Temperature-Dependent Kinetic Prediction for Reactions Described

Article pubs.acs.org/JPCA

Temperature-Dependent Kinetic Prediction for Reactions Described by Isothermal Mathematics L. N. Dinh,* T. C. Sun, and W. McLean II Lawrence Livermore NationalLaboratory, Livermore, California 94550, United States ABSTRACT: Most kinetic models are expressed in isothermal mathematics. This may lead unaware scientists either to the misconception that classical isothermal kinetic models cannot be used for any chemical process in an environment with a time-dependent temperature profile or, even worse, to a misuse of them. In reality, classical isothermal models can be employed to make kinetic predictions for reactions in environments with time-dependent temperature profiles, provided that there is a continuity/conservation in the reaction extent at every temperature−time step. In this article, fundamental analyses, illustrations, guiding tables, and examples are given to help the interested readers using either conventional isothermal reacted fraction curves or rate equations to make proper kinetic predictions for chemical reactions in environments with temperature profiles that vary, even arbitrarily, with time simply by the requirement of continuity/conservation of reaction extent whenever there is an external temperature change.

flow, and temperature.20,22,25 However, to the best of our knowledge, there is currently no kinetic report to offer the basics and fundamentals in making proper kinetic predictions for reactions that are well-described by isothermal mathematics but suffer from certain external/environmental temperature variations. From conventional isothermal rate equations, there are families of isothermal reacted fraction curves for every temperature involved. Because isothermal reactions reach different conversion levels for the same reaction time at different temperatures, when there is a temperature change in the environment, the kinetic prediction can be thought of as a switch from one isothermal reacted fraction curve to another. This necessitates a time correction to ensure the continuity/ conservation of the reaction extent upon switching between two isothermal reacted fraction curves. However, when the reaction rate (instead of reaction fraction) is used in kinetic prediction and the total reacted fraction is then derived by integration of the rate at each time step, any arbitrary temperature change with time can be much more easily incorporated. In this article, mathematical treatments of these tasks are described, guiding tables, and illustrating examples are given to help students and chemists making proper kinetic predictions for reactions that are theoretically well-described by isothermal mathematics but practically involves some sort of environmental temperature changes.

1. INTRODUCTION Traditionally, the kinetics of reactions such as reaction rate and reaction extent are expressed mathematically for isothermal processes.1−5 An excellent derivation of the most commonly used isothermal kinetic models is given in ref 5. When the nature of the chemical reaction is not known, some heating schedule (mostly constant ramp rate) in the temperatureprogrammed desorption/decomposition/reaction has been used in conjunction with isoconversional (model-free) analysis or deconvolution to extract kinetic parameters and even to make kinetic prediction at some constant temperature.6−14 However, even when the model-free approach is used, some isothermal reaction model is usually desired for a more detailed understanding of the kinetic process.11,15−17 But in practical applications, many clearly identified but slow reactions (such as those encountered in the degradation of rubber products in storage, unwanted corrosion of parts in a sealed device due to material incompatibility over service life, or outgassing from surfaces of vacuum/dry containers, etc.) may experience external temperature variations that might be a complex function of time. The questions of possibility and adequacy in using isothermal kinetic equations to make kinetic predictions for reactions in environments with time-dependent temperature profiles naturally arise in such situations. The most often encountered nonisothermal reactions reported in the literature are runaway reactions,18−25 in which the exothermic feature of the reaction drives up the reaction temperature, which in turn increases the reaction rate. The constant selffeedback between temperature and reaction rate is responsible for the runaway effect. Due to the complex nature of the runaway reactions, the main emphases in the literature are to accurately measure the reaction kinetics18,19,21 and to logically describe the multistage nature of the reactions20−22,24 usually through coupled differential equations for reacted fraction, heat This article not subject to U.S. Copyright. Published XXXX by the American Chemical Society

2. METHODS The conventional reaction rate of an isothermal reaction at T0 is usually written as Received: August 15, 2016 Revised: September 10, 2016

A

DOI: 10.1021/acs.jpca.6b08219 J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A dα = kf (α) = A e−E / RT0f (α) dt

