Thermotransport in Monatomic and Ionic Liquids

13 Thermotransport in Monatomic and Ionic Liquids A. LODDING

Downloaded by MONASH UNIV on March 2, 2016 | http://pubs.acs.org Publication Date: June 1, 1969 | doi: 10.1021/ba-1969-0089.ch013

Department of Physics, Chalmers University of Technology, Gothenburg, Sweden

The phenomenon of thermotransport in simple liquids is described in a generalized treatment which does not require the adoption of any particular existing model of liquid diffusion. The results of recent isotope experiments in liquid metals are used as illustrations of applicability. The treatment is extended to liquid salts, where electrotransport mobility data under favorable conditions may facilitate the interpretation of thermotransport experiments. The experimental results in the few liquid salt systems, where thermoas well as electrotransport of isotopes has been measured, are discussed in terms of the above treatment.

/

i

T hermotransport ("thermal diffusion," thermomigration) in solids has received considerable attention in recent years. From thermotransport studies it has been possible to derive information concerning—i.a., defect mechanisms, the role of electrons and phonons in atom transport and the energy distribution around an activated configuration. Since the special symposium on the topic i n Miinster in 1965, literature published on solid state thermotransport has been extensive. Good reviews have been given by A d d a and Philibert (1), Allnatt and Chadwick (2), and Oriani (24). As for the evidence of isotope effects in solid state thermotransport, only two instances have been reported (see Reference 16). The usefulness of thermotransport studies, especially isotope studies, in liquids has been pointed out by the present author at the Miinster conference and in a paper which appeared in 1966 (17). U n t i l recently the basis for a theory applicable to liquids has been rather restricted owing to the scarcity, inexactness, and heterogeneity of available experimental data. However, recent studies of isotope thermotransport i n liquid A

264

In Isotope Effects in Chemical Processes; Spindel, William; Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

13.

LODDING

Monatomic and Ionic Liquids

265


metals (18, 20, 25, 26) have allowed the formulation of a theoretical approach (17), which also may be applied to other classes of liquids. Further work i n this field is greatly encouraged. The most straightforward liquid thermotransport experiments are performed in leaving a mixture of chemical components or isotopes under a temperature gradient in a closed cylindrical separation tube. The components receive different migration velocities, which tend to enrich them at one or the other end of the tube. After a certain time, given by the length of the tube and the diffusivity of the liquid, a practical steady state is reached, i n which further enrichment is prevented by diffusion. The composition along the tube is then analyzed as a function of temperature. Actual experiments on pure liquid metals have yielded isotope separations of the order of one percent for a temperature interval of 100 degrees. Similar, or slightly smaller, isotope separations have been found in pure liquid salts. In mixtures of cations large chemical separations can be obtained, in optimal cases about a hundred times greater than for isotopes; i n other cases the separation can be almost undetectable. W h i l e in all liquids the light isotope migrates towards the higher temperatures, hitherto no simple tendency can be seen i n the investigated chemical mixtures. The purpose of the present paper is (a) to give a generalized version of the theoretical approach of Reference 17; ( b ) to see whether useful physical information can be extracted from the isotope experiments i n liquid metals; (c) to apply the formalism to the results of thermotransport experiments i n liquid salts (10, 11), which have not hitherto received satisfactory theoretical discussion. Theoretical Treatment In our discussion nothing w i l l be assumed initially about the nature of the migrating species, which may be an atom, an ion, or a cluster of particles. W e shall also avoid the frequently occurring postulate that the mean displacement length is of the same order as a nearest like-neighbor distance. While the treatment of Reference 17 employed a language adapted from solid-state arguments, the present approach w i l l be based on premises which are not bound to a definite model and do not violate the physical picture of any of the various existing theories of liquid diffusion. It w i l l simply be stated, that the probability of a displacement depends on the energy distribution in the vicinity of the migrating particle in such a way that it can be expressed as the product of an energy function f(T) at the particle site (coordinate x) before displacement, and another energy function, g(T - f A T ) , at an "imperfection" a definite


