1
Kine ic analysis o solid-s a e eac ions: P ecision o
he ac i a ion ene gy calcula ed by in eg al me hods
L.A. PÉREZ-MAQUEDA*, P.E. SÁNCHEZ-JIMÉNEZ AND J.M. CRIADO
Ins i u o de Ciencia de Ma e iales de Se illa. C.S.I.C.-Uni e sidad de Se illa. A d.
Ame ico Vespucio 41092 Se illa. Spain
Abs ac
The in eg al me hods a e ex ensi ely used o he kine ic analysis o solid-s a e
eac ions. The A henius in eg al unc ion [p(x)] does no ha e an exac analy ical
solu ion. Thus, di e en app oaches, accomplishing he condi ion ha ln g(α) is a linea
unc ion o ei he 1/T o a p ede e mined unc ion o T, ha e been p oposed o his
in eg al o de e mine he ac i a ion ene gy om a linea plo o he loga i hm o g(α)
e sus some unc ion o T. The i s app oach was p oposed by Van K e elen and a e
ha , a numbe o au ho s de eloped new app oaches, e y o en wi h he scope o
inc easing he p ecision o he A henius in eg al as checked om he s anda d
de ia ion o he p(xa) unc ion de e mined om hese app oxima ion wi h ega ds o he
ue alue o he p(x) unc ion. Besides his me hod, hose p oposed by Doyle, Ho owi z
and Me zge , Coa s and Red e n, MacCallum and Tanne and Gyulay and G eenhow
a e e y popula o de e mining ac i a ion ene gies. In ac , we ha e ound mo e han
4500 ci a ions (1300 in he las i e yea s) o he pape s we e hese me hods we e
p oposed. Howe e , a sys ema ic analysis o he e o s in ol ed in he de e mina ion o
he ac i a ion ene gy om hese me hods is s ill missing. A compa a i e s udy o he
p ecision o he ac i a ion ene gy as a unc ion o x and T compu ed om he di e en
in eg al me hods has been ca ied ou .
Keywo ds: A henius in eg al, in eg al me hods, solid s a e eac ions, e o s in
ac i a ion ene gy.
2
1. INTRODUCTION
The mally s imula ed solid-s a e eac ions, such as decomposi ions, solid-solid
eac ions, c ys alliza ions, e c, a e, in gene al, he e ogeneous p ocesses. The eac ion
a e o such p ocesses can be kine ically desc ibed, when i akes place unde condi ions
a om equilib ium, by he ollowing exp ession:1
)()(


T
d
d (1)
whe e is he ime and α is he ex en o eac ion anging om 0 be o e he p ocess
s a s o 1 when i is o e . Thus, he le hand side e m in eq. (1) is he eac ion a e.
The igh hand side e m in eq. (1) consis s o wo e ms, i.e. (T) and (α), being (T) a
unc ion ha desc ibes he dependence o he eac ion a e wi h he empe a u e (T).
Usually, his dependence is desc ibed by he A henius equa ion:
RTE
eAT /
)( 
 (2),
being A he p eexponen ial ac o o A henius, E he ac i a ion ene gy and R he gas
cons an . Addi ionally, (α) is a e m ha desc ibes he dependence o he eac ion a e
wi h he mechanism o he p ocess. Di e en unc ions ha e been p oposed in li e a u e
o desc ibing he kine ic mechanism o he solid-s a e eac ions. These mechanisms a e
p oposed conside ing di e en geome ical assump ions o he shape o he ma e ial
pa icles (sphe ical, cylind ical, plana ) and d i ing o ces (in e ace g ow h, di usion,
nuclea ion and g ow h o nuclei). Some o he mos common equa ions p oposed o
hese eac ions a e included in Table 1.
The mos common hea ing p o ile used o s udying solid-s a e eac ion is he
linea hea ing p og am. Unde hese expe imen al condi ions, T changes in a wide ange
o alues and a en i e α-T cu e is eco ded in a single expe imen . Fo linea hea ing
a e condi ions eq. (1) can be w i en
3
)(
/



e
A
dT
dRTE
 (3)
being β he hea ing a e.
Many o he expe imen al me hods used o pe o m kine ic analysis o solid-s a e
eac ions a e based in he measu emen o he e olu ion o an in eg al magni ude, i.e.
p opo ional o he ex en o eac ion, such as mass loss, eleased gas, amoun o
con ac ion, as a unc ion o empe a u e. To pe o m he e alua ion o such
expe imen al da a, i is necessa y ei he o nume ically di e en ia e he expe imen al
da a o o in eg a e eq. (3):



x
x
xp
R
AE
dx
x
e
R
AE
g)()( 2


(4),
being x=E/RT. This exp ession can be w i en in he loga i hmic o m:
)(lnln))(ln( xp
R
AE
g


