Diffusion thermoeffect measurements of heats of transport
in ternary liquid toluene-chlorobenzene-bromobenzene
mixtures at 250 and 35 c
G. Platt,8) T. Vongvanich, G. Fowler,b) and R. L. Rowley
Department a/Chemical Engineering, Rice University, Houston, Texas 77251
(Received 8 April 1982; accepted 6 May 1982)
Heats of transport in ternary nonelectrolyte liquid mixtures have been directly measured for the first time.
Diffusion thermoeffect measurements have been made for ternary mixtures of toluene--chlorobenzen
bromobenzene at 25 and 35 C. A new boundary sharpening technique was used in addition to a liquid gate
withdrawal method. Both techniques yield consistent, accurate values of the heat of transport, but the new
cell provides versatility in allowable densities and compositions of the initial phases. Heats of transport were
obtained by nonlinear least-squares fitting of calculated to experimental temperature differences measured at
positions symmetric about the initial diffusional interface. The two independent heats of transport, obtained
by performing two runs with different starting composition gradients for each mean composition, were each
fit to a polynomial in temperature and composition. The resultant analytical expression is valid only in the
region bounded by 0.1 w; 0.6(i = 1,2) and w3 0.I, where the composition is far enough from the binary
limits that both of the two heats of transport remain defined and independently determinable. The results
obtained for the heats of transport show a direct dependence on temperature and a strong dependence on the
relative amount of the heavy component, bromobenzene. These facts are consistent with a proposed
interpretive model of the diffusion thermoeffect.
I. INTRODUCTION
Not only will a temperature gradient give rise to a
heat flux in multicomponent mixtures, but so also will
composition gradients (more appropriately, isothermal
chemical potential gradients). This latter effect, known
as the diffusion thermoeffect or the Dufour effect, is
characterized by a transport property called the heat of
transport. The heat of transport of component i, Qt
relates the magnitude of the produced heat flux to the
diffusional flux of component i under isothermal conditions.
Just as there are only n -1 independent diffusional
fluxes in an n component system, there are also
only n -1 independent heats of transport. These transport
coefficients are of interest because they contain
significant information concerning the actual energydiffusional
relationship of molecular interactions. 1.2
Study of heats of transport should aid in the development
of a generalized theory of transport processes.
In the companion paper to this one, 3 hereafter referred
to as PVR1, the phenomenology of the diffusion
thermoeffect was described. Moreover, the composition
and energy partial differential equations describing the
diffusion thermoeffect in ternary systems were solved
SUbject to initial and boundary conditions designed to
maximize and facilitate experimental measurement of
the two independent heats of transport. Further modeling
was done in that paper to provide information relative
to thermocouple positioning, parameter optimization,
parameter decoupling, expected temperature responses
for different conditions and thermophysical properties,
and elimination of heat of mixing problems. This work
is based on the foundation provided in PVR1; it uses the
equations and experimental suggestions of PVR1 to dealpresent
address: E. D. H., Texaco, Inc., Houston, Texas
77052.
b)Present address: Baylor College of Medicine, Houston,
Texas 77025.
termine Qt and Qf as a function of composition and temperature
in ternary mixtures of toluene-chlorobenzenebromobenzene.
Heats of transport in gas mixtures have been measured
by several investigators4- 8; quantitative measurements
in liquid mixtures were thought to be quite impossible.
9 The work of Ingle and Horne10 and of Rowley and
Horne, 11,12 however, established the diffusion thermoeffect
as an accurate and reliable method of obtaining liquid
heats of transport in binary systems. The work reported
in PVR1 indicates that successful multicomponent diffusion
thermoeffect experiments can be carried out in an
adiabatic boundary cell with an initially sharp, isothermal
compositional gradient as the driving force if (1) temperature
differences of points symmetric about the initial
interface are measured as a function of time and (2) two
experiments with different initial starting gradients are
performed at each mean or bulk composition in order to
decouple the heats of transport.
The solution of the composition and energy equations
in PVR1 was obtained using a double perturbation technique.
The thermophysical properties were expanded
in a Taylor's series expansion about the mean composition
while the cross diffusion terms in the composition
equation were treated as a perturbation to the straight
diffusion terms. It was shown that the zeroth order
composition solution was sufficient for those systems
which have small cross diffusion coefficients relative
to the straight coefficients. Furthermore, the zeroth
order solution to the temperature equation is by far the
predominant term in the expansion unless the system
shows a large compositional dependence in its thermophysical
properties. For composition independent properties,
the zeroth order solution is in fact the exact
solution.
The toluene -chlorobenzene -bromobenzene system
was chosen for this work because of the availability
J. Chem. Phys. 77(4). 15 Aug. 1982 0021-9606/82/162121-09$02.10 1982 American Institute of Physics 2121
2122 Platt, Vongvanich, Fowler, and Rowley: Heats of transport in liquid
II. EXPERIMENTAL
Rowley and Horne12 developed a liquid-gate withdrawal
cell for measurement of heats of transport in binary
liquid mixtures. Highly accurate heats of transport can
be obtained by this technique as evidenced by the agreement
shown between Onsager coefficients obtained from
diffusion thermoeffect experiments and those obtained
from thermal diffusion factors. 11 The disadvantages of
this type cell are that the gate fluid must be immiscible
of diffusion coefficient data and the moderate composition
dependence of its thermophysical properties. For
this system, the zeroth order temperature solution is
adequate and should provide values for the heat of transport
within the combined precision limits of the thermophysical
properties, the experimental technique, and
the parameter estimation procedure; we estimate this to
be roughly 10%. Temperature differences were measured
between points symmetric about the interface to
eliminate the heat of mixing effect as discussed in PVR1.
