Communication pubs.acs.org/jchemeduc
Agreement, Complement, and Disagreement to “Why Are Some Reactions Slower at Higher Temperatures?” Yingbin Ge* Department of Chemistry, Central Washington University, Ellensburg, Washington 98926, United States S Supporting Information *
ABSTRACT: In the article “Why Are Some Reactions Slower at Higher Temperatures?” published in this Journal, Revell and Williamson explained, from the enthalpic and entropic aspects, why an A + B → P reaction may proceed more slowly at higher temperatures via an A + B ↔ C → P mechanism using the pre-equilibrium approximation. Their explanation is convincing but may be too abstract for undergraduate physical chemistry students to understand fully. In this communication, a numerical implementation and graphical demonstrations of their explanation are provided for students to use to “see” for themselves a negative activation energy. Although Revell and Williamson perfectly explained the negative temperature dependence of the A + B ↔ C → P reaction rate, great caution must be exercised when their explanation is applied to interstellar chemistry where the pre-equilibrium approximation may be invalid and statistical thermodynamics functions may be ill-defined. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Computer-Based Learning, Kinetics, Kinetic-Molecular Theory, Student-Centered Learning
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AGREEMENT In the article “Why Are Some Reactions Slower at Higher Temperatures?” published in this Journal, Revell and Williamson explained, from the enthalpic and entropic aspects, why an A + B → P reaction may proceed more slowly at higher temperatures via an A + B ↔ C → P mechanism.1 On the basis of Revell and Williamson’s analysis, two conditions must be satisfied to obtain a negative activation energy (Ea): (i) The A + B ↔ C reaction must have reasonably small forward and reverse enthalpies of activation (ΔH⧧). Meanwhile, the entropy of activation (ΔS⧧) of the C → P reaction must be negative enough to ensure that the rate constant of C → P is significantly smaller than the forward and reverse reaction rate constants of A + B ↔ C. This condition ensures that C → P is the rate-limiting step and that the pre-equilibrium approximation is valid. (ii) The A + B ↔ C reaction must be exothermic, and the magnitude of the negative reaction enthalpy must be sufficiently greater than the ΔH⧧ of the rate-limiting C → P step. This condition ensures that the transition state (TS) of C → P lies sufficiently lower than the reactants enthalpy-wise.
themselves a negative Ea. Since Revell and Williamson did not explicitly state the difference between the ΔH⧧ and the Ea obtained from the Arrhenius equation,1 a succinct proof for Ea ≅ ΔH⧧ + RT, where R is the gas constant and T is the temperature, is given as follows. The temperature dependence of reaction rate constant k can be represented by the empirical Arrhenius equation:2
COMPLEMENT Revell and Williamson’s explanation for the negative Ea is convincing, but may be too abstract for undergraduate physical chemistry students to understand fully.1 In this communication, a numerical implementation and graphical demonstrations of this explanation are provided for students to use to “see” for
In contrast to the Arrhenius equation, the pre-exponential factor in the Eyring equation is apparently temperature-dependent.
k = Ae−Ea / RT
where A is the temperature-independent pre-exponential factor. Thus, the slope of the 1/T versus ln k plot, using the Arrhenius equation, equals −Ea/R: d(ln k) d
1 T
()
= −T
2
(
d ln A − dT
Ea RT
) = −T ⎛
Ea ⎞ Ea ⎟ = − ⎝ RT 2 ⎠ R
2⎜
(2)
The temperature dependence of reaction rate constant can also be represented by the more accurate Eyring equation, which is based on quantum statistical thermodynamics:2 k=
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© XXXX American Chemical Society and Division of Chemical Education, Inc.