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to another isothermal curve at temperature Ti for the next time interval of Δti, he/she first needs to calculate the equivalent start time on the Ti curve that has the same level of reacted fraction after a time of Δt0 on the T0 curve to ensure the continuity/conservation of reaction extent. The corrected time, tcor Ti , defined to be the equivalent time since the start of reaction (at t = 0) through the last second on Ti curve is given by

(1)

In eq 1, t is time, α is the reacted fraction (0−1), k is the rate constant, A is the pre-exponential factor, E is the activation energy barrier for the rate controlling process, R is the gas molar constant, T0 is the isothermal reaction temperature in Kelvin, and f(α) is an analytical function determined by the rate-limiting reaction mechanism.1,5 With a known reaction mechanism,1,2,5 α can be written as an analytical function of t such as ⎛ ⎛ E ⎞⎞ α = ⎜⎜A exp⎜ − ⎟⎟⎟ ⎝ RT0 ⎠⎠ ⎝

∫0

t

α = 1 − exp[− (A e−E / RT0)t ]

1 1 + At e−E / RT

(2) for 1st‐order kinetics (n = 1) (3)

for 2nd‐order kinetics (n = 2) (4)

3 ⎛ At −E / RT0⎞⎟ α = 1 − ⎜1 − e ⎝ ⎠ 3

for 3D contracting reaction front (5)

α = [2At e

−E / RT0 1/2

]

for 1D diffusion

(6)

2.1. Treatment Based on Reacted Fraction Curves/ Mathematics. In general, for an isothermal reaction at T0 with known E and A, α can be expressed simply as a function of t only:

α = F (t )

(7)

αT0→ Ti = F(tTcor ) i

And if F(t) is invertible, then t can be written as a function of α: −1

t = F (α )

(9)

The first term on the right-hand side of eq 9 corresponds to the equivalent start time on the Ti curve that yields the same reaction level, αΔt0,T0, after a time Δt0 on the T0 curve (obtained by substituting into eq 8 the same reacted fraction produced from the T0 isothermal curve after a time of Δt0 {i.e., αΔt0,T0} and solving for the equivalent reaction time on the Ti isothermal curve). Because Δti is the time interval during which the reaction stays at Ti, the summation of the first and second terms in eq 9 gives the corrected time, tcor Ti , on the Ti curve, which is equivalent to the total reaction time since the start of the reaction through the last second of the Δti interval at Ti (as if the whole reaction was at a single temperature Ti since the beginning). Such a calculation of the corrected time, tcor Ti , should be made whenever one switches from an isothermal kinetic prediction curve at T0 to a kinetic prediction curve at temperature Ti to take into account the fact that the time it takes to reach the same reacted fraction are different for the different kinetic prediction curves at different temperatures. The total reacted fraction after durations of Δt0 at T0 and Δti at Ti is given in terms of an equivalent reaction at just the latter temperature for a corrected time, tcor Ti :

⎛ ⎛ E ⎞⎞ dt = ⎜⎜A exp⎜ − ⎟⎟⎟t ⎝ RT0 ⎠⎠ ⎝

for 0th‐order kinetics (n = 0)

α=1−

tTcor = F −1(αΔt0 , T0{Ti }) + Δti i

(8)

(10)

The above procedure can be repeated to cover as many temperature−time steps as there are variations in the environmental temperature and is summarized in Table 1. From this table, it is observed that the calculation of the total reacted fraction in a chemical reaction with a multistep temperature profile requires only one additional step in comparison with a true isothermal reaction: the calculation of the corrected time in column 3 to be used in the formula for the reacted fraction instead of real time. 2.1.1. Graphical Illustration. Is it always necessary to calculate the corrected time according to eq 9 and to use it in eq 10 to arrive at the correct conversion level for a multiple temperature−time step reaction? The answer to that question depends on the type of relationship between α and t. Such time correction when jumping from one isothermal curve to another one is absolutely necessary if the relationship between α and t is nonlinear (all kinetic models except for zeroth-order reaction) but is only optional if that relationship is linear (the case of zeroth-order reaction described by eq 2). An illustration is given in Figure 2a for a linear relationship between α and t (zeroth-order reaction) and in Figure 2b for a nonlinear relationship between α and t (all other kinetic models). In Figure 2a,b, each reaction starts out on the T0 curve at t = 0 for duration of Δt0 and is represented by the thicker solid line on the T0 curve. The corresponding reacted fraction after a time of Δt0, αΔt0,T0, is determined graphically by drawing a line perpendicular to the vertical axis from point A (representing the end of reaction after Δt0) on the T0 curve.

