A Universal Integrated Rate Equation for Chemical Kinetics

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A: Kinetics and Dynamics

A Universal Integrated Rate Equation for Chemical Kinetics Wesley D. Allen J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b02372 • Publication Date (Web): 16 Mar 2018 Downloaded from http://pubs.acs.org on March 17, 2018

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The Journal of Physical Chemistry

A Universal Integrated Rate Equation for Chemical Kinetics Wesley D. Allen* Department of Chemistry and Center for Computational Quantum Chemistry, University of Georgia, Athens, GA 30602 ABSTRACT The overarching analytic integrated rate equation for the chemical kinetics of any reversible or irreversible reaction involving an arbitrary number of species and any integral orders is shown to be

∏ i=1[1− r

f i −1ξ (t)]γ i = e(−1)

r

F0 t

, where ξ(t) is the extent of reaction variable, the fi are

roots of a polynomial of order r, the exponents are determined by γ i = ∏ k (≠i) ( f i − f k )−1 , and F0 is r

a factor involving the stoichiometric coefficients and rate constants (k±).

All integrated rate

equations of elementary reactions appearing in chemical kinetics are special cases of this universal solution. Not only does the solution provide insight into the analytical form of the exponents γi and F0 that govern the time evolution of the system, but it also provides an elegant framework for the pedagogy and application of kinetics in physical chemistry.

Journal of Physical Chemistry A

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INTRODUCTION The birth of chemical kinetics is often traced to 1850, when the rate of inversion of sucrose was studied by German physicist Wilhemy (1812-1864).1

For the first time a differential

equation was set up and integrated for the concentration of a chemical species as a function of time, and the resulting expression proved consistent with experimental measurements. In the years 1865-67, as part of a collaboration with Harcourt at Oxford investigating the HOOH + HI and KMnO4 + oxalic acid reactions, Esson wrote down and integrated several types of differential equations, including those for first- and second-order reactions, as well as consecutive first-order processes.1 The most outstanding early work on chemical kinetics was carried out by Van’t Hoff (1852-1911), whose seminal book in 1884 analyzed examples of unimolecular, bimolecular, and “polymolecular” reactions, integrated the associated differential equations, and determined rate constants.2

Ensuing advances in third-order reactions led

Germann3 in 1928 to clarify the proper use of integrated rate equations involving one secondorder and one first-order reactant. Such a rate law was discovered for the reaction of NO with O2 in 1918,4 although the reaction is not termolecular and the nature of the intermediates remained controversial for many years.5-7 While uncommon, even fourth-order reactions exist in liquid phases and were already under investigation by the 1920s and 1930s.8-10 The derivation of integrated rate equations for simple irreversible first- and second-order chemical reactions is now ubiquitous in modern physical chemistry textbooks. A survey of the types of reactions covered in a comprehensive list of 20 textbooks is given in Supporting Information. The pervasive approach is to integrate rate equations on a case-by-case basis, but covering all possibilities is considered too lengthy and tedious. To eschew mathematical complexity, “gory details of the integrations” are either omitted or left as an exercise for the reader.11 Even if realistic models are presented in which reverse reactions are included in


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determining the time evolution of the species, restrictions are made to simple paradigms such

!! ⇀ as first-order A ↽ ! ! B reactions. The overall structure of integrated rate equations for chemical kinetics was pointed out in a pedagogical article in 1994.12 However, a general analytical solution was not advanced, and the formalism was only demonstrated for the simplest cases. One research article from 1970 provides a compilation of integrated rate equations for forward reactions involving only two species with any combination of integral or half-integral orders up to a maximum of three.13 Continued interest in the mathematical form of integrated rate laws is shown in a 2011 article that solved two third-order cases by employing the Lambert function.14 Here a unified, compact analytic solution is presented for integrated rate equations of chemical kinetics. Any number of reactant and product species may be present; each species may be involved in the rate law to any integral order; and both forward and reverse reactions are incorporated. The reactions do not necessarily have to be elementary ones, as long as the observed rate laws take on the assumed form.

Our approach is amenable to automated

implementation in symbolic algebra programs.

Our extensive searching has not found a

unified solution of this type in any textbook or journal article.

The dissemination of this

solution in physical chemistry should lead to recognition that elegant analytic integration can fully solve the differential rate law for a single reversible or irreversible chemical reaction.

