Article pubs.acs.org/JPCA
Cite This: J. Phys. Chem. A 2018, 122, 4009−4014
A Universal Integrated Rate Equation for Chemical Kinetics Wesley D. Allen* Department of Chemistry and Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia 30602, United States S Supporting Information *
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 γi (−1)r F0t integral orders is shown to be Πri=1[1 − f −1 , i ξ(t)] = e where ξ(t) is the extent of reaction variable, the f i are roots of a polynomial of order r, the exponents are determined by γi = Πrk(≠i)( f i − f k)−1, and F0 is 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.
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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 1865−1867, 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 second-order 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 the 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 © 2018 American Chemical Society
included in 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.
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UNIFIED ANALYTIC SOLUTION Consider a generalized chemical equation for reactants Ri (i = 1, 2, ..., M) and products Pj (j = 1, 2, ..., N) with corresponding Received: March 15, 2018 Published: March 16, 2018 4009
DOI: 10.1021/acs.jpca.8b02372 J. Phys. Chem. A 2018, 122, 4009−4014
Article
The Journal of Physical Chemistry A stoichiometric coefficients ai and bj:
F(ξ) = k+Q [(ξ − c1)m1 (ξ − c 2)m2 ...(ξ − cM )mM
k+
a1R1 + a 2 R 2 + ... + aM R M ⇄ b1P1 + b2 P2 + ... + bN PN
− q(ξ + d1)n1 (ξ + d 2)n2 ...(ξ + dN )nN ]
(1)
k−
where
Reaction progress can be quantified in terms of the customary extent of reaction variable (ξ), which yields the reactant concentrations via
q = k −(k+Q )−1b1n1b2n2...bNnN
(2)
dξ = ( −1)r F0(ξ − f1 ) p1 (ξ − f2 ) p2 ...(ξ − fl ) pl dt
and those of the products by means of b1−1([P1]
ξ= =
− [P1]0 ) =
bN−1([PN ]
b2−1([P2]
− [P2]0 ) = ...
− [PN ]0 )
Accordingly, the consumption rates of the reactants are given by dξ 1 d[R1] 1 d[R 2] 1 d[R M] =− =− = ... = − dt a1 dt a 2 dt aM dt
∫0 (4)
ξ
dx = ( −1)r F0t (x − f1 ) p1 (x − f2 ) p2 ...(x − fl ) pl
1 = (x − f1 ) p1 (x − f2 ) p2 ...(x − fl ) pl
(5)
pi
l
∑ ∑ Cij ∫
l
⎡
i=1
⎣
⎛ f − ξ⎞ i ⎟⎟ − ⎝ fi ⎠
Di , j(ξ) =
N
i=1
i=1
(8ab)
respectively. For convenience, we assume here that the forward direction of the reaction is chosen so that r ≥ s and the quantity mM F0 = k+a1m1a 2m2...aM − ( −1)r k −δrsb1n1b2n2...bNnN
(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[P] i 0
Q=
pi − 1
⎤
j=1
⎦
∑ (−1) j Ci ,j + 1Di ,j(ξ)⎥⎥ = (−1)r F0t
1 [f j − (fi − ξ) j ] j fi (fi − ξ) j i j
⎛ f − ξ ⎞γi r ⎟⎟ = e(−1) F0t ∏ ⎜⎜ i fi ⎠ i=1 ⎝
(11)
(19)
(20)
(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
Hence, eq 7 can be written as dξ = F (ξ ) dt
(18)
r
(10ab)
and mM ( −1)r a1m1a 2m2...aM
(17)
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 [Ri] = 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 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 = γi are required. The exponential form of the final result is