−1

where F (α) is the inverse function of F(t). As can be deduced from eqs 1−6, the same type of isothermal chemical reaction carried out at different temperatures has different α vs t curves. This is illustrated in Figure 1

Figure 1. Plot of a first-order reaction with E = 90 kJ/mol and A = 108 s−1, at three temperatures: 305, 310, and 315 K.

for a first-order reaction (eq 3) with E = 90 kJ/mol and A = 108 s−1, at three temperatures, 305, 310, and 315 K. If there is a change in the reaction temperature imposed by a variation in the environmental temperature, then the kinetic prediction of the reaction extent has to take into account the same temperature variation profile. In general, when a researcher, working on the kinetic prediction for such a reaction, switches from an isothermal curve at temperature T0 after a time of Δt0 B


The Journal of Physical Chemistry A

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Table 1. Mathematical Summary of How To Calculate the Proper Time Correction and the Total Reacted Fraction in a Multi Temperature−Time Step Reaction T (K)

elapsed time

corrected time

0 Δt0

T0 T0

0 Δt0

Δt0 + Δt1

T1

−1 tcor T1 = F (αΔt0,T0{T1}) + Δt1

Δt0 + Δt1 + Δt2

T2

tcor T2

...

... Tn

... −1 tcor Tn = F (αT0→Tn−1{Tn}) + Δtn

∑n0Δti

reacted fraction 0 αT0 = F(Δt0) αT0→T1 = F(tcor T1 )

−1

= F (αT0→T1{T2}) + Δt2

αT0→T2 = F(tcor T2 ) ... αT0→Tn = F(tcor Tn )

Figure 2. Graphical illustration of the time correction upon switching from one isothermal curve to another isothermal curve at a different temperature for (a) linear relationship between α and t (zeroth-order reaction) and for (b) nonlinear relationship between α and t (non-zeroth-order reaction).

This line intercepts the Ti curve at point B. A line drawn from B and perpendicular to the time axis yields the equivalent time, F−1(αΔt0,T0{Ti}), for the same amount of reacted fraction as if the whole reaction happened on the Ti curve up to this time. If after the initial time of Δt0 at T0 the reaction continues at a higher temperature Ti for a time of Δti, the correct point on the Ti curve at which the reaction continues is point B and not point C, which is formed by the intersect of the line drawn from A and perpendicular to the time axis at the end of the initial Δt0 period. In Figure 2a, α is a linear function of t, and for such a constant reaction rate associated with zeroth-order reaction (eq 2), none of this graphical analysis for the corrected time is necessary, as can be seen below: The reaction extent after a time of Δt0 at T0 is ⎛ E ⎞ αΔt0 , T0 = A exp⎜ − ⎟ ⎝ RT0 ⎠

∫0

Δt 0

and 10. Does this result agree with the result obtained with a reference to the correction time on the Ti curve after Δt0? The time on the Ti curve corresponding to a conversion level of αΔt0,T0 after duration of Δt0 at T0 is labeled by the point B on the Ti curve in Figure 2a, and given by the first term in eq 9 (which is simply an expression of t as a function of αΔt0,T0 with a temperature of Ti): F −1(αΔt0 , T0{Ti }) =

∫Δt

(14)

And the total reaction is given by eq 10: ⎛ E ⎞ tTcori = αT0→ Ti = F(tTcor ) A exp dt ⎜− ⎟ i ⎝ RTi ⎠ 0 ⎤ ⎛ E ⎞⎡ ⎛ E ⎞ = A exp⎜ − ⎟⎢αΔt0 , T0A−1 exp⎜ ⎟ + Δti ⎥ ⎥⎦ ⎝ RTi ⎠⎢⎣ ⎝ RTi ⎠

dt

∫

0

⎛ E ⎞ ⎛ E ⎞ = A exp⎜ − ⎟Δt0 + A exp⎜ − ⎟Δti ⎝ RT0 ⎠ ⎝ RTi ⎠

(13)

