Charles E. Hecht
American University of Beirut Beirut, Lebanon
Chemical Kinetics and Thermodynamic Consistency
The requirements of thermodynamic consistency provide a very satisfactory way of obtaining the functional form of the specific reaction rate constants. They also lead a t once to the useful interpretation of these constants in terms of entropy and enthalpy of activation. Few elementary texts mention this approach and some advanced studies deal with it in a summary and often careless way. It is the aim of this paper to suggest the value of presenting these thermodynamic requirements and thus to re-emphasize how much can be deduced from them. Following Hollingsworth (I), let us first recall the rationale for usually setting the ratio of ki, the specific rate constant for the forward reaction, to k,, the specific rate constant for the reverse reaction, equal to the equilibrium constant written down by reference to the stoichiometric equation for the change. The stoichiometric equation very rarely displays the actual mech% nism involved. If it does, as in the famous case of HI equilibrium:
unity and which we will arbitrarily call g may be written
Here g is some function of the equilibrium concentrations and temperature. Thermodynamics gives us another function h([A],, [B],, [C],, [Dl., T) that is also equal to unity, namely
This is because the equilibrium constant of reaction (3) can be written
We are assuming ideal systems in replacing activities by concentrations in equation (6). A simple form that satisfies equations (5) and (6) and which seems satisfactory is to set g = h ~ = l
The equality of rates of the forward and reverse reactions leads to the familiar equilibrium expression:
In equation (2) we mean by [HZ]the concentration of Hz in moles per liter and we use subscript c to indicate that we are expressing concentration in these units. Note that we will, for simplicity, only treat of ideal systems in this paper. We will not belabor the point that seldom does the stoichiometric equation represent the mechanism. We (2-4) which refer the reader t,o articles in THIS JOURNAL have dealt with this point. Nevertheless we will almost always be correct in writing kf/k, = KO. Consider the reaction aA + b B s c C + d D
(3)
The rate forward, ratef, and the rat,e in reverse, rate,, may be quite complicated functions of any or all of the concentrations and the temperature so that in general rate, = rater ([A], [B], [C], [Dl, T) and rate, = rate, ([A], [B], [C], [Dl, T). The parentheses notation means that rate, etc. is a function of all the variables standing between the parentheses. If we represent the equilibrium concentrations at T by subscript,^ "en on the brackets we have ratedIAl., [BL, [Cl., [Dl., T)
=
rate,(IAl,, PI., [Cl., [Dl., T) (4)
so that the ratio of the left t o the right hand sides of equation (4) which is a function identically equal to
(7)
Where o is an integer or the ratio of small integers. This relation is clearly sufficient since if the function h is identically unity, h raised to any power, o, will still be unity. The function g equals unity by equation (5) and thus equation (7) follows. It is not possible to prove that the relation (7) is necessary. Thus n
=
rater([Al., [Bl., LC]., [Dl., T) = rated [A]., [Bl., [Cl., [Dl., TI ki[equilibrium concentrations]Powrls k,[equilihrium concentrationsl9o"e"s =
Kc~IAI,~=-[BleD~ = I ICler--ID]."*
in which we have written both (ratef and rate,) as simple algebraic products of specific rate constants times concentrations, raised to various powers. In general unless all the concentration terms cancel out in (81, the separate specific reaction rate constants will depend on concentrations of one or more single constituents at equilibrium, which is not empirically the case. The concentration of a single constituent at equilibrium is not a constant. Hence we have
We finally take o = 1 because this is rigorously true if our stoichiometric equation represents the mechanism. Although experimentally no system of opposing reactions has been observed to violate the use of o = 1, one can easily imagine such a system. One such mechanism, suggested by Manes (5), may be displayed here. Consider the reaction AP
+ 'hB2 = AnB
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with mechanism B z e B+ B A,
+B
$ A2B
(10.1) (10.2)
less the number of gaseous moles of reactants and R is the gas constant. We have the following two thermodynamic equations:
(K dT In K D)
and equilibrium constant
These equations lead to the following two requirements on the specific reaction rate constant:
If reaction (10.1) is the rate controlling step rater = kdBd
AHo = RT' -
(11.1)
and
where we have obtained rate, by using the usual steady state assumption to find the concentration of the intermediate, B. That is
and we neglect terms with small rate constants as factors so that
We see that in order to determine (11.2) from a howledge of the forward rate expression (11.1) only we need to use equation (8) with a = 2. Then
However if equation (10.2) is the rate controlling step, equation (12) shows us that
so that, 7ater =
kr[Anl [Bzl'/z
(13.1)
k,[AzB]
(13.2)
and rate,
=
In this case rate, can be found from (13.1) and equation (8) using the usual rule of o = 1 because
I n the rest of this paper we will restrict consideration to the case o = 1 only. It should be kept in mind that this is not a rigorous requirement of thermodynamics but rather the simplest requirement consistent with thermodynamics. Furthermore it is in accord with all the systems of opposing reactions yet studied. It should also be noted that if a is not equal to unity the left baud side of equation (19), which follows, should be written as (kt)'@and not merely as (k3. We are now ready to deal with the specific reaction rate constants themselves. I n the usual system of standard states, when gases enter an equation they appear as partial pressures or more precisely as fugacities in the expression for the equilibrium constant. This equilibrium constant may be denoted K,, and as is well known, K, = K,(RT)-Am
(14)
where An is the number of gaseous moles of products 31 2
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Journul of Chemicul Education
and
To satisfy these requirements it is necessary to introduce a non-thermodynamic concept, that of the activated state (subscript a), through which the reactants (subscript R) must pass on their way to forming products (subscript P) and inversely for the reverse reaction. We then write
in which 7 may be a function of the temperature but of course the same function for both forward and reverse rate constants. The function 7(T) is introduced in equation (18) because we anticipate its need in identifying certain factors that appear in the collisional and transition state model theories to be discussed. The occurrence of y is immaterial to the satisfaction of conditions (17) because there we have conditions on the ratio of forward t o reverse rate constants and any function, such as y, which is the same for these two rate constants will cancel. From equations (la), equation (17.2) follows immediately since HP- H R AS" = SP - SR An = nP - nn AH" and
=
in which the various n's refer always to numbers of gaseous moles of products, reactants, or activated state. We also satisfy (17.1) since
and the last two terms cancel by virtue of the thermodynamic relations H o = F o TSO and dFo/aT =
-so.