Under these conditions, Eq. (3.16) of PVRl applies:
2
6.T(z')= (4/a2Cp ) L (QtD11 + QrD21)[ (6.wI)/(ej} - 7-
1
)]
1=1
F ll =1T(-l)/{exp[- (21-1)2t/e ll ]
- exp[ - (21-1)2t/ 7 ]}/(21-1) , (1. 2)
7= pCpa2/1T 2k , (1. 3)
ell = a2/1T 2Dll , (1. 4)
z' is distance of either thermocouple probe from the initial
interface (both thermocouples are equidistant from
the interface), a is total cell height, Cp is specific heat,
DI } are mutual diffusion coefficients, t.wI is the initial
composition of component i in the upper phase minus
that of the lower phase, p is density, and k is thermal
conductivity.
Equations (1.1)-(1. 4) form the basis for obtaining Qt
and Qr from temperature measurements as a function
of time. The values for a particular composition can be
obtained by nonlinear least-squares fitting of calculated
and measured temper.ature differences by treating Qt
and Qr as two simultaneously adjustable parameters.
However, as was pointed out in PVR1, two experiments
must be done at each mean composition in order to decouple
the parameters. While there is actually a composition
nonuniformity in the system throughout the
duration of the experiment, the composition at which the
heats of transport are measured is rigorously defined
by the Taylor's series expansion performed in obtaining
the final solution. ThUS, all properties are to be evaluated
at the mean or average mass fraction located at
the center of the cell about which the expansion was
made. Hence, values of thermophysical properties,
including the obtained heats of transport, are evaluated
at wjA=(ww + w1L )/2 for each component, where Ww and
WIL represent the initial upper and lower phase compositions,
respectively, of component i.
where
..
xL(-l)IFll sin[(21 -l)1Tz'/a] ,
1=1
(1. 1)
A
B
0
= =C l [
1/\
F
FIG. 1. Schematic diagram of liquid gate withdrawal cell. A
Cell chamber. B. Thermocouples. C. Cell equatorial rim.
D. Vacuum jacket. E. Withdrawal ports. F. Ground glass
fittings for support of thermocouples and filling and venting of
cell.
in all other components and must further have a density
intermediate to the two initial phases. This becomes
a burdensome requirement for ternary systems because
two independent experiments must be performed at each
mean composition with different initial composition differences.
In order to fit both Qt and Qr the composition
differences must be large enough to decouple the parameters
which often requires mixtures out of the proper
density range.
A new type of cell was designed to proVide much more
versatillity in initial interface creation. The boundary
sharpening cell developed is based on the concept used
in Tiselius diffusion cells, sharpening an initially somewhat
diffuse interface by siphoning out fluid at the interfaciallevel.
13 To eliminate any possibility of a celldependent
systematic error, the two runs required for
analysis of Qt and Ql at each mean composition were
made in the two different types of boundary formation
cells.
The liquid-gate withdrawal cell used in this work was
similar to that used by Rowley and Horne. A schematic
diagram is shown in Fig. 1. Details of the liquid-gate
withdrawal technique and use of this type of cell are given
elsewhere. 12 This particular cell measured 3. 00 em
in height and 6.0 cm in diameter. The outer jacket was
evacuated following thermal equilibration to provide
adiabatic boundary conditions. Thermocouple leads
were secured, at positions symmetric about the interface,
with a "filled epoxy" into small ground glass fittings,
which also served as filling ports. Thermocouple
beads were made by arc welding precision 40
gage copper and constantan wires. Bead sizes were
consistently of radius 0.2 mm. Bead locations were
measured visually with a measuring cathetometer; uncertainty
in the temperature measurement loci is equivalent
to bead radius of 0.2 mm. A circumferential
rim or bulge at cell half-height contained a narrow ring
opening into the cell and thereby provided fixation of
the interface locus as well as a microreservoir for the
remainder of the liquid -gate fluid following contact of
the upper and lower phases. This rim has been shown
to be required for this type of cell in order to prevent
J. Chern. Phys., Vol. 77, No.4, 15 August 1982
Platt, Vongvanich, Fowler, and Rowley: Heats of transport in liquid
boundary formation at cell half-height.
2123
J
A
D
FIG. 2. Schematic diagram of boundary sharpening cell. A.
Cell chamber. B. Thermocouples. C. Teflon body. D. Teflon
end cap. E. Withdrawal ports. F. O-ring. G. Septum
and cap for thermocouple lead seal to the tubes attached to
micrometer heads. H. Brass securing ring. I. Alignment
pin. J. Filling port.
geometrical constriction of the diffusional area by any
remaining gate fluid. 12 Water was used as the liqUid
gate in all of the experiments performed in this cell due
to its insolUbility in the other components. The solubilities
of bromobenzene, chlorobenzene, and toluene
in 100 g water at 20 C are listed in Perry's Handbooku
as 0, 0.05, and 0.05 g, respectively. Water was withdrawn
from four equatorially placed ports at O. 2 cm3/
min. The cell was vibrationally isolated from the room
and air bath by layers of sand and foam pads. This ensured
a smooth, full-term contact between the two
phases. The upper phase, initially contained in a
syringe cothermostated in the air bath with the actual
cell, was brought into the upper portion of the cell
through a small diameter copper tube heat exchanger
as the liquid gate was withdrawn.