(5),
Unde linea hea ing a e p og am, eqs. 4 and 5 do no ha e an exac analy ical solu ion
o p(x) and, he e o e, he solu ion canno be exp essed in a closed o m.2 Al hough,
o he T- p o iles, such as pa abolic o hype bolic p og ams, yield o analy ical solu ions
o he A henius in eg al, hey a e e y seldom used. Thus, se e al app oxima ed
equa ions ha e been p oposed o p(x) unde linea hea ing p og am.
The app oxima ions o p(x) mos commonly used in he de e mina ion o he
ac i a ion ene gy a e hose p oposed by Coa s and Red e n,3,4 Doyle,5-7 Ho owi z and
Me zge ,8 MacCallum and Tanne ,9,10 Gyulai and G eenhow,11,12 and Van K e elen.13
All hese app oxima ions ha e been ob ained ei he by simpli ica ions o he se ies
exp essions o in an empi ical way. Fo a gi en kine ic model, he esul ing equa ions
lead o a linea co ela ion whe e he kine ic ac i a ion ene gy is easy ob ained om he
slope. The numbe o publica ions whe e hese in eg al me hods a e used o
4
de e mining ac i a ion ene gies is as . Thus, abou 4500 ci a ions can be ound in he
li e a u e o he o iginal pape s3-13 whe e hese equa ions a e p oposed. Besides, he
popula i y o hese in eg al me hods has no dec eased, as indica ed by hei mo e han
1300 ci a ions jus in he las yea s, i.e. 2000-2004. In hese las i e yea s, he app oach
wi h mo e ci a ions has been ha o Coa s and Red e n3,4 wi h abou 590 ci a ions,
ollowed by hose o Ho owi z and Me zne 8 and Doyle5-7 wi h 230 and 102 ci a ions,
espec i ely (in o ma ion on he numbe o ci a ions ha e been ob ained om ISI Web
o Science da a base). Ne e heless, independen ly o he app oxima ion used, e e y
g(

) leads o a high linea co ela ion coe icien and, he e o e, i is no possible o
disc imina e he kine ic model om a single expe imen al cu e. Addi ionally, he
esul ing ac i a ion ene gy alues a e e y much dependan on he g(

) unc ion
assumed o he analysis ( hese limi a ions a e ex ended no only o in eg al me hods bu
also o any p ocedu e ha uses a single linea hea ing a e cu e14,15 ). Thus, in
p inciple, he in eg al me hods should be only used unde he wo ollowing
ci cums ances: (i) when he kine ic model is al eady known o ob aining he ac i a ion
ene gy o (ii) when he ac i a ion ene gy is known o de e mining he kine ic model.
Ne e heless, a new ques ion a ises abou he p ecision o he ac i a ion ene gy alues
de e mined by hese popula in eg al me hods because, as men ioned abo e, hey a e
based in app oxima ions o he p(x) unc ion and hei p ecision o he es ima ion o
he kine ic pa ame e s a e s ill in doub , hus some au ho s ha e claimed ha hese
me hods a e imp ecise.16-19 Some s udies ha e es ima ed he e o s in he app oxima ed
p(x) unc ions by compa ing he esul ing alues wi h hose calcula ed by nume ical
in eg a ion, concluding ha he e o s a e qui e la ge. Fig. 1 shows as a way o example
he e olu ion o he ela i e e o o he Coa s and Red e n app oxima ion o he
es ima ion o he p(x) unc ion e sus x. This igu e indica es ha he e o dec eases
5
wi h x, being signi ican ly la ge o alues o x commonly ound in li e a u e o solid-
s a e eac ions. These indings ha e been used as an a gumen o in alida ing hese
app oxima ed equa ions in he es ima ion o he kine ic pa ame e s. Ne e heless, he
aim o he a o emen ioned app oxima ions is he de e mina ion o he ac i a ion ene gy
and no he accu a e compu a ion o p(x). Taking in o accoun ha he in eg al me hods
a e so widely ex ended and ha he e is some con o e sy in hei p ecision, i would be
o in e es o es ima e he p ecision o such me hods o he de e mina ion o he
ac i a ion ene gy. The aim o he p esen pape is o pe o m a compa a i e s udy o he
p ecision o he mos ex ensi ely used app oxima ions o p(x) in he de e mina ion o
he ac i a ion ene gy.
2. ERRORS IN THE ACTIVATION ENERGY
2.1. Coa s and Red e n me hod.
The Coa s and Red e n3,4 app oach o he A henius in eg al is he ollowing:











