# Master Plot Verification

Verification of the Master Plot in Model-Free Analysis

### Theory

Master plot is the curve in model-free analysis, which has the specific shape, depending on the type of reaction. The shape of the master curve for experimental data can point to the specific reaction type of the single-step reaction.

Master plot is used only for the single-step reactions with constant or almost constant dependence of activation energy on conversion.

Master plot is not-suitable for multi-step reactions and for reactions with significant changes of activation energy on conversion in model-free analysis.

The general theory of master plot can be found in the article [1] (https://doi.org/10.1016/j.tca.2020.178597)

Master plot can be calculated for the reactions with reaction rate depending on the conversion alpha and temperature T according to the equation:

$\frac{\mathrm{d\alpha }}{\mathrm{dt}}=k\left(T\right)f\left(\alpha \right)$

Where dependence on temperature is done according to Arrhenius:

$k\left(T\right)=A\mathrm{exp}\left(\frac{-E}{\mathrm{RT}}\right)$

Then the master plot should be calculated as y(alpha) according to the expression:

$y\left(\alpha \right)={\left(\frac{\mathrm{d\alpha }}{\mathrm{dt}}\right)}_{\alpha }\mathrm{exp}\left(\frac{{E}_{o}}{{\mathrm{RT}}_{\alpha }}\right)=Af\left(\alpha \right)$

However, this theory works for constant activation energy, but practically the activation energy could be not really constant for all conversion values, therefore in our software we use the actual activation energy E(alpha). Additionally, the theory in this form contains the value of pre-exponent A and y(alpha) must be normalized in order to have the values which can be easy read. In Kinetics Neo software we do the normalization at the half of reaction with the point of alpha = 0.5, and final master plot has always value 1 at alpha = 0.5.

Finally, we have master plot, calculated by the following way:

$\frac{f\left(\alpha \right)}{f\left(0.5\right)}=\frac{{\left(\frac{\mathrm{d\alpha }}{\mathrm{dt}}\right)}_{\alpha }\mathrm{exp}\left(\frac{{E}_{a}\left(\alpha \right)}{{\mathrm{RT}}_{\alpha }}\right)}{{\left(\frac{\mathrm{d\alpha }}{\mathrm{dt}}\right)}_{0.5}\mathrm{exp}\left(\frac{{E}_{a}\left(0.5\right)}{{\mathrm{RT}}_{0.5}}\right)}$

For single-step reaction with almost constant pre-exponential factor the master plot is proportional to f(alpha) and the reaction type can be estimated from the shape of the master plot.

For the Friedman analysis master plot can be found from the intercept of the current conversion and intercept for the conversion 0.5, if b is calculated for natural logarithms:

$\frac{f\left(\alpha \right)}{f\left(0.5\right)}=\frac{\mathrm{exp}\left({b}_{\alpha }\right)}{\mathrm{exp}\left({b}_{0.5}\right)}$

If for Friedman analysis decimal logarithms are used then the calculation of master plot is:

$\frac{f\left(\alpha \right)}{f\left(0.5\right)}=\frac{{10}^{{b}_{\alpha }}}{{10}^{{b}_{0.5}}}$

### Applicability: When to Use

The shape of master plot corresponds to the shape of reaction type only for the multi-point differential model-free analysis methods like

• Friedman
• Vyazovkin
• Numerical .

It can not provide correct information for the single-step methods like ASTM E698, ASTM E2890, ASTM E1641 and for the integral model-free methods like Ozawa-Flynn-Wall and Kissinger-Akahira-Sunose.

Master plot works good only for the experimental data of good quality without high noise. We definitely recommend using model-based method instead of model-free because it has more advantages, including:

• fit between model and experimental data, including visual comparison and R² value
• stable when the data are not high quality.

### Simulated Data for Master Plot Verification

In order to compare the theory with the master plot calculated by Kinetics Neo, we created the artificial data for known Reaction typesReaction type is the elementary mechanism of one individual reaction step in multi-step chemical reaction. Reaction type f(Cr, Cp) describes dependence of the reaction rate for individual reaction step on the concentrations of reactant Cr and product Cp for this step.reaction types. The data were simulated manually outside of the Kinetics Neo software, then loaded into software to see the result. You can find all simulated data and corresponding kinetic projects in the directory Alpha_Simulated for pre-installed samples (Kinetics Neo version 2.7.2 or later):

#### 1. First-Order Reaction F1

Please open pre-installed project F1_Simulated.kinx2 from Alpha_Simulated directory in pre-installed samples (Kinetics Neo version 2.7.2 or later):

Select Analysis-Model-Free - Friedman – Master Plot:

We know that for the first-order reaction the reaction type is described by equation: f(alpha)=(1-alpha), which presents the straight line. Therefore, according to theory, we expect also the straight decreasing line for the first-order reaction.