266

ISOTOPE EFFECTS IN CHEMICAL PROCESSES

distance away from the particle site—e.g., at the coordinate x + g i l l i e "imperfection/' the presence of which is thus necessary for the displacement of the particle, may be, for example, a void or another density fluctuation. The functions f ( T ) and g ( T + A T ) w i l l express the entire temperature-sensitive part of diffusion. Let us first consider isothermal diffusion. Einstein's diffusion law can be written D = 1/6 l r = 1/6 l / ( T ) g ( T ) 2

(1)

2


The activation energy—i.e., the slope of the Arrhenius plot (not necessarily a straight line) of D , w i l l be faln/(T)

_d\nD

F

alngfT)

W h e n a temperature gradient is imposed, the flux velocities i n the forward and reverse direction w i l l be, respectively, ^ l + i - i > = ( D i - D ) * ^ [ ( E D / R T ) — 2(d I n / ( T ) / d T ) ] ox 2

2

2

(6)

However, i n the steady state the gradient of the separation factor Q =

( (S^)

i s g i v e n b y

v -v 1

2

= D (d]nQ/dx) ett


(7)

13.

LODDING

267

Monatomic and Ionic Liquids

where D is the effective diffusion coefficient, consisting of convection as well as molecular self-diffusion. The combination of Equations 6 and 7 yields e{t

aheff

d

(

8

)

T

In reliable experiments convection is satisfactorily suppressed and one may (insofar as the mechanisms of self-diffusion and thermotransport can be assumed to be identical, see Reference 16) put D = D—i.e., the mean of D± and D . Introducing the "isotope factor" of diffusion, defined by e f f


2

D

-D D

l

2

/

M

I

l

- M M

2 K

1

where M denotes the isotope mass, we can express the variation of istotopic composition with temperature at steady state by d(ln Q) d(l/RT)

_

=

a ( A M / M

)^|-

E D

_ 2RT2 e In /(T)/dT]

(10)

(It should be pointed out that the Bardeen-Herring correlation factor has been left out of the present discussion. In rigorous treatment the factor a, as defined by Equation 9, should in Equation 10 be replaced by a = a/i, where / is the correlation factor (see References 5, 16, and 17). The continued use of the term a instead of a in this paper w i l l be justified when bearing in mind that the symbols D in our equations denote uncorrelated diffusion coefficients, as measurable—e.g., by N M R — a n d not tracer diffusion coefficients, as measured i n most actual studies.) The meaning of the term in brackets i n Equation 10 (which we w i l l denote by E * ) can be illustrated by two examples. The first is the solidstate model, with well-defined and constant activation energies of formation and motion, and f(T) = const, exp ( — E / R T ) . This gives a temperature-independent term in the brackets, so that the equation can be integrated from a "cold" to a "hot" temperature, which is convenient in many experiments, where the exact temperature variation along the cell is unknown. The experimentally determined entity w i l l be w

w

m

Qmax-1

TV

1

- TH

= 1

-Ra^^>[E M

Z )

-2E ]

(11)

m

This is the formula derived i n Reference 16. According to practically all recent theories of liquid diffusion, however, to treat E and E as constants is, at best, a reasonable approximation and actually the integration should be permissible only over short ranges of temperature. The expressions for D i n the different theories appear in many different forms (8, 22, 29, 32), depending on the various models employed. In all cases, D


m

268

ISOTOPE EFFECTS IN CHEMICAL PROCESSES

however, the temperature function at the particle site can be expressed by f(T) = const. T , where n ranges between 0 and + 3/2. This simplifies Equation 10 to n

a(lnQ)

A M ^

d(l/RT)

(

E

o

_

2

n

R

r

)

(

1

2

)

M


Discussion of Experimental Results in Liquid Metals Table I shows the results of isotope thermotransport experiments on a number of liquid metals. In the latest work (20) we have been able to plot l n Q vs. inverse temperature and obtained good straight lines, which indicates that the product

Thermotransport in Monatomic and Ionic Liquids

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