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calculated from LBG control flask and C SL is LBG concentration in supernatant (mg L - 1 ) after adsorption. The absolute amount of LBG, m L , adsorbed to the pulp (mg g - 1 o.d. fibre) was calculated as: m L ¼ C 0F þ C 0L C SL C 0P ð 2 Þ where C 0P is the initial o.d. pulp fibre concentration (g L - 1 ).
K e ½ C e ¼ 1 þ K e ½ C e ð Þ h
ð 9 Þ
K e ½ C e 1 þ K e C e ½ ¼
q e Q max
h ¼
ð
10 Þ
- 1 o.d. fibre) is equilibrium LBG
where q e (mg g
adsorption capacity and Q max (mg g - 1 o.d. fibre) is maximum adsorption capacity. From Eq. 10, when C e is large, h is approximately equal to 1, representing full coverage of substrate. In contrast when C e is small, h approaches zero, suggesting limited adsorption and surface coverage. Nonlinear regression of Eq. 10 was conducted by OriginLab 2016 to determine the Q max and K e (Tran et al. 2017). The rate equation for the Freundlich isotherm (Freundlich 1906; Foo and Hameed 2010) is: q e ¼ K f C 1 n e ð 11 Þ where K f is the Freundlich equilibrium constant (mg g - 1 (mg L - 1 ) - 1/n ), q e is the concentration of LBG(mg g - 1 o.d. fibre) adsorbed at equilibrium state, n is the Freundlich constant (dimensionless) related to adsorption intensity and C e is concentration of LBG (mg L - 1 ) in the aqueous phase (Foo and Hameed 2010; Bergmann and Machado 2015). Nonlinear regression of Eq. 11 was conducted by OriginLab 2016 to determine the K f and n (Tran et al. 2017). The isotherm fitting was assessed by the reduced chi- squared ( v 2 ) and R 2 adj as described in Eq. 12 and 13 (Bergmann and Machado 2015). v 2 ¼ X N i q i ; exp q i ; model 2 N p ð 12 Þ 13 Þ where q i,model is model fitted value of q , q i,exp is experimental value of q , N is the total number of experiments, and p is the number of parameters in the model. R 2 adj ¼ 1 1 R 2 N 1 N p 1 ð Pseudo-first-order and pseudo-second-order adsorp- tion kinetics (Lagergren 1898; Ho 1995, 2006; Ho et al. 1996; Blanchard et al. 1984; Ho and McKay 2000) are described by Eq. 14 and Eq. 15, respectively: dq t dt ¼ k q e q t ð Þ ð 14 Þ
Adsorption kinetics and isotherms
The Langmuir isotherm model assumes ideal mono- layer chemisorption on a smooth surface with a finite number of sites (Langmuir 1916; Laidler 1987; Foo and Hameed 2010). The Langmuir isotherm is derived from the equilibrium adsorption reaction of LBG to substrate NBSK pulp fibre: C e þ S e $ C e S e ð 3 Þ where C e is the equilibrium concentration of LBG (mg L - 1 ) in the aqueous phase, S e is the concentration of empty sites at equilibrium (mg g - 1 o.d. fibre) on the surface of pulp fibres, and C e S e is the equilibrium concentration of adsorbed LBG (mg g - 1 o.d. fibre) on fibre surface. When at equilibrium, k 1 is the adsorption rate constant (L g mg - 2 ) and k -1 is the desorption rate constant (g mg - 1 ). k 1 ½ C e S e ½ ¼ k 1 ½ C e S e ð 4 Þ Defining substrate surface coverage as h , then S e can be expressed as Eq. 6:
C e S e ½ S e ½ þ C e S e ½
h ¼
ð
5 Þ
S e ½ ¼ 1 h ð Þ S e ½ þ C e S e ½ ð Þ ð 6 Þ Equations 4, 5 and 6 can be combined to yield Eq. 7: k 1 ½ C e 1 h ð Þ¼ k 1 h ð 7 Þ The equilibrium constant K e (L mg - 1 ) is defined as: K e ¼ k 1 k 1 ¼ C e S e ½ C e ½ S e ½ ð 8 Þ Equation 7 and 8 can be combined and further rearranged as Eq. 9. Thus, h could be solved as Eq. 10:
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