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The literalized form of pseudo-first-order rate equation by Lagergren and Svenska (Lagergren 1898) is given as:

where q_e and q_t are the amount of metal ion adsorbed at equilibrium and at contact time t (min) and K_ad (min⁻¹) is the rate constant. From the plot of log (q_e−q_t) versus time t, the rate constant k₁ and correlation co-efficient R² are calculated and listed in Table 1.

Table 1

Kinetic models and other statistical parameters for metal ions sorption onto TDPI polymer

Kinetic model	Parameters	Concentration of lead(II) solution = 100 (mg/L)	Concentration of copper(II) solution = 100 (mg/L)	Concentration of nickel(II) solution = 100 (mg/L)	Concentration of cadmium(II) solution = 100 (mg/L)
Pseudo-first-order equation	k_ad (min⁻¹)	0.0716	0.0776	0.0672	0.0621
	q_e, cal (mg/g)	131.21	168.96	113.92	110
	R²	0.9891	0.9838	0.9867	0.9880
Pseudo-second-order equation	k (g mg⁻¹ min⁻¹)	8.58 × 10⁻⁴	7.06 × 10⁻⁴	7.79 × 10⁻⁴	7.14 × 10⁻⁴
	q_e, cal (mg/g)	108.69	109.89	105.26	104.1
	h(mg.g⁻¹ min⁻¹)	10.14	8.53	8.64	7.74
	R²	0.9966	0.9946	0.9956	0.9957
Intraparticle diffusion	K_p (mg/g. min^1/2)	6.2933	6.8158	6.4047	6.5874
	C	41.101	34.649	35.05	31.4
	R²	0.8790	0.8793	0.879	0.8898

Kinetic model	Parameters	Concentration of lead(II) solution = 100 (mg/L)	Concentration of copper(II) solution = 100 (mg/L)	Concentration of nickel(II) solution = 100 (mg/L)	Concentration of cadmium(II) solution = 100 (mg/L)
Pseudo-first-order equation	k_ad (min⁻¹)	0.0716	0.0776	0.0672	0.0621
	q_e, cal (mg/g)	131.21	168.96	113.92	110
	R²	0.9891	0.9838	0.9867	0.9880
Pseudo-second-order equation	k (g mg⁻¹ min⁻¹)	8.58 × 10⁻⁴	7.06 × 10⁻⁴	7.79 × 10⁻⁴	7.14 × 10⁻⁴
	q_e, cal (mg/g)	108.69	109.89	105.26	104.1
	h(mg.g⁻¹ min⁻¹)	10.14	8.53	8.64	7.74
	R²	0.9966	0.9946	0.9956	0.9957
Intraparticle diffusion	K_p (mg/g. min^1/2)	6.2933	6.8158	6.4047	6.5874
	C	41.101	34.649	35.05	31.4
	R²	0.8790	0.8793	0.879	0.8898

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