AT. K.
1854
NAGARAJAN AND
J. o
w BOCKRIS
Diffusion in Molten Salts at Constant Volume
by M. K. Nagarajan and J. O’M. Bockris T h e Electrochemistry Laboratory, T h e University of Pennsylvania, Philadelphia, Pennsylvania 19104 (Received December 2 , 1966)
Self-diffusion coefficients of NaZ2and Cs134ions in their respective nitrate melts have been measured as a function of applied pressure a t four arbitrarily selected temperatures. The activation energies a t constant volume (E,) were found to be about 1/5 of the value of the corresponding experimental activation enthalpies at constant pressure (E,). The “activation volumes” were found to be approximately equal to the most probable volume of holes in the liquid. It is concluded that the predominant contribution to the numerical value of E , is the enthalpy of formation of holes in the liquid and that the formation of holes constitutes the rate-determining step in the diffusion process at constant pressure. It was also found that as the size of the diffusing ion increases (in relation to the hole volume in the diffusion medium), a corresponding increase occurs in both E, and the activation volume for jumping.
Introduction The temperature dependence of diffusion of ions in molten salts is generally represented, at constant pressure, by the empirical equation
D
=
Doexp(-E,/RT)
(1)
where Dois a temperature-independent term and E, is the experimental activation enthalpy for the diffusion process. However, when the temperature increases, the volume of the system increases simultaneously. Therefore, the experimental activation enthalpy (E, of eq 1) is not a simple activation enthalpy for diffusion, since it contains a term related to the volume expansion work. The necessity for constant volume transport studies in liquids has long been recognized, but there have only been a few scattered references in the literature to constant volume transport data in liquid and (room temperature) organic liquids.3-5 The first measurements of diffusion coefficients in molten salts, at constant volume, have recently been reported from this laboratory.6 A comparison of the activation enthalpies at constant pressure (E,) and the activation energies a t constant volume (E,),7 for a few representative liquids, is made in Table I. An analysis of results in Table I shows that, in liquids, the activation energies for diffusion a t constant volume are usually smaller than the activation enthalpies at The Journal of Physical Chemistry
constant pressure. Alternatively, one can observe that the volume expansion work (vix., E, - E,) is of importance in deciding the value of experimental actiTable I: Comparison of Heats of Activation a t Constant Volume and Constant Pressure
Liquid
AIP, OK
CClr C6H6 Hg Ga NaNOa
250.3 278.6 234.2 302.9 580.0
E,,
Ev,
kcal
kcal
mole-’
mole -1
Ev/Ep
1.1
0.33 0.25 0.96 0.89 0.18
3.3 2.8 1.0 1.1 4.3
0.70 0.96 0.98 0.78
(1) N. H. Nachtrieb, Symposium on Liquid Metals and Solidification, Chicago, Ill., 1957. (2) K. Furukawa, Nature, 184, 1209 (1959). (3) H. Watts, B. J. Adler, and J. H. Hildebrand, J . Chem. Phys., 23, 659 (1955). (4) H. Hiraoka, J. Osugi, and 15‘. Jono, Rev. Phys. Chem. J a p a n , 28, 52 (1959). (5) 8 . Jobling and A . S. C. Lawrence, Proc. Roy. SOC. (London), A206, 257 (1951). (6) (a) S.B. Tricklebank, L. Nanis, and J. O’RI. Bockris, Reu. Sci. Instr., 35, 807 (1964); (b) M. K. Nagarajan, L. Nanis, and J. O’M. Bockris, J . P h y s . Chem., 68, 2726 (1964). (7) In this paper, the term activation enthalpy will be used only when the process being considered occurs at constant pressure and for the processes occurring at constant volume the term activation energy will be used (since ( A H ) p = AE PAV and ( A H ) ” = A E ) .
