Ronald W. Missen Deparlment of Chemical Engineering and Applied Chemistry University of Toronto Toronto. Ontario. Canada MSS 1A4
Applications of the L'Hbpkil-Bernoulli Rule in Chemical Systems
I t is now well established, following publication of correspondence between the Marquis de L'Hbpital and Johann Bernoulli ( I ) , that "L'HBpital's Rule" for the evaluation of "indeterminate" forms, such as 010, is actually due to Bernoulli. Nevertheless, L'HBpitaI made important contributions in eliciting and popularizing the result, and because of their joint efforts, perhaps it should realistically be called the L'HApital-Bernoulli Rule. I t is so designated here, in recognition of both historical practice and historical fact, in spite of the fact that a plea has been made for retention under the name of L'Hbpital alone (2). Regardless of the name, it is an extremely useful tool, and many applications abound that are of chemical interest. In spite of this, and in spite of its introduction via the calculus (or perhaps because of it), it does not seem to be a very sharp tool in the hands of most students, including graduate students, in my experience. The main purpose here then is to provide a collection of examples of the application of the rule in chemical situations, to illustrate and emphasize its use. These examples are drawn mostly from thermodynamics, kinetics, and reactor behavior. Some of them are well known and are treated explicitly in the literature. The L'HBpltal-Bernoulli Rule The L'HBpital-Bernoulli Rule usually enables us to determine the value of a function such as f(x) = g(x)/h(x), where g(x) and h(x) are known and each becomes zero at a particular value of x, say x = a. The rule rewires that and h(x) be differentiated independently, and the limit of the resulting ratio be examined as x a. The procedure may be, and may have to be, repeated two or more times before the limit, if i t exists, can be determined. More formally the rule is (3)
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P(X) = lim & lim ! = lim
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where R is the gas constant, and x = hulkT = a constantlT. The high-temperature ( T -1 and low-temperature ( T 0) limits of C,are indeterminate from eqn. (2) directly, since orx O), the right side reduces to in the former case ( T 010, and in the latter case (x - a ) , it reduces to ,I-. However, these limits may be obtained by double and triple applications, respectively, of the L'HBpital-Bernoulli Rule. Thus
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Similarly C, = 0 a t T = 0. The proof of the rule given originally by Bernoulli, in a letter to L'HBpital dated July 22, 1694 (la), was based on a geo488 1 Journal of Chemical Education
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U(X) - V ( T ) = [1/v(x)- ~Iu(x)II[(IIu(x)v~x)I (4) Illustration (2). For consecutive, first-order chemical reactions, represented by
where k l and k2 are rate constants, occurring at constant density in a batch or plug-flow reactor, the maximum concentration by intermediate species is 9) (6) C B .=~CAo ~ ml/('-m) ~ is the initial concentration of A, where rn = hl,k2, and cAO When = (hl = kz), this reduces to the indeterminate cAol-, To determine cBvmaX for situation, we
I
1im 1, m-1
gno = .. . h"(x1
h(x) .-#h'(x) z-n where g'(x) and gn(x) are the first and second derivatives, respectively, of g(x), and similarly for h'(x) and h"(x). The resulting limit, if it exists, may be finite (including zero) or infinite. The functions g(x) and h(x) must be differentiable at the point a as many times as required to determine the limit. The limit is obtained when the limit of the ratio of the n t h derivative of g(x) to the nth derivative of h(x) first becomes finite (includine zero) or infinite, as n increases from 1. Illustration (1). For a monatomic solid, the Einstein equation for the molar heat capacity a t constant volume is ( 4 )
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metrical construction together with the significance of the then-developing ideas on differentials. The proof given in elementary calculus books is usually baaed on Rolle's theorem and the mean-value theorem (5).A proof involving Taylorseries expansions of the numerator and denominator in eqn. (1) is given by Jenson and Jeffreys (6). Indeterminate Forms In add~tionto 010, indeterminate forms inrlude mlm, O(m), On, 1 ', m",~ n md - = (7). The form m/m can be put in the form O/O by use of the reciprocal of the variable, as indicated in IIlustration (1) abovr. Theothrrsare to be ~ u int the t'orm0/0 or -1- for use with the rule. For this purpose, O(-) is written as O/(l/-) or as mI(110). The forms 0°, 1- and m0 can each be treated by taking the logarithm of the ratio g(x)/h(x). A form resulting in m m can be treated by writing (8)
