330 Thernrochimica Acta, 7 (I 973) 330-332 ,$J Elsevier Scientific Publishing Company, Amjterdam - Printed in Belgium

Note

The role of constitutive equations in cherrical kinetics

3.43 KR4TOCHVfL and JAROSLAV $XSTAK

Irmitute of Solid 9are Phpics of The Cze~~oslocuk Acdemy of Sciences, Prague (Czechosiorukia)

(Received 31 May 19731

Recent artic!es on the non-isothermal rate equationls3 have initiated a rather extensive discussion I1 3_ Misunderstanding arose mainly about the meaning of the partial derivativ-es of the function f in the equation for the fractional conversitin Z, (a kinetic variabte conventionaIIy normaiized Q = 0 at t = 0 and 31 = 1 at t+n):

where T and t represent temperature and time, respectively-. Some authors claimed that onIy the partial derivative (?~Zt), can ce used so that the functionfis appro- priate for the description of an isothermally mez;sured rate of a chemical reaction1-3, whiIe others proposed that it may describe non-isothermal kinetics as weI1 if either the partia! derivative (ZJ~CT),~ or the tota1 differential dr5 are equal to zero. -4nother susgestion6 related this probIem to the non-uniform temperature within a solid sample. Xlthou~h some useful criticism on these inccrrect ideas ~3s already given7-’ 3 and the path function character off in eqn (1) emphasizeds*9 there still remain certain confusions_ Therefore we wouid like to cIa;ify this question from 3 more unifyins viewpoint_

We will mention two different ways in which the telation (1) may be understood: (A) If eqn (I) represents an equation of siate (calied “the constiturive

equation”) of a chemical system under consideratior then eqn (1) impIies that the vaiue cz at t depends on the time t, and the instantaneous vaIue of the temperature T at t regardtess of its previous temperature hist nry_ The zonstitutive equation of type (1) would describe the behaviour of a system controlled by an independent “internal clock”. Such an equation is e.g. the constitutive equation for the pressure p of a gas which is evoIved by a radioactive decay. The number of moIecuIes produced is con- troIIed by the radioactive mechanism and is not affected by the temperature. The pressure p is expressed by a function P of the number of molecules, i.e. a function of time t, and the temperature T at t only, p = P(T, 1).

331

It seems that ordinary chemical systems treated so far by the standard methods of chemicai kinetics 1 s do not represent the case of systems with an “internal clock”.

The controversy concerning the total differential dz and the partial derivatives

(CfiZT), and/or (Cfl?~)~ which appears in some preceding papers’-’ is a consequence

of the application of the described non-adequate interpretation of eqn (1) for ordinary

chemical systems_

(B) We are convinced that the constitutive equation for the fractional con-

version z determined in a chemical system is

5 = F(z, T) (3)

or in a more general form

.2= G(z, T, i, T, ___) (41

Equations of the eqn (3) type are well estabIished in isothermal kinetics’*’ **lZ_

Equation (4) which has the form known for non-equiIibrium situations could be more

appropriate in the case of hi&Iy non-isothermal kinetics.

Based on eqn (3) or (4) eqn (1) may be interpreted as foIIows. For the initial

conditions (r = 0 and T= T,, at t = 0) and a specsed temperature regime. T= G(t).

we denote the soIution of eqn (3) or (4) as

z=Z&)_ (51

Kate that rhe subscript CD indicates that the solution depends on rhe temperature

regime @(tj, i.e. x is p?th dependent_ In mathematical terms it means that r is a

functional of @.

For some special instances of “temperature” kinetics soIution (5) can be

converted into the form (1% XVe illustrate this for a simple case of the --Iinear” non-

isothermal kinetics, where

T=@(l)=jliT, (6)

and where q5 is the constant temperature rate. When substituting eqn (6) into eqn (3)

we obtain 5 = F(z_ &t-t T,,j or simiIarl_v from eqn (4) i = G!z_ (;5r i TO_ 4,. 0. _ _ _i_ The

solution (5) can then be written in the form z =gO(Q, l,,. I)_ Tt means that the

functicnal dependence (5) is reduced to the function dependence of r on two para-

meters $ and T,, v. hich specify the temperature regime @. If we compare the e-.oIution

of the fractional conversion x for multipIe runs accompiishcd at diKerent $ holding

the initial temperature T, fixed. we can write

x =g,,(+, T,, t) =g(@. t) =g((T- T&t, r> =f,,(T, To. r) =f(T. I) (7)

If we understand eqn (1) in the described sense. which is illustrated b_v eqn (7).

then the interpretation of the total differential dr and the partial derivatives, rxtmely

(CTXS7-),, represents no principal difXcuIty_ For a specified class of temperatkrc

regmes eqn (6) the eqn (7) can be properly represented as a surface in a three-

dimensiona diagram z-r+. TheI; it is easy to realize that within this specified class of

333

temperature regimes 0 the isochronal derivative (Cgi&$), indicates the change of z

measured at the same time z between two infinitesimally close re,oimcs differing by d+?

i-e_ between two paralle! z-r cm-x-es lvhich are the sections of the surface taken at 4 and

4-td+_ Similarly we can interpret the derivative (CxC;T), using the 2-r-T diagram.

A more comprehensive discussion of the meanin g of the partial derivatives was dealt

with in our previous pqer’ 3_

\Ve thank Dr. A. Bergstein, Dr. P. Holba, Dr_ P. Hrma, Dr. E_ KratochviIov5

and Dr. K. ZG3a for helpful comments.

REFFXEECES

1 J- R- !UacCallurn and J- Tanner. Sczrrrre. 225 (1970) 1127_ 2 A- L- Draper, in H. G. %qcAdie (ed.). Proc. 3rd Toronto Syntp. T’tcrntal =Imtiuis. Canada 1970.

p. 63. 3 J. R. MiacCalium. r%wre P&x Sri.. 232 (1971) 41. 4 R. A. W. Hiil, .Vurure, 227 (19’70) 703. 5 V. Xi. Gorbatchev 2nd V. A. I_ogvinenko. J. Thermui Anal.. 4 (1972) 475. 6 X1. ?cIzcCart~. V. R. PaiVemekcr and J. N. Xciaycok, in H. G_ Wiedemann (ed.). ThermpI .-inul_vsis,

Pruc. 3rd ICT.4 in Drtt-os, Vol. 1, Rirkh5uscr Verlag. Bawl-Stuttgrt, 1972, p_ 3.55 7 J. %st&, ref 6. VOI_ 9. p_ 3.

S R. 51. Feldtr and F. P. Stehcl. A-urttre. 275 (19iO) IOSS. 9 J. h-I. Gilks and H. Tompa. Satttre P/t_rs. Sci.. 229 (1971) 57.

10 G. Gbuiai and J. Greenhow. 7Xertmchint. Acra. 5 (1973) 451. II P. Holba and J. %stik. 2. Ph_vs. Citettt. (Frttttkfitrr ant .\faitt). SO (1972) I. 12 E. L. Simmons and W_ W. U’cndlandt. Them;ochinr. Acta. 3 (1972) 495. I3 J- %s:ik and J- Kratcch\k J_ ThcrmtI Anuf.. 5 (1973) 19X I-5 C- H.. Bamford and C_ F. H. Tipper (eds.1. Contprchensirr Chemical Kinetics. Vol. 2, 771~ Theory of

Kinerics. Elscvier. Amsterdam. 1969.