aa
x
axx
e
xp
a2
1)( (6)
he subsc ip a s ands o app oxima ed. In gene al, he exp ession mo e commonly
used is he simpli ied o m:
2
)(
a
x
ax
e
xp
a

 (7)
This app oach is named some imes in li e a u e as Fishe app oach.20 By in oducing eq.
(7) in o eq. (4), i ollows
RTE
a
aa
eT
E
RA
g/
2
)( 



(8)
By aking na u al loga i hms, eq. (8) esul s

6
RT
E
E
RTA
ga
a
a


2
ln))(ln( (9)
Thus, he ac i a ion ene gy could be easily ob ained om he slope o he line esul ing
o plo ing ln(g(

))-2ln(T) e sus 1/T.
The ela i e e o ε o he ac i a ion ene gy (Ea) calcula ed by he Coa s and
Red e n equa ion can be de ined by he ollowing equa ion:
1001100%















R
E
R
E
E
EE
a
a

(10)
By di e en ia ing eq. (9):
T
R
E
T
ga2
/1
)(ln 





(11)
and by di e en ia ing eq. (5):
x
xp
R
E
T
xp
T
g







))(ln(
/1
))(ln(
/1
)(ln

(12)
Thus, om eqs (11) and (12), i ollows
xx
xp
R
E
R
Ea2))(ln( 


 (13)
ha subs i u ing in eq. (10) leads o
1001
2))(ln(
%









 xx
xp

(14)
This equa ion indica es ha he alues o ε% depend on x=(E/RT), and, he e o e, on he
alue o he ac i a ion ene gy and o he ange o empe a u e o he p ocess. The
alues o ε% ha e been compu ed by means o he Ma hcad so wa e by nume ical
in eg a ion o he p(x) unc ion using a ole ance (p ecision in he calculus) o 10-5. The
esul ing ε% alues as a unc ion o he pa ame e x a e included in Table 2. The alues
7
included in Table 2 illus a e ha he e is a signi ican in luence o x in he p ecision o
he calcula ed ac i a ion ene gy alues. Thus, ε% anges om almos -20% o x=2 o
less han -1% o x alues la ge han 20; in he limi , o x=

, he e o is ce o.
2.2 Doyle me hod.
The Doyle app oach o he A henius in eg al is he ollowing:5-7
aa xxp 4567.0315.2))(log(



(15)
F om eq. (15) and eq. (5), i ollows
RT
E
R
EA
ga4567.0315.2log))(log( 


(16)
Thus, he ac i a ion ene gy can be ob ained om he slope o he line esul ing om
plo ing he le hand side o eq. (16) as a unc ion o 1/T:
R
E
T
ga
4567.0
/1
))(log( 



(17)
The ela i e e o ε% (eq. (10)) o he ac i a ion ene gy ob ained by he Doyle me hod
can be ob ained om eqs. (12) and (17):
1001
))(ln(
4567.0
4343.0
%















x
xp

(18)
The alues o ε% ha e been compu ed by he same p ocedu e as desc ibed in he la e
sec ion and he esul ing e o alues a e included in Table 2.
2.3. Ho owi z and Me zge me hod.
The in ege equa ion a e assuming he Ho owi z and Me zge app oach8 o he
p(x) unc ion is he ollowing:
8
2
052.1052.1
33.5ln))(ln(
s
a
s
aa
RT
E
TR
EA
g



 (19)
whe e

is a cha ac e is ic empe a u e such ha

=T-Ts, being Ts an a bi a y e e ence
empe a u e. F om eq. (19), i is clea ha he ac i a ion ene gy is ob ained om he
slope o he line esul ing o plo ing he le hand side o eq. (19) e sus