As usual, the rate coefficients contain temperature and pressure effects. The forward and reverse reactions are of total order
∑ ni
(x − fi ) j
where
(7)
s=
i=1 j=1
Cij
dx = ( −1)r F0t (x − fi ) j
∑ ⎢⎢Ci ,1ln⎜⎜
− k −([P1]0 + b1ξ)n1 ([P2]0 + b2ξ)n2 ...([PN ]0 + bN ξ)nN
and
pi
Integrating the partial fractions in eq 18 provides
dξ = k+([R1]0 − a1ξ)m1 ([R 2]0 − a 2ξ)m2 ...([RM]0 − aM ξ)mM dt
M
ξ
0
i=1 j=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
∑ mi
l
∑∑
where the Cij constants depend on all of the f i. From eqs 16 and 17,
The net rate of the reversible reaction is governed by the equation
r=
(16)
The integrand can be decomposed by partial fractions to yield
while the appearance rates of the products are determined from
dξ = k+[R1]m1 [R 2]m2 ...[RM]mM − k −[P1]n1 [P2]n2 ...[PN ]nN dt
(15)
where the pi values are the multiplicities (degeneracies) of the l distinct roots. The solution ξ(t) is determined by integrating differential eq 12 using dummy variable x:
(3)
dξ 1 d[P1] 1 d[P2] 1 d[PN ] = = = ... = dt b1 dt b 2 dt bN dt
(14)
involves the ratio of the reverse and forward rate coefficients. The roots of F(ξ) are denoted as f i. After extracting these roots, eq 12 can be transformed to
ξ = a1−1([R1]0 − [R1]) = a 2−1([R 2]0 − [R 2]) = ... −1 ([R M]0 − [R M]) = aM
(13)
r
(12)
γi =
∏ (fi k(≠ i)
in terms of the rth order polynomial 4010
− fk )−1
(22) DOI: 10.1021/acs.jpca.8b02372 J. Phys. Chem. A 2018, 122, 4009−4014
Article
The Journal of Physical Chemistry A for r > 1, while γ1 = 1 otherwise. In the mathematical space of the f i 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 f i of F(ξ) in eq 13. If the reverse reaction is omitted, then the roots f i 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
−F0t
)
Degenerate fourth-order reactions: (p1 , p2 , p3 ) = (1, 1, 2): 2
⎛ f − ξ34 ⎞(f2 − f3 ) ⎟⎟ ξ34(t ) = lim ξ(t ) ⇒ ⎜⎜ 1 f4 → f3 f1 ⎠ ⎝ 2
⎛ f − ξ34 ⎞−(f1 − f3 ) ⎛ f − ξ34 ⎞(f1 + f2 − 2f3 )(f1 − f2 ) ⎜⎜ 3 ⎟ ⎟⎟ × ⎜⎜ 2 f2 ⎠ f3 ⎟⎠ ⎝ ⎝ ⎡ ⎤ ⎢ ξ34(f1 − f2 )(f1 − f3 )(f2 − f3 ) ⎥ × exp⎢ ⎥ f3 (f3 − ξ34) ⎢⎣ ⎥⎦ 2
(p1 , p2 ) = (1, 3):
f1 f2 [1 − e
3
] (24)
⎛ ⎞ 1 ⎟⎟ ξ12(t ) = lim ξ(t ) = f1 ⎜⎜1 − f2 → f1 1 + f1 F0t ⎠ ⎝
⎛ f − ξ⎞ ⎜⎜ 3 ⎟⎟ ⎝ f3 ⎠
(25)
(p1 = 4):
(26)
ξ23(t ) = lim ξ(t ) ⇒ f3 → f2
⎡ (f − f )ξ23 ⎤ 2 1 ⎥ = e−(f1 − f2 ) F0t exp⎢ 2 f1 (f2 − ξ23) ⎢⎣ f2 (f2 − ξ23) ⎥⎦ f2 (f1 − ξ23)
(27)
⎞ 1 ⎟ ⎟ 2 1 + 2f1 F0t ⎠
3
(33)
⎤
pi − 1
∑ Ti ,p − 1 − jDi ,j(ξ)⎥⎥ j=1
⎦
i
r
= ( −1) F0t
(34)
where the asterisk denotes that the summation index runs only over the unique f i values, each of degeneracy pi, while
(28)
r
Fourth-order reactions (p1, p2, p3, p4) = (1,1,1,1):
Ti , n =
∑′ j1 ≥ j2 ≥ ... ≥ jn
⎛ f − ξ ⎞(f2 − f3 )(f2 − f4 )(f3 − f4 )⎛ f − ξ ⎞(f3 − f1 )(f1 − f4 )(f3 − f4 ) ⎜⎜ 1 ⎟⎟ ⎜⎜ 2 ⎟⎟ ⎝ f1 ⎠ ⎝ f2 ⎠
(fi − f j )−1(fi − f j )−1...(fi − f j )−1 1
2
n
(35)
In eq 35 the summation is not restricted to unique f i values, and the prime denotes that all singular terms are excluded. The generality of eq 34 warrants emphasis, because it is the 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 f i.