+
DIFFUSION IN MOLTENSALTS AT CONSTANT VOLUME
1855
vation energy for diffusion in liquids; in fact, the results for molten sodium nitrate suggest that E , is predominantly determined by the volume expansion work. The volume expansion work (or the energy to form free space) in liquids can be computed on the basis of structural modelss-10 for the liquid state. Hence, the experimental determination of E , at constant volume is of great significance in a critical evaluation of different structural models for liquids.” In particular, one of the most important criteria for the density fluctuation hole model15 is that its numerical predictions are based on the assumption that the activation energy for diffusion a t constant volume is small in comparison to the activation enthalpy at constant pressure. For the marked agreements between the model’s predictions and experiment to carry full weight, it is necessary to know that E , is indeed small compared with E p for molten salts.
a t a given temperature is chosen (e.g., 45 cc mole-1 a t 350” and 1 atm; cf. Figure 1) and a constant volume line drawn in the (V-P), diagram as shown by the dotted line in Figure 1. This line intersects the various (V-P). isotherms and each of these intersection points then gives the pressure and temperature a t which volume is 45 cc mole-’. For example, in Figure 1, these pressures are shown as Pz, Pa, and P4 at 370, 390, and 420°, respectively. One then refers to log D us. pressure plots (Figure 2) and picks out the value of log D corresponding to the “constant volume pressures,” viz., Pz, Pa, and Pd. The activation energy at constant volume is then obtained14bby plotting these interpolated log D values against the corresponding reciprocal absolute temperatures.
The Principle of Constant Volume Measurement and Calculation The experimental activation enthalpy is obtained from eq 1 as
E p = -R(b In D / b l / T ) ,
(2)
The activation energy a t constant volume can similarly be defined as
E,
=
-R(b In D / b l / T ) ,
Experimental Section ( a ) Technique. Diffusion coefficients of (i) E a + in molten n’aSO3, (ii) Cs+ in molten CsN03, and (iii) Cs+ in molten NaN03, were determined in the temperature range approximately 100” above the melting point of the respective salt and over the pressure range of 1-1200 atm. The diffusion measurement technique used was the capillary-reservoir method of Bockris
(3)
Of these two quantities, E , is easily evaluated from an experimental plot of log D us. 1 / T , a t constant pressure, e.g., atmospheric pressure. For the direct evaluation of EV,one should conduct diffusion studies in molten salts, keeping the material volume constant by the application of external pressure. However, direct measurement of diffusion coefficients in hightemperature liquids a t high pressures in closed containers involves great experimental difficulties (e.g., lack of suitable technique for measuring D , nonavailability of high-temperature pressure-transmitting fluids which are also stable in contact with molten salts, etc.) . Hence, the following alternative procedure was used in the evaluation of E,. Diffusion coefficients in a molten salt were first determined as a function of applied external pressure a t a series of constant temperatures (cf. Figure 2). Using known values of expansivitylz and ~ompressibilityl~of molten salts it is possible to construct (V-P), diagrams148at constant temperatures such as the one shown in Figure 1, for molten sodium nitrate. The isotherms of Figure 1 have been drawn a t the same temperatures as those a t which the log D vs. P data were obtained experimentally. From such a diagram, a certain constant volume
(8) H. Bloom and J. O’M. Bockris, “Fused Salts,” B. R. Sundheim, Ed., McGraw-Hill Book Co., Inc., New Tork, N. T., 1964, Chapter I. (9) R. A. Swalin, Acta Met., 7, 736 (1959). (10) F. H. Stillinger, “hlolten Salt Chemistry,” hf. Blander, Ed., Interscience Publishers, Inc., New York, N. Y., 1964, Chapter I. (11) A. Lunden, Transactions of the Chalmers University of Technology, Gothenburg, Sweden, No. 241 (1961). (12) H. Bloom, I. W. Knaggs, J. J. 310110y, and D. Welch, Trans. Faraday SOC.,49, 1458 (1953). (13) J. O’M. Bockris and N. E. Richards, Proc. Roy. SOC.(London), A241, 44 (1957). (14) (a) The recent work of Owens (B. B. Owens, private communication) shows that both the expansivity and compressibility of molten alkali nitrates are independent of pressure, up to at least 4000 atm. This fact is reflected in the linear relationships between V and P shown in Figure 1. (b) The graphical method for the evaluation of ( 3 In D/bl/T)v, outlined above, can also be expressed in the form of a rigorous differential equation (eq I V below), which is derived in the following manner
In D
=
fl(P,V,T)
(1)
hlaking use of the equation of state, fz(P,V,T) = 0, and eliminating one of the variables in (I),we have
In D = f 3 ( P , T )
(11)
Partial differentiation of (11)gives
b In D
b In
D
Differentiating (111)with respect to 1 / T at constant volume
or
Volume 70.Number 6 June 1966
AT. K. NAGARAJAN AND J. 0 '111.BOCKRIS
1856
"r
4310
350%
I 400
I 800 Pressure
I
I 400
1200
bfm.)
