Evaluation of Kinetic Data from Thermogravimetric
Measurements
Introduction
In thermogravimetric experiments the mass of a sample is
recorded during the heating program. The experiments can be static,
that means isothermal or dynamic, that means with a defined heating
rate.
The basic equation is familiar from formal reaction kinetics and
reads for a reaction A  products:

dwa
n
 k  wa
dt
(1)
wa = mass of the active component in the sample, t = time, k = reaction
constant, n = formal order of the reaction. The rate constant k is in
particular a function of the temperature and is according to the semiempirical Arrhenius equation given by:
 E 
k  A  exp  a 
 RT 
(2)
Formally, A can be interpreted as rate constant k(T = ). A is also called
"pre-exponential factor" or "collision factor", while the activation energy
Ea is sometimes called "efficiency factor". Physically, eq. 2 expresses that
in terms of the "simple" collision theory not all collisions A result in a
reaction but only a fraction that is given by the term 
Ea
that
RT
indicates the efficiency of the collisions with respect to a reaction. In this
sense Ea is the minimal energy required for a collision to result in a
reaction. R = 8.314 J mol-1 K-1 is the universal gas constant, T [K] is the
thermodynamic temperature.
Given this interpretation the rate constant k should have the same
dimension for all orders of reaction n, namely t –1. This is accomplished
by the factor
 n 1
M n 1  wa  wr 
n 1
that transforms k  k '. wa is a
function of the time (and the temperature), and with wr the mass of
unreactive material and  the density of the whole sample. M is the
molar mass of the active component and as such remains constant as
long as there is any reactive substance left, although the mass of active
substance decreases during the experiment. The initial sum
wa (t  0)  wr is therefore the initial (=starting) mass of material
(density  ). Finally, eq. 1 becomes:
wa 0  wr  n 1 dw
wa
a
n
 k ' dt
(3)
Considering eq. 2 and integration yields:
wa


wa 0
wa 0  wr 
wa
n
n 1
T
dwa 

A'
 E 
exp   a dT
 dT 
 RT 
 T
 dt 
(4)
0
dT
  heating rate HR
dt
(5)
With the definitions of the (fractional) conversion C and the residual
fraction of unreacted substance S:
C
wa 0  wa
w
1  a
wa 0
wa 0
w
S r
wa 0
(6a,b)
The mass-integral (C ) (left side of eq. 4) can be calculated for selected
(formal) orders of reaction n (see table 1), and the temperature-integral
(T ) (right side of eq. 4) can be approximated. The mass-integral (T )
is a function of n and S except for n =1 that is S – independent.
Table 1: integrals of the mass-integral for selected n
order of reaction, n
mass-integral (C)
0
1


ln 
1  C  S  C 

1  1  S 
2 ln 
 1  C  S  C   1  C  
 1 
ln 
1  C 
 1
1  S  C   
2

1  S  
 1  S 
0.5
1.0
1.5


1  S  


1  1  S 
2 ln 

 1  C  S  C   1  C  1  S  
S C

 1  
ln 



1  C   1  C 1  S 
2.0
The exponential temperature integral (T ) can be approximateda by:
g T   A'
Ea
E
 p X  with X   a
RT
RT
and  X  20
lg p X    2.315  0.4567
(7)
Ea
RT
The combination of the above equations result in the final eq. 8 that can
be used to determine the activation energy Ea, the pre-exponential factor
A and the order of the reaction n.
lg f C   lg
A  Ea
E
 lg   2.315  0.4567 a
R
RT
The Osawa-methodb for the evaluation
thremogravimetric results rearranges eq. 8:
a
b
C. D. Doyle, J. Appl. Polym. Sci. 6 (1962) 639
T. Osawa, Bull. chem. Soc. Japan 38 (1965) 1881
of
kinetic
(8)
data
from
A  Ea
E 1
lg   lg
 lg f (C )  2.315  0.4567 a  (8a)
R T
R 
intercept A'
slope m
Then the experiment is carried out at a number of different heating rates
. At different (rather freely chosen) fixed conversions C the
corresponding temperature is determined and lg  is plotted vs. 1/T.
From the slope m of the resulting more or less straight lines Ea (if n was
properly chosen and hence (C) – see Tab. 1 – is appropriate) is
obtained and from the intercept A' the pre-exponential factor A is
determined. Also information about the formal order of the reaction n
can be obtained.
McCallum and Tannerc have modified eq. 8 resulting in:
A  Ea
E
0.449  0.22Ea
lg f C   lg
 0.48 a 
 R
R
103  T
0.44
(9)
which gives a better approximation of the function p(X). The scaling of
eq.9 gives the unit of Ea in kcal mol-1 c.
In summary, the data processing should follow the following lines,
see also fig 1:
According to eq. 8a log  is plotted vs. 1/T trying different
assumptions for n (and hence for (C) ). Frequently a first-order
reaction (n = 1) fits for degradation in an inert atmosphere while
for degradation in a reactive atmosphere a second order reaction
(n = 2) could hold. Non-integer reaction orders can be a hint for
(radical) chain-reactions. The amount of unreacted material S has
to be determined – it is present in all but the first-order reaction –
and the reaction order can be found by trial and error.
Alternatively, eq. 9 can be used and the appropriate function (C)
is plotted vs. 1/T.
Eq. 1 is not applicable for heterogeneous reactions
solidproductsd, however, the final equations developed here (eq. 8 and
9) can be used for the calculation of kinetic data of the degradation of
polymers above their melting pointe.
the conversion is: 4.18 J = 1cal
T. A. Clarke, E. L. Evans, K. G. Robins, J. Thomas, Chem. Commun. 266 (1969)
e
R. McCallum, J. Tanner, Europ. Polym. J. 6 (1970) 1033. (This manuscripts is partly basing on this
publication in much of its theoretical part)
c
d
1-C
1
2
3
lg 
1
slope m = -0.457
Ea/R
2
3
T [K]
1T [K1]
Fig. 1: data-evaluation according to Osawab. Left figure: conversion vs. Temperature at different heating rates
123
Fig. 2: Schematic of a conventional TGA. The gas outlet can be
combined with DC, GC, MS, IR, (HPLC),…The oven with sample is on the
left, on the right there is the balance and the electromagnetic zero-mass
adjustment.