A simple program to determine the reaction rate and thermodynamic properties of reacting system

A simple program to determine the reaction rate and

thermodynamic properties of reacting system

Patrıcia R.P. Barretoa,*, Alessandra F.A. Vilelab, Ricardo Garganob

aLaboratorio Associado de Plasma—LAP, Instituto Nacional de Pesquisas Espaciais—INPE/MCT, CP515,

Sao Jose dos Campos, SP, CEP 12247-970, BrazilbInstituto de Fısica, Universidade Brasılia, CP04455, Brasılia, DF, CEP 70919-970, Brazil

Received 14 April 2003; revised 14 April 2003; accepted 13 August 2003

Abstract

We developed a simple program to determine the reaction rate by using conventional transition state theory with the Wigner

transmission coefficient and also the thermodynamic properties of the species. The hydrogen abstraction in the reaction

NH3 þ H ! NH2 þ H2 is used as a model to demonstrate the program usage. The rate constants have been computed for the

gas-phase chemical reaction over the temperature range of 200–4000 K. Our data are in a good agreement with the published

data for this reaction.

q 2003 Elsevier B.V. All rights reserved.

PACS: 82.20.Pm; 82.20.Db; 82.20.Wt; 82.30.Hk; 82.30.Lp; 51.30. þ i; 82.60. 2 s

Keywords: Kinetics; Thermal rate constants; Conventional transition state theory; Direct dynamics; Bimolecular abstraction; GAUSSIAN 98;

Thermodynamic properties

1. Introduction

There has been considerable interest in recent

years, in the growth of boron nitride thin films. Like

carbon, boron nitride has different allotropes, the most

common of which are the hexagonal (hBN) and cubic

(cBN) phases. The hBN, although electrically insulat-

ing, has properties that are very similar to graphite

while the cBN has properties comparable to diamond.

At present there is little understanding of the

chemical process which are involved in and which

control the synthesis of either hexagonal or cubic

boron nitride from the vapor phase. Theoretical

research found in the literature includes thermodyn-

amic equilibrium calculations for mixtures involving

B/F/N/H [1–5], B/Cl/N/H [4,5], and B/N/H [4], as

well as limited kinetics studies of the reactions

between BCl3 and NH3 [6,7].

P.R.P. Barreto et al. proposed a kinetic mechanism

to describe the growth of boron nitride films, for the

Ar/B/F/N/H system, using both CVD and arcjet

reactor [8–12]. The gas-phase mechanism includes

35 species and 1012 reactions and also extends a

previous mechanism that contained 26 species and

67 elementary reactions. Rate constants for 117

elementary reactions were obtained from published

0166-1280/$ - see front matter q 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2003.08.006

Journal of Molecular Structure (Theochem) 639 (2003) 167–176

www.elsevier.com/locate/theochem

* Corresponding author. Tel.: þ55-12-3945-6694; fax: þ55-12-

3945-6710.

E-mail address: [email protected] (P.R.P. Barreto).

experimental/theoretical data and those for the other

895 reactions should be estimated using transition

state theory (TST). As the number of reactions as

large we developed a simple code to determine the

reaction rate using conventional TST with the Wigner

transmission coefficient, since we normally work at

high temperature, 1000–3000 K, in which the tunnel-

ing effects decreases.

To show the capability of our code we applied it to

the well-known reaction NH3 þ H ! NH2 þ H2,

since it has been studies by several groups who

provided many theoretical [13–18] and experimental

results [19–22] in the temperature range of 500–

2000 K. Comparing the results obtained using our

code with published data, our data are in a good

agreement and now we can apply our code to the

reaction in a Ar/B/F/N/H system.

Following, we outline how the paper is organized.

In Section 2 we present the main characteristics of our

program. Our results and discussion are presented in

Section 3.

2. The program

Our program has basically two subroutines, one to

calculate the thermodynamic properties of the species

and the other one to determine the rate constants. The

input files are very easy to write down and the output

files are very simple to analyze, and they can be

exported, directly, to a graphical software. The

following sections will discuss the calculation of the

reaction rate using TST [23–26] and the determi-

nation of the thermodynamic properties [23,25].

2.1. Transition state theory

The TST was developed in the 1930s and has since

formed a framework for much of the discussion of rate

processes. It is a model to determine the rate constants

based in an interaction potential between reactant and

products with a statistical representation of the

dynamics. In addition to the Born–Oppenheimer

approximation it uses more three assumptions: there

is a surface in phase space that divides into a reactant

region and a product region; the reactant equilibrium

is assumed to maintain a Boltzmann energy distri-

bution and the transition state or activated complex is

assumed to have Boltzmann energy distribution

corresponding to the temperature of the reacting

system.