The second type of cell employed was the boundary
sharpening cell diagrammed in Fig. 2. The cell body
was machined from a 10 cm diameter Teflon rod such
that the inside cell diameter was 3. 0 cm. The end caps
were also machined from 3.0 cm thick slices of the Teflon
rod such that when both caps were seated againsttheir
respective O-rings by brass screw caps, inside cell
height was 2. 97 cm. In this type of cell, adiabatic
boundary conditions were maintained due to the thick,
low thermal conductivity end caps. Additionally, both the
inside and outside surfaces of the end caps were at essentially
the same temperature since the entire apparatus
was in a constant temperature environment thermally
equivalent to the initial cell temperature. Furthermore,
cell temperature responses were small so that
conduction through the end caps was negligible. Four
equatorial 1/64 in. holes were drilled to provide exact
Thermocouples were again made from 40 gage precision
copper-constantan thermocouple wire with 0.2
mm bead radius. Thermocouple positions were more
accurately controlled in this type of cell as the beads
were enclosed in 25 gage stainless steel tUbing which
was in turn connected to precision micrometer dials;
the cell-tubing seal being made with a silicon septum.
This arrangement allowed the thermocouple positions
to be fixed quite accurately from run to run. Again the
uncertainty in position was due to finite bead size 0.2
mm. The introduction of thermocouple leads into the
boundary sharpening cell provided less interference to
the diffusion thermoeffect than the liquid-gate cell
counterpart. There the leads run vertically through the
fluid at the radial center of the cell. This is also where
the response is to be measured in accord with Eq. (1. 1)
which was derived in PYR1 from the one-dimensional,
vertical, equations of change. Therefore, some conduction
loss through the leads would cause slightly modified
readings. In the boundary sharpening cell configuration,
the small stainless steel thermocouple sheath is secured
in a snug, narrow and shallow, groove in the cell wall
(see Fig. 2) where it is removed from the central portion
of the cell where the effect is to be measured. The
stainless steel tUbing has a sharp 90 bend in it to locate
the thermocouple bead at the cell's radial center. Because
of the diffusion thermoeffect's one-dimensional
nature in this type of cell, the temperature response is
independent of the radial coordinate, and little interference
due to thermocouple leads should occur. The
actual thermocouple lead extends from the RTY sealed
end of the tUbing about 3 mm to further minimize any
disturbance due to the support tUbing.
The principle of the boundary sharpening cell is the
same as that of the typical Tiselius diffusion cell. An
initial interface formed at a three -way stopcock is
lowered as the more dense fluid (which initially filled
the entire cell) is withdrawn through the side ports.
Siphoning of the fluid continues until the mixed fluid in the
diffuse boundary has been removed, sharpening the
boundary to a distinct sharp interface. Preliminary
experiments were conducted in an analogous glass visual
cell. Siphon times in the actual Teflon cell were determined
from these studies in conjunction with cell dimensions.
Interferometric studies have conclusively shown
that this type of initial interface formation is superior
to most other techniques especially with respect to turbulence.
15 Actual siphon rates were maintained at a constant
1.23 mljmin with a syringe pump until sufficient
time had elapsed for boundary sharpening. As with the
liquid gate cell, the less dense fluid entering the cell had
been cothermostated with the cell. Initiation of the diffusion
thermoeffect occurred when the siphoning action
of the syringe withdrawal pump was shut off.
Although the fluid is being siphoned at a rate faster
than diffusion in order to obtain an initial sharp boundary,
there is no guarantee that the isothermal initial
condition upon which Eq. (1. 1) was based will be maintained.
It is therefore imperative to look at the diffusion
thermoeffect response for nonuniform initial tem-
J. Chern. Phys., Vol. 77, No.4, 15 August 1982
2124 Platt, Vongvanich, Fowler, and Rowley: Heats of transport in liquid
perature profiles. Rowley and Horne16 have simulated
various nonuniform initial starting temperatures and
found that in all cases, the temperature difference measured
between symmetrically located thermocouples relaxes
back to that predicted by solution of the energy
equation based on the isothermal initial condition; e. g. ,
Eq. (1. 1). This is due to the competing effects of the
two portions of the heat flux
the z' values used in these experiments. The zero of
time for the boundary sharpening cell is precisely known;
it is the time at which the siphon pump is turned off.
All mixtures were prepared gravimetrically from
"Baker analyzed" reagent grade pure components without
further purification. Distilled, deionized water was
used for the liqUid-gate cell experiments.
