A novel linear mass–uptake model for Ni(II) adsorption on green-synthesized Prunus spinosa silver nanocalcite composite

Determining the optimal adsorbent dosage remains a critical yet unresolved challenge in adsorption science, as no established mathematical framework directly links adsorbent mass to equilibrium uptake capacity. This study proposes a linear mass–uptake model for the first time, expressed as m/qₑ = (1/qₘ)m + Kₒ/qₘ. The qₘ (mg g⁻ 1 ) represents the maximum adsorption capacity, and Kₒ (g) denotes the semi-saturation constant—the adsorbent mass required to achieve qₘ/2. A green-synthesized Prunus spinosa silver nanocalcite composite (PSSNCC) was prepared by combining biogenic silver nanoparticles (AgNPs) with nanocalcite and used as the model adsorbent for Ni(II) removal from aqueous solution. The composite was characterized by field emission scanning electron microscopy (FE-SEM), transmission electron microscopy (TEM), X-ray diffraction (XRD), energy dispersive X-ray spectroscopy (EDX), Fourier transform infrared spectroscopy (FT-IR), thermogravimetric analysis (TGA), and ultraviolet–visible (UV-Vis) spectroscopy. Application of the linear model yielded qₘ = 54.95 ± 0.42 mg g⁻ 1 and Kₒ = 3.423 ± 0.05 g with an exceptionally high coefficient of determination (R 2 = 0.9998). Equilibrium isotherm experiments conducted at 298, 308, and 318 K demonstrated that Ni(II) adsorption on PSSNCC follows the Langmuir model (R 2 = 0.986 at 298 K), yielding a maximum monolayer capacity of 70.42 mg g⁻ 1 . The proposed model eliminates trial-and-error adsorbent dosing, enabling rational prediction of optimal adsorbent quantities for any target uptake, and represents a significant advancement in adsorption process design.