In this paper, we investigate the averaging principle for stochastic delay differential equations (SDDEs) and SDDEs with pure jumps. By the Itô formula, the Taylor formula, and the Burkholder-Davis-Gundy inequality, we show that the solution of the averaged SDDEs converges to that of the standard SDDEs in the sense of pth moment and also in probability. Finally, two examples are provided to illustrate the theory.

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Keratin biosorbents were prepared by reprocessing the deposits produced from chicken feathers after soluble keratin extraction. Biosorption of Cr(VI) ions onto the obtained biosorbents from aqueous solution was studied. Experimental factors affecting biosorption process such as pH and initial concentration of Cr(VI) ions were investigated. Langmuir, Freundlich and Dubinin–Radushkevich (D–R) isotherm models were used to describe the biosorption process for Cr(VI) ions. The results showed that Langmuir model better fit the experimental data than the Freundlich and D–R models, and the maximum monolayer biosorption capacity for Cr(VI) ions was found to be 21.35 mg/g at pH 6, 200 mg/L initial concentration of Cr(VI) ions and 30°C. The free energy was calculated as 11.35 kJ/mol from the D–R model, indicating that the biosorption of Cr(VI) occurred chemically. This result was also confirmed by scanning electron microscopy and Fourier transform infrared regarding the changes in morphology and surface structure of the biosorbents before and after Cr(VI) biosorption. Overall, the deposits generated from chicken feathers after soluble keratin extraction, which currently treated as wastes, should be regarded as a favorable alternative for the removal of Cr(VI) ions from aqueous solution.

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