(2. 1) III. RESULTS
Qt =(MW1W3)[ -29.687 + O. 0706T + O. 08715Tw1
- (98.70 - O. 418T)w2+ (787. 3 - 2. 765T)W1W2
-21, 59u1-0.1071Tw ]Jlg, (3.1)
Qf = (MW1W2) [- 39.370 + O. 1302T - O. 01679Twt
- (120.6 - O. 4199T)w2 + (993. 7 - 3. 308T)wtW2
Equation (1, 1) was used to obtain Qt and Qf by simultaneous
nonlinear least-squares fitting of two independent
data sets, one obtained from an experiment performed
in the liquid-gate withdrawal cell, the other obtained in
the boundary sharpening cell. Both experiments were
performed at the same mean mass fraction for each
component but with different initial cOmpositions in the
two phases. This provides a stringent test on the accuracy
of the boundary sharpening cell since the data obtained
in two different cells must fit Eq. (1, 1) simultaneously.
In our experience, the goodness of the fit is
indicative of the quality of the experiments. A representative
example of the quality of data fit obtained from
this procedure is shown in Fig. 3. In the fitting procedure,
all thermophysical properties are evaluated at
the mean mass fraction as required by the perturbation
solution discussed in PVR1. The composition and temperature
dependence of the thermophysical properties
for the toluene(l )-chlorobenzene(2)-bromobenzene(3)
system as well as the literature source from which they
were obtained are shown in Table I. These equations
were directly used to obtain property values for the experimental
conditions of each run.
Experiments were performed at various ternary compositions
at 25 and 35 e. No values could be obtained
close to the binary limits where the fitting technique
could not provide values for both Qt and Qf. This is
to be expected since there is only one independent heat
of transport in binary mixtures. eare was also taken to
insure that proper density differences were maintained
between the upper and lower phases so that sharp initial
inter faces could be formed in one or the other of the two
types of cells. However, this limited the composition
region over which the experiments could be performed.
Experimental results are shown in Table II.
The compositional dependence of Qt and Qf is not immediately
obvious from Table II, therefore Qt and Qr
were separately fit as a polynomial in mass fractions.
The compositional behavior of Q1* and Ql at 25 and at
35 e are displayed in Figs. 4-7 and are analytically
given by
where qc is the conduction flux due to thermal conduction
down the temperature gradient and qt is the transported
heat flux due to heat of transport down the composition
gradient. As long as the initial condition for
the composition profile matches that used to obtain Eq.
(1. 1) the conductive flux will relax back to the proper
balance. If a temperature gradient larger than that
which should balance the transported heat flux is initially
present, more heat is conducted than would be otherwise,
and the local temperature relaxes back to that
predicted by the diffusion thermoeffect with isothermal
initial conditions. If a temperature difference less than
that predicted by Eq. (1. 1) exists at short times due to
initial temperature nonuniformities then qc is smaller
than otherwise, and again AT comes back to that predicted
by Eq. (1, 1). This results in a "steady -state"
AT with respect to the heat being conducted and transported.
The steady state AT is not constant, however,
as the composition, hence the measured temperature
difference, slowly changes in time due to diffusion.
Temperature response data from the boundary sharpening
cell were only used for times greater than 500 s
for which Eq. (1.1) applies even when some temperature
nonuniformities exist at the time of boundary
formation. Rowley and Horne16 have shown that 500 s
is sufficient time for thermal relaxation back to the
steady state AT even for quite large initial temperature
nonuniformities.
Both cells were mounted on a vibrationally isolated
stand in an air bath maintained at the desired run temperature
to 0.01 e. Temperature differences were
measured directly by using the lower thermocouple as
the reference junction and the upper thermocouple as
the measuring junction. As temperature differences were
normally less than 0.3 e, the Seebeck effect potential
difference was amplified by a factor of 500 through a
Leeds-Northrup model 9829 linear amplifier. The
amplified signal was then periodically logged with a
Hewlett-Packard data logger and continuously monitored
with a strip chart recorder. In this manner, potential
differences of 0.03 J1.V were measurable with a
corresponding temperature difference resolution of
slightly better than O. 001 e.
The exact zero of time in the liquid-gate cell was unknown
because no visual observation of phase contact
could be made due to the smooth layering process. However,
the initial AT response was obtained from the
strip chart recording. A time lag parameter which depends
on z' was fit with each run and was added to the
initial response time shifting the time coordinate to
zero for initial phase contact. The time lag parameter
seldom exceeded 15 s and was usually less than 10 s for + 7. 006w - O. 04601Tw ]Jlg (3.2)
J. Chern. Phys., Vol. 77, No.4, 15 August 1982
Platt, Vongvanich, Fowler, and Rowley: Heats of transport in liquid 2125
AT
mK
100
FIG. 3. Simultaneous fit of data at 25 C,
wtA =0.397, and wZA =0.303 taken with
liquid gate withdrawal cell, -0-, and
with the boundary sharpening cell, -e-. The points represent the measured
AT in millikelvins while the
curves are predicted AT based on best
fit Q: and Q;. Run conditions in the
liquid gate cell were Awl =+0.795,
wz = - O. 195 while in the boundary
sharpening cell Awl = +0. 794, and
Awz= - O. 406.
OL--------....L.---------'------- o 3600 7200
tIs
The goodness of the fit of the data to empirical Eqs.
(3. 1) and (3. 2) is also shown in Table II.