Figure 1. Evaluation of constant volume conditions (molten NaN03). Standard of reference: 1 atm, 350" (PI,VI).
and Hooper,16 as modified for use a t high pressures by Tricklebank, Nanis, and Bockrisss of this laboratory. The details of the pressure vessel, the technique of diffusion measurement, and the associated calculations have already been dealt with in ref 6a. (b) Materials. Both the sodium nitrate and cesium nitrate used in this work were of reagent grade and were dried a t 110". The radioisotopes used were Na22 and C S ' (both ~ ~ y emitters).
Results ( a ) Difusion Coeficients. At each temperature and pressure, between four and twelve capillaries were used. The results given in Tables 11-IV are the mean values of diffusion coefficients a t the indicated temperatures and pressures.
I
t 800
I200
I 1600
Pressure (aim.) Figure 2. Effect of pressure on t8hediffusion of C S I ion ~ ~ in molten XaN03.
Table 111: Effect of Pressure on Diffusion Coefficients of Cs134in NaN03 Pressure, atm
1 400 800
--------D
x
105 cm2 sec-l-------
350'
370'
390'
420'
2.31 2.05 1.83
2.62 2.35 2.07
2.90 2.64 2.42
3.39 3.03
...
Table IV: Effect of Pressure on Diffusion Coefficients of Cs'34 in CsN03 Pressure, atm
1 200 400 800
-----440'
1.84 1.79 1.75 1.67
D X 106 cmZ seo-l------470' 500'
2.16 2.10 2.02
...
2.63 2.55 2.48 2.33
530'
3.06 2.99
... 2.72
Table 11: Effect of Pressure on Diffusion Coefficients of Na*z in NaN03 Pressure, atm
1 400 600 800 1200
---350°
D X 10s cm* aec-l-------370° 390°
2.14 1.91
2.38 2.16
...
...
1.74 1.58
1.99
2.61 2.32 2.21 2.08
...
...
The Journal of Physical Chemistry
420°
Figure 2 is a typical plot of log D os. P , in this case for the diffusion of Cs+ ions in molten sodium nitrate.
3.04 2.65
In all three systems studied, the value of the diffusion coefficient decreases linearly with increasing pressure,
... 2.50
...
(15) J. O'M. Bockris and G . W. Hooper, Discussions Faraday Soc., 32, 218 (1961).
DIFFUSIONIN MOLTENSALTSAT CONSTANT VOLUME
at a particular temperature. The error associated with the value of D is about f 9%. (b) Activation Energies. The experimental activation enthalpies for diffusion were obtained using eq 2 . Table V lists the experimental activation enthalpies and the preexponential factors, at constant pressure, given by eq 1, for the three systems investigated. The error in the value of E, is about f7%. Figure 3 illustrates a typical plot of log D vs. 1/T at three different, but constant, pressures. It was found that the variation of E , with external pressure was small compared to the experimental error in E , of f7o/n, and hence E, can be considered to be independent of pressure in this pressure range. A typical plot of log D us. 1/T, a t constant volume, is shown in Figure 4. The values of activation energies a t constant volume and the corresponding preexponential factors are given in Table V. Since an interpolation method was used to obtain the values of EV, the error in these values is about f20%.
1857
n
5.151.4
Ep, kcal mole-’ D~ x 103
Ev,kcalmole-l
x
105~
EvIEP
Na22 in NaNOa
Cs”4 in NaNOa
1.7
1.6
'air%
Table V : Diffusion Parameters a t Constant Pressure and Constant Volume Diffusion parameters
1.5
Figure 3. Activation energy for diffusion a t constant pressure. (Cs134 ion in molten NaN03). Cat84 in CsNOa
Constant pressure 4.30 f 0.30 4.69 f 0.21 0.55 f 0 . 1 3 1 . 2 4 f 0 . 4 0
6.47 f 0.32 1.79 f 0 . 2 5
Constant volume 0.78&0.18 1.30f0.17 3.9 f 0 . 9 5 8.3 f 2 . 3 0.18 0.27
1.83f0.26 6.9 f l . 1 0.28
(Do)vis the preexponential factor a t constant volume. ~~
7 535
(c) Activation Volumes. From the results of pressure dependence of diffusion coefficients, one can obtain another important parameter called “activation volume,” AV*. Activation volume is defined as the isothermal pressure variation of the activation free energy (AG*) for diffusion AV*
(bAG*/dP)T
(4) Since diffusion in liquids is essentially a rate process, eq 1 can be written, for constant pressure, as =
D = Do’exp(-AG*/RT) (5) The variation of the preexponential factor, DO’,with pressure is known16 to be small up to a pressure of 6000 atm and hence, from eq 4 and 5 AV* = -RT@ In D / d P ) ,
(6)
5.30
14 .