⎛ E ⎞ −1 −1 tTcor = F ( α { T }) + Δ t = α A exp ⎜ ⎟ + Δti Δ t , T i i Δ t , T i 0 0 0 0 ⎝ RTi ⎠

For the next duration of Δti, the reaction continues at Ti and the total reacted fraction is αT0→ Ti = αΔt0T0

E

So the corrected time (the total reaction time since the start of the reaction through Δti as if the whole reaction so far happened at only Ti) is given by eq 9:

(11)

Δt 0 +Δti

( )

A exp − RT

i

⎛ E ⎞ dt = A exp⎜ − ⎟Δt0 ⎝ RT0 ⎠

⎛ E ⎞ + A exp⎜ − ⎟ ⎝ RTi ⎠

αΔt0 , T0

(12)

Clearly, the result in eq 12 can be obtained without any reference to the correction time calculation prescribed by eqs 9

From eq 11 for αΔt0,T0: C



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⎤ ⎛ E ⎞⎡ ⎛ E ⎞ ⎛ E ⎞ αT0→ Ti = A exp⎜ − ⎟⎢A exp⎜ − ⎟Δt0A−1 exp⎜ ⎟ + Δti ⎥ ⎥⎦ ⎝ RTi ⎠⎢⎣ ⎝ RT0 ⎠ ⎝ RTi ⎠ ⎛ E ⎞ ⎛ E ⎞ αT0→ Ti = A exp⎜ − ⎟Δt0 + A exp⎜ − ⎟Δti ⎝ RT0 ⎠ ⎝ RTi ⎠ (15)

The value in eq 15 obtained by referencing to the correction time when switching from the T0 isothermal curve to the Ti isothermal curve is the same as that deduced from eq 12 without any such reference and is an exceptional result of a constant reaction rate in zeroth-order reaction as mentioned in the paragraph above. However, in Figure 2b, α is a nonlinear function of t, and therefore, the reaction rate, which is dα/dt, varies with the curvature of the isothermal α vs t curve. And a time correction is needed upon switching from a T0 curve to a Ti curve. Mathematically, after a duration of Δt0 at T0, the reacted fraction, αΔt0,T0, for a nonlinear α vs t relationship (all kinetic models except zeroth-order reaction) cannot be expressed in a form containing ∫ dt (such as seen in eq 12 for a zeroth-order reaction). It has to be written, instead, in the form αΔt0 , T0 =

∫0

Δt 0

Figure 3. Graphical illustration of the concept of time correction for a zeroth-order reaction with E = 90 kJ/mol and A = 108 s−1 at T0 = 310 K for the first 1000 h and then at T1 = 315 K for another 1000 h.

at point A. The rest of the reaction is shown by the thicker solid line on the 315 K, starting at point B (where A and B are defined in the previous section). The reacted fraction after Δt0 of 1000 h at 310 K, αΔt0,310K, is calculated from eq 11: ⎛ E ⎞ αΔt0 ,310K = A exp⎜ − ⎟Δt0 ⎝ RT0 ⎠

F (t ) d t

For the next duration of Δti, the reaction continues at Ti and the total reacted fraction is

⎛ 90000 ⎞ ⎟(1000 × 3600) = 108 exp⎜ − ⎝ 8.314 × 310 ⎠

αT0→ Ti = αΔt0T0 + αΔti , Ti =

∫0

Δt 0

F (t ) d t +

∫F

= 0.246

tf −1

(αΔt0 , T0{Ti})

The equivalent time to reach the same conversion level of 0.246 on the 315 K curve, F−1(αΔt0,310K{315K}), as labeled by point B is calculated from eq 13:

F (t ) d t (16)

with Δti = tf − F −1(αΔt0 , T0{Ti })

⎡ ⎛ E ⎞⎤ F −1(αΔt0 ,310K {315K}) = αΔt0 ,310K /⎢A exp⎜ − ⎟⎥ ⎢⎣ ⎝ RTi ⎠⎥⎦

(17)