⎛ f − ξ ⎞(f1 − f2 )(f1 − f4 )(f2 − f4 )⎛ f − ξ ⎞(f1 − f2 )(f3 − f1 )(f2 − f3 ) ⎟⎟ ⎜⎜ 4 ⎟⎟ × ⎜⎜ 3 f ⎝ 3 ⎠ ⎝ f4 ⎠ = e(f1 − f2 )(f1 − f3 )(f1 − f4 )(f2 − f3 )(f2 − f4 )(f3 − f4 )F0t
⎞ ⎟ ⎟ 3 1 + 3f1 F0t ⎠ 1
⎡ ⎛ f − ξ⎞ ⎟⎟ − ∑* γi(−1) pi − 1⎢⎢Ti ,pi − 1ln⎜⎜ i ⎝ fi ⎠ i=1 ⎣
(p1 = 3): ⎛ ξ123(t ) = lim lim ξ(t ) = f1 ⎜⎜1 − f2 → f1 f3 → f2 ⎝
f2 → f1 f3 → f2 f4 → f3
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 integrated rate equations can be cast into the form
Degenerate third-order reactions: (p1 , p2 ) = (1, 2):
(32)
ξ1234(t ) = lim lim lim ξ(t ) ⇒
⎛ ξ1234(t ) = f1 ⎜⎜1 − ⎝
f1 − f2
= e(f1 − f2 )(f2 − f3 )(f3 − f1 )F0t
f3 → f1 f4 → f2
⎤ ⎡1 = exp⎢ (f1 − f2 )3 F0t ⎥ ⎣2 ⎦
Third-order reactions (p1, p2, p3) = (1,1,1): ⎛ f − ξ⎞ ⎜⎜ 2 ⎟⎟ ⎝ f2 ⎠
ξ13,24(t ) = lim lim ξ(t ) ⇒
f1 (f2 − ξ13,24) ⎡ (f1 − f2 )ξ13,24[f12 + f22 − (f1 + f2 )ξ13,24] ⎤ ⎥ exp⎢ f2 (f1 − ξ13,24) ⎢⎣ 2f1 f2 (f1 − ξ13,24)(f2 − ξ13,24) ⎥⎦
Degenerate second-order reactions (p1 = 2):
⎛ f − ξ⎞ ⎜⎜ 1 ⎟⎟ ⎝ f1 ⎠
(31)
(p1 , p2 ) = (2, 2):
f2 − f1 e
f3 − f1
f3 → f2 f4 → f3
= e(f1 − f2 ) F0t
(f1 − f2 )F0t
f2 − f3
ξ234(t ) = lim lim ξ(t ) ⇒
⎡ ξ (f − f )(2f f − 4f 2 − f ξ + 3f ξ ) ⎤ 234 1 2 1 2 2 1 234 2 234 ⎥ exp⎢ f1 (f2 − ξ234) ⎢⎣ ⎥⎦ 2f22 (f2 − ξ234)2
Second-order reactions (p1, p2) = (1, 1): ξ(t ) =
(30)
f2 (f1 − ξ234)
(23)
(f1 − f2 )F0t
2
= e(f1 − f2 )(f1 − f3 ) (f2 − f3 ) F0t
(29) 4011
DOI: 10.1021/acs.jpca.8b02372 J. Phys. Chem. A 2018, 122, 4009−4014
Article
The Journal of Physical Chemistry A and 3 c(1 − q)−1(q2/3 − q1/3)
w=
(39)
Although f1 and f 2 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 ⎜ 2 ⎟e ⎝ ξ − 2vξ + v 2 + w 2 ⎠ ⎛ u 2 ⎞ −3Q [(u − v)2 + w 2]k t + =⎜ 2 ⎟e ⎝ v + w2 ⎠
(40)
where tan θ =
ξ(t ) = u(1 ± Ge−αt )
■
α=
3 k+Q [(u − v)2 + w 2] 2
)
). If + = 0, then the roots are real but not
distinct; specifically,
ξ(t ) = a1−1[R1]0 (1 − e−a1k+t )
(45)
[R1] = [R1]0 e−a1k+t
(46)
Second order irreversible: From Cases (2,3,4), and Cases (5,6) with stoichiometric initial proportions (f1 = f 2 = f± = c1):
⎫ ⎛ 1 ⎞1/3 1 −1⎧ A ⎨B − ⎜ T ⎟ , B + (4T )1/3 ⎬ ⎝2 ⎠ 3 ⎩ ⎭ ⎪
If + < 0, then one real and two complex roots exist. The complex roots are f1,2 = 16 A−1[2B − U1/3 − V1/3 ± 31/2 i(U1/3 − V1/3)], and
ξ12(t ) =
1
the real root is f3 = 3 A−1(B + U1/3 + V1/3). 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) = Qk+[(ξ − c)3 − q(ξ + c)3 ]