I5
1.6
17
&K Figure 4. Activation energy of “jumping” (Nazz ion in molten NaN03).
Activation volumes obtained from the results given in Tables 11-IV, by the use of eq 6, are given in Table VI. The change in AV* with a change in temperature was small when compared with the probable error in AV* (viz., f15%), and hence AV* is assumed to be inde(16) N. H. Nachtrieb and J. Petit, J . Chem. Phys., 24, 746 (1956).
Volume YO, Number 6
June 1966
19.K. NAGARAJAN AND J. O’M. BOCKRIS
1858
pendent of temperature for the purposes of discussion in this paper.
formally treated as a chemical reaction, with ions and holes considered to be in equilibrium. Thus Nh/N
Table VI: Activation Volumes for the Diffusion Process Na22 in
AI“*, cc mole-’
cc mole-’ A v j * , cc mole-’
Nvh,
CsIa’ in NaNOs
Cs18‘ in
NaNOs
10.7 f 1 . 7 9.8 f 1.0 0.9
14.9 f 1 . 4 9.8 f l . 0 5.1
18.2 f 2 . 2 15.6 f 1 . 6 2.6
Discussion
CsNOs
=
exp(-AGh/RT)
(7)
where Nh and N are the equilibrium number of holes and ions per mole of liquid, respectively, and AGh is the free energy of formation of holes. The diffusion process can be visualized as the net result of a density fluctuation in the salt, which provides, momentarily, a local cavity, and the jumping of a sufficiently activated ion into such a cavity. Let the probability of a successful collision between an ion and a hole, which is adjacent to the ion, be given by Since the jumping process (which is equivalent to a successful collision between ion and hole) is a rate process, the term A in the probability factor is a free energy of activation for jumping and is denoted by AGj*. On this basis, the diffusion coefficient can be written as
( a ) Phenomenological Trends. (i) At constant temperature, increase of pressure decreases the value of the diffusion coefficient by approximately 15-25% for a 1000 atm pressure increase. (ii) The activation energy a t constant volume (E,) is always smaller than the D = Do’ exp[(A& ASj*)/R] X experimental activation enthalpy at constant pressure exp [- ( A H h AEj*)/RT] (8) ( E p ) . The value of EV is approximately 15-25% of E p . (iii) The activation volume is approximately and comparing this equation with eq 1, one ha$ equal to one-half of the gram-ionic volume in the melt, Ep = A H h AEj* (9) i.e., AV* = 0.5 (molar volume/2). In solids,” the activation volume is approximately equal to one-half and of the gram-atomic volume. (iv) In the same diffusing Do = Do’exp[(ASh ASj*)/R] (10) medium (e.g., molten NaNOa) the ratio EV/EP increases with increasing size of the diffusing ion [e.g., Na+ Since the change in the interionic distance^,^^^^^ with (r = 0.95 A) and Cs+ ( r = 1.69 A) 1. the melting process in a molten salt is zero or negative (b) Expectations f r o m the Density Fluctuation Hole and since the change of interionic distances with temModel. (i) Activation Energy. A model for molten perature is negligible for ionic liquids, it is reasonable salts, to have value, must predict as many observed to suppose that the change in volume of the melt properties as possible with reasonable numerical with a change in the temperature is entirely due to