.; Lin,J. J . Solution Chem. 1976, 3, 183. (8) See for example: Conway, B. E. Ionic Hydration in Chemistry and Biophysics; Elsevier: New York, 1984.
2082 The Journal of Physical Chemistry, Vol. 93, No. 5, 1989 TABLE I: Single-Ion Heat of Transport at Infhite Dilution 10-'2Q*o /Z:Di, ion 109Di, m2/s Qi*O, lo3 J/mol J s/(mol m2) H+ 9.31 13.3 1.43 Lit 0.5 1 0.53 1.03 3.46 Na+ 1.33 2.60 2.59 1.96 1.32 K+ 2.07 3.91 1.89 Rb' 2.08 4.01 cs+ 1.93 1.73 1.95 NH4' 0.89 10.00 1.20 Me4Nt 8.33 Et4Nt 16.43 14.29 0.87 18.36 0.62 29.61 n-Pr4NC 20.79 n-Bu4Ni 0.5 1 40.76 3.86 6.37 1.65 "%+ 2.18 4.33 TI+ 1.99 9.04 3.20 0.7 1 Mg2+ 9.8 3.10 Ca2+ 0.79 3.52 11.1 Sr2+ 0.79 3.66 12.4 0.85 Ba2+ 9.3 Ni2+ 3.30 0.7 1 3.14 17.2 OH5.32 2.67 3.93 1.47 F 0.26 0.53 2.03 c10.29 0.60 2.08 B r1-1.55 -0.76 2.04 -0.33 -0.63 1.90 NO,-0.17 1.81 -0.31 C1042.00 1.45 10,1.38
properties of electrolyte solutions. Take for instance the Frank and Wen t h e ~ r y . The ~ second hydration cosphere (region B) is called the structure-breaking zone because the water molecules in this region is less organized than the normal water molecules polarized in the ordinary way by the ionic field. Accordingly, one may take the value of Qi*" computed on the basis of the Born theory as a reference value and discuss the structural properties of the ion by comparing the experimental value with the Born value. To facilitate this comparison Riis first replaced by the self-diffusion coefficient Di. For H 2 0 at 25 "C the solvent viscosity = 0.8904 X 10-3 kg m-I s-I. Thus from eq 4.8 and 4.3 and in SI units,
Qi*"/Zi2Di = 2.48
X
= 2.20 x 1012
10I2 (slip)
(stick)
(4.9a) (4.9b)
In eq 4.9 the two hydrodynamic boundary conditions yield values differed by 10%. Although this difference will have no major consequences in the present discussion, it seems reasonable nonetheless to expect that the stick ion will have a larger positive value of (Qi*"/Di) than the slip ion. To compare the results of Born-Stokes calculations with experiment, the ionic heat of transport must be estimated from the experimental molar heat (9) Frank, H. S.; Wen, W. Y . Discuss. Faraday SOC.1957, 24, 133.
Agar et al. of transport. Table I lists values of (Qi*"/DiZ?) for some univalent and divalent ions obtained by using the standard single-ion heat of transport calculated by the reduction rule of Takeyama and Nakashima.lo The reduction rule yields QCI-*"= 0.53 kJ/mol which is close to Qcr*" = 0.37 kJ/mol based on the Gurney scale" by assigning a value of -23 J K-' mol-' to the standard partial molal entropy of the hydrogen ion. Experimental data for the alkaline earth metal, nickel, ammonium, and tetraalkylammonium ions are from Petit, Hwang, and Lin." Others are from Takeyama and Nakashima. The self-diffusion coefficient Di is calculated from the standard ionic equivalent conductance Xi which may be found in Harned and OwenI3 and Lobo and Quaresma.14 Several interesting features may be identified by comparing the results of eq 4.9 and those listed in Table I. According to the Soret effect, Li' is a structure breaker and the only structure maker among the alkali metal ions is Na+. Tetraalkylammonium ions are strong structure makers. The Hf ion is a structure breaker where (Qi*"/Di)is only slightly more than half of a Born-Stokes ion. The only anions listed that are structure makers are OHand F.The divalent ions listed are structure makers. In concluding this section, we emphasize again that the above discussion on the structural properties of ions is based on a Born-Stokes reference ion which has no influence on water structure other than the direct action of the ionic charge on water as a delectric medium. The predictions based on the standard single-ion heat of transport agree in most cases with predictions based on other methods such as the viscosity B coefficient. Two important exceptions are Li+ and H'. They are structure breakers according to this classification. The explanation of this lies in the fact that the principal contribution to the Soret effect comes primarily from the hydration structure beyond the immediate vicinity of the ion. Thus, in spite of the existence of the hydronium ion H(H20),+, the Soret effect may see H+ as a structure breaker. This explanation is also consistent with the interpretation of the Li+ ion as being a structure breaker found in Chakraborty and Lin. Among the effects ignored in the Born model, inhomogeneity and electrostriction are perhaps the two most important ones. Calculations of the heat of transport incorporating these effects will be considered in our future articles.
Acknowledgment. J.L. thanks Professor U. Mohanty for many helpful discussions during the course of this investigation. This research is supported by N S F grants CHE-8415541 and CHE87 19039. (10) Takeyama, N.; Nakashima, K. J. Phys. Soc. Jpn. 1983, 52, 2692, 2699; J. Solution Chem. 1988, 17, 305. (11) Lin, J. J . Solution Chem. 1979, 8, 125. (12) Petit, C. J.; Hwang, M. H.; Lin, J. Int. J. Thermophys. 1986, 7 , 687; J. Solution Chem. 1988, 17, 1. (13) Harned, H. S.; Owen, B. B. The Physical Chemistry of Electrolytic Solutions; Reinhold: New York,1958. (14) Lobo, V. M. M.; Quaresma, J. L.Electrolyte Solutions: Literature Data on Thermodynamics and Transport Properties; Coimbra: Lisbon, Portugal, 1981.