(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. First order irreversible: From Case 1 ( f1 = f+ = c1):
1 1 fn + 2 = 3 A−1 B + 2W 1/2cos⎡⎣ 3 (2nπ + β)⎤⎦ for n = (−1, 0, 1), where
⎪
(43)
and
2
{f1 = f2 , f3 } =
(u − v)2 + w 2 w−1(u − v)arctan[uw(v 2 + w 2 − uv)−1] e v2 + w2
G=
3A(−3+ )1/2], V = 1 [T − 3A(−3+ )1/2], and W = B2 − 3AC. 2 If + > 0, then W > 0 and the three roots are real and distinct: β=
(42)
where
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 Af 3 − Bf 2 + Cf − 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 + = B2C2 − 4AC3 − 4B3D − 27A2D2 + 18ABCD and the quantities T = 2B3 + 27DA2 − 9ABC, U = 1 [T +
( (
(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
Figure 1. Logarithmic plots of the integrated rate eqs 23−33 when f1 = 0.1, f 2 = 0.2, f 3 = 0.3, and f4 = 0.4. Each curve is labeled with the corresponding set of pi values.
1 cos−1 2 TW −3/2
wξ v + w 2 − vξ 2
1 1 − = k+a1t [R1] [R1]0
ξ(t ) =
The quantities for finding the roots of F(ξ) are + = −1728 c6q2, T = 216 c3q(q+1), U = 216 c3q2, V = 216 c3q, and W = 36 c2q. The exact roots of F(ξ) can then be expressed as f1 = v + iw, f 2 = v − iw, and f 3 = u, where (37)
v = c(1 − q)−1(1 − q1/3 − q2/3 + q)
(38)
(47)
(48)
From Cases (5,6) with nonstoichiometric initial proportions [( f1, f 2) = (c1, c2)]:
(36)
u = c(1 − q)−1(1 + 2q1/3 + 2q2/3 + q)
k+[R1]20 t 1 + k+a1[R1]0 t
c1c 2[1 − e−(c1− c2)a1a2k+t ] c1 − c 2e−(c1− c2)a1a2k+t
[R1][R 2]0 = e−(a1[R 2]0 − a2[R1]0 )k+t [R1]0 [R 2]
(49)
(50)
Third order irreversible: From Case 7 with nonstoichiometric initial proportions [( f1, f 2, f 3) = (c1, c2, c3)]: 4012
DOI: 10.1021/acs.jpca.8b02372 J. Phys. Chem. A 2018, 122, 4009−4014
Article
The Journal of Physical Chemistry A Table 1. Fn(ξ) Polynomials of Order n and Their Roots (f) for 12 Cases of Chemical Kineticsa F3(ξ) = Qk+(Aξ3 − Bξ2 + Cξ − D)
Notation: ( f1, f 2) = ( f+, f−) Case 1. One reactant of order 1, one product of order 1 (R1 ⇌ P1)
(1 − q)f+ = c1 + qd1
F1(ξ) = − k+a1(ξ − c1) − k −b1(ξ + d1)
Case 2. One reactant of order 2, one product of order 1 (2R1 ⇌ P1)
2f± = 2c1 + q ± [q(q + 4c1 + 4d1)]1/2
F2(ξ) = k+a12(ξ − c1)2 − k −b1(ξ + d1)
Case 3. One reactant of order 2, one product of order 2 (2R1 ⇌ 2P1)
F2(ξ) =
k+a12(ξ
− c1)2 − k −b12(ξ + d1)2