the agreement, without the use of adjustable or calibratory change in the total volume of holes, Le.. J ~ J out ~ - ~a t ~ the constants. The W O ~ ~ ~ ~ carried VT*- VT,= (Nhvh)Tz- (NhVh)T1 (11) University of Pennsylvania during the past few years has resulted in a particular version of a general model where Uh is the most probable volume of holesI3 and entitled the hole model for liquids,22 which serves the number of holes, NL, is given by eq 7. It is further particularly well t o interpret the properties of simple molten salts, when compared with other structural (17) R. W. Keyes, J . Chem. Phys., 29, 467 (1958). models. It may be called the Density Fluctuation (18) J. O’M. Bockris, S. Yoshikawa, and S. R. Richards, J . Phys. Hole Model and is distinguished from other v e r s i o n ~ ~ ~Chem., , ~ ~ 68, 1838 (1964). (19) L. Nanis and J. O’M. Bockris, ibid., 67, 2865 (1963). of the hole model for liquids by the fact that its holes (20) J. O’M. Bockris and S.R. Richards, ibid., 69, 671 (1965). are thermally distributed in size. The value of the (21) J. O’M. Bockris, S. R. Richards, and L. Nanis, ibid., 69, 1627 heat of hole formation,15 on the basis of the density (1965). fluctuation hole model, is (contrary to an often-held (22) R. Furth, Proc. Cambridge Phil. Soc., 37, 281 (1941). misapprehension) independent of the measured surface (23) H. Eyring, J . Chem. Phys., 4, 283 (1936). free energy. (24) J. Frenkel, Acta Physicochim. URSS, 3, 633, 913 (1935). (25) J. O’M. Bookris and A. K. N. Reddy, “A Brief Course in ElecIn the hole model for molten salts, the physical trochemistry,” Vol. I, Plenum Press Inc., New York, N. Y . , 1966. picture25 of hole formation involves a “breathing” (26) A. Bondi, J . Chem. Phys., 14, 591 (1946). motion of ions in the liquid, resulting in a cooperative (27) H. Levy, P. A. Agron, hl. A. Bredig, and M. D. Danford, Ann. “backing away” of ions from a central point, thus A‘. Y . Acad. Sci., 79, 762 (1960). creating a hole. Hole formation in a liquid can be (28) K. Furukawa, Discussions Faraday SOC.,32, 53 (1961).
+
+
+
+
T h e Journal of Physical Chemistry
DIFFUSION IN MOLTEN SALTSAT CONSTANT VOLUME
1859
assumed that the change in the temperature of the system a t constant volume has no effect on uh. Combining eq 7 and 11, one has (Ti, - T’*J = Nuhie- AGhIRTz - e(12)
has been neglected.21 The density fluctuation model expression for Do is21,30
- VTl = 0 and hence
where u and p are the surface tension and the density of liquid, respectively. Using eq 16 in eq 8, one has
AGh’RT1l
However, at constant volume T,i e-
AGh‘RTz
-e
-4Gh
RTI =
0
(13)
Equation 13 implies that the change of e-AGh’RT with temperature a t constant volume is zero. At constant volume, then, the experimentally obtained activation energy (E,) is equal to the energy of activation for jumping (AE,*),Le.
E,
=
( b In D/bl/T)
=
AE,*
(14)
Bockris and Hooper’5 showed, on the basis of the density fluctuation hole model, that the enthalpy of formation of holes ( A H h ) is 3.74RTm and, a s s u m i n g AE,* to be small in comparison with AHh, concluded that E , is approximately equal to 3.74RTm,i.e.
