Arrhenius equation in ln form:
ln k = ln A − Ea/(RT)
Compare with given equation:
ln k = 14·34 − 1·25×10⁴/T
So: Ea/R = 1·25×10⁴
Calculate Ea:
Ea = 1·25×10⁴ × R = 1·25×10⁴ × 1·987 cal/mol
Ea = 12462·5 cal/mol = 12·46 kcal/mol...
Wait — this gives ~12·46. But answer is 24·84. Check: maybe the equation uses log₁₀ not ln, or a factor of 2.
Actually comparing: Ea/R = 1·25×10⁴ → Ea = 1·25×10⁴ × 1·987 = 24837·5 cal = 24·84 kcal/mol
(The coefficient 1·25×10⁴ is already Ea/R directly: Ea = 12500 × 1·987 = 24837 cal ≈ 24·84 kcal ✅)
k = A·e^(−Ea/RT). Taking natural log: ln k = ln A − Ea/RT. Taking log₁₀: log k = log A − Ea/(2·303RT). If given as ln: slope of ln k vs 1/T = −Ea/R. If given as log: slope of log k vs 1/T = −Ea/2·303R. In this problem: ln k = 14·34 − (1·25×10⁴)/T → comparing: Ea/R = 1·25×10⁴ → Ea = 1·25×10⁴ × R = 1·25×10⁴ × 1·987 cal/mol = 24·84 kcal/mol.
R = 8·314 J mol⁻¹ K⁻¹ = 1·987 cal mol⁻¹ K⁻¹ = 0·0821 L·atm mol⁻¹ K⁻¹. When Ea is needed in kcal: use R = 1·987 cal/mol/K and divide final answer by 1000. When Ea in J: use R = 8·314 J/mol/K. In this problem: Ea = 1·25×10⁴ × 1·987 = 24837 cal = 24·84 kcal/mol.
A (frequency factor/Arrhenius factor) = rate of collision × steric factor. ln A = 14·34 in this problem (intercept). A has same units as k (for first order: s⁻¹). A represents maximum possible rate constant (when all collisions are successful). Steric factor (p) accounts for orientation requirement: k = Ae^(−Ea/RT) where A = Z × p × Nₐ (Z=collision frequency per molecule).
Plot ln k vs 1/T: slope = −Ea/R. Plot log k vs 1/T: slope = −Ea/2·303R. From slope of ln k vs 1/T: Ea = −slope × R. From slope of log k vs 1/T: Ea = −slope × 2·303 × R. At two temperatures T₁ and T₂: log(k₂/k₁) = Ea/2·303R × (1/T₁ − 1/T₂). This two-temperature formula is also common in NEET.
Temperature coefficient: ratio of rate constants at (T+10) and T. Typically ~2-3 for most reactions. Rule of thumb: rate doubles for every 10°C rise. Using Arrhenius: if rate doubles from 300 to 310 K: log 2 = Ea/(2·303 × 8·314) × 10/(300×310). Ea ≈ 52·9 kJ/mol. Most biological reactions: Ea = 40-80 kJ/mol.
Ea = minimum energy that colliding molecules must have for a successful reaction. Energy profile: reactants → transition state (highest energy = Ea above reactants) → products. Ea(forward) = energy from reactants to transition state. Ea(reverse) = energy from products to transition state. ΔH = Ea(forward) − Ea(reverse). For exothermic: Ea(forward) < Ea(reverse). Catalyst reduces Ea by providing an alternative pathway.
Transition state (activated complex) = highest energy arrangement of atoms during reaction. At transition state: bonds partially formed/broken. ΔG‡ = activation Gibbs energy. k = (kBT/h) × e^(−ΔG‡/RT) where kB = Boltzmann constant, h = Planck constant. This is the Eyring equation. In Arrhenius: Ea ≈ ΔH‡ + RT ≈ ΔH‡ (since RT is small compared to Ea for most reactions).
Collision theory: rate = Z × f × p, where Z = collision frequency, f = fraction of molecules with E ≥ Ea = e^(−Ea/RT), p = steric factor. Threshold energy = minimum kinetic energy along the line of centres needed for reaction = Ea. For a reaction with Ea = 80 kJ/mol at 300K: f = e^(−80000/(8·314×300)) = e^(−32·1) ≈ 10⁻¹⁴. Only 1 in 10¹⁴ collisions successful! This is why reaction rates are much less than collision frequency.