(1 − q)f± = c1 + qd1 ± (c1 + d1) q
Case 4. One reactant of order 2, two products each of order 1 (2R1 ⇌ P1 + P2)
F2(ξ) = k+a12(ξ − c1)2 − k −b1b2(ξ + d1)(ξ + d 2)
(1 − q)f± = c1 −
⎡ ⎤1/2 1 1 q(d1 + d 2) ± ⎢q(c1 − d1)(c1 − d 2) + q2(d1 − d 2)2 ⎥ ⎣ ⎦ 2 4
Case 5. Two reactants each of order 1, one product of order 1 (R1 + R2 ⇌ P1)
2f± = c1 + c 2 + q ± [(c1 − c 2)2 + 2q(c1 + c 2 + 2d1) + q2]1/2
F2(ξ) = k+a1a 2(ξ − c1)(ξ − c 2) − k −b1(ξ + d1)
Case 6. Two reactants each of order 1, two products each of order 1 (R1 + R2 ⇌ P1 + P2) F2(ξ) = k+a1a 2(ξ − c1)(ξ − c 2) − k −b1b2(ξ + d1)(ξ + d 2)
2(1 − q)f± = c1 + c 2 + q(d1 + d 2) ± [q2(d1 − d 2)2 + 2q(c1 + c 2)(d1 + d 2) + 4q(c1c 2 + d1d 2) + (c1 − c 2)2 ]1/2 Case 7. Three reactants each of order 1, three products each of order 1 (R1 + R2 + R3 ⇌ P1 + P2 + P3) F3(ξ) =− k+a1a 2a3(ξ−c1)(ξ−c 2)(ξ − c3) − k −b1b2b3(ξ + d1)(ξ + d 2)(ξ + d3) A = 1 − q , B = c1 + c 2 + c3 + q(d1 + d 2 + d3), C = c1c 2 + c1c3 + c 2c3 − q(d1d 2 + d1d3 + d 2d3), D = c1c 2c3 + qd1d 2d3
Case 8. Three reactants each of order 1, two products of order (1, 2) (R1 + R2 + R3 ⇌ P1 + 2P2) F3(ξ) = − k+a1a 2a3(ξ − c1)(ξ − c 2)(ξ − c3) − k −b1b22(ξ + d1)(ξ + d 2)2 A = 1 − q , B = c1 + c 2 + c3 + q(d1 + 2d 2), C = c1c 2 + c1c3 + c 2c3 − qd 2(2d1 + d 2), D = c1c 2c3 + qd1d 22
Case 9. Three reactants each of order 1, one product of order 3 (R1 + R2 + R3 ⇌ 3P1)
F3(ξ) = − k+a1a 2a3(ξ − c1)(ξ − c 2)(ξ − c3) − k −b13(ξ + d1)3 A=1−q , B=c1+c 2+c3 + 3qd1, C =c1c 2 + c1c3 + c 2c3 − 3qd12 , D=c1c 2c3 + qd13 Case 10. Three reactants each of order 1, two products each of order 1 (R1 + R2 + R3 ⇌ P1 + P2) F3(ξ) = − k+a1a 2a3(ξ − c1)(ξ − c 2)(ξ − c3) − k −b1b2(ξ + d1)(ξ + d 2) A=1, B=c1+c 2 + c3+q , C =c1c 2 + c1c3 + c 2c3 − q(d1 + d 2), D=c1c 2c3 + qd1d 2
Case 11. Three reactants each of order 1, one product of order 2 (R1 + R2 + R3 ⇌ 2P1)
F3(ξ) = − k+a1a 2a3(ξ − c1)(ξ − c 2)(ξ − c3) − k −b12(ξ + d1)2
A=1, B=c1 + c 2 + c3 + q , C = c1c 2 + c1c3 + c 2c3 − 2qd1, D = c1c 2c3 + qd12 Case 12. Three reactants each of order 1, one product of order 1 (R1 + R2 + R3 ⇌ P1) F3(ξ) = − k+a1a 2a3(ξ − c1)(ξ − c 2)(ξ − c3) − k −b1(ξ + d1) A = 1, B = c1 + c 2 + c3 , C = c1c 2 + c1c3 + c 2c3 − q , D = c1c 2c3 + qd1 a
For each case an example is given in parentheses of a corresponding elementary reaction.