Ep
=
AHh
+ AE,*
N
AHh = 3.74RTm
(15)
This prediction from the density fluctuation hole model, Le., E , ‘v 3.74RTm, has been demonstrated to be remarkably widely true for self-diffusi~n~~ in liquid inert gases, ordinary liquids, molten salts, and liquid metals and for viscous flowz0 in a similar group of liquids. The assumption of Bockris and Hooper, i.e., AE,* is small compared with AHh, has not, before the present work, been subjected to test. The results given in Table V show that the enthalpy of hole formation ( A H h ) is indeed the predominant factor in determining the value of the so-called “activation energy” (E,) for self-diffusion in molten alkali nitrates a t constant pressure. The present results at constant volume do not supportz9a microjump modelg for diffusion in liquids, which considers diffusion as the result of small (-0.1 A) and variable movements of particles in the liquid. Since the microjump model assumes that no holes comparable in size to the particle size are present, the expectation from this model would be that the energy of activation for j u m p i n g that obtained from the constant volume work-would be near the total enthalpy of activation, i.e., that obtained at constant pressure. The value of the ratio E,/E, is 0.18 in the case of sodium ion self-diffusion and 0.28 in the case of cesium ion self-diffusion. The values of E , thus represent 18 to 28% of E,. This is in fair agreement with the expectation from the density fluctuation model expression, E , ‘v AHh = 3.74RTm. I n arriving a t this relation the variation of preexponential factor, Do,
The final two terms in eq 17 need not compensate each other (as had been assumed earlierz1); in actual fact,
and -1220 cal mole-’ in the case of KaNOs (at 350”) and CsN03 (at 414’)) respectively. These values are seen to be about equal to the values of E , (or AE,*) given in Table V. It is this compensating effect of aEj* and the temperature coefficient of Do which leads to an almost exact prediction of E , according to E, = 3.74RTm, in the two cases mentioned. The neglect of the temperature coeffcient of Do while discussing E, is, perhaps, not wholly justified. According to eq 17, E , for sodium ion self-diffusion should be
E,
=
3.73RTm
+
‘/2RT
AEj*
-
->
+ RT2(a + -1 dc
4310
+
=
dT 780 - 827 = 4263 cal mole-’ u
which is in good agreement with the experimental value of 4300 cal mole-’ given in Table V. Similarly, in the case of cesium ion self-diffusion, the values of E , (experimental) and E , (calculated according to eq 17) are 6470 and 5748 cal mole-’, respectively. Thus, the present values of E , or AE,* are consistent with the predictions of the density fluctuation hole model. (ii) Activation Volume. The activation volume (AV*),given by eq 4 and 6, is experimentally obtained from eq 6. Comparison of eq 5 and 8 shows that the so-called “free energy of activation” (AG*)a t constant pressure is the sum of the free energy of formation of holes (A(&) and the free energy of activation for jumping (AG,*),i.e. (29) The monotonic free-path distribution obtained by Alder and Einwohner (B. 3. Alder and T. Einwohner, J. Chem. Phys., 43, 3399 (1965)) from their statistical studies with close-packed hard spheres, are also not consistent with the results reported in this paper. (30) S. R. Richard, Ph.D. Thesis, University of Pennsylvania, 1984.
Volume 70, Sumber 6 June 1966
M. K. NAGARAJAN AND J. O’LI. BOCKRIS
1860
AG*
= AGh
+ AGj*
(18)
From eq 18 and 4 AV*
=
(bAG*/bP),
=
(bAGh/bP)T f (bAGj*/bP). = AVh
+ AVj*
(19)
where AVh and AV,* are to be interpreted. The first term, AVh, is the rate of change of the free energy of formation of holes with pressure and hence can be equated to the most probable volume of holes in the liquid (NUh for a mole of holes). The second term, the volume of activation for the jumping process, ATj*, was defined by Bockris and Hooper15as AVj*
=
[volume of (ion
+ hole) with ion
at the saddle point of jump] [volume of ion
+ volume of hole]
(20)
They considered that AVj*, to a first approximation, is zero, although small values on either side of zero could arise from the distortion or dilation of holes during jumping.I5 If AVj* is zero (or near zero), the density fluctuation hole model expectation for AV* is that its value be approximately equal to the most probable hole volume (NUh). The experimentally determined values of AV* listed in Table VI can now be compared with the most probable volume of holes in molten NaN03 and CsNO3 also given in Table VI. The most probable volume of holes has been calculated from the (density fluctuation) hole model expression13 NZ’h = 0.68N(kT/rr)”‘?
