A Simplified Integration Technique for Reaction Rate Laws of Integral Order in Several Substances J. G. Eberhart University of Colorado, Colorado Springs, CO 80933
E. Levin Eloret Institute, Palo Alto, CA 94033
Several years ago we presented a technique for simplifying the integration of the rate law for reactions that are first-order in each of two reactants (1). The approach was based on a combination of the form of the rate law itself and the material-balance condition for the two reactants. A more powerful method is presented here that permits a simplified integration based only on the material-balance condition and generalization to other orders and to several reactants or products. The traditional approach to this problem presented in most physical chemistry texts (e.g., Levine ( 2 ) or Atkins (3))involves the separation of the time variable from the several concentration variables, followed by the expression of all the concentration variables in terms of one selected variable. The integration of the resulting function of one concentration variable can be lengthy and, for integral orders, involves the method of partial fractions. In the approach presented here, the various concentration variables are not reduced to one but are further separated from one another via the material-balance conditions. As a result, rate laws with positive-integer orders can be reduced to a sum of terms whose forms are identical to those found in reactions that are first- or second-order in one reactant. Thus, this technique reduces rather complicated integrals in one concentration variable to several very elementary integrals (one for each concentration variable). We will illustrate the technique for reactions that are second- or third-order in several substances and begin with a reaction that is first-order in each of two reactants. (This is the only case covered in our previous paper (I), and the separation of concentration variables and the resulting key integration formula is obtained here without invoking the form of the rate law itself.) Material Balance The heart of the approach taken here to rate-law integration lies in the judicious use of the material-balance condition. We assume a general reaction of the form aA + bB + cC +other reactants +pP
+ other products
(1)
The material-balance condition in differential form is then
whereAo is the concentration of A a t zero time, andA is the concentration of A a t some later time t . From eq 4 the material-balance relationship between A and B can then be written as bA - bAo = aB - aBo, or alternatively 6AB= bA - aB = bAo - aBo= canst
Even though A and B depend on time, f i is~ a ~constant becauseAo and BOare constants. (This notation is slightly different from that of Levine (41.1 Similar results are obtained for the relationships betweenA and C and between B and C,namely
For eqs 5-7 reversing the order of the subscripts changes the sign of delta. Thus, for example,
Similar material-balance relationships exist between reactant and product concentrations. Thus, from eq 4, -(PA -pAo) = a P - aPo, or
We use 6 to represent a difference and o to represent a sum. With o the interchanging of subscripts has no effect and
First-Order Kinetics in Each of Two Reactants The simplest rate law to be considered here is for the reaction aA + bB + other reactants
where A, B, C, ...,P, ... are the concentrations of the reactants and products, and a , b, c, ..., p , ... are the associated stoichiometric coeff~cients.The rate of the reaction can then be written in any of the alternative forms
(5)
+ products
with first-order kinetics in both A and B. The rate law is thus
where k is the reaction-rate constant. The concentration variables, A and B, are easily separated from t , giving where t is time. Integration of eq 2 yields Volume 72 Number 3 March 1995
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The traditional approach, then, is to express B as a funca, tion of A using eq 5, which yields B = (bA - 6 ~ ~ ) land substitute the result into eq 12, giving The substitution of a d P = -pdA from eq 2 separates P from A and yields The left-hand side (LHS) of eq 13 is then integrated using partial fractions. Our approach will be first to separate the variables A and B on the LHS of eq 12 before carrying out the integration. This will be accomplished only through the use of the material-balance condition relatingA and B. Thus, we will obtain a formula for the integral of - (lla)(dA/AB) that will be valid for anv rate law.. . ~rovidedonlv that A and B are both reactants. We begin with multiplication of eq 12 by (bA - a B ) 1 6 ~ ~ , which is unity and yields
Integration of eq 23 then provides the rate-law solution,
and i n the process the general integration formula PIA f %el,PA %A PoIAo P,