UNIFIED ANALYTIC SOLUTION Consider a generalized chemical equation for reactants Ri (i = 1, 2, …M) and products Pj (j = 1, 2, …N) with corresponding stoichiometric coefficients ai and bj: k+ ⎯⎯ ⎯ → b P + b P +...+ b P a1R 1 + a2 R 2 + ... + a M R M ← ⎯ 1 1 2 2 N N k

(1)

−

Reaction progress can be quantified in terms of the customary extent of reaction variable (ξ), which yields the reactant concentrations via

(

)

(

)

(

)

⎡ R M ⎤⎦0 − ⎡⎣ R M ⎤⎦ , ξ = a1−1 ⎡⎣ R 1 ⎤⎦0 − ⎡⎣ R 1 ⎤⎦ = a2−1 ⎡⎣ R 2 ⎤⎦0 − ⎡⎣ R 2 ⎤⎦ = ... = a −1 M ⎣


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(2)

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and those of the products by means of

(

)

(

)

(

)

ξ = b1−1 ⎡⎣ P1 ⎤⎦ − ⎡⎣ P1 ⎤⎦0 = b2−1 ⎡⎣ P2 ⎤⎦ − ⎡⎣ P2 ⎤⎦0 = ... = bN−1 ⎡⎣ PN ⎤⎦ − ⎡⎣ PN ⎤⎦0 .

(3)

Accordingly, the consumption rates of the reactants are given by

dξ 1 d[R 1 ] 1 d[R 2 ] 1 d[R M ] =− =− = ... = − , dt a1 dt a2 dt a M dt

(4)

while the appearance rates of the products are determined from

dξ 1 d[P1 ] 1 d[P2 ] 1 d[PN ] = = = ... = . dt b1 dt b2 dt bN dt

(5)

The net rate of the reversible reaction is governed by the equation

dξ m m m n n n =k [R ] 1 [R 2 ] 2 ...[R M ] M − k− [P1 ] 1 [P2 ] 2 ...[PN ] N , dt + 1

(6)

in which k+ and k– are the forward and reverse rate coefficients, respectively, the orders of the reactants are m1, m2, ..., mM, and those of the products are n1, n2, ..., nN. The corresponding differential equation for ξ(t) is

(

) ( ⎡⎣ R ⎤⎦ − a ξ ) ...( ⎡⎣ R ⎤⎦ − a ξ ) − k ( ⎡⎣ P ⎤⎦ + b ξ ) ( ⎡⎣ P ⎤⎦ + b ξ ) ...( ⎡⎣ P ⎤⎦ + b ξ )

dξ = k+ ⎡⎣ R 1 ⎦⎤0 − a1ξ dt

m1

m2

2 0

2

n1

−

1 0

M

M

0

n2

1

2 0

nN

2

N

0

mM

.

(7)

N

As usual, the rate coefficients contain temperature and pressure effects.

The forward and

reverse reactions are of total order M

N

r = ∑ mi

s = ∑ ni ,

and

i=1

(8ab)

i=1

respectively. For convenience, we assume here that the forward direction of the reaction is chosen so that r ≥ s and the quantity

F0 = k+ a1 1 a2 2 ... a MM − (−1)r k−δ rs b1 1 b2 2 ... bNN m

m

m

n

n

n

(9)

is positive. In eq 9, δrs is the Kronecker delta symbol. The stoichiometric coefficients and initial concentrations define the constants

ci = ai−1 ⎡⎣ R i ⎤⎦0 ,

di = bi−1 ⎡⎣ Pi ⎤⎦0 ,

(10ab)

and m

m

m

Q = (−1)r a1 1 a2 2 ... a MM .

(11)

Hence, eq 7 can be written as


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dξ =F ξ , dt

()

(12)

in terms of the r-th order polynomial

()

(

F ξ = k+ Q ⎡ ξ − c1 ⎣⎢ where

) (ξ − c ) ...(ξ − c )

(

q = k− k+ Q

m1

m2

2

)

−1

n

mM

M

n

(

− q ξ + d1

) (ξ + d ) ...(ξ + d ) n1

n2

2

N

nN

⎤, ⎦⎥

(13)

n

(14)

b1 1 b2 2 ... bNN

involves the ratio of the reverse and forward rate constants. Denote the roots of F(ξ) as fi. After extracting these roots, eq 12 can be transformed to

dξ = (−1)r F0 ξ − f1 dt

(

) (ξ − f ) ...(ξ − f ) p1

p2

2

r

pr

,

(15)

where the pi values are the root multiplicities (degeneracies). The solution ξ(t) is determined by integrating differential eq 12 using dummy variable x:

∫

dx

ξ

0

( x − f ) ( x − f ) ...( x − f ) p1

p2

1

2

pr

= (−1)r F0 t .

(16)

r

The integrand can be decomposed by partial fractions to yield r

1

( x − f ) ( x − f ) ...( x − f ) p1

p2

1

2

pr

pi

= ∑∑

r

i=1 j=1

Cij

(x − f )

j

,

(17)

i

where the Cij constants depend on all of the fi. From eqs 16 and 17, r

pi

∑∑ C ∫ i=1 j=1

dx

ξ

ij 0

(

x − fi

)

j

= (−1)r F0 t .

