Kissinger Analysis

Kissinger Analysis is the model-free (isoconversional) method of kinetic analysis calculating dependence of activation energy value E at the point of maximum conversion rate α_m for dynamic experiments with different constant heating rates β.

It is always necessary to check if this model-free method is valid to be used and is applicable because of restrictions of model-free methods.

Kissinger analysis belongs to the group of single-point model-free methods where activation energy value is found from general kinetic equation Eq. (1):

\frac{d\alpha}{dt} = A(\alpha) \bullet f(\alpha) \bullet exp\left( \frac{- E(\alpha)}{RT} \right)

only at the point of maximum conversion rate α_m , where time derivative is equal to zero, Eq.(2):

$\frac{d}{dt}\left( \frac{d\alpha}{dt} \right)_{m} = 0$

After substitution of Eq.(1) into Eq.(2) and rearrangement the dependence of maximum conversion rate α_m over heating rate β is found, Eq. (3):

$\frac{\beta}{{T_{m}}^{2}} = \frac{AR}{E} \bullet \frac{- df(\alpha_{m})}{d\alpha}\exp\left( \frac{- E}{RT_{m}} \right)$

Taking logarithm of Eq. (3) the linear dependence is found, Eq (4):

\ln\frac{\beta}{{T_{m}}^{2}} = ln\left( \frac{AR}{E} \bullet \frac{- df(\alpha_{m})}{d\alpha} \right) - \frac{E}{R} \bullet \frac{1}{T_{m}}

If the points of maximum conversion rate α_m with the assumed same degree of conversion (isoconversional points) will be taken from experiments performed at different heating rates, then values ln[(AR/E)*(-df(α)/dα)] will be the same for all of experiments and Eq. (4) will look like the straight line, EQ. (5):

y = b + ax

Where

y=ln(β/T_m²),
b= ln[(AR/E)*(-df(α_m)/dα)],
a=E/R,
x=-1/T

The Kissinger plot y(x) looks like the straight line for values α_m, where the activation energy can be found from the slope and pre-exponent from the intercept for known (or assumed) f(α).

Often in Kissinger method the assumption of the first order reaction is done f(α)=1- α, then (-df(α)/dα)=1, and pre-exponential factor can be found from intersect b for known activation energy E.

Example

Dimerization of cyclopentadiene:

Experimental data (Fig.1),
Kissinger plot(Fig.2),
Activation energy E and pre-exponent A at the point of maximal reaction rate for the assumption of first-order reaction (Fig.3)
Comparison between experiments and simulation for Kissinger values E and A under assumption of the first-order reaction(Fig.4):

Fig.2 Kissinger plot containing straight line for points at maximal reaction rate.

Fig.3 Values of activation energy E and pre-exponential factor A found by Kissinger method for the assumption of first-order reaction

Fig.4 Comparison between experiments(symbols) and simulation (solid lines) using Kissinger values E and A for the assumption of first-order reaction

It is seen that the simulation according to the model-free method of Kissinger has the peak points at the same position as the peak points of experimental data. However, the shape of simulated curve differs from the experimental curve if the reaction type is not equal to the assumed first-order reaction. It is always necessary to simulate the curves by Kissinger method and compare them with experiment. This comparison helps to check if Kissinger method is suitable for analysis of the current reaction.

Reference:

Vyazovkin, S., et al., ICTAC Kinetics Committee Recommendations for Performing Kinetic Computations on Thermal Analysis Data, Thermochimica Acta520 (2011) 1–19. https://doi.org/10.1016/j.tca.2011.03.034