IV. DISCUSSION
It is evident from the standard deviations shown in Table
II that the analysis procedure for diffusion thermoeffect
data for this system yields more reliable Qt results
than Qt. This appears to be due to the relative
magnitudes of Qt and Qt. The simulation of the diffusion
thermoeffect done in PVR1 indicated that for this
system and under conditions similar to the runs performed
in this work, a change in the value of Qt produced
a much larger variation in the measured AT than
an eqUivalent change in the value of Qt. It was also
shown, and is evident from Eq. (1. 1), that Qt and Qt
affect the magnitude of the AT response but not the time
dependent shape. Actually it is not Qt that directly affects
AT, but rather the product QTDIJAwJ' Since the
cross diffusion terms are small, the main effect is from
QTDjjAWf In the case of the numerical simulations, the
larger sensitivity of AT to Qt is due to larger input values
of AWt. In the case of the actual experiments, the
larger uncertainty in fit Qt values is due to the smaller
magnitude of Qt relative to Qt. Experimentally, this effect
was reduced in several cases by beginning the experiments
with larger values of AWz However, the
range of AWf values were limited by density considerations;
phase densities were necessarily limited by consistency
with experimental techniques. In future work,
consideration should always be given to the probable
magnitude of each QT and attempts should be made to
design experiments which optimize the parameter estimation
precision for both QT. The fit smooth curves
of Eqs. (3.1) and (3.2) represent the parameters Qt
and Qf fairly well on an absolute basis. Percent deviations
from these equations are large at times due to the
small absolute value of Qt and the resultant lesser
sensitivity to the AT fitting procedure.
It should be noted that the fit polynomial equations for
Qt and Qt were obtained at the compositions displayed
in Table II. This represents a triangular region of composition
defined by O. 10:s wf:S O. 60 for i = 1, 2 and W3
2: O. 1. Data reduction on experiments performed outside
of this region did not lead to independent values for Qt
and Qt because in the limit of the three binary systems,
only one heat of transport is independent. ThUS, Eqs.
(3. 1) and (3.2) should be used only within the region defined
above as is the case with Figs. 4-7. Within this
region, interpolative values of Qt and Qt can be expected
to be quite accurate; extrapolative estimates from these
analytical expression should not be made.
0.60
"I 0.10
Q' --i7
3
-5
0.60 -13
-40
-54
0.60 ----,--
-26
FIG. 4. Compositional dependence of Q; at 25 C from Eq.
(3.1).
FIG. 5. Compositional dependence of Qi at 25 C from Eq.
(3.2).
J. Chern. Phys., Vol. 77, No.4, 15 August 1982
2126 Platt, Vongvanich, Fowler, and Rowley: Heats of transport in liquid
TABLE I. Thermophysical properties for the toluene (1)-bromobenzene(2)-chlorobenzene(3) system.
Part A-defining equations
(1). Density (kg/m3
)
p=iJII+,x, V + t tv",); V =7Jlexp(7J2T)+7JT2+7J4T3]
/\jj J=1 k=J
3
V:k=xJXk BIJk txJ-Xk) 1-1; BlJJi=!3 1 +!32T +!33 T2
(2). Diffusion coefficients (m2Is)
Du =D13 [(1 -Xl)Dt'2+X1D2 ]/1/'; D12 =XtV;,(Vt. -Vil)/1/'
D22 =D;3 [(1 -x2)D21 +x2D i3]/1/'; D21 =x2Di3W{3 -D21 )/1/'
T,I, =U XIDJk'(i j k=l,2,3); DI,J=DJ,I = -.!:.L- DQIJ+-x - DJ0I
/=1 xJ+XI xJ+ 'l:1
Ref.
17
18
(3)
D J =(01 xl0-10) (T) exp(02 IT)
Specific heat (kJ/kg 'K)
3 3 3
c = f;i WI C .I + f.f xJ"kFJkl(MJxJ+MkXk) (j k)
2 3
j ='Y1+12 T +1a T3; F Jk = L; , {xJ -Xk)1-1
, I
19-25
(4). Thermal conductivity (W1m K)
3 3 3 3
k = '5: Wi k + L; WI L; wJG JI (kJI -k )1 '5: WIG Ii
t;t 1=1 J=1 t;1
Part B-parameter values
Index
i=l, j=l, k=2
i=2, j=l, k=2
i=3, j=l, k=2
i=l, j=l, k=3
i=2, j=l, k=3
i=3, j=l, k=3
i=l, j=2, k.,3
i=2, j=2, k=3
i=3, j=2, k=3
i=l
i=2
i=3
i=l, j=2
i=l, j=3
i=2, j=l
i=2, j=3
i=3, j=l
i=3, j=2
i=l
i=2
i=3
j=l, k=2
j=l, k=3
j=2, k=3
j=l, k=2
j=l, k=3
j=2, k=l
j=2, k=3
j=3, k =1
j=3, k=2
i=l
i=2
i=3
Parameter 1=1
-4.9011 x 10-3
-2.6431X10-3
-1.6452 X10-1
5.596 9x 10-2
-3.3617xI0-2
1. 453 2 x 10-1
1. 348 7 x 10-3
-2.3303X10-4
1. 2961 x 10-3
8,35773X10-2
7. 98155x 10-2
7.87366 x 10-2
2.13001
L 947 59
2.37125
1,87952
2.28082
2.08268
1.486
1.111
0.879
2.'H
7.91
-8.24
76.951 J/mol
304.6918 J/mol
-482.7839 J/mol
341.6425 J/mol
- 572. 402 0 J/mol
- 238.0783 J/mol
0.2153
0.1991
0.1658
1=2
2.5064 x 10-5
1.6793x10-5
1. 042 9 x 10-3
-3.5858x10-4
2.1313x10-4
-9.2044x10-4
-8.1481 X10-6
1.5887 x 10-6
-8.2872X10-6
5.22475x10-4
6.82333 x 10-4
1.24020 X10-3
-1083.34880
-1130.16898
-1033.82635
-1130.16898
-1033.82635
-1083.34880
- 3.861 x 10-6
o
o
1.3
-6.5
-6.7
0.4219
0.2771
0.5889
0.3533
0.7245
0.6413
2.88 x 10-4
2.51 X10-4
1. 98 X 10-4
1=3
-3.,40><10-8
-2.66><10-8
-1.'35X10-6
5.68 X10-1
-3.38><10-1
1.46)( 10'"
1.24 ><10-8
-2.70x10-9
1. 32 X 10-8
1. 22126x 10-6
4. 90x 10-1
-1. 46378 x 10-6
1. 93 X 10-8
6. 985x 10-9
3.253x10-9
-2.. 2
12.1
6.0
0.5889
0.7245
0.4219
0.6413
0.2771
0.3533
1=4
- 6. 667 X 10-10
o
2.0 X 10-9
J. Chern. Phys., Vol. 77, No.4, 15 August 1982
Platt, Vongvanich, Fowler, and Rowley: Heats of transport in liquid
TABLE II. Heats of transport of toluene(1)-chlorobenzene(2)-bromobenzene(3) ternary liquid
mixtures at 1 atm.