(18)

Integrating the partial fractions in eq 18 provides r

⎡

∑ ⎢C i=1

⎢⎣

i,1

⎤ ⎛ f − ξ ⎞ pi −1 − ∑ (−1) j Ci, j+1 Di, j (ξ ) ⎥ = (−1)r F0 t , ln ⎜ i ⎟ ⎝ f i ⎠ j=1 ⎥⎦

(19)

where

Di, j (ξ ) =

1 j fi ( fi − ξ ) j j

(

)

⎡ f j − f −ξ j⎤ . i ⎢⎣ i ⎥⎦

(20)

Nonlinear eq 19 determines ξ as a function of t, or more directly t as a function of ξ.

(

)

(

The

)

species concentrations can then be obtained from ⎡⎣ R i ⎤⎦ = ai ci − ξ and ⎡⎣ Pi ⎤⎦ = bi di + ξ . In general, the roots of F(ξ) will be nondegenerate, being determined by a complicated balance of stoichiometric coefficients, reaction orders, initial concentrations, and the ratio of


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the forward and reverse rate constants. However, even if degeneracies arise, we can perform the partial fraction decomposition assuming that all roots are distinct and subsequently take the appropriate limit of the resulting expression as one or more roots coalesce. Therefore, we proceed by setting all pi = 1 in eq 19, which means that only the constants Ci,1 = γ i are required. The exponential form of the final result is

⎛ fi − ξ ⎞ ∏ ⎜ f ⎟ ⎠ i=1 ⎝ i r

γi

=e

(−1)r F0 t

,

(21)

which constitutes a universal integrated rate equation for chemical kinetics. To complete the derivation, we prove in the Appendix that the general formula for the exponents is r

(

γ i = ∏ fi − fk k (≠i)

)

−1

.

(22)

for r > 1, while γ1 = 1 otherwise. In the mathematical space of the fi roots, the appearance of degeneracies in eq 21 will only generate point discontinuities that are removable. The evaluation of eq 21 for overall orders r = 1-4 can be used to generate explicit integrated rate equations for the most fundamental cases encountered in chemical kinetics. It should be emphasized that these results hold for either reversible or irreversible reactions, the forward reaction corresponds to the highest overall order, the reverse direction can be of equivalent or lower order, and degeneracies refer in a general sense to the number of occurrences (pi) of each root fi of F(ξ) in eq 13. If the reverse reaction is omitted, then the roots fi are equivalent to the initial concentration ratios ci of eq 10a, and degenerate cases occur when either a reactant has order mi > 1 in eq 6 or some reactants happen to be in stoichiometric proportion. With this understanding, the compact solutions for all cases through fourth order are as follows: First-order reactions (p1 = 1):

(

ξ (t) = f1 1− e− F0 t

(23)

f −f F t f1 f 2 ⎡1− e( 1 2 ) 0 ⎤ ⎣ ⎦ ξ (t) = f −f F t f 2 − f1e( 1 2 ) 0

Second-order reactions (p1, p2)= (1,1):


)

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⎛ ⎞ 1 ξ12 (t) = lim ξ (t) = f1 ⎜ 1− ⎟ f 2 → f1 ⎝ 1+ f1 F0 t ⎠

Degenerate second-order reactions (p1 = 2):

(25)

Third-order reactions (p1, p2, p3) = (1,1,1):

⎛ f1 − ξ ⎞ ⎜ f ⎟ ⎝ 1 ⎠

f 2 − f3

⎛ f2 − ξ ⎞ ⎜ f ⎟ ⎝ 2 ⎠

f3 − f1

⎛ f3 − ξ ⎞ ⎜ f ⎟ ⎝ 3 ⎠

f1 − f 2

f −f f −f f −f F t = e( 1 2 )( 2 3 )( 3 1 ) 0

(26)

( f2 − f1 )ξ23 2 f 2 ( f1 − ξ 23 ) f2 ( f2 −ξ23 ) − f −f F t e = e ( 1 2) 0 f1 ( f 2 − ξ 23 )

(27)

Degenerate third-order reactions: (p1, p2)= (1,2):

ξ 23 (t) = lim ξ (t) ⇒

(p1 = 3):

⎛ ⎞ 1 ⎟ ξ123 (t) = lim lim ξ (t) = f1 ⎜ 1− f 2 → f1 f3 → f 2 ⎜⎝ 1+ 2 f12 F0 t ⎟⎠

f3 → f 2

(28)

Fourth-order reactions (p1, p2, p3, p4) = (1,1,1,1):

⎛ f1 − ξ ⎞ ⎜ f ⎟ ⎝ 1 ⎠

( f 2 − f3 )( f 2 − f 4 )( f3 − f 4 )

⎛ f3 − ξ ⎞ ⎜ f ⎟ ⎝ 3 ⎠

⎛ f2 − ξ ⎞ ⎜ f ⎟ ⎝ 2 ⎠

( f1 − f 2 )( f1 − f 4 )( f 2 − f 4 )