+
Introducing the definitions of enthalpy of activation and entropy of activation AH* = H a - H E AS* = S. - S,
we have
This form for the specific reaction rate constant has been obtained without recourse to any detailed molecular model such as that of collision theory or transition state theory. It is quite important to note how
this single rate constant varies with temperature. Use the thermodynamic relations:
in centimeters, Mi is the molecular weight of molecule i, and N is Avogadro's number. (Concentration units of moles per liter have been used.) Hence in this case Ea,.. = AE*
in which equation (21.2) follows by neglecting the change in the (pV) product for condensed phases. Then we have
+ '12RT
using- eauation (25) . . . for a reaction in a constant volume system. I t is clear that the right hand sides of equations (26) do not adequately allow for changes in entropy upon formation of the activated state and this indeed is whs collision theory fails so badly for reactions between polyatomic molecules. The so-called transition state theory of general reactions introduces a universal frequency factor for the function y, namely
and where h is Planck's constant. Thus AE* is the theoretical energy of activation. I t should not be confused with the experimental energy of activation, E*,., found by using the empirical Arrhenius form k,
= A~-E*..~ET
(24)
I n fact, under the usual experimental conditions of constant volume,
and if y is proportional to Tm Of course E*,,. and AE* will not differ greatly since even a t 500°K, R T is only 1 kcal/mole. For reactions in condensed phases both n, and n~ will be zero. For gaseous r~actionsthe activated state is assumed to be a single complex molecule and thus n. will be unity. If the reaction is unimolecnlar nn is one, if bimolecular nR is two, etc. It should be noted that by virtue of equations (21) there is no contribution to the temperature dependence of the rate constant from the temperature dependence of the entropy of activation, AS*. Simple bimolecular collision theory for gaseous reactions puts
E*,., = AE*
+ RT(nn)
The complete expression for k t as given by transition state theory is where K*. is the equilibrium constant for the reaction forming the activated complex. Equation (28) can then assume precisely the form of k given in equation (19). However equation (19), derived from considerations of thermodynamic consistency, is more general. Only for gaseous reactions between very simple molecules does transition state theory lead to a more detailed model of reaction kinetics by providing estimates of K*. a priori in terms of the partition functions of the known reactants and intclligent guesses about the partition function of the complex. For more complicated gaseous reactions and for all reactions in solution the estimation of part,ition functions is extremely dubious at best and the language of entropy and enthalpy of activation can best be used in connection with the more general equation (19). I t should be noted that experimentally it is not really possible to separate out eAS*lRfrom the total pre-exponential factor,
and the equating of y t,o a universal frequency function, as in equation (27), gives us no particular advantage. . Literature Cited
for reactions between unlike and like molecdes respectively in which aiiis the molecular diameter,
(1) HOLLINGSWORTH, C. A,, J. Chem. Phys., 20, 921 (1952). (2) FROST,A. A,, J. CHEM. EDUC.,18, 272 (1941). (3) MYSELS,K. J., J. CHEM.EDUC.,33, 178 (1956). (4) GUGGENHEIM, E. A,, J. CHEM.EDUC.,33, 544 (1956). (5) MANES,M., ET AL., J . ,Chem. Phys., 18, 1355 (1950).
More Mnemonics For the 10 indispensable amino acids required for normal growth of the immature rat, threonine, arginine, tryptophan, histidine, wline, isoleucine, leucine, methimine, lysine and phenylalanine: Three arduous trips to historic valleys of Iceland loosened the metallic license plates. BENJAMIN BECKER RUTGERS, THESTATEUNIVERSITY NEWJERSEY NEWBRUNSWICK, Volume 39, Number 6 , June 1962
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