Reference ID: MET-72BB | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Arrhenius equation quantifies how the rate constant k of an elementary reaction varies with absolute temperature T. In process engineering it is indispensable for:
Scaling-up kinetic data from laboratory reference conditions to plant operating temperatures.
Estimating thermal acceleration of degradation, spoilage or sterilisation reactions in food, pharma and petrochemical units.
Performing quick order-of-magnitude checks before running rigorous reactor simulations.
Evaluate the pre-exponential factor A from a known reference state (kref, Tref):
\[ A = k_{\text{ref}} \exp\left(\frac{E_{\text{a}}}{R\,T_{\text{ref}}}\right) \]
Compute the rate constant at any new temperature T:
\[ k(T) = A \exp\left(-\frac{E_{\text{a}}}{R\,T}\right) \]
where R is the universal gas constant (supplied in kJ kmol-1 K-1).
Parameter
Empirical Range
Remark
Operating temperature
90 °C – 150 °C
Outside this window, additional transport or phase effects may invalidate the simple Arrhenius form.
Activation energy Ea
200 000 – 400 000 kJ kmol-1
Typical for thermal sterilisation and many first-order decomposition reactions.
Extrapolation distance
ΔT ≤ 30 K
Reduces error when kinetic parameters are fitted only at Tref.
Start with the ratio form of the Arrhenius equation: k2/k1 = exp[(Ea/R)(1/T1 – 1/T2)].
Convert temperatures to kelvin: T1 = 353.15 K, T2 = 383.15 K.
Insert your activation energy Ea (in J mol-1) and R = 8.314 J mol-1 K-1.
Calculate the exponent term; for a typical Ea = 50 kJ mol-1 the ratio k2/k1 ≈ 2.9, so the rate almost triples.
Use the new rate constant in your rate law to update conversion or residence-time targets.
Run a short pilot trial or use plant data at two controlled temperatures; rearrange the Arrhenius ratio to solve for Ea. If data are scarce, choose the highest reported Ea for conservative scale-up; this gives the largest rate increase with temperature and prevents under-designing heat-removal systems.
The equation describes the intrinsic rate constant for a single reaction. Once competing or degradation pathways activate, each reaction has its own Ea. You must fit separate Arrhenius expressions for the main and side reactions and then solve the coupled kinetic model; otherwise the simple single-rate extrapolation will over-predict yield.
Accuracy drops quickly; a 10 °C extrapolation can add ~5% error, while a 50 °C extrapolation may exceed 30% if the true Ea varies with temperature (e.g., diffusion limitations or heat-capacity changes). Re-measure k or verify with a high-temperature lab reactor before full-scale implementation.
Worked Example – Estimating Sterilisation Rate Increase in a UHT Pre-Heat Exchanger
A dairy plant is upgrading its ultra-high-temperature (UHT) line from 121 °C to 130 °C to improve sterilisation throughput. The quality manager needs to know how much faster the microbial destruction rate constant will become at the new temperature. The Arrhenius model is used with the literature activation energy for Bacillus stearothermophilus spores.
Compute the pre-exponential factor A from the reference condition:
\[
A = k_{ref}\ \text{exp}\!\left(\frac{E_a}{R\ T_{ref,K}}\right)
= 1.0\ \text{exp}\!\left(\frac{300\,000}{8.314 \times 394.15}\right)
= 5.739 \times 10^{39}\ \text{s}^{-1}
\]
Determine the rate constant at the new temperature:
\[
k_{new} = A\ \text{exp}\!\left(\frac{-E_a}{R\ T_{new,K}}\right)
= 5.739 \times 10^{39}\ \text{exp}\!\left(\frac{-300\,000}{8.314 \times 403.15}\right)
= 7.719\ \text{s}^{-1}
\]
Final Answer
The microbial destruction rate constant increases from 1.000 s-1 to 7.719 s-1 when the sterilisation temperature is raised from 121 °C to 130 °C, giving a 7.7-fold acceleration in kill rate.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