( f3 − f1 )( f1 − f 4 )( f3 − f 4 )

×

⎛ f4 − ξ ⎞ ⎜ f ⎟ ⎝ 4 ⎠

(29)

( f1 − f 2 )( f3 − f1 )( f 2 − f3 )

=e

( f1 − f 2 )( f1 − f3 )( f1 − f 4 )( f 2 − f3 )( f 2 − f 4 )( f3 − f 4 ) F0 t

Degenerate fourth-order reactions: (p1, p2, p3) = (1,1,2): ξ34 (t) = lim ξ (t) ⇒ f 4 → f3

⎛ f1 − ξ34 ⎞ ⎜ f ⎟ ⎝ ⎠ 1

( f 2 − f3 )2

⎛ f 2 − ξ34 ⎞ ⎜ f ⎟ ⎝ ⎠ 2

−( f1 − f3 )2

⎛ f3 − ξ34 ⎞ ⎜ f ⎟ ⎝ ⎠ 3

( f1 + f 2 −2 f3 )( f1 − f 2 )

e

ξ34 ( f1 − f 2 )( f1 − f3 )( f 2 − f3 ) f3 ( f3 −ξ34 )

=e

( f1 − f 2 )( f1 − f3 )2 ( f 2 − f3 )2 F0 t

(30)

(p1, p2) = (1,3): ξ 234 (t) = lim lim ξ (t) ⇒ f3 → f 2 f 4 → f3

f 2 ( f1 − ξ 234 ) e f1 ( f 2 − ξ 234 )

ξ 234 ( f1 − f 2 )(2 f1 f 2 −4 f 22 − f1ξ 234 +3 f 2ξ 234 ) 2 f 22 ( f 2 −ξ 234 )2

=e

( f1 − f 2 )3 F0 t

(31)

f1 − f 2 )3 F0 t

(32)

(p1, p2) = (2,2): ξ13,24 (t) = lim lim ξ (t) ⇒ f3 → f1 f 4 → f 2

f1 ( f 2 − ξ13,24 ) f 2 ( f1 − ξ13,24 )

( f1 − f 2 )ξ13,24 [ f12 + f 22 −( f1 + f 2 )ξ13,24 ]

e

2 f1 f 2 ( f1 −ξ13,24 )( f 2 −ξ13,24 )

1(

= e2

⎛ ⎞ 1 ⎟ (p1 = 4): ξ1234 (t) = lim lim lim ξ (t) = f1 ⎜ 1− 3 f 2 → f1 f3 → f 2 f 4 → f3 ⎜⎝ 1+ 3 f13 F0 t ⎟⎠


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Figure 1. Logarithmic plots of the integrated rate equations (23)-(33) when f1 = 0.1, f2 = 0.2, f3 = 0.3, and f4 = 0.4. Each curve is labelled with the corresponding set of pi values.

In the Supporting Information, detailed derivations of the limits of the logarithmic form of eq 21 are given for the general cases of one double degeneracy, one triple degeneracy, one quadruple degeneracy, and two double degeneracies; eqs 25, 27, 28, 30, 31, 32, and 33 are shown therein to be special cases of the general expressions.

Remarkably, all of the

logarithmic forms of the integrated rate equations can be cast into the form

∑

*

i=1

⎡ ⎤ ⎛ f − ξ ⎞ pi −1 − ∑ Ti, p −1− j Di, j (ξ ) ⎥ = (−1)r F0 t , γ i (−1) pi −1 ⎢Ti, p −1 ln ⎜ i ⎟ i i ⎝ f i ⎠ j=1 ⎢⎣ ⎥⎦

(34)

where the asterisk denotes that the summation index runs only over the unique fi values, each of degeneracy pi, while

Ti,n =

r

∑′

( f − f ) ( f − f ) ...( f − f )

j1 ≥ j2 ≥...≥ jn

−1

i

j1

−1

i

j2

i

jn

−1

.

(35)

In eq 35 the summation is not restricted to unique fi values, and the prime denotes that all singular terms are excluded.


The generality of eq 34 warrants emphasis, because it is the

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logarithmic form of eq 21 in which all possible point discontinuities arising from root degeneracies have been analytically removed. A demonstration of eq 34 is given in Figure 1, which plots solutions for all the cases of eqs 23-33 for an arbitrary choice of the roots fi. APPLICATIONS To facilitate the application of eqs 21 and 34, Table 1 lists the F(ξ) polynomials and their roots for 12 cases of chemical kinetics. In Cases 7–12 the roots of F(ξ) satisfy a cubic equation of the form A f 3 − B f 2 + C f − D = 0 . Analytic solutions15,16 for the roots of a cubic polynomial exist but are not commonly known. An efficient procedure for obtaining these solutions is as follows. First calculate the discriminant D = B2C2 – 4AC3 – 4B3D –27A2D2 + 18ABCD and the quantities T = 2B3 + 27DA2 – 9ABC, U = ½[T + 3A(–3D)½], V = ½[T – 3A(–3D)½], and W = B2 – 3AC. If D > 0, then W > 0 and the three roots are real and distinct:

(

)

f n+2 = 13 A−1 B + 2W 1/2 cos ⎡⎣ 13 (2nπ + β ) ⎦⎤ for n = (–1, 0, 1), where β = cos −1 roots are real but not distinct; specifically, one

real

and

two

complex

{f

1

(

1 2

{

}

)

TW −3/2 . If D = 0, then the

}

= f 2 , f3 = 13 A−1 B − ( 12 T )1/3 , B + (4T )1/3 . If D < 0, then

roots

exist.

The

complex

(

)

roots

are

f1,2 = 61 A−1 ⎣⎡ 2B − U 1/3 − V 1/3 ± 31/2 i(U 1/3 − V 1/3 ) ⎦⎤ , and the real root is f3 = 13 A−1 B + U 1/3 + V 1/3 .

Table 1. Fn(ξ ) polynomials of order n and their roots (f) for 12 cases of chemical kineticsa Notation:

1

+

2

(

()

( f , f )=( f , f )

F3 ξ = Qk+ Aξ 3 − Bξ 2 + C ξ − D

−

)

⇀ !! Case 1. One reactant of order 1, one product of order 1 ( R 1 ↽ ! ! P1 )

()

(

)

(

F1 ξ = −k+ a1 ξ − c1 − k− b1 ξ + d1

)

(1− q ) f

+

= c1 + qd1

⇀ !! Case 2. One reactant of order 2, one product of order 1 ( 2R 1 ↽ ! ! P1 )

()

(

)

2

(

F2 ξ = k+ a12 ξ − c1 − k− b1 ξ + d1

)

(

)

1/2

2 f ± = 2c1 + q ± ⎡⎣ q q + 4c1 + 4d1 ⎤⎦

⇀ !! Case 3. One reactant of order 2, one product of order 2 ( 2R 1 ↽ ! ! 2P1 )

()

(

)

2

(

F2 ξ = k+ a12 ξ − c1 − k− b12 ξ + d1


)

2

(1− q ) f

±

(

= c1 + qd1 ± c1 +d1

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)

q

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!! ⇀ Case 4. One reactant of order 2, two products each of order 1 ( 2R 1 ↽ ! ! P1 + P2 )

()

(

)

(

2

)(

F2 ξ = k+ a12 ξ − c1 − k− b1b2 ξ + d1 ξ + d2

) (1− q ) f

±

(

)

(

)(

)

(

)

1/2

2 = c1 − 12 q d1 + d2 ± ⎡ q c1 − d1 c1 − d2 + 14 q 2 d1 − d2 ⎤ ⎥⎦ ⎣⎢

!! ⇀ Case 5. Two reactants each of order 1, one product of order 1 ( R 1 + R 2 ↽ ! ! P1 )

()

(

)(

)

(

F2 ξ = k+ a1a2 ξ − c1 ξ − c2 − k− b1 ξ + d1

)

(

2 f ± = c1 + c2 + q ± ⎡ c1 − c2 ⎢⎣

)

2

(

1/2

)

+ 2q c1 + c2 + 2d1 + q 2 ⎤ ⎥⎦

!! ⇀ Case 6. Two reactants each of order 1, two products each of order 1 ( R 1 + R 2 ↽ ! ! P1 + P2 )

()

(

)(

)

(

)(

(

)

F2 ξ = k+ a1a2 ξ − c1 ξ − c2 − k− b1b2 ξ + d1 ξ + d2

(

)

(

)

2 1− q f ± = c1 + c2 + q d1 + d2 ± ⎡ q 2 d1 − d2 ⎣⎢

2

)

(

)(

)

(

) (

)

1/2

2 + 2q c1 + c2 d1 + d2 + 4q c1c2 + d1d2 + c1 − c2 ⎤ ⎦⎥

!! ⇀ Case 7. Three reactants each of order 1, three products each of order 1 ( R 1+R 2+R 3 ↽ ! ! P1+P2 +P3 )

()

(

)(

)(

)

(

)(

)(

F3 ξ = −k+ a1a2 a3 ξ − c1 ξ − c2 ξ − c3 − k− b1b2 b3 ξ + d1 ξ + d2 ξ + d3

(

)

(

)

)

A = 1− q , B = c1 + c2 + c3 + q d1 + d2 + d3 , C = c1c2 + c1c3 + c2 c3 − q d1d2 + d1d3 + d2 d3 , D = c1c2 c3 + qd1d2 d3

