Journal of Colloid and Interface Science 313 (2007) 423–427 www.elsevier.com/locate/jcis Modeling the sorption kinetics of divalent metal ions onto mineral adsorbent using integral method Nidhi Singhal a , Murari Prasad b,∗ , Neelmani Gupta a , Vineet Kumar a a Department of Chemical Engineering, Thapar Institute of Engineering and Technology, Patiala, India b Environmental Chemistry Division, Advanced Materials and Processes Research Institute (CSIR), Hoshangabad Road, Bhopal, India Received 21 February 2007; accepted 29 April 2007 Available online 3 May 2007 Abstract A mathematical model has been developed that could predict kinetic parameters for the adsorption of divalent cations (lead, copper and zinc) onto low-grade rock phosphate using experimental data. The experiments were conducted with the initial concentrations of metal ions ranging from 10 to 100 mg/L. The mathematical model is based on application of Freundlich isotherm to mass transfer across the film surrounding the adsorbent. A code in C programming is used to numerically integrate the model equation, and to obtain the best simulated values of Freundlich constants K, N , order of reaction n, and film transfer coefficient, α. It is observed that the adsorption of metal ions on rock phosphate is more sensitive to N,n, and α in comparison to K, and lead is adsorbed more favorably than copper and zinc.  2007 Elsevier Inc. All rights reserved. Keywords: Adsorption; Divalent metal ions; Low-grade rock phosphate; Mathematical model; Trapezoidal method; Gradient descent; Least square analysis; Freundlich constant; Mass transfer; Film transfer constant 1. Introduction Since the renaissance of the industrial revolution, industries have remained the chief source of pollution of all types. The discharge water from various industries contains many toxic el- ements including cations of heavy elements. Several techniques are being used for the treatment of wastewater containing metal ions. Sorption is considered a better technique for the removal of metal ions from high volumes and/or high concentration ef- fluents. The most commonly used adsorbent is activated carbon. However, activated carbon is not cost effective for the removal of metal ions such as lead, copper, zinc, etc., due to its high cost and difficulty in regeneration. With this in view, a number of other sorbents such as montmorillonite [1], slag [2], humic acid [3], glauconite [4], kaoline [5], peat [6,7], fly ash [8] and other carbonaceous substrates [9] have also been studied. In the present study, rock phosphate is used as an adsorbent. This is a low-grade (<12% P 2 O 5 ) phosphatic rock found in * Corresponding author. Fax: +91 0755 2587042/2488985. E-mail address: prasadmurari56@yahoo.co.in (M. Prasad). Jhabua, Madhya Pradesh, India. The reserves of this mineral are not considered for the manufacture of phosphatic fertilizer. However, our earlier studies [10–15] have highlighted the util- ity of this very low cost and environmental friendly mineral’s use as an adsorbent for divalent cations from aqueous solu- tions. Large number of experimental and theoretical studies have been reported on the adsorption of cations, particularly heavy metal ions on various adsorbents [16–20]. Mathematical mod- els for sorption of heavy metal ions have been reported by dif- ferent investigators using surface complexation [17,20], cation exchange [18], triple layer model [19] and finite difference and Simpson’s 1/3rd rule [21]. Due to complex nature of the adsorption process, several simplifying assumptions have been made by various authors to solve the problem. Lee et al. [2] suggested a SPNSM (sta- tic, physical, non-equilibrium sorption model) for phosphorous sorption onto slag. This model assumes first order rate of re- action (n = 1) and used the Freundlich isotherm to develop the mathematical expression. It also assumed the Freundlich isotherm constant (N ) as unity, which reduced the isotherm to S = KC . The mathematical expression obtained from SPNSM 0021-9797/$ – see front matter  2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2007.04.068