(1)
⧧ ⧧ kBT −ΔG⧧ / RT kT e = B eΔS / R e−ΔH / RT h h
(3)
Received: March 20, 2017 Revised: May 20, 2017
A
DOI: 10.1021/acs.jchemed.7b00208 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Communication
The slope of the 1/T versus ln k plot, using the Eyring equation, equals −(ΔH⧧ + RT)/R: d(ln k) d
1 T
()
(
d ln = −T 2
kB h
+ ln T +
ΔS‡ R
−
ΔH ‡ RT
)
dT
⎛1 ΔH ⧧ ⎞ ΔH ⧧ + RT ⎟ = − T 2⎜ + = − R RT 2 ⎠ ⎝T
(4) ⧧
Equations 2 and 4 suggest that Ea ≅ ΔH + RT and that a negative Ea indicates the negative temperature dependence of the reaction rate constant. Note that A and Ea are assumed to be approximately independent of T in eq 2, whereas ΔH⧧ and ΔS⧧ are assumed to be approximately independent of T in eq 4. Because of these approximations, eqs 2 and 4 are inexact. However, for reactions within a moderate temperature range such as tens of Kelvins, eqs 2 and 4 are nearly exact. For simplicity without losing generality, numerical kinetic simulations for an A ↔ C ↔ P mechanism were carried out on an Excel spreadsheet (included in the Supporting Information) to illustrate a negative Ea. The relative enthalpy and entropy were assigned values (Table 1) that satisfy the conditions for observing a negative Ea.
Figure 2. [A], [C], and [P] vs time of the A ↔ C ↔ P reaction mechanism at 600 K. The relative values of H and S of A, C, P, and the two transition states are given in Table 1. [A], [C], and [P] at various time intervals are calculated using eq 8. The calculations are implemented in the Excel spreadsheet in the Supporting Information.
C → P reaction rate constant and thereby ensure that the C → P reaction is the rate-limiting step. The P → C reaction rate is negligible because of its very large ΔG⧧ of 95, 101, or 107 kJ/mol at 540, 600, or 660 K, respectively. The forward and reverse reaction rate constants k1,f and k1,r for A ↔ C and k2,f and k2,r for C ↔ P were calculated using eq 3, the Eyring equation. The rate of the concentration change of each species was calculated using the following equations:
Table 1. Relative Enthalpy and Entropy of All Species and Transition States in the A ↔ C ↔ P Complex Mechanism Thermodynamic Function a
H (kJ/mol) S (J/mol K)a
A
TS1b
C
TS2b
P
0 0
20 0
−10 0
−9 −100
−50 0
a
Reactant A is used as the zero reference for relative enthalpy and entropy. bTS1 and TS2 are the transition states of the A ↔ C and C ↔ P reactions, respectively.
d[A] = −k1,f [A] + k1,r[C] dt
(5)
d[C] = k1,f [A] − k1,r[C] − k 2,f [C] + k 2,r[P] dt
(6)
d[P] = k 2,f [C] − k 2,r[P] dt
(7)
In the spreadsheet-enabled kinetic simulations, [A] at the nth time interval, [A]n, was calculated using the following equation:
The relative enthalpy profile and the Gibbs energy profiles at 540, 600, and 660 K are plotted side by side in Figure 1 to stress the entropic effect in the C ↔ P step. Because the entropy of activation (ΔS⧧) is zero for the A ↔ C reaction, its ΔH⧧ and ΔG⧧ are exactly the same in both forward (20 kJ/mol) and reverse (30 kJ/mol) directions. Whereas due to the largely negative ΔS⧧ of the C → P reaction, its ΔG⧧ is 55, 61, or 67 kJ/mol at 540, 600, or 660 K, respectively, despite its small ΔH⧧ of only 1 kJ/mol. The large values of ΔG⧧ diminish the
⎧ d[A] ⎫ ⎬ Δt [A]n ≅ [A]n − 1 + ⎨ ⎩ dt ⎭ n − 1 = [A]n − 1 + {−k1, f [A]n − 1 + k1, r[C]n − 1 } ·Δt
(8)
where Δt is the time step interval. The changes of [C] and [P] were calculated similarly. To achieve reasonable numerical accuracy and avoid too many simulation steps, Δt was set to be
Figure 1. (a) Enthalpy profile of the A ↔ C ↔ P mechanism and (b) the corresponding Gibbs energy profiles at 540, 600, and 660 K. B
DOI: 10.1021/acs.jchemed.7b00208 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Communication
Figure 3. (a) 1/T vs ln(k) and (b) 1/T vs ln(k/T) plots to determine Ea and ΔH⧧. The slopes of both plots are in the unit of Kelvin (K).
may try using much less negative values for ΔS⧧ of C → P [e.g., −20 J/(mol K)] to see that the Ea becomes positive.