Units=kJ!kg
Deviation Deviation
from from
T!K WiA w2A -Q; Eq. (3.1) -Qi Eq. (3.2)
298 0.100 0.500 18.2 0.5 -3.1 9.3 0.1 -0.6
298 0.200 0.200 20.0 0.1 8.6 5.9 0.2 3.6
298 0.200 0.400 16.5 0.2 2.5 10.3 0.1 -0.2
298 0.200 0.600 20.1 0.1 1.3 1l.4 0.3 0.3
298 0.362 0.317 15.9 0.2 0.9 6.1 0.7 1.0
298 0.397 0.303 18.1 0.2 -1.7 9.7 0.4 -3.8
298 0.450 0.174 16.8 0.05 -4.3 9.8 0.2 -1.4
298 0.600 0.200 19.3 0.3 -2.6 -4.1 0.6 1.6
308 0.200 0.200 23.0 0.2 -3.1 -0.2 0.3 -2.0
308 0.200 0.400 19.8 0.1 -2.4 9.7 0.1 -1.7
308 0.200 0.600 21.1 0.1 1.3 13.0 0.3 0.3
308 0.325 0.175 16.4 1.2 0.9 9.7 2.4 1.3
308 0.326 0.274 20.9 0.1 1.3 14.3 0.2 3.4
308 0.397 0.303 25.9 0.2 5.3 22.6 0.3 3.4
308 0.400 0.100 21. 3 0.1 -8.7 16.6 0.4 -4.8
308 0.450 0.450 33.9 0.4 -3.5 21.2 0.6 -0.8
308 0.600 0.250 29.6 0.6 5.7 22.4 0.8 -0.1
aLinear estimate of standard deviation of the fit parameter.
2127
As Fig. 3 indicates, runs performed in both types of
cell agree well. This indicates that no systematic errors
occurred and that the boundary sharpening cell can
be used, exclusively if desired, to obtain accurate heats
of transport. This should permit much more extensive
diffusion thermoeffect work as experiments can be performed
on those systems for which the liquid gate withdrawal
cell is not appropriate. Much more versatility
is available with respect to density and constituency
when the boundary sharpening cell is used for both runs.
The predominant feature of Qt displayed in Figs. 4-7
is the rapid increase in absolute value as the mass fraction
of component three is increased. Why this occurs
is not known, but it is evidently correlated with the much
larger density and/or higher molecular weight of bromobenzene
relative to the other components. Toluene and
chlorobenzene have densities which are more similar
to each other than they are to bromobenzene and so the
compositional effects along lines of constant w3 are not
nearly as large as those along either constant W1 or W2'
The other significant feature of Qt and Qr is that the
absolute value of both generally increases with increasing
temperature. This is more clearly observed by
comparing actual values in Tables II than from comparison
of Figs. 4-7.
There is a known sensitivity of cross transport coefficients,
such as the heat of transport, to intermolecular
potential parameters in gases. 1.2 Thus, obtaining
intermolecular potential parameters using some cross
transport coefficient data can lead to more appropriate
values than the sole use of diffusivities or thermal conductivities.
While theories of liqUid mixtures are not
as advanced as those of the gas phase, heats of transport
should, by the very nature of the phenomenon which
they characterize, yield significant information concerning
intermolecular interactions. The heat of transport
represents the heat transported by the isothermal
molecular diffusion of a molecule from a region of one
intermolecular environment to a new environment and
should therefore provide valuable information about
those interactions.
While the data obtained to date for heats of transport
Qj
J7
10
-3
-16
-28
0.60
WI 0.10
0.10
0.60
FIG. 6. Compositional dependence of Qt at 35 C from Eq.
(3.1).
FIG. 7. Compositional dependence of Q; at 35 C from Eq.