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3 %- lim ~i 1 Cno - m-, 1 - m
n rn = lim l l m = -1 m-1 -1
(7)
That is, Ca,.,lC~o = e-I, and C .B . =~ CA& ~ = 0.3679 CAO. The limiting value of C B ,as~rn ~ m can also be obtained by use of the rule, but that as m 0 can be obtained directly from eqn. (6) by inspection. When rn = m, the right side of eqn. (6) becomes Cnomo. The treatment contained in eqn. (I), ~ h2). conhowever, shows that CB.,,,~~ CAOas m m ( k >> sistent with expectations on kinetics grounds when the second stage is suppressed. Further Illustrations The L'HBpital-Bernoulli Rule is useful in a number of situations of chemical interest. Further illustrations are eiven below, some of which have been treated directly or indirectly in the literature. The first three (illus. (3) to (5))relate toPuT behavior and thermodynamics, the next three (illus. (6) to (8)) relate to kinetics, and the last three (illus.(O) to (11)) relate to chemical reactor behavior. For brevity, the solutions are not given in detail, and there is no attempt to be exhaustive. Illustration (3). I t can be shown (1O)by means of the rule that the limiting value of the residual volume, a = (RTIP) o, as P 0 a t a given temperature T (or reduced temperature TR) is
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~
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lim a = -(RTIPJ lim (anlaP,&,
P-0
P-0
(8)
where the second factor on the right is the limiting value of the slope of the plot of compressibility factor (z = PoIRT)
against reduced pressure, PR = PIP,, and subscript c denotes the critical value. This illustration can he carried one sten further. without use of the rule, to show that thwe limiring slmes and anv correlation for the second virial coeffirirnt n. in PUIRT = i+ Blu + . . . ,such as that given by Pitzer and Curl (ll), must be consistent, since lim ol = -B, and then from eqn. (8)(IOa)
Illustration (4). Consider any molar excess function (12), such as excess enthalpy, hE, for a binary solution of components 1and 2. The functions (1) hElxlxz, (2) hElxl, and (3) hElxz, where x is mole fraction, are used to represent compositional dependence in terms of x l for various purposes. The rule can he used to show that the intercept a t X I = 0 in plot (1) or (2) is the excess-partial molar enthalpy of component 1a t infinite Glution, hlE", and the intercept a t x = 1in plot (1) or (3) is hzE" (lob). In this example it is necessary to recall that h E 0 as x l or x2 0, SO that the indeterminate form 0/0 results directly in each of the four cases cited, and that x l xz = 1. The interpretations given ahove are then ohtained by means of the rule together with the relation (lob)
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+
-hlE = hE - xldhEldx~
(10) (or similarly for idzE3ith su_bscripts 1and 2 interchanged), 0 (or similarly for and realization that hlE hlEL as X I id2E). Illustration (5).If we adopt, as a statement of the Third Law of Thermodynamics, the Nernst hypothesis (13) that the curves of AA (change in Helmholtz function, A = U - TS) and of AU (change in internal energy) for a chemical reaction involving pure liquids andlor solids at temperature T approach each other asymptotically as T 0, then it is possible to show, by considering the limit of (AU - AA)IT as T 0, that for such a reaction the limiting value of the entropy change is (13a)
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lim AST = 0
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E = RT In IAI[Al exp (-EIIRT) Az exp (-E2IRT)]) (16) The rule can he used to determine the high- and low-temperature limits of E , and the limiting slope d E l d T as T 0
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(~1 7,)
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Illustration (9). For a first-order reaction (A products) occurring a t constant density in N continuous-flow, stirredtank reactors (CSTR's) of equal size in series, the space time, r = Vlu, where V is the total volume of the tanks and u is the steady-state flow rate through the system, is (9a) r = (N/k~)[(CndCn)'l"- 11 (17) where CAOis the concentration of A at the inlet t o the first tank, and CAis that a t the outlet from the N t h tank, and k~ is the rate constant. As N increases indefinitely, application of the rule shows that the limiting value of r becomes i= ( l l k 3 In ( C A ~ C A ) (18) which is the exoression for a olue-flow reactor of volume V. IUustration (i0). For a cont
