An Integrated Approach to the Analysis of Chemical Reaction Data: A Basic Equation
	Kenneth M. Maloney Chemistry and Physics Division, Baton Rouge Community College, 201 Community College Drive, LA 70806
	A basic equation for the nonlinear least squares regression analysis of reaction rate data is presented. The equation is derived solely from fundamental stoichiometric considerations. The facile manner in which this basic relationship is applied to a wide range of reaction systems and empirical models demonstrates the overall utility of the approach. The fact that this approach is phenomenologically based, without any predetermined models or assumptions, emphasizes the simplicity of the consistency requirements/tests which derives naturally from the basic equation. These consistency requirements are shown to be quite instrumental in the deconvolution of complex reaction systems.
Introduction The basic chemical equation provides a quantitative relationship from which one can express generalized proportional gains and losses of reactants and products. A simple synthesis reaction can be written as

x × A + y × B ¾® AxBy

(I)

In the specific case where the progress of reaction(I) can be measured gravimetrically, the weight gain of A_xB_y can be determined from the weight loss of B as

wt[B] = Sample wt[AxBy] × {1 - x × (Molar wt[A]/Molar wt[AxBy])}

(1)

where wt indicates the weight of the substance represented in brackets{i.e., [-]}. From eq 1, the fractional loss of B is
WF¢ = (1 – RF¢)	(2)
where

WF¢ = wt[B]/Sample wt[A_xB_y]

and

RF¢ = x × (Molar wt[A]/Molar wt[A_xB_y])

Therefore, the reduced dependent variable is WF(= WF¢/WF*)[WF* is a representative value of WF¢ selected to reduced the sequence] and the corresponding reduced independent variable is RF(= RF¢/RF*)[RF* is the corresponding value of the RF¢sequence].

Since the loss of B is directly proportional to the gain of A_xB_y[i.e., on a molar basis, the loss of B = (1/y) × (gain of A_xB_y)], eq 2 also expresses the proportional gain in A_xB_y. For very small values of W, eq 2 can be expressed as

WF = 1 - RF ~ (1/[1 + RF]) (RF<<1)

(3)

Eq 3, although limited to small values of RF, arises out of the basic form of the chemical equation. This limitation can be removed by the definition of the order based parameters, g and pf, as follows

WF = 1 - RF = ¡_r×(1/[1 + RF^g])^pf

(4)

where ¡ is a scaling factor. Eq 4 is well behaved in that, for positive values of g, WF ® ¡ as g ® ¥ and, for negative values of g, WF ® 0 as pf ® ¥. The decomposition reaction,

A_xB_y ¾® x × A + y × B

(II)

which is the reverse of (I), yields the same general form. Namely,

WF = (1 – RF)

(5)

where WF is the fractional loss of A_xB_y which is also directly proportional to the fractional gain of B[i.e., on a molar basis, the loss of A_xB_y = y×(gain of B)] and

RF = y × (Molar wt[B]/Molar wt [A_xB_y])

(6)

Thus, reaction types (I) and (II) can both be represented by eq 4.

Convergence Considerations and Consistency Tests

Eq 4 is a transcendental function where the exponential and the related logarithmic forms are, by definition, the inverse of the other. The recursive properties of eq 4 are maintained and the initialization convergence requirement(WF = RF = 0) is ensured through the order index, h, by the the following reformulation

WF= SF× {1/[1 + RF^g]}^{[pf + h]} × RF^{[h × (g -1)]}

(7)

Therefore,

pf + h = log_b{ (1/[1 + RF^g])^{[pf + h]} }

where the base, b, is defined as

b = 1/(1 + RF^g)

Transformation of eq 7 from base b to base e yields

WF = a×SFס_r×RF^{[h(g -1)]} exp(-RF^g) = ¡×RF^p × exp(-S_L×RF)

(8)

The combined scaling factor ¡ = a×SFס_r(a is the logarithm conversion constant), exp(-RF^g) = exp(-S_L×RF), and p = h(g -1). Thus, for a given reaction sequence, eq 8 defines a uniquely characteristic set of reaction field factors(RxnF-factors) ¡, p, and S_L.

An important consequence of eq 8 is the following set of consistency tests;

1^st derivative:

From eq 8, one has

d(WF)/d(RF) = [(p/RF) - S_L]×WF

(9)

The necessary condition, for a minimum or maximum [i.e., eq 9 = 0], is fulfilled when

S_L = p/RF_L

(10)

Thus, from eqs 8 and 10, the maximum or minimum value of RF(= RF_L) is attained at S_L = p for which RF_L = 1. For RF_L = 1, the corresponding value of the dependent variable WF is also defined to be WF = 1.