Then, from eq 2, adB is substituted into eq 14 for bdA, yielding the separated form
First-Order Kinetics in Each of Three Reactants With consideration of a reaction t h a t is first-order i n each of three reactants, we move on to a significantly more complicated rate law and have a n opportunity to see how our knowledge of the two-reactants solution will simplify the solution of the three-reactant problem. The reaction can be written as
aA+ bB + eC +other reactants + products and the reaction rate law is Equation 16 is now separated in the variables A and B and can also be easily integrated from a knowledge of how to integrate first-order kinetics in one reactant. The result is
or, in more compact form,
This general integration formula, when applied to eq 12, yields
as the solution to the original rate law, eq 11 First-Order Kinetics in One Product and One Reactant: Autocatalysis A reaction with first-order kinetics in one product and one reactant can be treated in very similar fashion. The reaction aA+ other reactants +pP + other products
has a rate law
Separating the three concentration variables from time yields
6~ in Multiplying the LHS of eq 27 by (bA - ~ B 1 1 results
Then substituting bdA = adB into eq 28 gives
Equation 29 does not yet represent a full separation of the concentration variables A, B, and C, but the separation process can be stopped a t this point because the integration of M A C and dBlBC can be carried out immediately with the aid of eq 18. With this general result, the integration of eq 29 yields
This is the solution of the rate law, and the comparison with eq 27 provides another general integration formula. AIC
Separation of t h e concentration variables from time yields I dP - kdt P PA Multiplication of the LHS of eq 21 by (UP+pA)lop~then gives
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Journal of Chemical Education
b
BIC
In -
Although we will not carry out the process of rate-law integration to reactions with a total order larger than 3, i t is clear that eq 31 would be very useful in integrating r =M C D
If one were to reach the stage of eq 29 and be unaware of eq 18, the process of variable separation could simply be continued by multiplying the first differential by (cA aC)l& and the second differential by (cB - bC)lG~c.The result is
Then substitution of cdA = a d C and cdB = bdC in the first and third differential expressions gives
or, more simply,
Separation of the two concentration variables from time results in
Multiplying the LHS by (bA - aB)lGas gives
Then, substitution ofadB for bdA, yields
The first differential is easily integrated from eq 18, whereas the second one i s the same a s that encountered in second-order reactions i n one variable. Thus, integration of eq 42 yields
In eqs 33 or 34 the variables A, B, and C are completely separated and can easily be integrated without eq 18, yielding
Because A, B, and C enter the rate law with the same orders, a symmetrical form of eq 35 must exist. The coefficient of the C term in eq 35 is
Substitution of eq 36 into eq 35 gives
Then, simplifying and using the symmetry properties of eq 8 , we get
which is symmetrical i n A, B, and C and equivalent to the earlier solutions presented in eqs 30 and 35. Second-Order Kinetics in One Reactant and First-Order in Another The last reaction example to be considered is aA+ bB + other reactants +products
with a rate law of
Again, in the process of solving the rate law, we have obtained a general integration formula, namely
t h a t would be useful for integrating r a t e laws of t h e form r = kA2BC, r = kA2B2,or r = kA3B. Summary We have shown here a number of features of rate laws with integral orders in several substances. The rate laws can be integrated quite easily if the process of separation of variables is extended to all the concentration variables. The separation of the concentration variables can be accomplished using only the material-balance condition. The integration formulas obtained for simple reaction kinetics can be used to facilitate the integration of more complicated rate laws. The resulting integrations are simpler than the conventional method of using material balance to express all of the concentration variables in terms of one of the concentration variables. Com~licatedrate laws can ultimately be decomposed into the Bame differential expressions as those found in first- or second-order reactions in one variable. Literature Cited 1. Lerin, E.: Eberhalt. J. G . J Chpni. Edur 1989.6fi. 705. Chsrnisir.~, 3rd ed.: MeGraw-Hill: New Yolk. 1988; pp 519-520. 2. Leuine. I . N . Pl~.~~ivricnl 3. Atkms. P W.Pixvsicnl Chemistiv. 3rd ed.: Freeman: New York. 1986 o 695.
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