!! ⇀ Case 8. Three reactants each of order 1, two products of order (1, 2) ( R 1+R 2 +R 3 ↽ ! ! P1 + 2P2 )

()

(

)(

)(

)

(

)(

F3 ξ = −k+ a1a2 a3 ξ − c1 ξ − c2 ξ − c3 − k− b1b22 ξ + d1 ξ + d2

(

)

)

2

(

)

A = 1− q , B = c1 + c2 + c3 + q d1 + 2d2 , C = c1c2 + c1c3 + c2 c3 − q d2 2d1 + d2 , D = c1c2 c3 + qd1d22

!! ⇀ Case 9. Three reactants each of order 1, one product of order 3 ( R 1+R 2 +R 3 ↽ ! ! 3P1 )

()

(

)(

)(

)

(

F3 ξ = −k+ a1a2 a3 ξ − c1 ξ − c2 ξ − c3 − k− b13 ξ + d1

)

3

A = 1− q , B = c1 + c2 + c3 + 3qd1 , C = c1c2 + c1c3 + c2 c3 − 3qd12 , D = c1c2 c3 + qd13

!! ⇀ Case 10. Three reactants each of order 1, two products each of order 1 ( R 1+R 2 +R 3 ↽ ! ! P1 + P2 )

()

(

)(

)(

)

(

)(

F3 ξ = −k+ a1a2 a3 ξ − c1 ξ − c2 ξ − c3 − k− b1b2 ξ + d1 ξ + d2

(

)

)

A = 1 , B = c1 + c2 + c3 + q , C = c1c2 + c1c3 + c2 c3 − q d1 + d2 , D = c1c2 c3 + qd1d2 ⇀ !! Case 11. Three reactants each of order 1, one product of order 2 ( R 1 +R 2 +R 3 ↽ ! ! 2P1 )

()

(

)(

)(

)

(


)

2

A = 1 , B = c1 + c2 + c3 + q , C = c1c2 + c1c3 + c2 c3 − 2qd1 , D = c1c2 c3 + qd12 ⇀ !! Case 12. Three reactants each of order 1, one product of order 1 ( R 1+R 2 +R 3 ↽ ! ! P1 )

()

(

)(

)(

)

(


)

A = 1 , B = c1 + c2 + c3 , C = c1c2 + c1c3 + c2 c3 − q , D = c1c2 c3 + qd1 a

For each case an example is given in parentheses of a corresponding elementary reaction.


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To demonstrate our methodology, consider an example of Case 7 in which a1a2a3=b1b2b3,

(

)

and c=c1=c2=c3=d1=d2=d3. This choice of parameters yields F0 = −Qk+ 1− q and

()

(

)

(

)

F ξ = Q k+ ⎡ ξ − c − q ξ + c ⎤ . ⎢⎣ ⎥⎦ 3

3

(36) 6 2

3

The quantities for finding the roots of F(ξ) are D = –1728 c q , T = 216 c q(q+1), 3 2

3

2

U = 216 c q , V = 216 c q, and W = 36 c q. The exact roots of F(ξ) can then be expressed as

f1 = v + i w, f2 = v – i w, and f3 = u, where

( ) (1+ 2q + 2q + q ) , v = c (1− q ) (1− q − q + q ) ,

u = c 1− q

−1

1/3

−1

and

2/3

1/3

(

w = 3 c 1− q

) (q −1

(37)

2/3

2/3

(38)

)

− q1/3 .

(39)

Although f1 and f2 are complex roots, the integrated rate equation given by eq 26 can be simplified to a form involving only real quantities:

⎛ ξ 2 − 2uξ + u 2 ⎞ 2θ ( v−u )/ w ⎛ u 2 ⎞ −3Q ⎡⎢⎣(u−v )2 +w2 ⎤⎥⎦k+t e =⎜ 2 , 2⎟ ⎜ ξ 2 − 2vξ + v 2 + w2 ⎟ e ⎝ ⎠ ⎝v +w ⎠

(40)

where

tan θ =

wξ . v 2 + w2 − vξ

(41)

To illustrate the insight that can be gained from such an analytic form, eq 40 can be expanded about ξ = u to ascertain that the approach of the chemical system to equilibrium is governed by

(

ξ (t) = u 1± Ge−α t

)

(42)

where

G=

(u − v )

2

+ w2

v 2 + w2

e

w−1(u−v ) arctan ⎡⎣uw(v 2 +w2 −uv)−1 ⎤⎦

(43)

and

(

)

α = 23 k+ Q ⎡ u − v + w2 ⎤ . ⎣⎢ ⎦⎥ 2

(44)

When the reverse reaction is excluded in eq 1 by setting k– = 0, our formulas provide standard integrated rate equations for common cases, with parameters extracted from Table 1.


