A Homework Problem Demonstrating the Importance of Assumptions Made While Integrating Chemical Rate Equations

George D. J. Phillies The University of Michigan. Ann Arbor, MI 48109

Many physical chemistry texts illustrate the integration of a macroscopic rate equation by examining the bimolecular combination reaction

whose rate equation is assumed to be

Here k is the rate constant, and [A] = NA/V and [B] = NBIV are the concentrations of A and B. resoectivelv. If the volume V and temperature Tare assumed to be cone.&t, eqn. (2) can he integrated, showing

The subscr i~ts 0 and t denote concentrations a t time 0 and the later time t.

The familiar discussion of this intenation does not alwaw make entirely clear why the volume must he assumed to be held constant. Canagaratna' has exhibited a textbook error which arises from the neglect of this condition. The purpose of the homework problem presented hrre is to show the sig- nificance of the constant vblume assumption by having the student figure out what would happen if the volume were to change as the reaction proceeds. The student is asked to cal- culate N A and other quantities and is supplied with a series of increasingly leading hints. In the interests of brevity, the full details of the calculation are left for the instructor to du- p l i ~ a t e . ~

Problem Let us consider the special use of the limolrcular combination re-

action defined l,y eqns. (1) and [?) above. Here Aand Bare presumed to be ideal gases, so that

RT ~ = - [ N A + N B ]

P (4)

We denote the initial values of the variables by subscript or super- script "0": V., To, Po, NA', NB'.

(a) Suppose the reaction vessel is so arranged that Tand P, rather than T and V, are held constant. What is the number of A molecules left at time t?

Hint 1. In a fixed volume of 1 L, increasing NA by 0.01 mole or [A] by 0.01 Mare the same, so l /VdN~/dt = d[A]ldt. Ifthevolume is not constant, one can change [A] without changing NA, or vice versa, so 1/V d N ~ / d t and d[A]ldt differ by a quantity proportional to dV1dt. Equation (2) was written for a fixed volume. Even if dV/dt f 0, the reaction rate depends on the instantaneous concentration [A]. However,chernical reactions fundamentally create molecules,

not concentration units, so for this problem eqn. (2) is properly written

Hint 2. The reaction changes the total numher of molecules in the system,suasrhe reactiungnesun. V willdecreaue. InIAJ = NdV, N A and I' both d e ~ m d on time. A convenitnt line of attark 1s to introduce a progress variable x

and eliminate NA, NB, and V in its favor.

Hint 3. The answer is

(b) For the same system, calculate how NB depends on time. Ea- press your solution in the fonn t = ~(NBA'B~,~'A~). If NB 2 NBO + N A ~ / ~ , what value oft do you calculate? Explain this perhaps sur- prising result. [Hint: What does stoichiometry say about NB 2 N B ~ + Na0/2? .. . -

(c) Using the arguments and results of pan (a), ralculate how V depends on time. (Hint: The convenient furm of the solution is I = FI 1'8 .\. 1.

(d) Suppose that the reaction vessel also contains an inert carrier gas C. If A, B, and C form an ideal mixture, and if an amount NcO of C is initially present, solve part (a) again.

(e) If the 2A - B reaction is collision-rate-limited. the use of ean. (5). ;ather than eqn. (2) can be given a microscopic justifieati&. Provide this justification.

Remark Equations (3) and (7) are directly conparahle, since they differ only in the reaction conditions. Bv running the reaction a t constant pressure rather than constant vol;me, one gains a factor of 2 in the k t term as well as the term in l n [ N ~ / N a ~ ] . RTo/P~(NAO + 2N& would be the initial volume of the sys- tem if all the B were initially present as A. One observes that the reaction proceeds faster a t constant pressure than a t constant volume, as ought to be expected on physical grounds. As A is converted t o B, the number of gss molecules in the system falls. In order to maintain constant pressure a t con- stant temperature, the volume of the system must also he reduced, thereby increasing [A] (and thereby dNAldt) over its value in the corresponding constant-volume system.

' Canagaratna, S. G., J. CHEM. EDUC., 50,200 (1973). This note has been condensed from a substantiallv lonoer manu-

scrip (copes oi whim are available from the author) which also treats reactions n which Tis not qu'te constant and examines details of the integration for tne lam liar (1. V) = constant reacrion.

Volume 59 Number 12 December 1982 1029