2^nd derivative:

The sufficient condition, from eq 9, is

d²(WF)/d(RF)² = {(p/RF²)×(p -1) + S_L×(S_L - [2×p/RF])}× WF = 0

(11)

Therefore, for WF = 1 and RF = RF_L = 1

p×(p -1) - S_L×(S_L - 2×p) > 0 minimum

(12)

or

p×(p -1) - S_L×(S_L - 2×p) < 0 maximum

(13)

In accordance with the necessary[eq 9] and sufficient[eq 11] conditions, the baseline Rxnf-factors (¡ = 1.65, p = S_L = 0.5) yield a reaction sequence with a maximum (WF = 1, RF = 1) at WF = RF = 1. In effect, the nature of the interdependence of the scaling factor(¡), the curvature factor(p) and the reaction terminus factor(S_L), for a given reaction sequence, are strictly defined by the consistency requirements embodied in eqs 9, 10 and 11(Figure 1a).

Figure 1a. The proportion reacted vs reduced time for the necessary and sufficient baseline conditions of WF = 1, RF = 1, ¡ = 1.65, and p = S_L = 0.50.

Importantly, for complex reaction sequences, these consistency requirements establish guidelines from which the appropriate translations can successfully deconvolute multiple sequences(see next section).

If one considers a reaction system where the independent variable is time(t) and t* is any value of t arbitrarily chosen as the value of the representative divisor for the reaction sequence, then RF = t_r (i.e., reduced time) where t_r = t/t*. Equation 8 becomes

WF = ¡×t_r^p ×exp(-S_L×t_r)

(14)

The reaction rate, for a reaction sequence, transforms into the following traditional rate equation[eqs 9 and 10]

rate = d(WF)/dt_r = [(p/t_rL) - S_L]×WF

(15)

where t_rL(= RL_L )= t_L/t and t_L is the actual value of the characteristic time for the reaction sequence. The relative invariance of the curvature of each reaction sequence is represented by the order factor, p. The points of initiation(¡) and termination(S_L) are unique values for any sequence. The criterion S_L = p, together with eqs 10 and 14, establishes the required self consistent relationship[i.e., eq 15 = 0]

WF = ¡×t_r^p ×exp(-[p/SP]×t_r)

(16)

where ôSPô = t_rL. Thus, the specific rate constant is

k = t_SP^-1 = (t_rL/ t* )^-1

(17)

In essence, eqs 8, 14, and 16 are equivalent nonlinear regression equations for a reaction sequence which result solely from the form and stoichiometry of the basic chemical reaction. Although the formulation as presented is based on gravimetric systems, eqs 8, 14, and 16 apply equally to any reaction system where the relative changes in the dependent (WF) and independent (RF) variables can be determined by other comparable methods (next section).

Selected Applications

The applications, presented herein, were selected to demonstrate the facile manner in which eqs 8, 14, and 16 can be employed as regression tools in the determination of the RxnF-factors(¡,p,S_L). Additionally, the deconvolution of complex reaction sequences that result from different regions of temperature and ambient atmospheres/environments are evaluated for both decomposition and synthesis type reactions.

CASE 1. Single Sequence/Alternate Independent Variable.

A). Reduced Time Independent Variable. Consider the generalized isothermal decomposition reaction curve(fig 1b)

employed by Criado and Morales¹ to demonstrate significant shortcomings in existing thermogravimetric models and approaches in discerning between first order reactions and those taking place through the Avrami-Erofeev mechanism.

Figure 1b. The proportion reacted vs time curve for the generalized decomposition reaction isotherm(350 °C)[Ref 1]. The kinetic parameters are A^1/3 = 104 min^-1 and E_act = 176 kJ for a heating rate(b) = 10 °C min^-1.

Nonlinear least squares regression of this generalized isotherm with eq 14(i.e., in the absence of any specific set of boundary conditions or consistency requirements) yields a value of p = 0.52 with a corresponding value of S_L = -0.33 (Table A).