(3.2).
J. Chem. Phys., Vol. 77, No.4, 15 August 1982
2128 Platt, Vongvanich, Fowler, and Rowley: Heats of transport in liquid
are insufficient for development of quantitative correlations
of theories for the diffusion thermoeffect, we
would like to suggest a descriptive model which is consistent
with the features of Qt and t observed in this
study as well as those of Qt measured in binary systerns
by Rowley and Horne. 12 ,16 Kinetic interpretations
of the heat of transport formulated by Wirtz, 27 Wirtz
and Hiby, 28 Denbigh, 29 Prigogine et al. , 30 Dougherty
and Drickamer, 31 and Rutherford and Drickamer, 32 treat
the liquid structure as quasilatticelike. The heat of
transport is assumed to be the difference in the local
energy of a molecule at one lattice site and its energy
at the next site after an energetic jump between the two.
This concept, however, is inconsistent with the more
fundamental view of diffusion as a randomizing process
due to the collisions between molecules in constant
thermal motion. Moreover, the diffusion thermoeffect
must be related to the actual diffusional process not
just the environment in which the molecule finds itself.
We propose instead that the heat transported by the diffusional
process is inherently coupled to the diffusion
mechanism.
In an initially isothermal liquid mixture, the velocities
of each molecule vary but obey a distribution law, hence
there are molecules which at anyone time have velocities
larger than the average velocity characteristic of
the system temperature. For convenience of discussion,
these molecules have an "excess energy" over and
above that of their average counterparts. Now when two
isothermal phases at different compositions are brought
into contact such that mutual diffusion begins, the more
energetic molecules will diffuse more rapidly than their
average or below average energy counterparts. Although
the collisions continuously randomize the energy and
velocity of the molecules, the net effect is the diffusion
of molecules carrying the excess energy. This same
phenomenon occurs in the opposite direction, but the
excess energy carried from the two separate original
phases is not equivalent because the breadth of the distributions
depends on the various properties of the
phases. Since the two phases were initially isothermal,
the mean energy of the distributions were initially
equivalent, but the distribution widths differed because
of the constituent component properties. This means
that diffusion occurs principally by the more energetic
molecules which in turn carry excess energy. The
difference between the excess energy carried in the two
directions is directly related to the difference in the
relative widths of the energy distributions in the two
phases and corresponds to the temperature rise and
decrease in the two phases as the diffusion thermoeffect
occurs.
This picture of the heat of transport is suggested as
a model for understanding the trends in the heats of
transport as measured in this work. A quantitative
theory is not implied. Molecular dynamics simulations
of heat effects involved in the diffusional process
should be perfoormed to begin development of a theory.
Nevertheless, several features of our diffusion thermoeffect
are now explicable.
First, the temperature above the interface always
warmed due to diffusion while the lower fluid cooled.
This is a density effect. The more dense or lower fluid
apparently has the widest distribution of energies about
the expectation value. Hence, the faster molecules
from the more dense phase carry more excess energy
than those from the less dense phase. The net result,
as the energies of the diffusing molecules are randomized'
is an elevation of the average energy of the upper
portion of the cell and a lowering of the average energy
of the lower portion of the cell. The diffusional process
driven by the nonuniformity of chemial potential
thereby acts as a natural Maxwell's demon allowing an
"energetic" molecule (relative to the lower phase distribution
for that component) to move into the top phase
and an energetic molecule (relative to the upper phase
distribution but "not quite as energetic" relative to the
lower phase) to move into the bottom phase.
Secondly, our measurements show a very large density
or molecular weight dependence as Qt and Qt change
dramatically due to an increased amount of bromobenzene.
It must be remembered that Qt is the heat flux
per isothermal mass flux of component i under conditions
such that all other component mass fluxes except jn are
zero. Thus, two heats of transport are required for
ternary systems to describe the transfer of energy due
to the two interdiffusional processes between components
one and three and between components two and
three. The increased bromobenzene concentration has
the effect of widening the upper and lower phase energy
distributions, due to its larger molecular weight, by
differing amounts, thereby increasing the magnitude of
QT
Thirdly, an increase in temperature would increase
the breadth of the energy distributions, but again unequally.
One would expect the wider distribution to spread
out more than the narrower distribution for a given temperature
increase. This corresponds to an increase
in the diffusion thermoeffect measured t::. T, consistent
with our experiments.
The critical exponent for QT measured by Rowley and
Horne l6 indicates that the heat of transport goes to zero
as the liquid-liquid critical temperature is approached
along the critical composition line. This is also consistent
with the proposed model. As the critical point
is approached, the intermolecular correlation length
diverges. As the molecules become more correlated,
the distribution of energies necessarily narrows. In
the limit of perfect correlation, the distributions become
spikes at the same mean temperature value. In
this idealized limit, all molecules would have the same
energy and QT would be zero.
Although this interpretation of the diffusion thermoeffect
views the t::.T response to diffusion as a function
of the initial composition differences t::.w., it does not
imply that the QT are dependent upon the composition
gradients. Indeed, they cannot depend upon t::.w. and
still be consistent with the assumption of linear forceflux
relations. Instead, the QT must be thought of as
the energy differences in the energy distribution widths
evaluated in the limit as t::.w 1 approaches zero.