In eqs 16 and 17, F−1(αΔt0,T0{Ti}) and tf are the equivalent start point and end point, respectively, for the Δti interval on the Ti curve. Due to the nonlinearity in the relationship between α and t presented in Figure 2b, every point on the curve has a different slope, and therefore a different reaction rate (for example, the slope, and therefore reaction rate, at point B are much higher than at point C). And, so, an incorrect determination of the starting point, F−1(αΔt0,T0{Ti}), on the Ti curve causes an incorrect location for the end point, tf, resulting in an incorrect value for αΔti,Ti, and therefore a faulty value for the total reaction. So, for all kinetic models besides the zerothorder reaction, when switching from one isothermal curve at T0 to another at Ti, one needs to obtain the proper starting point on the Ti curve as well as the corrected time for the total reaction up through Δti, as presented in eqs 9 and 10. 2.1.2. Illustrating Examples. 2.1.2a. Zeroth-Order Reaction (n = 0) with Two Temperature−Time Steps. Assume a zerothorder reaction (eq 2 with E = 90 kJ/mol and A = 108 s−1), at T0 = 310 K for the first 1000 h and then at T1 = 315 K for another 1000 h, as presented graphically in Figure 3. As explained in detail in the previous section, it is not absolutely necessary to invoke a time correction for a zeroth-order reaction to obtain the correct total reaction. However, it is still a good practice to do so, in particular when a proper graphical presentation is to be obtained. In Figure 3, the reaction for the first Δt0 = 1000 h is shown by the thicker solid line on the 310 K curve and ends

⎡ ⎛ 90000 ⎞⎤ ⎟⎥ = 0.246/⎢108 exp⎜ − ⎝ ⎣ 8.314 × 315 ⎠⎦ F −1(αΔt0 ,310K {315K}) = 2.07 × 106 s = 574.6 h

So, the proper starting point after the first 1000 h at 310 K is point B on the 315 K curve with a corresponding coordinate of time 574.6 h and of reacted fraction 0.246. The corrected time for 1000 h on the 310 K curve and another 1000 h on the 315 K curve as if the equivalent reaction happened on the 315 K curve only is calculated from eq 9: cor t315K = F −1(αΔt0 ,310K {315K}) + Δt315K

= (574.6 + 1000) × 3600 s = 1574.6 h

The total reacted fraction is calculated from eq 10: ⎛ E ⎞ cor cor ) = A exp⎜ − α310K → 315K i = F(t315K ⎟t315K ⎝ RT1 ⎠ ⎛ 90000 ⎞ ⎟(1574.6 × 3600) = 108 exp⎜ − ⎝ 8.314 × 315 ⎠ = 0.674

So, 67.4% reaction is reached after 1000 h at 310 K and another 1000 h at 315 K. D



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2.1.2b. First-Order Reaction (n = 1) with Three Temperature−Time Steps. Assume the same first-order reaction in Figure 1 (E = 90 kJ/mol and A = 108 s−1) but with a temporal variation, as illustrated in Figure 4 with the first 2000 h (Δt0) at

The start point on the 310 K is labeled B and corresponds to a coordinate of 1126.5 h and a conversion level of 0.242 (Figure 5). The corrected time after 2000 h at 305 K and 2000 h at 310 K, tcor 310K, as if the reaction so far happened on only the 310 K curve, is calculated from eq 9: cor t310K = F −1(αΔt0 ,305K {310K}) + Δt1

= (1126.5 + 2000) × 3600 s = 3126.5 h

And the reaction extent after 2000 h on the 305 K curve and 2000 h on the 310 K curve is calculated from eq 10: cor α305K → 310K = F(t310K )

= 1 − exp[−(108e−90000/8.314 × 310)(3126.5 × 3600)] = 0.537

Figure 4. Temporal variation of temperature for a first-order reaction with E = 90 kJ/mol and A = 108 s−1 with the first 2000 h (Δt0) at T0 = 305 K, the next 2000 h (Δt1) at T1 = 310 K, and T2 = 315 K thereafter.