(21)
where u is the surface tension of the liquid and N is the Avogadro’s number. I n the case of self-diffusion of sodium and cesium ions (cf. Table VI) in their respective nitrate melts, AV* is indeed seen to be approximately equal to the most probable volume of holes (“uh). The fact that the magnitude of the volume change (AV*) associated with the difusion process at constant pressure is approximately equal to the most probable volume of holes is a clear and direct demonstration of the importance of the part played by the holes in diffusive transport. The values of AVj* for the selfdiffusion of sodium and cesium ions, given in Table VI, indicate that AB,* is indeed near zero, as had been assumed by Bockris and H00per.l~ The experimental finding, Le., AV* N Nvh, js in contradiction with that expected from the microjump model,g which assumes that there are no holes comparable in size to the ion size in the liquid. The most probable hole size on the density fluctuation hole model is of the same order* as the ion size in a molten salt. The Journai of Physical Chemistry
(c) Efect of Size of Digusing Ion on Transport Parameters. The diffusion of cesium ions (in tracer quantities) in molten NaN03, a t constant volume, was studied with the purpose of finding the effect of size on the diffusion process. Cesium ion has a larger radius (1.69 A) than the most probable hole radius in molten NaN03, which is about equal to sodium ion radius (0.95 A). The results of Tables V and VI show that when the diffusing ion size increases (e.g., from Xa+ to Cs+) there is a corresponding increase in the values of AEj* (from 0.78 to 1.30 kcal mole-’), whereas AHh does not depend on the size of diffusing ion. The value of AV1* also changes (from 0.9 to 5.1 cc) as the diffusing ion size increases, which is consistent with the density fluctuation hole model. The size of the diffusing ion affects the parameters associated with the “jumping” process, whereas the hole size (Nub) and the enthalpy of hole formation (AHh) are solely the property of the diffusing medium.
Summary of Concepts Concerning Transport in Molten Salts The model for transport in molten salts which these results indicate can be summarized as follows. The essential aspect of the liquid is the fluctuating position of one particle with respect to another, within an overall volume of the liquid salt (which allows ca. 20% of free space). These fluctuations lead to temporary holes of varying sizes and lifetimes. The making of these holes is preliminary to the individual act of transport, i.e., the jumping of the particle into the hole. If it is assumed that the time taken for a single jump is smaller than the mean lifetime of the hole, the two processes (i.e., formation of holes and jumping of ion) can be separately considered. The present measurements show that the energy of activation for the jump process (at least in the simple molten salts) is only about 20% of the total experimental enthalpy of activation for transport a t constant pressure. The values of the enthalpy of formation of holes are well p r e d i ~ t e d ’ ~ * * ~ by the density fluctuation hole model. When the cations and anions are comparable in size for a pure salt or when the intruder ion is smaller than the host ion for mixtures, the over-all enthalpy of activation a t constant pressure differs very little from the enthalpy of formation of holes, and the enthalpies of activatiorl, at constant pressure, of cations and anions in a given salt are about equal. When the cations and anions differ greatly in size, the enthalpy of formation of holes remains a property of the salt (and has the numerical value of 3.74BTm), but the observed enthalpy of activation at constant pressure is (secondarily) ion-dependent; i.e., the larger ion has the larger en-
DIFFUSIONIN MOLTEN SALTSAT CONSTANT VOLUME
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thalpy of activation for transport a t constant pressure. This secondary .,ariation arises due to the small but significant nature of AEj* for such systems. These conclusions have been established for simple molten salts. The fact that the equation E, = 3.74 RT, applies to rare gases, ordinary liquids, and molten metals as well, encourages the speculation that it may apply to all liquids in which the rate-determining act in transport does not involve intermolecular bondbreaking (e.g., as it does in transport for, e.g., liquid SiOz or H20). The rate-determining step in the above liquids is hole formation and the following step, the diffusive jump, is “fast” in the kinetic sense ( i e . , the specific rate constant is relatively high compared with that for hole formation).
mation although it plays the essential part in the control of the rate of the process. 2. The activation energy for self-diffusion at constant volume is small in comparison to that at constant pressure ( i e . , E , < E,). Hence, E , can be predicted well by calculations which neglect E , (but which take account of the temperature coefficient of the preexponential factor, Do). 3. The experimental findings that: (i) the enthalpy of formation of holes is about 80% of E,, and (ii) the activation volume is approximately equal to the most probable hole volume, lend support for the applicability of the model of ion-sized holes in ionic liquids. 4. When the diffusing ion has a larger size than the hole into which it is jumping, the activation barrier to jumping is greater than when sizes of ion and hole are about equal.
Conclusions 1. The experimental activation enthalpy in selfdiffusion at constant pressure consists of two contributions: the work done in making a hole and that in jumping into it. The first term is an enthalpy of for-
Acknowledgments. The authors are grateful to the Atomic Energy Commission for support of this work under Grant No. AT-30-1-1769.
Volume 70.Number 6 June 196‘6
