Arrhenius Equation
Why Arrhenius matters
The Arrhenius equation quantifies how reaction rates increase with temperature. Even modest temperature shifts can change k by orders of magnitude.
\[k = A · exp\left(\frac{-Ea}{R·T}\right)\]
Higher activation energy Ea means stronger temperature sensitivity. This is essential for process design, kinetic modeling, and shelf-life estimation.
- Compare rates at room and elevated temperature
- Estimate degradation and stability behavior
- Optimize chemical process conditions
Formulas
\[k = A · exp\left(\frac{-Ea}{R·T}\right)\]
\[E_a = -R·T·ln\left(\frac{k}{A}\right)\]
\[ln\left(\frac{k₂}{k₁}\right) =\frac{-Ea}{R} · \left(\frac{1}{T₂} - \frac{1}{T₁}\right)\]
R = 8.314 J·mol⁻¹·K⁻¹; Ea in kJ/mol is internally converted to J/mol.
Where:
- k = reaction rate constant
- A = pre-exponential factor (Arrhenius factor)
- Eₐ = activation energy
- R = universal gas constant
- T = absolute temperature in Kelvin
- e = Euler's number (≈ 2.718)
Detailed examples
Example 1: k at 25°C
A = 1.0·10¹² s⁻¹, Ea = 75 kJ/mol, T = 298.15 K
Result: k ≈ 0.72 s⁻¹
A = 1.0·10¹² s⁻¹, Ea = 75 kJ/mol, T = 298.15 K
Result: k ≈ 0.72 s⁻¹
Example 2: Temperature increase
Same reaction at T = 318.15 K
k grows sharply because the exponential damping becomes weaker.
Same reaction at T = 318.15 K
k grows sharply because the exponential damping becomes weaker.
Example 3: Back-calculate Ea
With measured k and known A, Ea can be estimated directly.
This is widely used for kinetic parameter fitting.
With measured k and known A, Ea can be estimated directly.
This is widely used for kinetic parameter fitting.
Practical note
Always use Kelvin. Unit mismatch (°C vs K, or kJ vs J) is the most common source of major error.
Always use Kelvin. Unit mismatch (°C vs K, or kJ vs J) is the most common source of major error.
Deeper context
Why does k increase so quickly?
As temperature rises, a larger fraction of molecules can cross the activation barrier. That increases the number of effective collisions per unit time.
Model limitations
Arrhenius behavior is highly useful but can deviate for complex multi-step mechanisms or diffusion-controlled processes. It remains the practical standard in most engineering and lab workflows.
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