⎛ c1 − ⎜ ⎝ c1
ξ⎞ ⎟ ⎠
c 2 − c3
⎛ c2 − ⎜ ⎝ c2
ξ⎞ ⎟ ⎠
c3− c1
⎛ c3 − ⎜ ⎝ c3
ξ⎞ ⎟ ⎠
c1− c 2
= e−(c1− c2)(c2 − c3)(c1− c3)k+a1a2a3t
which is equivalent to ξ23 = f2 +
(51)
ξ23 − f2
−1
= W ( − f1 f 2−1 e−f1 f 2
+ F0(f1 − f2 )2 t
+ F0(f1 − f2 )2 t
)
(54)
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(52)
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 in these equations are the roots ( f i) of the polynomial F(ξ) (eq 13), exponents γi that depend on products of differences of the f i (eq 22), and a factor F0 (eq 9) involving the stoichiometric coefficients and rate coefficients (k±). Explicit equations for F(ξ)
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 f1 − ξ23
−1
1 + W ( − f1 f 2−1 e−f1 f 2
Note that eq 54 gives ξ23 explicitly as a function of t, even in the case of a reversible reaction. Making the replacements ξ23 → x(t), f 2 → δ, f1 → γ, F0(f1 − f 2)2 → −β, and − f1 f −1 2 → α within eq 54 yields eq 5 of Williams for the corresponding irreversible reaction.
⎛ [R1] ⎞c3− c2 ⎛ [R 2] ⎞c1− c3⎛ [R3] ⎞c2 − c1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎝ [R1]0 ⎠ ⎝ [R 2]0 ⎠ ⎝ [R3]0 ⎠ = ea1a2a3(c2 − c1)(c3− c2)(c1− c3)k+t
(f1 − f2 )
) (53) 4013
DOI: 10.1021/acs.jpca.8b02372 J. Phys. Chem. A 2018, 122, 4009−4014
The Journal of Physical Chemistry A
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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.
and the f i roots are collected in Table 1 for the 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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APPENDIX Decomposition by partial fractions takes the form 1 = (x − f1 )(x − f2 )...(x − fr )
r
∑ i=1
(A1)
assuming that all the f i are distinct. Multiplying eq A1 by the product in the denominator on the left side yields r
1=
∑ γj gj(x)
(A2)
j=1
in which r
∏ (x − fk )
g j (x ) =
k(≠ j)
(A3)
Note that r
∏ (fi
gj(fi ) = δij
− fk )
k(≠ i)
(A4)
where δ is the Kronecker delta symbol. Evaluating eq A2 for x = f i then provides r
1=
r
∑ γjδij ∏ (fi j=1
− fk )
k(≠ i)
(A5)
which immediately yields r
γi =
∏ (fi
− fk )−1
k(≠ i)
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 Sauerstoff. 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-HJReaktion. II. Z. Phys. 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. (13) 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.
γi (x − fi )
Article
(A6)
and thus proves eq 22 of the text.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.8b02372. 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 (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The author declares no competing financial interest. 4014
DOI: 10.1021/acs.jpca.8b02372 J. Phys. Chem. A 2018, 122, 4009−4014