The rest of the reaction is at 315 K, and the equivalent start time on the 315 K curve after 2000 h at 305 K and 2000 h at 310 K, F−1(α305K→310K{315K}), is labeled as point D on the 315 K curve in Figure 5 and obtained from

T0 = 305 K, the next 2000 h (Δt1) at T1 = 310 K, and T2 = 315 K thereafter. The reacted fraction after the first 2000 h at 305 K, αΔt0,305K, is shown in Figure 5 by the thicker solid line on the 305 K curve (ended at point A) and is calculated from eq 3:

F −1(α305K → 310K {315K}) = −

ln(1 − α305K → 310K )

( ) E

A exp − RT

2

αΔt0 ,305K = 1 − exp[− (A e−E / RT0)t ]

=−

8 −90000/8.314 × 305

= 1 − exp[−(10 e

)(2000 × 3600)]

= 0.242

The form of eq 3 for first-order reaction when applied to eq 8 yields ln(1 − α)

( ) E

A exp − RT

(18)

0

The next 2000 h is at 310 K, and the equivalent start time on the 310 K curve after the first 2000 h at 305 K, F−1(αΔt0,305K{310K}), is obtained from F −1(αΔt0 ,305K {310K}) = −

ln(1 − αΔt0 ,305K )

( ) E

A exp − RT

(

=−

(

90000

108 exp − 8.314 × 310

(

)

= 6.47 × 106 s = 1799 h

) changes only by a constant (e.g., A exp(− RTE ) 0

( ) by just a constant) if the reacted

6

)

E

A exp − RT

1

ln(1 − 0.242)

90000

108 exp − 8.314 × 315

The readers can follow Table 1 for guidance on kinetic processes involving three or more temperature time steps. Note that the total reacted fraction in a chemical reaction with a multistep temperature profile requires only one additional step in comparison with a true isothermal reaction: the calculation of the corrected time in column 3 to be used in the formula for the reacted fraction instead of real time in the next step. 2.2. Treatment Based on Reaction Rate Mathematics. The purpose of the time correction is to ensure the continuity/ conservation of the reacted extent when switching from one isothermal reacted fraction curve at one temperature to another one at a different temperature. As described and illustrated in section 2.1, when there is a continuity/conservation of the reacted extent upon switching isothermal reacted fraction curves, the proper reaction rate (which is the slope or time derivative of the reacted fraction curves in Figures 1, 2, 3, and 5) is also automatically warranted. Now that the importance of the continuity/conservation of reacted extent has been well explored, the readers might ask what happens if α = F(t) is not invertible? In the case that α = F(t) is not invertible, t cannot be expressed as a function of α. It is, thus, impossible to obtain the time correction needed to ensure the continuity/conservation of the reacted fraction (and therefore the proper reaction rate) upon switching isothermal reacted fraction curves. Luckily, there is an alternative treatment for making kinetic predictions for chemical reactions having time-dependent temperature profiles: working exclusively with reaction rate eq (eq 1) instead of reacted fraction (i.e., eqs 2−7). By examining eq 1, readers can see, when there is a temperature change,

Figure 5. Graphical illustration of the concept of time correction for a first-order reaction with E = 90 kJ/mol and A = 108 s−1 at T0 = 305 K for the first 2000 h, then at T1 = 310 K for the next 2000 h, and finally at 315 K thereafter.

t=−

ln(1 − 0.537)

= 4.06 × 10 s = 1126.5 h

is different from A exp

E − RT 1

fraction α is the same (hence, f(α) is the same). However, even E



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Table 2. Mathematical Summary of How Reaction Rate Used in Conjunction with the Integrated Reacted Fraction at Each Time Step for Kinetic Predictions of Reactions Having Time-Dependent Temperature Profilesa elapsed time

α

T (K) T0

0

Δt0

T0

⎡ ⎤ E α0 = ⎢A exp − RT f (0)⎥Δt0 ⎣ ⎦ 0

Δt0 + Δt1

T1

Δt0 + Δt1 + Δt2

T2

...

...

...

Tn

⎡ ⎤ E αn = αn − 1 + ⎢A exp − RT f (αn − 1)⎥Δtn ⎣ ⎦ n−1

0

∑n0Δti

( ) ⎡ ⎤ α = α + ⎢A exp(− )f (α )⎥Δt ⎣ ⎦ ⎡ ⎤ α = α + ⎢A exp(− )f (α )⎥Δt ⎣ ⎦

f(α)

dα/dt

f(0)

A exp − RT f (0)

f(α0)

1

0

E RT0

0

1

f(α1)

2

1

E RT1

1

2

f(α2)

(

)

( A exp(− A exp(− A exp(−

E

0

) )f (α ) )f (α ) )f (α )

E RT0

0

E RT1

1

E RT2

2

...