Considering a bimolecular reaction, such as:

A þ BC ! AB þ C ð1Þ

where A, B or C are atoms or group of atoms, the

thermal rate constants is given by:

kTST ¼kBT

h

QX‡

QAQBC

exp 2VG‡

a

RT

!ð2Þ

where QX‡ ; QA and QBC are the partition functions of

the transition state, X† (saddle point), and the reactant,

A and BC, respectively, kB Boltzmann constant, h

Planck constant, T temperature and R universal gas

constant, VG‡a is the barrier:

VG‡a ¼ V‡ þ 1ZPE ð3Þ

where V† is the classical potential energy of the saddle

point measured from the overall zero of energy and

1ZPE is the harmonic zero-point energy—ZPE.

The partition functions, QA; QBC and QX; are given

by:

Qj ¼ Qtrans £ Qrot £ Qvib £ Qelect;

j ¼ A; BC; X

ð4Þ

where Qtrans; Qrot; Qvib and Qelect mean the component

of the translation, rotation, vibration and electronic

partition functions.

The translation partition function is:

Qtrans ¼2pmkBT

h2

� �3=2

ð5Þ

where m gives the mass of the atom or group of atom.

The rotation partition function depends on the

molecule structure, and for a linear molecule the

rotation partition function, Qrot22D; is:

Qrot22D ¼8p2I kBT

seh2

!ð6Þ

whereas for a nonlinear molecule the rotation partition

function, Qrot23D; is

Qrot23D ¼p1=2

se

8p2ImkBT

h2

!3=2" #ð7Þ

P.R.P. Barreto et al. / Journal of Molecular Structure (Theochem) 639 (2003) 167–176168

https://www.researchgate.net/publication/291783864_Evaluation_of_chemical_thermodynamics_and_rate_parameters_for_use_in_combustion_modeling?el=1_x_8&enrichId=rgreq-8100c3563437e6a8e9b42af18499f958-XXX&enrichSource=Y292ZXJQYWdlOzIyODA4NTA2NztBUzo5NzM0NDEyMDk1MDc5M0AxNDAwMjIwMDU1NDky

where I and Im ¼ IAIBIC denote the moment of inertia

and the product of inertia, respectively, se is the

symmetry number of the molecule.

The vibrational partition function and zero-point

energy, in the harmonic approximation, are given by:

Qvib ¼Y

i

1 2 exp2hvi

kBT

� �� 2di

ð8Þ

1ZPE ¼1

2hX

i

divi ð9Þ

where vi; is the frequency, i ¼ 1;…;F and F is the

degree of freedom (F ¼ 3N 2 5 for linear molecule,

3N 2 6 for nonlinear molecule and F 2 1 for the

transition state) and di is the mode degeneracy.

The electronic partition function is given by:

Qelect ¼X

i

gi exp2e i

kBT

� �ð10Þ

where gi is the degeneracy, and e i is the energy of

electronic state g measured from the overall zero of

energy.

We have also included the transmission coefficient,

kðTÞ; in Eq. (2):

kWTSTðTÞ ¼ kðTÞkTSTðTÞ ð11Þ

where the transmission coefficient is used to account

for the tunneling effect along the reaction coordinate.

We have decided to estimate the transmission

coefficient by using the Wigner correction [13,23,

26] instead of some semi-classical tunneling approxi-

mation [26]. The Wigner correction for tunneling

assumes a parabolic potential for the nuclear motion

near the transition state and therefore cannot be

considered as an accurate correction. The Wigner

transmission coefficient is given by:

kðTÞ ¼ 1 þ1

24

"v‡

kBT

��2

ð12Þ

where the imaginary frequency at the saddle point is

denoted by v‡:

For more precise rate constants calculation the

variational TST should be used and also semi-

classical methods for the tunneling effect [26], yet in

this case more information about the potential energy

surface is required, but it is not employed in the

present work. A useful way to verify the importance

of using the variational transition state instead of

conventional transition state is the curvature of the

reaction path. A constraint on reaction-path curvature

is provided by the skew angle [26]:

b ¼ Arc CosmAmC

ðmA þ mBÞðmB þ mCÞ

� 1=2

ð13Þ

where mA; mB and mC are the masses of the A, B and

C moieties, respectively, for the schematic reaction

shown by Eq. (1). Large reaction-path curvature is

often encountered in the tunneling region in system

with small skew angles, so the tunneling effects

should be smaller.