J. Chern. Phys., Vol. 77, No.4, 15 August 1982
Platt, Vongvanich, Fowler, and Rowley: Heats of transport in liquid 2129
V. CONCLUSIONS
Heats of transport have been measured in ternary
mixtures of toluene -chlorobenzene -bromobenzene
as a function of composition at 25 and 35 C. In so doing,
a new type of diffusion thermoeffect cell has been
designed and tested which allows a much wider flexibility
in mixture measurements than the previously used
liquid gate withdrawal cell. The new cell is based on a
boundary sharpening technique for the creation of the
initial diffusional interface and yields results in excellent
agreement with the liquid gate withdrawal cell.
The measured heats of transport show a direct dependence
on the temperature and a very strong dependence
upon the composition of the mixtures in regions of high
bromobenzene concentration. This is presumed due to
the much larger density and molecular weight of bromobenzene
compared to the relatively close densities of the
other two components. These facts are in agreement
with a new interpretive model of the diffusion thermoeffect.
As heats of transport are of interest in developing
a consistent liquid mixture model and in understanding
the liquid mixture structure, more ternary data are required.
Additionally, binary heats of transport and
their relation to their mUlticomponent counterparts need
to be studied. Compositional and temperature dependencies
also require further study. Additionally, several
ideal and nonideal systems should be examined to determine
the effect chemical consistency and mixture nonidealities
have upon the magnitudes of QT. Finally, it
is suggested that molecular dynamics calculations be
used to test the appropriateness of the molecular explanation
of the diffusion thermoeffect presented here.
ACKNOWLEDGMENT
This material is based upon work supported by the
National Science Foundation under Grant No. ENG7907999.
IJ. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular
Theory of Gases and Liquids (Wiley. New York, 1954).
2E A. Mason. R. J. Munn. and F. J. Smith. Discuss. Fara.day
Soc. 40, 27 (1965); E. A. Mason, M. Islam, and S.
Weissman, Phys. Fluids 7, 1011 (1964).
3G. Platt, T. Vongvanich, and R. L. Rowley, J. Chern. Phys.
77, 2113 (1982).
4R. P. Rastogi and G. L. Madan. Trans. Faraday Soc. 62,
3325 (1966).
SA. Boushehri and S. Afrashtehfar. Bull. Chern. Soc. Jpn. 48,
2372 (1975).
SA. Boushehri, J. Chern. Eng. Data 19, 313 (1974).
7B. L. Sawford, T. H. Spurling, and D. S. Thurley, Aust. J.
Chern. 23, 1311 (197Q).
8E A. Mason. L. Miller. and T. H. Spurling, J. Chern. Phys.
47, 1669 (1967).
8S. R. deGroot and P. Mazur, Nonequilibrium Thermodynamics
(North-Holland. Amsterdam, 1962).
lOS. E. Ingle and F. H. Horne, J. Chern. Phys. 59, 5882 (1973).
IIR L. Rowley and F. H. Horne. J. Chern. Phys. 68, 325
(1978).
12R. L. Rowley and F. H. Horne. J. Chern. Phys. 72, 131
(1980).
13H. J. V. Tyrell. Diffusion and Heat Flow in Liquids (Butterworths,
London, 1961).
Up. E. Liley and W. R. Gambill, Chemical E7ffineers' Handbook,
5th ed edited by R. H. Perry and C. H. Chilton
(McGraw-Hill. New York. 1973).
ISO. Bryngdahl, Acta Chern. SCand. 12, 684 (1958).
I . L. Rowley and F. H. Horne. J. Chern. Phys. 71, 3841
(1979).
I7R K. Nigam and P. P. Singh. Trans. Faraday Soc. 65,. 950
(1969).
18J. K. Burchard and H. L. Toor, J. Phys. Chern. 66, 2015
(1962).
18J. W. Williams and F. Daniels. J. Am. Chern. Soc. 47, 1490
(1925).
2 . Tanaka and G. C. Benson. J. Chern. Eng. Data 21, 320
(1976).
2IR'. K. Nigam. P. P. Singh. and N. N. Maini. Indian J. Chern.
8, 908 (1970).
22J. Canning and G. H. Cheesman. J. Chern. Soc. 1955, 1230.
23S N. Bhattacharyya. A. V. Anantaraman. and S. R. Palit.
Physica (Utrecht) 28. 633 (1962).
24B. S. Harsted and E. S. Thomsen. J. Chern. Thermodyn.
7, 369 (1975).
25K. Amaya. Bull. Chern. Soc. Jpn. 34, 1349 (1961).
2lR. L. Rowley. Chern. Eng. Sci. 37, 897 (1982).
27K. Wirtz. Ann. Phys. Leipzig 36, 295 (1939).
28K. Wirtz and J. W. Hiby. Phys. Z. Leipzig 44, 369 (1943).
28K. G. Denbigh. Trans. Faraday Soc. 48, 1 (1952).
3ot. Prigogine. L. deBrouckere. and R. Armand. Physica
(Utrecht) 16, 577 (1950).
31E L. Dougherty and H. G. Drickamer. J. Phys. Chern. 59,
443 (1955); J. Chern. Phys. 23, 295 (1955).
32W. M. Rutherford and H. G. Drickamer. J. Chern. Phys. 22,
1157 (1954).
J. Chern. Phys., Vol. 77, No.4, 15 August 19