...

f(αn)

A exp − RT f (αn)

( ) E

n

a

Note that each isothermal period in this table can be further subdivided into many smaller time steps, at the end of which the reaction rate and the corresponding reacted fraction are refreshed for better accuracy in kinetic predictions.

obtained from Figure 5 (namely, the reacted fraction at the end of the first time step of 2000 h at 305 K is 0.242 and at the end of the second time step of 2000 h at 310 K is 0.537). So, the reaction rate treatment presented here is equivalent to switching from one isothermal reacted fraction curve to another one for temperature change, but it is much easier to deal with because it does not require α = F(t) to be invertible.

on a single isothermal reacted fraction curve (e.g, the 315 K curve in Figure 5), the reaction rate (equivalently, the slope or time derivative of the reacted fraction curve) keeps changing as the reaction extent α increases. This is the direct result of the evolution of the f(α) term with α. In other words, proper operation of the rate equation can ensure the continuity/ conservation of the reacted fraction and proper reaction rate upon any change in temperature. Indeed, eq 1 can be used for obtaining the reaction rate at every time step if the expression for f(α) is known. The total reacted fraction up to any particular time step can be obtained numerically by summing up the products of previous reaction rates and the corresponding time steps and then used as the input into the f(α) term in eq 1 for the determination of the proper reaction rate in the next step. This methodology that automatically combines the proper reaction rate calculation with the continuity of reacted fraction at every temperature− time step is illustrated in Table 2. As an illustration, the example with three time steps and temperatures presented in Figures 4 and 5 is now solved by the reaction rate approach. Again, this is a first-order reaction with E = 90 kJ/mol and A = 108 s−1. The time steps are 2000 h at 305 K, then 2000 h at 310 K, and the rest of reaction is at 315 K (Figure 4). For the first-order reaction, f(α) = 1 − α, and the rate eq 1 becomes dα = kf (α) = A e−E / RT0(1 − α) dt

3. CONCLUSIONS The kinetic expressions of most known chemical reactions are usually expressed in isothermal mathematics. This translates into a family of reacted fraction curves for different temperatures. To make kinetic predictions for reactions involving many time steps at different temperatures, researchers must ensure the continuity/conservation of reacted fraction at every temperature−time step. This translates to either employing a time correction to switch from one isothermal reacted fraction curve to another one or equivalently working with the reaction rate while updating the reacted fraction in the kinetic expression f(α) term at the end of every time step. The general treatment for proper inclusion of time-dependent temperature profiles in isothermal reactions and the accompanying tables and illustrations presented in this article can be used as a guide for making proper kinetic predictions for reactions in environments with time-dependent temperature profiles.

■

(19)

AUTHOR INFORMATION

Corresponding Author

Figure 6 shows the results from the reaction rate approach with the help of a spreadsheet (Excel, Kaleidagraph, etc. with Table 2 as the template), and they are the same as those

*L. N. Dinh. E-mail: [email protected]. Phone: (925) 422-4271. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

■

REFERENCES

(1) Thermal Decomposition of Ionic Solids; Galwey, A. K., Brown, M. E., Eds.; Elsevier: Amsterdam, 1999. (2) The Mathematics of Diffusion; Crank, J., Ed.; Oxford University Press Inc.: New York, 1975. (3) Chemistry of the solid state; Garner, W. E., Ed.; Academic Press: New York, 1955.

Figure 6. Results of reacted fraction vs time for the case represented in Figures 4 and 5, but with the reaction rate approach. F