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Page 12 of 18

First order irreversible From Case 1 ( f1 = f + = c1 ):

(

ξ (t) = a1−1 ⎡⎣ R 1 ⎤⎦0 1− e− a1k+ t

)

(45)

⎡⎣ R 1 ⎤⎦ = ⎡⎣ R 1 ⎤⎦ e− a1k+ t 0

(46)

Second order irreversible From Cases (2,3,4), and Cases (5,6) with stoichiometric initial proportions ( f1 = f 2 = f ± = c1 ): 2

ξ12 (t) =

k+ ⎡⎣ R 1 ⎤⎦0 t

(47)

1+ k+ a1 ⎡⎣ R 1 ⎤⎦0 t 1 1 − = k+ a1 t ⎡⎣ R 1 ⎤⎦ ⎡⎣ R 1 ⎤⎦

(48)

0

From Cases (5,6) with non-stoichiometric initial proportions [ ( f1 , f 2 ) = (c1 ,c2 ) ]: − c −c a a k t c1c2 ⎡1− e ( 1 2 ) 1 2 + ⎤ ⎣ ⎦ ξ (t) = −( c1 −c2 )a1a2 k+ t c1 − c2 e

(49)

⎡⎣ R 1 ⎤⎦ ⎡⎣ R 2 ⎤⎦ − a [R ] −a [R ] k t 0 = e ( 1 2 0 2 1 0) + ⎡⎣ R 1 ⎤⎦ ⎡⎣ R 2 ⎤⎦

(50)

0

Third order irreversible From Case 7 with non-stoichiometric initial proportions [ ( f1 , f 2 , f3 ) = (c1 ,c2 ,c3 ) ]:

⎛ c1 − ξ ⎞ ⎜ c ⎟ ⎝ 1 ⎠

c2 −c3

⎛ ⎡R ⎤ ⎞ ⎣ 1⎦ ⎜ ⎟ ⎜⎝ ⎡⎣ R 1 ⎤⎦ ⎟⎠ 0

c3 −c2

⎛ c2 − ξ ⎞ ⎜ c ⎟ ⎝ 2 ⎠

c3 −c1

⎛ ⎡R ⎤ ⎞ ⎣ 2⎦ ⎜ ⎟ ⎜⎝ ⎡⎣ R 2 ⎤⎦ ⎟⎠ 0

⎛ c3 − ξ ⎞ ⎜ c ⎟ ⎝ 3 ⎠

c1 −c3

c1 −c2

⎛ ⎡R ⎤ ⎞ ⎣ 3⎦ ⎜ ⎟ ⎜⎝ ⎡⎣ R 3 ⎤⎦ ⎟⎠ 0

− c −c c −c c −c k a a a t = e ( 1 2 )( 2 3 )( 1 3 ) + 1 2 3

(51) c2 −c1

a a a c −c c −c c −c k t = e 1 2 3( 2 1 )( 3 2 )( 1 3 ) + .

(52)

The solution for the third-order irreversible reaction that is second order in reactant A and first order in reactant B has been given in terms of the Lambert function W(x) in eq (5) of Williams.14 Because W(x) = y implies that y ey = x, our eq (27) can be written as 2 f1 − ξ 23 − f f −1 + F f − f t = W ⎛ − f1 f 2−1e 1 2 0 ( 1 2 ) ⎞ , ⎝ ⎠ ξ 23 − f 2

(53)

which is equivalent to


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ξ 23 = f 2 +

(f − f ) 1

1+ W ⎛ − f f e ⎝

2

(

)2 ⎞

−1 −1 − f1 f 2 + F0 f1 − f 2 t 1 2

.

(54)

⎠

Note that eq 54 gives ξ 23 explicitly as a function of t, even in the case of a reversible reaction.

(

Making the replacements ξ 23 → x(t) , f 2 → δ , f1 → γ , F0 f1 − f 2

)

2

→ − β , and − f1 f 2−1 → α within eq

(54) yields eq (5) of Williams for the corresponding irreversible reaction.

SUMMARY A compact, universal integrated rate equation (eq 21) for chemical kinetics has been presented, along with specific forms of this solution applicable to first-, second-, third-, and fourth-order reactions (eqs 23-33). The only quantities appearing these equations are the roots (fi) of the polynomial F(ξ) (eq 13), exponents γi that depend on products of differences of the fi (eq 22), and a factor F0 (eq 9) involving the stoichiometric coefficients and rate coefficients (k±). Explicit equations for F(ξ) and the fi roots are collected in Table 1 for all types of elementary reactions generally encountered.

These results are applicable to systems involving a single

extent-of-reaction variable, but not reaction networks involving multidimensional systems of coupled differential equations.

The elegant solutions derived here provide insight into the

analytic form of elementary chemical kinetics and provide a concise and comprehensive framework for understanding integrated rate equations germane to physical chemistry.