Table A: Reaction Factors for Selected Experimental Chemical Reaction Data.^a
Reaction System		System Energetics	Temp Range, °C	RxnF-Factors^b
				¡r	p	SL
Single Sequence
Baseline(Fig 1) (WF = 1, RF = 1)
				1.65	0.50	0.50
CASE 1
A(Ref 1, Fig 1) Generalized Decomposition Curve(Table B)
		Isothermal	350	0.62	0.52	-0.33
B(Ref 2, Fig 2)		Photocatalysis
Alternate Independent Coordinate
Q-sized TiO2 Dechlorination (CCl4/[CH3OH])
· Doped(0.5% Fe^+3)				0.86	0.05	-0.15
· Undoped				1.78	0.81	0.57
C(Ref 3, Fig 3)		Isothermal
Decomposition of Ammonium Dinitramide
· Sequence no.
1			132	1.64	0.92	0.49
2			138	2.40	1.10	0.87
3			143	2.05	0.95	0.72
4			150	2.44	1.02	0.90
Single/Multiple Sequences
CASE 2
A(Ref 13, Figs 4a-d) Decomposition of Solid Na2CO3×NaHCO3×H20		Isothermal
· Nitrogen atmosphere
%N2
1) 100%(Fig 4a)			88 - 187	42.46	4.53	3.75
2) 96%(Fig 4b)			134	3.23	1.16	1.18
3) 42%(Fig 4b)			134	3.04	1.08	1.12
4) 23%(Fig 4b)			134	1.11	0.55	0.11
5) 15%(Fig 4c)			134	0.96	0.34	-0.03
(Composite- Fig 4c) Deconvolution Results- Fig 4d)
Segment(1)			134	2.45	0.80	0.93
Segment(2)			134	1.06	0.93	0.06
· Carbon dioxide atmosphere (100
1(Fig 4a)			122 - 154	0.96	1.03	0.33
2(Fig 4a)			171 - 214	1.22	0.90	0.41
B(Refs 14-16, Figs 5-7)
		Isothermal
Nonirradiated UO2 Oxidation · Single
Sequences
1) Sample	150A		150	3.31	1.38	1.20
2) Sample	150B		150	4.77	1.78	1.56
3) Sample	150C		150	2.39	0.94	0.89
4) Sample	190		190	3.57	1.34	1.31
· Multiple Sequences
1) Composite Results
(Fig 7a)			230	1.92	0.71	1.89
2) De-convoluted Results
(Fig 7b)
Segment(1)			230	9.43	2.36	2.24
Segment(2)			230	2.04	0.93	0.72
^aThe regression correlation coefficients were, in all cases, greater than 0.999. ^bIt is important to note that, whereas the p-RxnF factor for a reaction sequence is independent of t*, the ¡ and SL RxnF factors are not.

The relative magnitudes of this combination of values for p and S_L necessarily limits eq 14 to a quasi-linear relationship(Figure 1b). Regression of the data with eq 16 with p = 0.52 gives the value of the specific rate constant as k = 0.117 min^-1. This is almost identical to the value of 0.118 min^-1 previously reported by Criado and Morales¹ for the first order rate constant.

Additionally, from the conventional definition of a first-order reaction, at 50% reaction(i.e., WF = 0.5),

k = 0.693/t_1/2

Since the proportion reacted[eq 14] is

WF = 0.5 = (0.62) × t_r^0.52 × exp(+0.33)

which yields t_rL = 0.50 and k = 0.693/5.88 min = 0.118 min^-1. Again, this result illustrates complete consistency. For these specific quasi-linear conditions, eq 14 gives the same set of RxnF-factors for both the experimental data(Table A) and the Avrami-Erofeev model(Table B).

Table B. Reaction Factors for Selected Models.^a
Reaction Model	RxnF-Factors
	¡	p		SL
Generalized Decomposition Isotherm(Ref 1) (CASE 1A-TABLE A)
Avrami-Erofeev	0.62	0.52	-0.33
Decomposition of Ammonium Dinitramide(Ref 3) (CASE 1C-Table A)
Contracting disc	1.98	1.11	0.82
Contracting sphere	1.73	1.13	0.69
Power law(α^n) n – values 1/4	0.62	3.61	-0.35
1/3	0.68	2.74	-0.26
1/2	0.78	1.87	-0.13
3/2	0.91	0.67	0.03
One-dimensional
Diffusion	0.91	0.50	0.02
First order
(Mampel)	1.90	0.92	0.76
Avrami-Erofeev [-ln(1 - a)^n] n – values
1/4	145.2	6.67	5.10
1/3	23.3	4.16	3.27
1/2	6.24	2.42	1.95
Two-dimensional
Diffusion	1.19	0.43	0.29
Contracting sphere	1.98	1.11	0.82
Contracting disc	1.76	1.13	0.69
^aThe regression correlation coefficients were, in all cases, greater than 0.999.