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(4) Sestak, J.; Berggren, G. Thermochim. Acta 1971, 3, 1−12. (5) Khawam, A.; Flanagan, D. R. Solid-State Kinetic Models: Basis and Mathematical Fundamentals. J. Phys. Chem. B 2006, 110, 17315− 17328. (6) Friedman, H. L. Kinetics of Thermal Degradation of CharForming Plastics from Thermogravimetry: Application to a Phenolic Plastic. J. Polym. Sci., Part C: Polym. Symp. 1964, 6, 183−195. (7) Akahira, T.; Sunose, T. Joint convention of four electrical institutes, Research report (Chiba Institute of Technology). Sci. Technol. 1971, 16, 22−31. (8) Flynn, J. H.; Wall, L. A. A quick, direct method for the determination of activation energy from thermogravimetric data. J. Polym. Sci., Part B: Polym. Lett. 1966, 4, 323−328. (9) Vyazovkin, S. Modification of the integral isoconversional method to account for variation in the activation energy. J. Comput. Chem. 2001, 22, 178−183. (10) Vyazovkin, S.; Burnham, A. K.; Criado, J. M.; Pérez-Maqueda, L. A.; Popescu, C.; Sbirrazzuoli, N. ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data. Thermochim. Acta 2011, 520, 1−19. (11) Burnham, A. K.; Dinh, L. N. A comparison of isoconversional and model-fitting approaches to kinetic parameter estimation and application predictions. J. Therm. Anal. Calorim. 2007, 89, 479−490. (12) Dinh, L. N.; Burnham, A. K.; Schildbach, M. A.; Smith, R. A.; Maxwell, R. S.; Balazs, B.; McLean, W., II Measurement and prediction of H2O outgassing kinetics from silica-filled polydimethylsiloxane TR55 and S5370. J. Vac. Sci. Technol., A 2007, 25, 597−600. (13) Moine, E. C.; Tangarfa, M.; Khachani, M.; El Hamidi, A.; Halim, M.; Arsalane, S. Thermal oxidation study of Moroccan oil shale: A new approach to non-isothermal kinetics based on deconvolution procedure. Fuel 2016, 180, 529−537. (14) Prasad, T. P.; Kanungo, S. B.; Ray, H. S. Non-isothermal kinetics: some merits and limitations. Thermochim. Acta 1993, 217, 329−330. (15) Khawam, A.; Flanagan, D. R. Role of Isoconversional Methods in Varying Activation Energies of Solid-State Kinetics − I. Nonisothermal Kinetic Studies. Thermochim. Acta 2005, 429, 93−102. (16) Khawam, A.; Flanagan, D. R. Role of Isoconversional Methods in Varying Activation Energies of Solid-State Kinetics − II. Nonisothermal Kinetic Studies. Thermochim. Acta 2005, 436, 101− 112. (17) Khawam, A.; Flanagan, D. R. Complementary Use of ModelFree and Modelistic Methods in the Analysis of Solid-State Kinetics. J. Phys. Chem. B 2005, 109, 10073−10080. (18) Lin, C. P.; Chang, Y. M.; Tseng, J. M.; You, M. L. Comparison of the Isothermal and Non-isothermal Kinetics for Predicting the Thermal Hazard of Tert-butyl Peroxybenzoate. Adv. Mater. Res. 2011, 328−330, 124−127. (19) Guo, S.; Wan, W.; Chen, C.; Chen, W. H. Thermal Decomposition Kinetic Evaluation and Its Thermal Hazards Prediction of AIBN. J. Therm. Anal. Calorim. 2013, 113, 1169−1176. (20) Tseng, J. M.; Shu, C. M.; Gupta, J. P.; Lin, Y. F. Evaluation and Modeling Runaway Reaction of Methyl Ethyl Ketone Peroxide Mixed with Nitric Acid. Ind. Eng. Chem. Res. 2007, 46, 8738−8745. (21) Bou-Diab, L.; Fierz, H. Autocatalytic Decomposition Reactions, Hazards and Detection. J. Hazard. Mater. 2002, 93, 137−146. (22) Kossoy, A. A.; Koludarova, E. Specific Features of Kinetics Evaluation in Calorimetric Studies of Runaway Reactions. J. Loss Prev. Process Ind. 1995, 8, 229−235. (23) Kotoyori, T. The self-accelerating decomposition temperature (SADT) of solids of the quasi-autocatalytic decomposition type. J. Hazard. Mater. 1999, 64, 1−19. (24) Kumpinsky, E. Lumped Kinetics of Self-Heating Runaway Reactions: A Statistical Treatment. Ind. Eng. Chem. Res. 2008, 47, 7570−7579. (25) Melcher, A.; Ziebert, C.; Rohde, M.; Seifert, H. J. Modeling and Simulation of the Thermal Runaway Behavior of Cylindrical Li-Ion Cells-Computing of Critical Parameters. Energies 2016, 9, 292.

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Temperature-Dependent Kinetic Prediction for Reactions Described

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