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Elovich (Roginsky-Zeldovich) Chemisorption Kinetics Formula: Uptake q(t) & Rate Decay
Model activated chemisorption on energetically heterogeneous catalysts, biochars, and soils using the integrated Elovich kinetic equation q(t) = (1 / beta) * LN(1 + alpha * beta * t) and instantaneous rate dq/dt = alpha * EXP(-beta * q).
Why chemisorption slows exponentially with coverage: activation energy rises linearly as low-energy sites fill first!
Formulated by S. Z. Roginsky & Ya. B. Zeldovich (1934) for carbon monoxide chemisorption on manganese dioxide and popularized by S. Yu. Elovich, the Elovich kinetic law describes adsorption where activation energy E_a increases linearly with surface coverage q—so instantaneous uptake rate decays exponentially via dq/dt = alpha * EXP(-beta * q), where alpha (mg/(g*min)) is the initial sorption rate as q -> 0 and beta (g/mg) is the desorption/activation-barrier constant!
Integrating from q(0) = 0 yields the exact closed-form Elovich equation (Chien & Clayton, 1980 — NEVER drop the "1 +" inside LN at short contact times!): q(t) = (1 / beta) * LN(1 + alpha * beta * t), with instantaneous rate dq/dt = alpha / (1 + alpha * beta * t)! At alpha = 18.0 mg/(g*min), beta = 0.12 g/mg, and t = 30 min, q(30) = (1 / 0.12) * LN(1 + 18 * 0.12 * 30) = 34.8885 mg/g while dq/dt slows 65.8-fold to 0.2736 mg/(g*min)!
| Contact Time t (alpha = 18 mg/(g*min), beta = 0.12 g/mg) | Exact Elovich Uptake q(t) = LN(1 + alpha*beta*t)/beta | Instantaneous Rate dq/dt (mg/(g*min)) |
|---|---|---|
| t = 0 min (Fresh Clean Surface) | 0.0000 mg/g | 18.0000 (Maximum Initial Rate alpha) |
| t = 5 min | 20.5816 mg/g | 1.5254 |
| t = 30 min | 34.8885 mg/g | 0.2736 |
Exact Integrated Elovich Chemisorption Uptake q(t) = (1 / beta) * LN(1 + alpha * beta * t)
Computes cumulative sorbed loading q(t) (mg/g) from initial rate alpha (B2), surface constant beta (B3), and contact time t (B4, min) without the high-time Chien-Clayton truncation error.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid Elovich Kinetic Parameters", ROUND((1 / B3) * LN(1 + B2 * B3 * B4), 4))Instantaneous Sorption Rate dq/dt = alpha / (1 + alpha * beta * t)
Evaluates the instantaneous chemisorption rate at time t.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0), "Invalid Elovich Kinetic Parameters", ROUND(B2 / (1 + B2 * B3 * B4), 4))Try the example
Use the original values or change them to test your understanding.