### Chemical reagents and adsorbent

The CBAC powder was purchased from Guoqing Water Purification Material Co. Ltd. in China and employed as a sorbent for the following lead adsorption experiments. According to the National Standard of China for Activated Nutshell Carbon. Testing, the CBAC pore structure and pore size distribution were determined by ASAP 2020 (Micromeritics). The t-plot and Barrett–Joyner–Halenda (BJH) methods were used to calculate the microporosity and the mesoporosity of CBAC, respectively. Boehm titration method was applied in the characterization of the surface functional groups of CBAC (Boehm 1966). FT-IR spectroscopy was used to detect vibration frequency change in the CBAC. The spectra were collected by a NICOLET iS10 (Thermo Scientific) within the range 500–4000 cm^{−1} using a KBr window. The structural morphology of CBAC surface was characterized via SEM (Model S-4800 Hitachi, Tokyo, Japan) observation.

The Pb(II) stock solution (1000 mg/L) was obtained by dissolving lead nitrate in distilled water. This stock solution was then diluted to those required concentrations and their pHs were adjusted to desired values with 0.1 or 1.0 mol/L of NaOH or HCl solution. All chemicals in this research were of analytical grade and were used as received without any further treatment.

### Batch adsorption procedure

Batch adsorption experiments were carried out in a series of 250 mL Erlenmeyer flask to explore the effects of the aforementioned process variables on Pb(II) removal. Preliminary experiments were also performed to make certain the minimum and maximum levels of each variable. In general, about 100 mL of Pb(II) solution was mixed with a known amount of CBAC powder. Thereafter, the flasks were agitated at 140 rpm on a thermo controlled rotary shaker. Finally, the equilibrated solutions were withdrawn and the adsorbent was separated from them via centrifugation. The residual Pb(II) concentration in the solution was quantified through a standard microtitration method proposed by Li et al. (2002). All the experiments were repeated twice or thrice to confirm the results and the average values are recorded.

The Pb(II) removal efficiency and adsorption capacity of CBAC powder were calculated by using the following equations:

$$Ad\% = \frac{{C_{0} - C_{t} }}{{C_{0} }} \times 100$$

(1)

$$Q_{t} = \frac{{\left( {C_{0} - C_{t} } \right) \times V}}{W}$$

(2)

$$Q_{e} = \frac{{\left( {C_{0} - C_{e} } \right) \times V}}{W}$$

(3)

where *Ad* % is the Pb(II) removal efficiency; *Q*
_{
e
} and *Q*
_{
t
} are the adsorption capacity (mg/g) at equilibrium and at time *t* (min), respectively; *C*
_{0}, *C*
_{
t
} and *C*
_{
e
} are the initial Pb(II) concentration, liquid-phase Pb(II) concentration at time *t*, and equilibrium Pb(II) concentration (mg/L), respectively; *V* is the volume of the aqueous solution (L); *W* is the mass of the adsorbent (g).

### Adsorption kinetics models

Pseudo-first-order and pseudo-second-order kinetics models are usually adopted in kinetics investigations. The pseudo-first-order equation is a simple kinetics model describing the kinetics process of liquid–solid phase adsorption which was put forward by Lagergren (1898). Its nonlinear formula is given as follows:

$$Q_{t} = Q_{e} (1 - e^{{ - k_{1} t}} )$$

(4)

where *k*
_{1} is the rate constant of the pseudo-first-order sorption (min^{−1}). Obviously, *Q*
_{
e
} and *k*
_{1} can be figured out by plotting *Q*
_{
t
} versus *t* and by further nonlinear regression analysis.

The pseudo-second-order model based on the adsorption equilibrium capacity may be expressed as the following linear form (Ho and McKay 1999):

$$\frac{t}{{Q_{t} }} = \frac{t}{{Q_{e} }} + \frac{1}{{k_{2} Q_{e}^{2} }}$$

(5)

where *k*
_{2} is the rate constant of pseudo-second-order adsorption [g/(mg·min)]. Obviously, *Q*
_{
e
} and *k*
_{2} can be determined experimentally by plotting t/*Q*
_{t} versus t.

### Adsorption isotherm models

Langmuir and Freundlich equations are commonly adopted to describe the adsorption isotherms. Langmuir model assumes adsorption homogeneity, such as uniformly energetic adsorption sites, monolayer surface coverage, and no interactions between adsorbate molecules on adjacent sites (Langmuir 1918). Freundlich isotherm is applicable to nonideal sorption onto heterogeneous surfaces involving multilayer adsorption (Li et al. 2012). In this study, the Langmuir and Freundlich adsorption equations were both used to correlate the obtained isotherm data.

The linearized Langmuir equation can be expressed as follows:

$$\frac{{C_{e} }}{{Q_{e} }} = \frac{1}{{Q_{\hbox{max} } }}C_{e} + \frac{1}{{bQ_{\hbox{max} } }}$$

(6)

where *Q*
_{max} represents the maximum monolayer adsorption capacity (mg/g), and *b* represents the Langmuir adsorption constant which is related to the adsorption bonding energy (L/mg).

Based on further analysis of the Langmuir equation, the Langmuir adsorption isotherm can be described using an equilibrium parameter (*R*
_{L}) calculated by the following equation (Szlachta and Wojtowicz 2013):

$$R_{L} = \frac{1}{{1 + b \times C_{0} }}$$

(7)

where *C*
_{
0
} is the initial Pb(II) concentration (mg/L), *b* is the Langmuir constant (L/mg) mentioned previously, and *R*
_{L} parameter is a useful indicator for estimating whether the adsorption is unfavorable (*R*
_{L} > 1), linear (*R*
_{L} = 1), favorable (0 < *R*
_{L} < 1), or irreversible (*R*
_{L} = 0).

The linearized Freundlich equation can be described as follows:

$${\text{In}}Q_{e} = \ln K_{F} + \frac{1}{n}\ln C_{e}$$

(8)

where *K*
_{
F
} is the Freundlich constant indicative of the adsorption capacity of the adsorbent (mg·(L/mg)^{1/n}), and *n* is the Freundlich exponent depicting adsorption intensity (dimensionless). In the Freundlich model, both monolayer and multiple-layer adsorptions are considered to take place during the adsorption process.

### Thermodynamics of adsorption

Thermodynamics parameters can be calculated out by using distribution coefficient, *K*
_{
d
}, which is dependent on temperature. The change in free energy (Δ*G*
^{0}), enthalpy (Δ*H*
^{0}) and entropy (Δ*S*
^{0}) related to the adsorption process can be worked out with the following three equations (Zhang et al. 2014):

$$\Delta G = - RT\ln K_{d}$$

(9)

$$K_{d} = \frac{{Q_{e} }}{{C_{e} }}$$

(10)

$$\ln K_{d} = \frac{{\Delta S^{0} }}{R} - \frac{{\Delta H^{0} }}{RT}$$

(11)

where *R* is the gas constant (8.314 J·mol/K), and *T* (K) is the absolute temperature. In terms of Eq. 11, Δ*H*
^{0} and Δ*S*
^{0} parameters can be deduced from the slope and intercept of the plot of ln*K*
_{
d
} against 1/*T*.