, o e sus T
ha yields he same slope:
2
))(ln(
s
a
RT
E
T
g



(20)
F om eqs. (12) and (20), he ela i e e o ε% (eq. (10)) in he ac i a ion ene gy
ob ained by he Ho owi z and Me zge 8 esul s:
1001
))(ln(
%






 dx
xp

(21)
Table 2 includes he e o s es ima ed by eq. (21) o he ac i a ion ene gy calcula ed by
he Ho owi z and Me zge app oach. 8
2.6. Van K e elen me hod
Conside ing he Van K e elen e al app oxima ion13 o he exponen ial in eg al
o A henius, eq. (5) has he loga i hmic o m:

T
TR
E
T
TE
A
ga
RT
E
a
a
a
ln
)1(
1
368.0
ln))(ln(
max
max
max
max
























(22),
whe e Tmax is he empe a u e a he maximum he mog a ime ic a e. The ac i a ion
ene gy is de e mined om he slope o he line esul ing om he plo o ln(g(α)) as a
unc ion o lnT:
)1(ln
))(ln(
max 



TR
E
T
ga

(23)
9
Thus, he Van K e elen e al me hod,13 e en hough i is a in ege equa ion, o he
de e mina ion o he ac i a ion ene gy, i equi es o he di e en ial expe imen al cu e
o ob ain he Tmax alue o eq. (23).
As in he p e ious sec ions, he ela i e e o can be calcula ed om eq. (12) and
(23), esul ing:
1001
1))(ln(
%














 xx
xp

(24)
The esul ing alues o he e o a e included in Table 2.
2.4. MacCallum and Tanne me hod
The decimal loga i hmic o m o he in ege equa ion (eq. (5)) using he
app oach p oposed by MacCallum and Tanne 9,10 o he p(x) unc ion esul s:
T
E
E
R
EA
ga
a
aa 217449
4828.0log))(log( 4351.0




(25)
Thus, he ac i a ion ene gy can be calcula ed om he slope o he line esul ing om
he plo o log(g(α)) as a unc ion o 1/T:
a
E
T
g217449
)/1(
))(log( 



(26)
The ela i e e o ε% (eq. (10)) o he ac i a ion ene gy can be calcula ed om eqs. (12)
y (26):
1001
217
449))(ln(
·
217
4343.0
%







xRTx
xp
R

(27)
In his case he e o depends bo h on x and T. Table 3 includes he e o s in he
ac i a ion ene gy as es ima ed by means o eq. (27).
16
TABLE 1. (

) and g(

) kine ic unc ions
Mechanism
Symbol
()
g()
Phase bounda y con olled eac ion
(con ac ing a ea)
R2
21
)1(


21 1 12
()

Phase bounda y con olled eac ion
(con ac ing olume)
R3 32
)1(

 31 1 13
()

Random nuclea ion ollowed by an
ins an aneous g ow h o nuclei.
(A ami-E o ee eqn. n =1)
1F1 )1(


)1ln(


Random nuclea ion and g ow h o
nuclei h ough di e en nuclea ion
and nucleus g ow h models. (A ami-
E o ee eqn.)
An


n
n11
)1ln()1( 



n/1
)1ln(


Two-dimensional di usion
D2 11


ln( )

()ln()11



Th ee-dimensional di usion
(Jande equa ion)
D3








3/1
3/2
112
)1(3



2
3/1
11 






Th ee-dimensional di usion
(Gins ling-B ounsh ein equa ion)
D4 3
21 1
13
()
/





12 3 1 23


()
1This equa ion ep esen s an A ami-E o ee kine ic model wi h n=1 ins ead o a i s o de eac ion.
The symbol A1 would be mo e p ope .

