Arrhenius Equation

The Arrhenius equation shows how reaction rate depends on temperature. It relates the rate constant (k) to activation energy (Ea) and temperature (T), explaining why reactions speed up when heated.

The Arrhenius Equation

k = Ae-Ea/RT

Linearized form (more common for calculations):

ln k = ln A - Ea/RT

Variable Definitions

k = Rate Constant

Units: Vary by reaction order (e.g., s⁻¹, M⁻¹s⁻¹)

Reaction rate constant at temperature T

A = Frequency Factor (Pre-exponential)

Units: Same as k

Meaning: Collision frequency and orientation factor

💡 Represents maximum possible rate constant (if Ea = 0)

Ea = Activation Energy

Units: J/mol or kJ/mol

Minimum energy needed for reaction to occur

Higher Ea = slower reaction, more temperature-sensitive

R = Gas Constant

Value: 8.314 J/(mol·K)

T = Temperature

Units: Kelvin (K) — MUST use Kelvin!

Conversion: K = °C + 273.15

Two-Point Form (Most Practical)

When you have rate constants at two different temperatures, use this form to find Ea:

ln(k₂/k₁) = (Ea/R) × (1/T₁ - 1/T₂)

This eliminates the unknown A, making it easier to calculate Ea from experimental data.

Step-by-Step Example

Problem: A reaction has k = 0.025 s⁻¹ at 300 K and k = 0.100 s⁻¹ at 350 K. Calculate Ea.

Given:

  • k₁ = 0.025 s⁻¹ at T₁ = 300 K
  • k₂ = 0.100 s⁻¹ at T₂ = 350 K
  • R = 8.314 J/(mol·K)

Step 1: Use two-point form

ln(k₂/k₁) = (Ea/R) × (1/T₁ - 1/T₂)

Step 2: Calculate ln(k₂/k₁)

ln(0.100/0.025) = ln(4) = 1.386

Step 3: Calculate (1/T₁ - 1/T₂)

1/300 - 1/350 = 0.003333 - 0.002857 = 0.000476 K⁻¹

Step 4: Solve for Ea

1.386 = (Ea / 8.314) × 0.000476
Ea = 1.386 / 0.000476 × 8.314
Ea = 24,200 J/mol = 24.2 kJ/mol

Answer: Ea = 24.2 kJ/mol

k quadruples when T increases 50 K, indicating moderate activation energy.

How Temperature Affects Rate

🔥 Higher Temperature

  • • Larger k (faster reaction)
  • • More molecules exceed Ea
  • • e^(-Ea/RT) closer to 1
  • • Rule of thumb: k doubles every 10°C

❄️ Lower Temperature

  • • Smaller k (slower reaction)
  • • Fewer molecules exceed Ea
  • • e^(-Ea/RT) closer to 0
  • • Reactions "freeze" at low T

Common Mistakes

❌ Using Celsius instead of Kelvin

Temperature MUST be in Kelvin. Using °C gives completely wrong results. Always add 273.15.

❌ Unit mismatch for Ea and R

If Ea is in kJ/mol, convert to J/mol OR use R = 0.008314 kJ/(mol·K). Units must match!

❌ Wrong sign in two-point form

It's (1/T₁ - 1/T₂), NOT (1/T₂ - 1/T₁). Order matters! Choose T₁ as lower temperature.

❌ Using log instead of ln

Arrhenius equation uses natural log (ln), NOT log₁₀. Check your calculator mode!

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Frequently Asked Questions

What is the Arrhenius equation?

k = Ae^(-Ea/RT). It describes how reaction rate constant (k) depends on temperature (T) and activation energy (Ea).

What does activation energy (Ea) mean?

Ea is the minimum energy molecules need to react. Higher Ea means fewer molecules can react, making the reaction slower.

Why do reactions speed up with temperature?

Higher temperature gives more molecules enough energy to overcome Ea. The fraction e^(-Ea/RT) increases exponentially.

How to find Ea experimentally?

Measure k at multiple temperatures, plot ln(k) vs 1/T. The slope equals -Ea/R, giving Ea.

What is the frequency factor A?

A represents collision frequency and proper molecular orientation. It's the rate constant if Ea = 0 (no energy barrier).

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