This simulation shows why increasing the temperature of a free-radical polymerization increases the rate of polymerization, \(R_p\), but decreases the number-average molecular weight, \(M_n\). The rate constants for initiator decomposition (\(k_d\)), propagation (\(k_p\)), and termination (\(k_t\)) each follow the Arrhenius equation with their own activation energy. Because these rate constants combine differently in \(R_p\) and in \(M_n\), the overall activation energy is positive for \(R_p\) and negative for \(M_n\).

Use the sliders to change the temperature, the initiator concentration, and the activation energies \(E_d\), \(E_p\), and \(E_t\). Select whether chains terminate by combination or by disproportionation.

The plot on the left shows \(R_p\) (left axis) and \(M_n\) (right axis) versus temperature on logarithmic axes; the dashed vertical line marks the selected temperature. Click "Arrhenius plot" to see the natural log of each quantity divided by its value at 60 °C versus \(1000/T\); the slope of each line is \(-E/R\).

The bar chart on the right shows how the three activation energies add to give the overall activation energies for \(R_p\) and for \(X_n\). One-half of \(E_d\) is added for \(R_p\) but subtracted for \(X_n\). Decrease \(E_d\) toward zero (as for photoinitiation) to see when the molecular weight increases with temperature.

Each rate constant follows the Arrhenius equation, written relative to a reference temperature \(T_\mathrm{ref}\) = 60 °C:

\[ k_i = k_{i,\mathrm{ref}}\exp\!\left[-\frac{E_i}{R}\left(\frac{1}{T}-\frac{1}{T_\mathrm{ref}}\right)\right], \qquad i = d,\,p,\,t \]

Initiation, propagation, and termination steps:

\[ \mathrm{I}\xrightarrow{k_d}2\,\mathrm{R}\cdot \] \[ \mathrm{R}\cdot+\mathrm{M}\xrightarrow{k_p}\mathrm{R}\cdot \] \[ \mathrm{R}\cdot+\mathrm{R}\cdot\xrightarrow{k_t}\text{dead polymer} \]

The steady-state approximation sets the rate of radical initiation equal to the rate of termination:

\[ R_i = 2fk_d[\mathrm{I}] = 2k_t[\mathrm{R}\cdot]^2 \] \[ [\mathrm{R}\cdot] = \left(\frac{fk_d[\mathrm{I}]}{k_t}\right)^{1/2} \]

The rate of polymerization is:

\[ R_p = k_p[\mathrm{M}][\mathrm{R}\cdot] = k_p\left(\frac{fk_d}{k_t}\right)^{1/2}[\mathrm{M}][\mathrm{I}]^{1/2} \]

The kinetic chain length is the rate of propagation divided by the rate of initiation:

\[ \nu = \frac{R_p}{R_i} = \frac{k_p[\mathrm{M}]}{2\left(fk_dk_t[\mathrm{I}]\right)^{1/2}} \]

The number-average degree of polymerization and molecular weight are:

\[ X_n = 2\nu \quad \text{(combination)}, \qquad X_n = \nu \quad \text{(disproportionation)} \] \[ M_n = X_n M_0 \]

Collecting the rate constants shows how each activation energy enters:

\[ R_p \propto k_p\,k_d^{1/2}\,k_t^{-1/2} \] \[ E_{R_p} = E_p + \tfrac{1}{2}E_d - \tfrac{1}{2}E_t \] \[ X_n \propto k_p\,k_d^{-1/2}\,k_t^{-1/2} \] \[ E_{X_n} = E_p - \tfrac{1}{2}E_d - \tfrac{1}{2}E_t \]

For thermal initiators such as AIBN or benzoyl peroxide, \(E_d\) ≈ 120–150 kJ/mol, much larger than \(E_p\) ≈ 20–40 kJ/mol and \(E_t\) ≈ 5–20 kJ/mol. Thus \(E_{R_p}\) is large and positive (≈ 80–90 kJ/mol), so \(R_p\) increases rapidly with temperature, whereas \(E_{X_n}\) is negative, so \(X_n\) and \(M_n\) decrease. A higher temperature generates radicals faster, so \([\mathrm{R}\cdot]\) increases. More radicals consume monomer faster, but they also terminate each other more often, so each chain adds fewer monomer units before it terminates.

\(E_{X_n}\) is positive only if \(E_d < 2E_p - E_t\), as for photoinitiation or radiation initiation, where \(E_d\) ≈ 0.

The parameters at \(T_\mathrm{ref}\) = 60 °C are representative of bulk styrene with AIBN initiator:

\(k_{d,\mathrm{ref}}\)	9.5 × 10−6 s−1
\(k_{p,\mathrm{ref}}\)	341 L mol−1 s−1
\(k_{t,\mathrm{ref}}\)	6.0 × 107 L mol−1 s−1
\(f\)	0.6
\([\mathrm{M}]\)	8.35 mol/L
\(M_0\)	104.15 g/mol

The model assumes low conversion (constant \([\mathrm{M}]\) and \([\mathrm{I}]\)), a temperature-independent \(f\), no chain transfer, and no gel effect.

Symbols:

\(E_d\)	activation energy for initiator decomposition (kJ/mol)
\(E_p\)	activation energy for propagation (kJ/mol)
\(E_t\)	activation energy for termination (kJ/mol)
\(E_{R_p}\)	overall activation energy for the rate of polymerization (kJ/mol)
\(E_{X_n}\)	overall activation energy for the degree of polymerization (kJ/mol)
\(f\)	initiator efficiency
\([\mathrm{I}]\)	initiator concentration (mol/L)
\(k_d\)	initiator decomposition rate constant (s−1)
\(k_p\)	propagation rate constant (L mol−1 s−1)
\(k_t\)	termination rate constant (L mol−1 s−1)
\(k_{i,\mathrm{ref}}\)	rate constant \(i\) at \(T_\mathrm{ref}\)
\([\mathrm{M}]\)	monomer concentration (mol/L)
\(M_0\)	monomer molar mass (g/mol)
\(M_n\)	number-average molecular weight (g/mol)
\(R\)	ideal gas constant, 8.314 J mol−1 K−1
\([\mathrm{R}\cdot]\)	radical concentration (mol/L)
\(R_i\)	rate of initiation (mol L−1 s−1)
\(R_p\)	rate of polymerization (mol L−1 s−1)
\(T\)	temperature (K)
\(T_\mathrm{ref}\)	reference temperature, 333.15 K (60 °C)
\(X_n\)	number-average degree of polymerization
\(\nu\)	kinetic chain length

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.