Temkin & Dubinin-Radushkevich (D-R) Adsorption Isotherm Energy Formula

Diagnose physisorption vs. chemisorption in activated carbon & zeolite water treatment!

Langmuir assumes every surface site is energetically identical and neighbors never interact—an assumption that breaks down on real heterogeneous catalysts and microporous activated carbons! Mikhail Temkin (1940) modeled adsorbate-adsorbate repulsion where heat of adsorption decreases linearly with coverage, yielding the logarithmic Temkin isotherm: q_e = ((R * T) / b_T) * LN(A_T * C_e), where b_T (J/mol) is the Temkin heat constant and A_T (L/mg) is the equilibrium binding constant!

Meanwhile, Mikhail Dubinin and L. V. Radushkevich (1947) modeled Gaussian pore-potential filling via ln(q_e) = ln(q_m) - B_DR * epsilon^2, where the Dubinin-Radushkevich constant B_DR (in mol^2/kJ^2) directly unlocks the Mean Free Energy of Adsorption: E_DR(kJ/mol) = 1 / SQRT(2 * B_DR)! Environmental chemists use E_DR as a sharp mechanistic litmus test: E_DR < 8.0 kJ/mol indicates reversible van der Waals physisorption (e.g., B_DR = 0.012 mol^2/kJ^2 -> E_DR = 6.45 kJ/mol), whereas 8.0 to 16.0 kJ/mol diagnoses chemical ion exchange (e.g., B_DR = 0.0025 -> E_DR = 14.14 kJ/mol)!

Dubinin-Radushkevich Mean Free Adsorption Energy E_DR = 1 / SQRT(2 * B_DR) (kJ/mol)

Evaluates 1 / SQRT(2 * B_DR) for B_DR (cell B2) entered in mol^2/kJ^2.

=IF(B2 <= 0, "Invalid B_DR Constant", ROUND(1 / SQRT(2 * B2), 2))

Temkin Equilibrium Surface Uptake q_e = ((R * T) / b_T) * LN(A_T * C_e) (mg/g)

Evaluates ((8.31446 * T_K) / b_T_J_mol) * LN(A_T * C_e) when A_T * C_e > 1.

=IF(OR(B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0, (B5 * B6) <= 1), "Outside Temkin Domain", ROUND(((8.31446 * B3) / B4) * LN(B5 * B6), 2))

Further reading

• Microsoft Support — LN function