For a generic bimolecular reaction:

A þ B Okf

krC þ D ð14Þ

where the rate constant in the forward direction, kf ;

determined by TST, may be conveniently writing in

the Arrhenius form as:

kf ¼ ATn exp2Ea

RT

� �ð15Þ

where A is the pre-exponential factor, n is the

temperature power factor and Ea is the activation

energy.

The rate constants in the reverse direction, kr; can

be determined using the forward constant rate and the

equilibrium constant, Kc; as:

kr ¼kf

Kc

ð16Þ

Remembering:

Kc ¼ Kp

p

RT

� �Sns

ð17Þ

and:

DG ¼ 2RT ln Kp ð18Þ

so, we can rewrite the reverse reaction rate as:

kr ¼ kf

RT

p

� �Sns

expDS

R

� �exp 2

DH

RT

� �ð19Þ

where Sns is the stoichiometric factor that is defined

to be positive for products and negative for reactants

and p is the pressure.

P.R.P. Barreto et al. / Journal of Molecular Structure (Theochem) 639 (2003) 167–176 169

2.2. Thermodynamic properties

To determine the reverse rate constants, Eq. (16), it

is necessary to know the thermodynamic properties

such as entropy and enthalpy for the reactants and

products.

The thermodynamic properties, internal energy,

entropy, and heat capacity, can be written in terms of

the partition functions as:

E ¼ kBT› ln Q

›lnT

� �v

ð20Þ

S ¼ kB ln Q þ kB

› ln Q

› ln T

� �v

ð21Þ

cv ¼ kB

› ln Q

› ln T

� �vþkB

›2 ln Q

›ðln TÞ2

!v

: ð22Þ

The partition functions are given by Eqs. (4)–(10).

Others thermodynamic properties such as enthalpy,

Gibbs free energy, can be obtained combining

the previous properties as:

H ; E þ pV ð23Þ

G ¼ H 2 ST : ð24Þ

Our code prints out the entropy and sensible enthalpy,

Hsen ¼ H 2 DfH0; for all species, reactants, products

and saddle point, as a functions of temperature.

3. Results and discussion

To apply TST, we must know the geometries,

frequencies, and the potential energy for reactants and

saddle point. These properties are obtained from

accurate electronic structure calculation performed

using the GAUSSIAN 98 program [27]. First, we

determined the reactants and saddle point geometry

and frequencies using the second order Møller–

Plesset perturbation theory and the triple-zeta plus

polarization 6-31G(d). Furthermore, we calculated

Table 1

Geometrical parameters (bond distance in Angstroms and bond angle in degree) of all stationary points of test reaction NH3 þ H ! NH2 þ H2

Parameter This worka Reference

Bb Cc Dd Ee Ff

NH3

RNH 1.0168 (1.0124g) 1.007 1.012 1.01

AHNH 106.4004 (106.67g) 107.4 106.1 106.1

NH2

RNH 1.0283 (1.024g) 1.023 1.02

AHNH 103.3089 (103.4g) 102.7 102.7

H2

RHH 0.7375 (0.7414g) 0.734 0.73

TS

RNH 1.019 1.015 1.023 1.023 1.013 1.02

RNH11.290 1.247 1.305 1.305 1.286 1.30

RH1H2 0.920 0.944 0.869 0.869 0.861 0.87

AHNH 106.40 103.5 103.5 103.5

AHNH1100.20 100.2 99.1 99.1 99.1

ANH1H2 162.59 162.6 158.5 158.5 177.7 158.5

DHNH1H2 57.48 53.8 52.7 52.7

a MP2/6-31G(d).b HF/SSANO [13].c MP2/6-31G(d) [14].d MP2/6-31p [15].e J. C. Corchado et al. [16].f MP2/6-31G(d,p) [17].g JANAF Table [28].


the energies for all species, reactants and saddle point,

at the MP2/6-311þþG(3df,3pd), MP4(SDTQ)/6-

31G p p and B3LYP/6-311þþG(3df,3pd) level.

The geometries computed at MP2/6-31G(d) for

reactant, product and saddle point are shown in

Table 1. These geometries are also compared with

reference data. The interatomic distance and bond

angle absolute errors for the reactants and products

are smaller than 0.0044 A and 0.38, respectively,

when comparing with the experimental results and

0.0098 A and 1.08, respectively, for the others

calculated results. While the interatomic distance

and bond angle absolute errors for the transition

state are 0.058 A and 4.788, respectively, exception

for the angle ANH1H2where this error is 15.118,

between our data and the Corchado and Espinosa-

Garcia one [16]. Fig. 1 shows the optimized

geometry of the saddle point, and also the notation

used in Table 1.