10% of the reciprocal of the rate constant of the fastest elementary reaction: Δt = 10% ×
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1 max(k1, f , k1, r , k 2, f , k 2, r )
DISAGREEMENT Revell and Williamson indicated that their explanation may be applied to reactions in interstellar space where the molecular density may be extremely low.1 However, when the molecular density is extremely low, the pre-equilibrium approximation is often invalid, and statistical thermodynamics functions such as entropy may be ill-defined. In this situation, the collision theory2 better explains why some interstellar reactions are slower “at higher temperatures”, or more properly stated, “when the reactants are more energetic”, since temperature may also be ill-defined.
(9)
Kinetic simulations were run for 100 time intervals at 540, 560, 580, 600, 620, 640, and 660 K. The initial concentrations of A, C, and P were set to be 1023, 0, and 0 molecules/m3 in a 1 m3 reaction vessel for simplicity without losing generality. The rates of the concentration changes of A, C, and P were calculated using eqs 5−7. The concentration of A at each time interval was calculated using eqs 8 and 9, and so are [C] and [P] in a similar manner. Figure 2 shows the concentrations of A, C, and P versus time for the kinetic simulation of the A ↔ C ↔ P mechanism at 600 K. Pre-equilibrium between A and C appears at ∼50 time intervals. The simulation results obtained at all other temperatures exhibit a similar pattern. The effective rate constant at the nth time interval was calculated using the following equation: kn =
(d[P]/dt )n ([P]n − [P]n − 1 )/Δt ≅ [A]n − 1 [A]n − 1
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The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.7b00208. Spreadsheet that carries out the kinetic simulations and calculates the effective Ea and ΔH⧧ for the A ↔ C → P reaction mechanism (XLSX)
(10)
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At all tested temperatures, the last effective rate constant at the 100th time interval of the numerical simulation agrees to three digits with the effective rate constant kpre‑eq calculated using eff the following equation with the pre-equilibrium approximation: pre ‐ eq keff
=
d[P] dt
[A]
≅
[A]
≅
k 2,f k [A] 1,r
[A]
= k 2,f
*E-mail:
[email protected]. ORCID
k1,f k1,r
AUTHOR INFORMATION
Corresponding Author
k1,f
k 2,f [C]
ASSOCIATED CONTENT
S Supporting Information *
Yingbin Ge: 0000-0001-5315-9312
(11)
Notes
The author declares no competing financial interest.
The values of the effective rate constants at the 100th time interval obtained via the numerical simulations at 540−660 K were used to calculate the effective Ea and ΔH⧧ using the 1/T versus ln(k) and 1/T versus ln(k/T) plots in Figure 3. Since the Eyring equation can be rearranged to ln(k/T) = [ln(kB/h) + ΔS⧧/R] − (ΔH⧧/R)(1/T), the slope of the 1/T versus ln(k/T) plot is equal to − ΔH⧧/R with the assumption that both ΔS⧧ and ΔH⧧ are independent of T. Figure 3 shows that the slopes on the 1/T versus ln(k) and 1/T versus ln(k/T) plots are both positive. These two positive slopes being multiplied by (−R) result in Ea = −4.0 kJ/mol and ΔH⧧ = −9.0 kJ/mol. As expected, the obtained ΔH⧧ equals the enthalpy of the transition state of the C → P rate-limiting step relative to reactant A, and the obtained Ea is greater than ΔH⧧ by ∼RT at T ≅ 600 K. The spreadsheet calculations and plots provided a numerical implementation and graphical demonstrations for students to better understand Revell and Williamson’s explanation, from the enthalpic and entropic aspects, for the negative Ea of the A ↔ C ↔ P reaction mechanism.1 Interested readers
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REFERENCES
(1) Revell, L. E.; Williamson, B. E. Why Are Some Reactions Slower at Higher Temperatures? J. Chem. Educ. 2013, 90 (8), 1024−1027. (2) Houston, P. L. Chemical Kinetics and Reaction Dynamics; Dover Publications: Mineola, NY, 2006.
C
DOI: 10.1021/acs.jchemed.7b00208 J. Chem. Educ. XXXX, XXX, XXX−XXX