B). Reduced Concentration Independent Variable(C_r). If the independent variable is expressed as a concentration, the resultant order would, of course, be relative to concentration. An important example of concentration, as the independent variable in a reaction system, is seen in the influence of metal ion dopants on the photocatalytic reactivity of quantum(Q)-sized TiO₂ colloids by acting as electron(or hole) traps. A systematic study² of the role of metal ion dopants in the photocatalytic reactivity of Q-sized TiO₂ establishes correlations with charge carrier recombination dynamics.

In the dechlorination of CCl₄ in a CCl₄/CH₃OH system, CH₃OH is a hole scavenger. The relative influence of this effect on the photocatalytic properties of Q-sized TiO₂ colloidal particles was systematically investigated as a function of CH₃OH concentration². The relative contributions of the hole scavenger(CH₃OH), in the photocatalytic dechlorination of CCl₄ by doped(0.5% Fe⁺³) and undoped Q-sized TiO₂, are illustrated in Figure 2.

Figure 2. The reduced rate ratio vs the reduced CH₃OH concentration ratio(C_r) for the relative contribution of the hole scavengering of CH₃OH in the photcatalytic dechlorination of CCl₄ by doped(0.5% Fe⁺³) and undoped Q-sized TiO₂(Ref 2).

The independent variable is represented as reduced methanol concentration(i.e., RF = C_r {= [CH₃OH]/[CH₃OH]*} where [CH₃OH]* is the specific value of [CH₃OH] selected to reduce the sequence). The results, from the nonlinear least squares regression of the experimentally determined reduced relative rate ratio(k_r) vs C_r, are presented in Table A.

The RxnF-factors easily reflect the substantial differences in the enhanced rate of dechlorination between doped and undoped. In fact, the difference, in the curvature related p – factors alone, is quite pronounced for the different sequences(Table A).

C). Decomposition of Ammonium Dinitramide[NH₄N(NO₂)₂]. The systematic investigation(132-150 °C) of the decomposition of ammonium dinitramide(ADN) has been reported by Vyazovkin and Wright³. Regression of these experimental results yield p-RxnF-factors that are reasonably invariant over the 132-150 °C range presented(Table A). The representative value is p = 1.00 ± 0.08 with an average deviation of 0. Of the selected models(Table B), only the RxnF-factors for the contracting disc and sphere(i.e., p = 1) are in reasonable agreement with experiment(Figure 3).

Figure 3. The proportion reacted vs reduced time-temperature series for the decomposition of ammonium dinitramide at 132 °C(p = 0.92), 138 °C(p = 1.10), 143 °C(p = 0.95), 147 °C(p = 0.98), and 150 °C(p = 1.02)[Ref 3]).

CASE 2. Single and Multiple Sequences.

A). The Decomposition of Solid Sodium Sesquicarbonate. In addition to the characterization of various chemical and

physical properties of sodium sesquicarbonate(Na₂CO₃×NaHCO₃×2H₂O), intermediates in the thermal decomposition such as wegsheiderite(NaHCO₃×Na₂CO₃) and sodium carbonate monohydrate(Na₂CO₃×H₂O) have also been investigated.^4-12 The thermal decomposition of solid Na₂CO₃×NaHCO₃×2H₂O was systematically investigated at temperatures between 350 K(77 °C) and 487 K(214 °C) in nitrogen and carbon dioxide atmospheres.¹³

a)

3(Na₂CO₃ × NaHCO₃ × 2H₂O) ¾® 3NaHCO₃ × Na₂CO₃ + 2Na₂CO₃ + 6H₂O ® 2(3NaHCO₃ × Na₂CO₃) ¾® 5Na₂CO₃ + 3CO₂ + 3H₂O

Depending on experimental conditions, solid Na₂CO₃×NaHCO₃×2H₂O was found to decompose by both single and multi-stage reaction sequences. The trends in the RxnF-factors are evaluated in the order of increasing temperature with 407 K(134 °C) as a transition temperature(Figures 4a –d and Table A).

Figure 4a. The proportion reacted vs time for the decomposition of sodium sesquicarbonate in (i) nitrogen(100%), 88-187 °C; (ii) carbon dioxide(100%), 122-154 °C; and (iii) carbon dioxide(100%), 171-214 °C[Ref 13].

Figure 4b. The effect of atmosphere composition on proportion reacted vs reduced time at the transition temperature of 134 °C for the atmospheres of (i) 96, (ii) 42, and (iii) 23% N₂ respectively.

