OpenSpecEngineering

Finds how a reaction’s rate constant changes with temperature. Give the pre-exponential factor and activation energy to get the rate constant at any temperature. Or give rate constants measured at two temperatures to get the activation energy and pre-exponential factor. Chemists and reaction engineers use it to scale lab kinetics to plant temperatures and to judge how sensitive a reactor is to a hot spot.

Equations

k = A·exp(−E_a/(R·T))

E_a = R·ln(k₂/k₁) / (1/T₁ − 1/T₂)

k is the rate constant, A the pre-exponential (frequency) factor, E_a the activation energy in kJ/mol, R the gas constant, 8.314 J/(mol·K), and T the absolute temperature in kelvin [1], [2]. The exponential, exp(−E_a/RT), is the fraction of collisions energetic enough to react; the page shows it and its exponent. In the second mode, k₁ and k₂ are rate constants measured at T₁ and T₂. The page fits E_a from the two points and then A = k₁·exp(E_a/(R·T₁)).

Inputs and limits

• k and A carry the same units, which depend on the reaction order (1/s for first order, L/(mol·s) for second). The page doesn’t convert them, so keep them consistent.
• Enter temperatures in any unit; the page works in kelvin. Never put degrees Celsius into exp(−E_a/RT) by hand.
• Two points give an exact fit but no check on scatter. With more data, fit ln k against 1/T instead.
• Extrapolating far beyond the measured range is risky. A change in mechanism, or diffusion control, bends the Arrhenius line.
• A rough rule says rates double for every 10 °C rise near room temperature. That holds only for E_a near 50 kJ/mol.

Worked example

A lab measures a first-order rate constant of 2.0 × 10⁻⁴ s⁻¹ at 40 °C and 1.5 × 10⁻³ s⁻¹ at 70 °C. The plant reactor will run at 85 °C. What rate constant should the design use?

1. Choose Solve For Ea and A (given k at two temperatures).
2. Enter Rate Constant k1 0.0002, Temperature T1 40 °C, Rate Constant k2 0.0015 and Temperature T2 70 °C.
3. Press Solve. The Solution shows: Activation energy Ea: 60.0072 kJ/mol
   Pre-exponential factor A: 2,043,020 (units of k)
   k ratio k2/k1: 7.5
4. Choose Solve For k (given A, Ea, T). Enter Pre-exponential Factor (A, same units as k) 2043020, Activation Energy (Ea, kJ/mol) 60.0072 and Temperature (T) 85 °C, and press Solve. The Solution shows: Exponent −Ea/RT: −20.1514
   Boltzmann factor exp(−Ea/RT): 1.77165e-9
   Rate constant k: 0.00361952 (units of A)

So k = 0.00362 s⁻¹ at 85 °C. Check by hand: ln 7.5 = 2.015, and 1/313.15 − 1/343.15 = 2.792 × 10⁻⁴ K⁻¹, so E_a = 8.314 × 2.015/2.792 × 10⁻⁴ = 60.0 kJ/mol. The 15 °C step from 70 to 85 °C multiplies k by 2.4.