Table 3 Mathematical equations.

Equation (1):

Particle size calculation from XRD

D_XRD = 0.9λ/(βcos θ)

D_XRD is the average particle diameter, λ is the Cu k_α wavelength, β is the full-width at half-maximum (FWHM) of the diffraction peak and θ is the diffraction angle.

Equation (2):

QDs and NPs capacity

\(\:{Q}_{e}=\left({C}_{0}-{C}_{e}\right)\frac{V*{10}^{3}}{M}\:\:\)

C₀ and C_e (mol L^− 1) are the initial and equilibrium concentrations, respectively. V: the volume solution and M: the used mass of adsorbent. The concentrations of Cr(VI)/Cur were detected by spectrophotometry (Unico UV/Vis-7200) at λ_max = 480 nm.

Equation (3):

Pseudo-first order

\(\:{q}_{t}={q}_{e\:}(1-{e}^{-kt})\)

q_t (mg g^− 1), amount of adsorbate adsorbed at time t.

k₁ (sec ^− 1), pseudo-first order rate constant.

Equation (4):

Pseudo-second order

\(\:{q}_{t}=\frac{{\text{k}}_{2}\:{{q}_{e}}^{2}t}{{1+\text{K}}_{2}{{\text{q}}^{2}}_{\text{e}}t\:\:}\)

k₂ (g mg^− 1 sec^− 1), pseudo-second order rate constant

Equation (5):

Intraparticle diffusion

\(\:{\text{q}}_{\text{t}}={\text{K}}_{\text{i}\text{d}}{\text{t}}^{\frac{1}{2}}+\text{C}\)

K_id (mg g^− 1 sec^− 1/2), the intra-particle diffusion rate constant.

C (mg g^− 1), constant.

Equation (6):

Elvoich

\(\:{q}_{t}=\raisebox{1ex}{$1$}\!\left/\:\!\raisebox{-1ex}{$\beta\:$}\right.ln(\alpha\:\beta\:t+1)\)

α (mg g^− 1 sec^− 1), the initial adsorption rate.

Β, is related to the extent of surface coverage and the activation energy for chemisorption

Equation (7):

Langmuir

\(\:{q}_{e}=\frac{{q}_{m}{\text{k}}_{L}\:{C}_{e}}{{1+\text{K}}_{L}{C}_{e}\:\:}\:\:\:\:\:\)

q_e (mg g^− 1), equilibrium adsorption capacity.

q_max (mg g^− 1), maximum adsorption capacity.

b (L mg^− 1), Langmuir constant.

C_e (mg L^− 1), equilibrium adsorbate concentration in solution.

Equation (8):

Langmuir, separation factor

\(\:{\:\:\text{R}}_{\text{L}}=1/(1+\text{b}{C}_{o})\)

R_L, separation factor.

b (L mg^− 1), Langmuir constant.

Equation (9):

Freundlich

\(\:{\text{q}}_{e}={\text{K}}_{f}{{C}_{e}}^{\frac{1}{n}}\)

K_f (L mg^− 1), Freundlich constant.

n, heterogeneity factor.

Equation (10):

D-R

\(\:{q}_{e}={q}_{s\:}\left({e}^{-\beta\:{\epsilon\:}^{2}}\right)\)

K_ad (mol² J^− 2), Dubinin-Radushkevich constant.

ε, Polanyi potential.

Equation (11):

D-R, Polanyi potential

\(\:{\upepsilon\:}=\text{R}\text{T}\text{l}\text{n}\:(1+\frac{1}{{\text{C}}_{\text{e}}})\)

R, universal gas constant (8.314 J mol^− 1 K^− 1).

T (K), absolute temperature.

C_e (mg L^− 1), Cr(VI)/Cur equilibrium concentration.

Equation (12):

Temkin

\(\:{\text{q}}_{e}=\left(\frac{RT}{b}\right)\text{l}\text{n}{a}_{T}{C}_{e}\)

A_T (L mg^− 1), Temkin adsorption potential.

b_T (J mol^− 1), Temkin constant.

q_e (mg g^− 1), theoretical maximum capacity.

Equation (13):

Gibbs free energy

ΔG^o=-RT ln Kd

ΔG^o (kJ mol^− 1), Gibbs free energy.

K_d, equilibrium constant.

T (K), absolute temperature.

R, universal gas constant (8.314 J mol^− 1 K^− 1).

Equation (14):

Equilibrium constant

\(\:{\:\:\text{K}}_{\text{d}}=\frac{{\text{q}}_{\text{e}}}{{\text{c}}_{\text{e}}}\)

K_d, equilibrium constant.

q_e (mg g^− 1), equilibrium adsorption capacity.

C_e (mg L^− 1), Cr(VI)/Cur equilibrium concentration.

Equation (15):

Vant`hoff

\(\:{\text{ln}\text{K}}_{\text{d}}=\frac{{\Delta\:}{\text{S}}^{\text{O}}}{\text{R}}-\frac{{\Delta\:}{\text{H}}^{\text{O}}}{\text{R}\text{T}}\)

K_d, equilibrium constant.

ΔH^o (KJ mol^− 1), enthalpy change.

ΔS^o(J mol^− 1 K^− 1), entropy change.

R, universal gas constant (8.314 J mol^− 1 K^− 1).