INTRODUCTION
Water pollution by heavy metal contamination is a serious threat to mankind and aquatic life. The present study is aimed to focus the contamination by chromium. Tanning and leather industries, electroplating industries, catalyst and pigments manufacturing industries, fungicides, ceramics, crafts, glass, photography and corrosion control application are main sources for Chromium^{13}. These industrial effluents can contain Cr (VI) from 10100 mg L^{–}^{1} which is higher than the standard limit of 0.1 mg L^{–}^{1} in industrial waste water^{46}. Chromium is a transition metal. It transpires in nine different forms of its oxidation states. The range is found to be Cr (II) up to Cr (+VI). The Cr (III) and Cr (VI) oxidation states are the most commonly observed in chromium compounds. In aqueous systems contamination with chromium is mainly found as Cr (III) and Cr (VI). The remaining forms are rare states^{1}. Cr (III) is considered as a bioelement since it plays an important role in the metabolic activity of plants and animals at its low concentrations^{79}. But Cr (VI) is highly toxic^{1013}.
According to report submitted by IARC^{1416}, Cr (VI) is classified in Group 1 (carcinogenic to humans) and chromium (III) is classified in Group 3 (not carcinogenic to humans). The maximum permissible limit of Cr (VI) for discharge to land surface water is 0.1 mg L^{–}^{1} and in potable water 0.05 mg L^{–}^{1} ^{1720}. Hence utmost attention is emerged in the removal of chromium (VI) from wastewater before discharge into the aquatic system. Ion exchange, membrane separation, dialysis and electro dialysis reduction followed by chemical precipitation, electro coagulation and adsorption/filtration are the conventional treatment technologies to remove Cr (VI) from water and wastewater^{2124}. Among the various conventional methods, adsorption is documented as the most efficient, promising and widely used fundamental technique. It is simple and cost effective for recovering and eliminating heavy metal ions from dilute solutions^{2528}. Understanding the basic mechanism of adsorption is also simple. It is a surface phenomenon in which one or more components present in a multicomponent fluid (gas or liquid) mixture is attracted to the surface of solid adsorbents via physical or chemical bonds^{2932}. The presence of metal ions in aqueous media and subsequent development of pollution measures have resulted in the success of adsorption process. Studies with various bio sorbents are abundant in literature which includes seaweeds, moulds, yeast, bacteria, crab shells and agricultural products so on^{3335}.
Marine macroalgae are harvested or cultivated in many parts of the world and are therefore readily available in large quantities for the development of highly effective bio sorbent materials^{13}. The microalgae which exist in the freshwater environment and in the oceans are important in global ecology, extremely efficient and taxonomically diverse^{3639}.
The investigation of adsorption capacities of various adsorbents for the removal chromium has been made conventionally with study of two parameter models. Extensive investigation have been made by various researchers in evaluating the characteristics of specific adsorbentadsorbate systems and evaluating the degree of applicability of two parameter models for the systems concerned specifically. The mechanism of chromium adsorption from relevant sources with three, four and five parameter models in literature is very limited^{40,41}. Hence it is essential to formulate the adsorption phenomena with welldesigned models describing adsorption isotherms.
Description with high parameter models may lead to provide very clear and apt information about the adsorption process under equilibrium condition. Then the applicability of one and two parameter isotherm models have very limited^{4144}. The high parameter model would overcome the limitations of the simple models with twoparameter^{4447}. The objectives of the present research were to study the single metal removal efficiency and biosorption capacity of BGMA for the removal of Cr (VI) metal ions from the synthetically prepared stock solution. More emphasis was given to analyze the isotherm datatwo, three, four and five parameter models.
MATERIALS AND METHODS
Blue Green Marine Algae (BGMA) used in this investigation to recover the hexavalent chromium metal ion from its respective synthetic solution under equilibrium condition. Preparation of adsorbent, synthetic stock solution and determination optimum experimental parameters viz., pH, temperature and adsorbent dosage under batch condition were provided here along with the effect initial metal ion concentration of equilibrium metal uptake. Analyses of effluent concentration to calculate the equilibrium metal uptake experimentally and theoretically with isotherm models were discussed.