We see here that the master plot is the straight line with value 1 at alpha=0.5, and this corresponds to the expected shape.

It means that if for experimental data the master plot looks like the straight decreasing line then the possible reaction type is the first-order reaction.

#### 2. Second-Order Reaction F2

Please open pre-installed project F2_Simulated.kinx2 from Alpha_Simulated directory in pre-installed samples (Kinetics Neo version 2.7.2 or later):

Select Analysis-Model-Free - Friedman – Master Plot:

For the second-order reaction the reaction type is described by equation f(alpha)=(1-alpha)^2 , which presents the decreasing half of parabola with minimum at the end of reaction. Therefore, according to theory, we expect the decreasing half of parabola for the second-order reaction.

Here master plot is the decreasing half of parabola with value 1 at alpha=0.5, and this corresponds to the expected shape.

It means that if for experimental data the master plot looks like the decreasing half of parabola then the possible reaction type is the second-order reaction.

#### 3. Prout-Thompkins Reaction with Autocatalysis Bna

Please open pre-installed project Bna_Simulated.kinx2 from Alpha_Simulated directory in pre-installed samples (Kinetics Neo version 2.7.2 or later).

Select Analysis-Model-Free - Friedman – Master Plot:

For the Prout-Thompkins reaction of the first order the equation for reaction type is f(alpha)=alpha*(1-alpha) , and for this equation we expect the symmetric parabola starting from zero with maximum at 0.5.

The shape of calculated master plot for the artificial data corresponds to the expected shape, and looks like the symmetric parabola starting from zero with maximum 1.0 at alpha=0.5.

It means that if for experimental data the master plot looks like the symmetric parabola starting from zero then the possible reaction type is the reaction with acceleration, possibly autocatalytic reaction.

#### 4. Nucleation Reaction of Avrami Type An

Please open pre-installed project An_Simulated.kinx2 from Alpha_Simulated directory in pre-installed samples (Kinetics Neo version 2.7.2 or later)

Select Analysis-Model-Free - Friedman – Master Plot:

For the simulated reaction A2 of nucleation type the known equation is f(alpha)=2*(1-alpha)*(-ln(1-alpha))^0.5, which describes the curve starting from zero with maximum before 0.5.

Here master plot is the non-symmetric function starting from zero with maximum 1.0 for alpha < 0.5, like expected from the theory.

It means that if for experimental data the master plot looks like the non-symmetric parabola starting from zero then the possible reaction type is the reaction with acceleration, possibly nucleation reaction of Avrami type .

#### 5. Reaction with Autocatalysis Cmn

Please open pre-installed project Cmn_Simulated.kinx2 from Alpha_Simulated directory in pre-installed samples (Kinetics Neo version 2.7.2 or later):

Select Analysis-Model-Free - Friedman – Master Plot:

For this simulated reaction Cmn with autocatalysis has equation  f(alpha)=(1-alpha)*(1+10*alpha), and we expect for it the non- symmetric parabola starting above zero with maximum before 0.5.

Here master plot is the non-symmetric parabola starting above zero with maximum 1.0 for alpha<0.5, like expected.

It means that if for experimental data the master plot looks like the non-symmetric parabola starting above zero then the possible reaction type is the reaction with acceleration, possibly autocatalytical  reaction Cmn.

#### 6. Phase-Boundary Reaction R3

Please open pre-installed project R3_Simulated.kinx2 from Alpha_Simulated directory in pre-installed samples (Kinetics Neo version 2.7.2 or later)

Select Analysis-Model-Free - Friedman – Master Plot:

For the phase-boundary reaction the reaction type is described by equation f(alpha)=3*(1-alpha)^(2/3) , which presents the concave decreasing curve with minimum at the end of reaction. Therefore, according to theory, we expect such a shape for the phase boundary reaction.

Here master plot is the concave decreasing curve with value 1 at alpha=0.5 and the minimum at the end of reaction, and this corresponds to the expected shape.

#### 7. Reaction with Diffusion D1

Please open pre-installed project D1_Simulated.kinx2 from Alpha_Simulated directory in pre-installed samples (Kinetics Neo version 2.7.2 or later)

Select Analysis-Model-Free - Friedman – Master Plot:

For the reaction with diffusion D1 the reaction type is described by equation f(alpha)=0.5/alpha, which presents the hyperbola. Therefore, according to theory, we expect such a shape for the diffusion reaction D1.

Here master plot is the hyperbola with value 1 at alpha=0.5, and this corresponds to the expected shape.

### References

[1] Sergey Vyazovkin et al. ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data, Thermochimica Acta 520 (2011) 1-19. https://doi.org/10.1016/j.tca.2020.178597