The harmonic vibrational frequencies calculated

from ab initio methods are overestimated, in this case,

in about 8%, then we have used a scaling factor of

92%. The scaled frequencies are also compared with

published data, as shown in Table 2. The absolute

error among our calculated and experimental fre-

quencies is less than 118 cm21, except for the HH

stretching where this error is 235 cm21. Comparing

the absolute error among our calculated frequencies

with published calculated frequencies for the reactants

and products we observe for the NH stretching an

error of 480 cm21, when for the other frequencies this

error is less than 189 cm21. The absolute error in

frequency for the transition state is less than

435 cm21, except for the imaginary frequency

where the minimum error is 181 cm21 for the

Henon data [13] and the maximum error is

1119 cm21 for the Corchado data [16].

3.1. Thermodynamic properties calculations

The first subroutine in our code calculates the

thermodynamic properties, particularly the entropy

and enthalpy, Eq. (21) and (23). Fig. 2 compares the

calculated properties with the reference data from the

JANAF tables [28] for the reactants, NH3 and H,

products, NH2 and H2 and, also, gives the thermo-

dynamic properties for the transition state

(TS ¼ NH4).

Compared with reference data [28], the maximum

absolute errors in our calculation and the reference

data are 4.71 kcal K21 mol21 for the NH3 entropy at

4000 K and 2.41 kcal mol21 for the NH3 enthalpy at

298.15 K. Fig. 3 shows the maximum, minimum andFig. 1. Atomic numbering of the reaction system for reaction

NH3 þ H ! NH2 þ H2.

Table 2

Harmonic vibrational frequencies (cm21) and zero-point energy

(kcal mol21) of reaction NH3 þ H ! NH2 þ H2

This worka Reference

Bb Cc Dd Ee Ff

NH3

1068 (950g) 1146 1067 1067

1614(2) (1627g) 1803(2) 1642(2) 1642(2)

3224 (3337g) 3709 3386 3386

3368(2) (3444g) 3841(2) 3535(2) 3535(2)

NH2

1501 (1497g) 1516 1516

3147 (3219g) 3307 3307

3259 (3301g) 3417 3417

H2

4170 4301 4301

TS

622 695 683 723 563 683

652 726 694 732 598 694

1033 1209 1089 1258 1038 1089

1310 1408 1249 1485 1327 1249

1490 1633 1530 1541 1640 1530

1538 1684 1973 1731 1902 1973

3243 3666 3314 3647 3383 3314

3366 3761 3425 3750 3446 3425

2536i 2717i 1920i 2833i 1417i 1920i

a MP2/6-31G(d).b HF/SSANO [13].c MP2/6-31G(d) [14].d MP2/6-31 p [15].e J.C. Corchado et al. [16].f MP2/6-31G(d,p) [17].g JANAF Table [28].


average error in the entropy and enthalpy for each

species analyzed here.

3.2. Rate constants calculations

The total energies for reactants, products and

saddle point as well as the reaction enthalpy

and potential barrier calculated by different methods

and basis set are shown in Table 3, in which we

applied the ZPE correction in the reaction enthalpy

and potential barrier. We found a large range of

values for the potential barrier, 14.39 –

25.37 kcal mol21, for the methods and basis set

used, as well as, for the reaction enthalpy, where

this range is 0.12–8.68 kcal mol21. The same

behavior, for the potential barrier, is found in the

literature, range of 7.6–22.6 kcal mol21 [13–15,17].

We would like to emphasize the calculation made in

Fig. 2. Comparisons of thermodynamic properties, enthalpy and entropy, obtained by our code (symbols) with experimental data from JANAF

Table [28] (lines).


MP4(SDTQ)/6-31G p p , where the absolute error

in the potential barrier was 0.1 kcal mol21 when

compared with the Garrett et al. [15] results for the

same methods and basis set.

The skew angle in this case is 46.78, as it is a mid-

value angle, we should expect same tunneling effects.

For reactions with intermediate/high-value of the

skew angle the Wigner transmission coefficient is not

the best correction. On the other hand, we are

interested in a high temperature range, 1000–

3000 K, where this problem is greatly reduced.

Fig. 4 compares the reaction rate obtained in this

work, considering conventional TST and also apply-

ing the transmission coefficient of Wigner (TST/W)

with data from the Refs. [13–22]. Several reaction

rates were calculated, each one considering the

different energy calculated by the four methods and

basis set. One can see from Fig. 4 that the rate

constants obtained for MP4/6-31G p p level both

agree with experimental and theoretical results, in the

temperature range, 500–2000 K.