AdsorbentBGMA: BGA used in this investigation for the study of biosorption of Cr (VI) ions from its synthetic stock solution. The algae were collected from ponds in and around Chidambaram town and Veeranam Lake. The algae was washed with distilled water and dried at room temperature (above 30°C) then it was powdered with a uniform size of 150200 microns. The dried and powered algae was immobilized on silica gel using standard methods for continuous adsorption^{48} which involves wetting a mixture of silica gel and biomass with ultra pure water followed by drying at 150°C for 20 min.
AdsorbateCr (VI): Analytical grade salt Potassium Dichromate K_{2}Cr_{2}O_{7} is used to prepare the synthetic hexavalent chromium Cr (VI) solution. The stock solution of Cr (VI) was prepared by dissolving a pre calculated quantity of respective salt in double distilled water. This stock solutions are further diluted to obtain desired concentration in the range of 25200 ppm. The pH of each solution is adjusted 26. The pH adjustments were done by using 0.1 N nitric acid (HNO_{3}) and 0.1 N sodium hydroxide (NaOH) solutions.
Analysis: The batch bio sorption study was carried out in a 500 mL conical flask. The initial metal ion concentrations are varied to 25, 50, 100, 125, 150, 175, 200 and 250 ppm. The pH adjustments are done by using 0.1 N nitric acid (HNO_{3}) and 0.1 N sodium hydroxide (NaOH) solutions. The biomass to be added to the conical flask is varied from 0.5, 1.0, 1.5, 2.0 and 2.5 g. All the experiments are performed at the room temperature. The flasks are kept in a rotary shaker at rpm of 120 for 24 h. This is more than sufficient for adsorption equilibrium. The samples are taken regularly with predetermined time intervals. The percentage removal and specific uptake of metal is then calculated.
The bio sorption of Cr (VI) metal ions from its respective synthetic solutions by the BGMA is performed under shaking conditions in rotary shaker (REMI12, India). The experiments are performed under batch operation mode. To each 400 mL of metal solution a desired quantity of the blue green microalgae is added in 500 mL conical flasks. The mixture is agitated in the rotary shaker at the room temperature and at 120 rpm for predetermined time intervals.
The initial and final concentrations of metal solutions are predicted by double beam Atomic Adsorption Spectrophotometer (AAS SL176Elico Limited India). The percentage removal of metal ions is calculated from the initial concentration (C_{i}) and the analyzed final concentration (C_{eq}) of the metal ion solution according to the following Eq. 1:
The equilibrium metal uptake is calculated from the initial concentration (C_{i}) and the analyzed final concentration (C_{eq}) of the metal ion solution according to the following Eq. 2:
where, V is the volume of liquid sample in liter and M is the weight of bio sorbent. The same procedure is repeated for another initial metal ion concentration of 50, 75, 100, 125, 150, 175, 200 and 250 ppm of all the metals. The experimental variables namely pH and biomass loading are optimized and at these optimized conditions the effect of Initial metal ion concentration on percentage removal and specific uptake of metal ions are studied.
EQUILIBRIUM ADSORPTION ISOTHERMS
Two parameter models (Langmuir, Freundlich, DubininRadushkevich, Temkin, Hillde Boer, FowlerGuggenheim, FloryHuggins, Halsey, HarkinJura, Jovanovic and Elovich, Kiselev), Three parameter models (Hill, RedlichPeterson, Sips, LangmuirFreundlich, FritzSchlunderIII, RadkePrausnitsI, RadkePrausnitsII, RadkePrausnitsIII, Toth, Khan, KobleCorrigan, Jossens, JovanovicFreundlich, BrouersSotolongo, ViethSladek, Unilan, Holl–Krich and Langmuir–Jovanovic), Four parameter models (FritzSchlunderIV, Baudu Webervan Vliet and MarczewskiJaroniec) and Five parameter model (FritzSchlunder5) are used to analyze the experimental equilibrium adsorption data.
Cftool and goodnessoffit statistics: Applicability of these models to fit the experimental data in predicting the mechanism of adsorption is accomplished using cftool kit available in MATLAB R2010a software. This toolkit aids in estimating the model parameters along with the nonlinear Regression coefficient (R^{2}), Sum of Squares due to Error (SSE) and Root Mean Squared Error (RMSE).
Isotherm modelstheoretical knowledge: The isotherms of adsorption indicate the distribution of molecules between the liquid and solid phase when the adsorption process reaches equilibrium. It is employed to establish the maximum capacity of adsorption of metals on adsorbents, expressed in terms of quantity of metal adsorbed per unit mass of adsorbent used (mg g^{–}^{1}). Insight knowledge on adsorption mechanism provides proper understanding and interpretation on the phenomena of adsorption of metal ions on the surface of adsorbents. This enables to improvise the adsorption pathways to facilitate effective design of adsorption^{49,50}.