Figure 4c. The composite proportion reacted vs reduced time at the transition temperature of 134 °C and 15% N₂ atmosphere.

Figure 4d. The deconvoluted proportion reacted vs time results for 15% N₂

In nitrogen atmospheres, although the decomposition of Na₂CO₃×NaHCO₃×2H₂O was found to separate into two temperature regimes(350-390 and 390-460 K) with two different activation energies(i.e., 58 and 25 kJ mol^-1, respectively), both reaction sequences were consistent with the Avrami-Erofeev model¹³ for n = 2[Figure 4a, curve(i)](Tables A and B). In CO₂ atmospheres, however, the decomposition is bifurcated where one sequence(395-427 K)(122-154 °C) is consistent with a first-order model[Figure 4a, curve(ii)] and the other sequence(444-487 K)(171-214 °C) is consistent with a contracting disc model¹³[Figure 4a, curve(iii)]. Additionally, at the transition temperature of 407 K(134 °C), as the decompositions are performed in atmospheres where the compositions of N₂ are systematically diminished from 96% to 15% N₂(i.e., the corresponding increases in CO₂ are 4% to 85%), the progressive emergence of another stage becomes evident at 15% N₂(Figures 4c and 4d).

As seen in Figures 4a – 4d, the experimental results from all stages of the decomposition of Na₂CO₃×NaHCO₃×2H₂O, are regressed in a facile manner by eq 14. Specifically, the systematic decrease in the %N₂ is manifested by a corresponding decrease in the p-RxnF factors. This suggest that the regression with eq 14 can add a significant level of quantification to the analysis and interpretation of reaction rate data. Interestingly, the results at 15% N₂(i.e., p = 0.34, Figure 4c) are easily deconvoluted into two distinct segments(Figure 4d) with p-RxnF factors of 0.80 and 0.93, respectively.

B). Oxidative Behavior of Nonirradiated Uranium Dioxide(UO₂). Depending on reaction conditions, UO₂ may form

hyperstoichiometric and hypostoichiometric compounds¹⁴.

b)

α-UO₂(s) + O₂(g) ¾® UO_2+x(s) ¾® U₄O₉(s) ¾® α-U₃O₇(s) ¾® γ-UO₃(s)

Additionally, as a reactor fuel, the buildup of fission products alters the lattice through the occupation of interstitial lattice sites. The fact that eq 14 enables the quantitative regression and deconvolution of such complex multifaceted reaction sequences is significant and highly desirable.

As a baseline illustration, the oxidative behavior of nonirradiated UO₂ is evaluated at 150, 190, and 230 °C and presented^15,16 in Table A as well as Figures 5-7.

Figure 5. The proportion reacted vs reduced time for the oxidation behavior of UO₂(sample A) at 150 °C(curves for samples B and C not shown).

Figure 6. The proportion reacted vs reduced time for the oxidative behavior of UO₂ at 190 °C.

Figure 7a. The composite proportion reacted vs reduced time for the oxidative behavior of UO₂ at 230 °C.

Figure 7b. The deconvoluted proportion reacted vs reduced time results for the oxidative behavior of UO₂ at 230 °C(Fig. 7a).

Not only are the RxnF-factors quite different for each temperature, they are also different for different samples at the same temperature(as an example, note the 150 C series in Table A). Fortunately, once the self consistent set of RxnF factors for a reaction sequence has been determined, the point of termination can be projected. In effect, the consistency test requirement, p = S_L, defines the end of the sequence. For example, consider that regression of the complete data base for the oxidation of UO₂ at 230 °C yields p = 0.71 with a value of S_L = 1.90(Table A, Figure 7a). The p = S_L requirement deconvolutes the data base into a minimum of two segments(Table A, 7b). The fact that p ¹ S_L(even without deconvolution) is a valuable indicator.

Conclusions

Equation 8 is presented as a basic tool for the nonlinear regression analysis of chemical reaction data. The basic equation is derived from fundamental stiochiometric considerations without the need for a predetermined set of isothermal or nonisothermal conditions. The consistency requirements are the natural outcome of the gains/losses inherent to the transformation of reactants to products. The variety of reaction systems evaluated and presented herein illustrates the value of eq 8 as a diagnostic tool. In essence, the RxnF factors derived from the application of this basic relationship allows a critical assessment and objective comparison of proposed models with experimental results.

References

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             Copyright © • Kenneth M. Maloney, Ph.D.              

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