17
TABLE 2. Values o he ela i e e o (ε%) o he ac i a ion ene gy calcula ed by
means o he Coa s and Red e n, Doyle, Ho owi z and Me zge , and Van K e elen e al.
equa ions as a unc ion o he pa ame e x (E/RT).
x Coa s and
Red e n Doyle Ho owi z and
Me zge Van K e elen
2 -19.72 71.43 80.28 30.28
5 -4.76 28.60 35.26 15.23
10 -1.47 12.72 18.53 8.53
20 -0.42 4.20 9.58 4.58
30 -0.20 1.25 6.47 3.13
50 -0.07 -1.17 3.92 1.92
100 -0.02 -3.02 1.98 0.98
 0 -4.90 0 0
TABLE 3. Values o he ela i e e o (ε%) o he ac i a ion ene gy calcula ed by
means o he MacCallum and Tanne equa ion as a unc ion o he pa ame e x (E/RT)
and he empe a u e (T).
x 400 600 800 1000 1200
2 -48.55 -5.14 16.56 29.59 38.27
5 -15.80 1.56 10.24 15.45 18.92
10 -6.59 2.09 6.43 9.03 10.77
20 -2.59 1.75 3.92 5.22 6.09
30 -1.38 1.51 2.96 3.83 4.40
50 -0.47 1.26 2.13 2.65 3.00
100 0.17 1.04 1.47 1.73 1.91
 0.78 0.78 0.78 0.78 0.78
TABLE 4. Values o he ela i e e o (ε%) o he ac i a ion ene gy calcula ed by
means o he Gyulai and G eenhow equa ion as a unc ion o he pa ame e x (E/RT) and
he empe a u e (T).
x 400 600 800 1000 1200
2 50.25 52.93 54.86 56.37 57.61
5 15.84 17.9 19.39 20.55 21.51
10 4.04 5.89 7.23 8.27 9.14
20 -1.21 0.55 1.82 2.81 3.63
30 -2.43 -0.69 0.56 1.54 2.35
50 -2.72 -0.99 0.26 1.24 2.04
100 -1.70 0.05 1.31 2.30 3.12
 18.90 18.90 18.90 18.90 18.90
18
TABLE 5. Values o he ac i a ion ene gies (Ea) and e o s (ε%) ob ained o he
analysis o he simula ed cu es included in Figs 2a and 2b by means o he di e en
in eg al me hods
Simula ed cu e Fig. 2a
(x

5)* Simula ed cu e Fig. 2b
(x

20)*
Ea (kJ mol-1)ε% Ea(kJ mol-1) ε%
Coa s and Red e n 33.5 -4.4 99.6 -0.352
Doyle 44.4 27.0 103.4 3.40
Ho owi z and
Me zge
48.5 38.7 114.2 14.20
MacCallum and
Tanne
38.4 9.7 100.8 0.75
Gyulai and
G eenhow
41.7 19.1 100.6 0.63
Van K e elen e al. 41.9 19.8 106.9 6.93
*The a e age alue o x has been ob ained om E/RT

=0.5, whe e T

=0.5 is he
empe a u e co esponding o

=0.5.
TABLE 6. Values o he ac i a ion ene gies (Ea) and e o s (ε%) ob ained o he
analysis o he expe imen al esul s o he he mal decomposi ion o BaCO3 ob ained
unde high acuum (Fig. 3) by means o he di e en in eg al me hods.
Me hod Co ela ion ac o
Ea

%*
Coa s and Red e n 0.999 212.5 -0.42
Doyle 0.999 218.6 2.39
MacCallum and
Tanne
0.999 222.9 4.45
Gyulai G eenhow 0.999 220.1 3.14
Ho owi z Me zge 0.999 230.3 7.92
Van K e elen e al. 0.999 222.6 4.31
F iedman 0.999 213.4 -
* The e o s o he ac i a ion ene gies ha e been calcula ed using he ac i a ion ene gy
de e mined by he F iedman equa ion as he accu a e alue (because no app oxima ion
is in ol ed in he me hod).
19
Figu e Cap ions
Fig. 1. E olu ion o he ela i e e o o he Coa s and Red e n app oach o he
es ima ion o he p(x) unc ion e sus he alue o x. The ela i e e o has been de ined
by he exp ession: (pa(x)-p(x)/ p(x))·100, being pa(x) he alue ob ained by he Coa s
and Red e n app oxima ion and p(x) he alue ob ained by nume ical in eg a ion.
Fig. 2. Simula ed cu es (a) β= 10 K min-1, an A2 kine ic model and he ollowing
kine ic pa ame e s: E=35 kJ mol-1 and A=10 min-1; and (b) β= 1 K min-1, an F1 kine ic
model and he ollowing kine ic pa ame e s: E=100 kJ mol-1 and A=108 min-1
Fig. 3. Expe imen al TG cu e ob ained o he BaCO3 unde high acuum a a hea ing
a e o 0.2 K min-1.
20
400 600 800 1000
0.0
0.2
0.4
0.6
0.8
1.0
b

T / K
a
Fig. 1
21
800 900 1000 1100 1200
0.0
0.2
0.4
0.6
0.8
1.0

T / K
Fig. 2