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Page 14 of 18

APPENDIX Decomposition by partial fractions takes the form

( x − f )( 1

r γi 1 =∑ x − f 2 ... x − f r i=1 x − f i

) (

)

(

)

,

(A1)

assuming that all the fi are distinct. Multiplying eq A1 by the product in the denominator on the left side yields r

1 = ∑ γ j g j (x) ,

(A2)

j=1

in which

g j (x) =

r

∏ (x − f ) k

k (≠ j )

.

(A3)

Note that r

g j ( f i ) = δ ij ∏ ( f i − f k ) ,

(A4)

k (≠i)

where δ is the Kronecker delta symbol. Evaluating eq A2 for x = fi then provides r

r

j=1

k (≠i)

1 = ∑ γ j δ ij ∏ ( f i − f k ) ,

(A5)

which immediately yields r

(

γ i = ∏ fi − fk k (≠i)

)

−1

(A6)

and thus proves eq (22) of the text.

SUPPORTING INFORMATION A tabulation of the types of chemical kinetics integrated rate equations presented in a comprehensive list of 20 textbooks; detailed derivations of the limits of the logarithmic form of the universal integrated rate equation for various cases of root degeneracies. AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]


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ACKNOWLEDGMENTS Research in first-principles chemical kinetics in the Center for Computational Quantum Chemistry at the University of Georgia is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Combustion Program, Grant No. DE-SC0018412. Alexander T. Winkles and Xueying Li are thanked for performing some of the literature searches for this paper as part of an undergraduate class project. REFERENCES 1.

Laidler, K. J. Chemical Kinetics and the Origins of Physical Chemistry. Archive for History of Exact Sciences 1985, 32, 43-75.

2.

Van’t Hoff, J. H. Études de Dynamique Chimique; F. Muller: Amsterdam, 1884.

3.

Germann, F. E. E. Chemical Reactions of the Third Order. J. Phys. Chem. 1928, 32, 1748-1750.

4.

Bodenstein, M. Die Geschwindigkeit der Reaktion Zwischen Stickoxyd und Sauerstorr. Z. Elektrochem. 1918, 24, 183-201.

5.

Galliker, B.; Kissner, R.; Nauser, T.; Koppenol, W. H. Intermediates in the Autooxidation of Nitrogen Monoxide. Chem. Eur. J. 2009, 15, 6161-6168.

6.

Beckers, H.; Zeng, X.; Willner, H. Intermediates Involved in the Oxidation of Nitrogen Monoxide: Photochemistry of the cis-N2O2⋅O2 Complex and of sym-N2O4 in Solid Ne Matrices. Chem. Eur. J. 2010, 16, 1506-1520.

7.

Gadzhiev, O. B.; Ignatov, S. K.; Gangopadhyay, S.; Masunov, A. E.; Petrov, A. I. Mechanism of Nitric Oxide Oxidation Reaction (2 NO + O2 → NO2) Revisited. J. Chem. Theory Comput. 2011, 7, 2021-2024.

8.

Rieder, R.; Skrabal, A. Nochmals die Landoltsche Reaktion. Die Beschleunigung der Landoltschen Reaktion durch Bromide und Chloride. Z. Elektrochem. 1924, 30, 109-124.

9.

Abel, E.; Hilferding, K. Revision der Kinetik der HJO3-HJ-Reaktion. II. Z. Physik. Chem. 1928, 136, 186.

10.

Young, H. A.; Bray, W. C. The Rate of the Fourth Order Reaction between Bromic and Hydrobromic Acids. The Kinetic Salt Effect. J. Am. Chem. Soc. 1932, 54, 4284-4296.

11.

Levine, I. N. Physical Chemistry, 1st ed.; McGraw-Hill: New York, 1978, p. 483.

12.

Tan, X.; Lindenbaum, S.; Meltzer, N. A. A Unified Equation for Chemical Kinetics. J. Chem. Educ. 1994, 71, 566-567.


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13.

Page 16 of 18

Robinson, P. J. Integrated Rate-Equations in Chemical Kinetics. Acta Chim. Acad. Sci. Hung. 1970, 66, 407-410.

14.

Williams, B. W. Alternate Solutions for Two Particular Third Order Kinetic Rate Laws. J. Math. Chem. 2011, 49, 328-334.

15.

King, R. B. Beyond the Quartic Equation; Birkhäuser: Boston, 1996.

16.

Handbook of Mathematical Functions, Abramowitz, M.; Stegun, I. A. eds.; Dover: New York, 1965, p. 17.


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GRAPHICAL ABSTRACT


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Page 17 of 17


Page 18 of 18

(1)

0.8

(1,1) (2) (1,1,1) 0.6

(1,2)

ln 1

(1,1,1,1) (1,1,2)

0.4

(3) (1,3)

0.2

(2,2) (4) 0.0

0

10

20


F0t

30

40

50

A Universal Integrated Rate Equation for Chemical Kinetics

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