Fig. 5 compares the reaction rate obtained in this

work considering the energy determined by MP4/6-

31G p p level, with the data from the POLYRATE 8.7

program [29], considering TST, TST/W, canonical

variational transition state theory (CVT) and also

CVT with zero curvature tunneling (CVT/ZCT). It is

possible to see that the POLYRATE curves from TST

and CVT are coincident. In the region of low

temperature, T , 500 K; the reaction rate obtained

by CVT/ZCT considerably differs from our calcu-

lation and also the TST and CVT calculation, since

the skew angle has a moderate-value, 46.78, it is

excepted a tunneling effects that is not well deter-

mined by the Wigner transmission coefficient. The

ratio between the POLYRATE rate constant (TST or

TST/W) and ours (TST or TST/W) are 22 times at

Fig. 3. Absolute error for calculates thermodynamic properties,

enthalpy and entropy, obtained by our code with experimental data

from JANAF Table [28].

Table 3

Total energy ðEaÞ; potential barrier and reaction enthalpy (kcal mol21) including ZPE correction, where A ¼ MP2/6-31g(d), B ¼ MP2/6-

311þþg(d,p), C ¼ MP4/6-31g p p , D ¼ B3LYP/6-311þþg(3df,3pd)

A B C D

NH3 256.354211 256.4567673 256.4013892 256.4284264

H 20.4982329 20.4998179 20.4982329 20.4998098

NH2 255.6908562 255.7728291 255.7298852 255.7544436

H2 21.1441408 21.1649448 21.1645717 21.1683161

TS 256.8096988 256.9196000 256.8693051 256.9030052

Reaction enthalpy 7.82 8.68 0.12 0.32

Potential barrier 25.37 21.76 17.58 14.39

B3LYP/6-311þþG [13] V‡ ¼ 7.6 kcal mol21; CAS(3,3)/SANO [13] V‡ ¼ 22.6 kcal mol21; PMP4/6-311 þ G(d,p) [14] V‡ ¼

15.7 kcal mol21; MP2/6-31p [15] V‡ ¼ 21.59 kcal mol21; MP4/6-31pp [15] V‡ ¼ 17.68 kcal mol21; MP2/6-31G(d,p) [17] V‡ ¼

10.8 kcal mol21.


Fig. 4. Comparisons of Arrhenius plot of k against the reciprocal temperature ðKÞ in the range of 400–4000 K for reactions NH3 þ

H ! NH2 þ H2 obtained in this paper with reference data where C, D, H, J and K are theoretical data from Espinosa-Garcia [14], Corchado

[16], Marshall [18], Garrett [15] and Espinosa-Garcia [17], respectively, and E, F, G and I are experimental results from Marshall [21], Hack

[22], Michael [20] and Ko [19], respectively.

Fig. 5. Comparisons of Arrhenius plot of k against the reciprocal temperature ðKÞ in the range of 200–4000 K for the reactions NH3 þ

H ! NH2 þ H2 obtained in this paper with data from the POLYRATE program, considering TST, TST/W, TST/MEPSAG, CVT, CVT/ZCT.


500 K and drops for seven times at 800 K and drops

more while the temperature increases. It does not

differ considerable if we consider the TST or the TST/

W rate constants.

4. Conclusion

Our purpose was to develop a simple code to

determine the reaction rate for the B/F/H/N system

easily. We opted to write our own code instead of

using the available ones such as POLYRATE

program, for example, since in that way we can

simplify the inputs files and write output files that

can be used directly by some graphical software.

Our code allows us to determine several reaction

rates at once since we have one input file for all

the reactants, one input file for the transition state,

and one input file describing which and how many

reactions we want to analyze.

The analysis of reaction NH3 þ H ! NH2 þ H2

allowed us to check the capability of our code.

Our code reproduce the reaction rate from the

reference data and from the POLYRATE program in

a high temperature range, T . 500 K: Now we

can extend this methodology for other reactions

and systems.

Acknowledgements

The authors are grateful to the Centro Nacional

de Computacao Cientıfica de Alto Desempenho de

Campinas-SP (CENAPAD-SP) for the provision of

computational facilities. A.F.A. Vilela acknowl-

edges the Conselho Nacional de Desenvolvimento

Cientıfico e Tecnologico (CNPq) for the

fellowship which provided her the opportunity to

perform this work.

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A simple program to determine the reaction rate and thermodynamic properties of reacting system

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Transcript of A simple program to determine the reaction rate and thermodynamic properties of reacting system