Two parameter models: In this section, more attention is paid to accrue theoretical insight knowledge on models containing two parameters to explain the adsorption mechanism. Essential theoretical approach for a total of twelve models viz., Langmuir, Freundlich, DubininRadushkevich, Temkin, Hillde Boer, FowlerGuggenheim, FloryHuggins, Halsey, HarkinJura, Jovanovic, Elovich and Kiselev are provided.
Langmuir isotherm model: The Langmuir model assumes that the surface as homogeneous. This model clearly indicates that the adsorption sites have equal sorbate affinity. Also the adsorption at one site does not affect sorption at the adjacent site. The formation of monolayer coverage of the adsorbate at the outer surface of the adsorbent is well explained by this model. It accounts the surface coverage by balancing the rate of adsorption and rate of desorption relatively under equilibrium condition.
This model sates that the rate of adsorption is proportional to the fraction of the adsorbent surface that is open while desorption is proportional to the fraction of the adsorbent surface that is covered^{51,52}. The model is given in Eq. 3:
where, b_{L} is Langmuir constant related to adsorption capacity (mg g^{–}^{1}), which correlates the variation of the suitable area and porosity of the adsorbent. The features of the Langmuir isotherm can be explained by a dimension less constant called the Langmuir separation factor R_{L} which is calculated as Eq. 4:
where, R_{L} values indicate the adsorption to be unfavorable when R_{L}>1, linear when R_{L} = 1, favourable when 0<R_{L}<1 and irreversible when R_{L} = 0.
Freundlich isotherm model: The Freundlich adsorption isotherm model indicates the extent of heterogeneity of the adsorbent surface. The adsorptive sites are made up of small heterogeneous adsorption sites each of which is homogeneous^{28}. The model is given in Eq. 5:
where, a_{F} is Freundlich adsorption capacity (L mg^{–}^{1}) and n_{F} is adsorption intensity. Relative distribution of the energy and the heterogeneity of the adsorbate sites are well indicated by this model. The larger the value of the adsorption capacity a_{F}, the higher the adsorption capacity is. The magnitude of 1/n_{F} ranges between 0 and 1, is an indicative of favourable adsorption, becoming more heterogeneous as its value tends to zero^{53,54}.
Dubininradushkevich isotherm model: This empirical model assumes a multilayer character involving Van Der Waal’s forces applicable for physical adsorption processes. It is often used to estimate the characteristic porosity in addition to the apparent free energy of adsorption^{55}. This empirical model is initially conceived for the adsorption of subcritical vapours onto micro pore solids following a pore filling mechanism. This model provides insight knowledge on the adsorption of gases and vapours on micro porous sorbents^{56}.
This isotherm is applicable for intermediate range of adsorbate concentrations because it exhibits unrealistic asymptotic behaviour and does not predict Henry’s laws at low pressure. Also it is suitable to distinguish the physical and chemical adsorption of metal ions with its mean free energy. Another unique feature of the DubininRadushkevich isotherm is the fact that it is temperature dependent^{57,58}. DubininRadushkevich isotherm model is given in Eq. 6:
where, ε is called as Polanyi Potential shown in Eq. 7. The energy of activation or mean free energy E (kJ mol^{–}^{1}) of adsorption per molecule of the adsorbate when it is transferred to the surface of the solid from infinity in the solution and can be calculated by the Eq. 8.
The value of E is used to predict whether an adsorption is physisorption or chemisorptions. If E<8 KJ mol^{–}^{1}, the adsorption is physisorption and if E = 816 KJ mol^{–}^{1}, the adsorption is chemisorptions in nature^{59}.
Temkin isotherm model: Temkin adsorption isotherm model is highly suitable for predicting the gas phase adsorption equilibrium. On the other hand, complex adsorption systems including the liquidphase adsorption isotherms are usually not suitable to be described. Moreover, Temkin isotherm model is valid only for an intermediate range of ion concentrations. It illustrates the effects of indirect adsorbate/adsorbate interactions on the adsorption process. The main assumption in this model is the heat of adsorption decreases linearly with increasing coverage and the adsorption is characterized by a uniform distribution of binding energies^{60,61}. The Temkin adsorption isotherm model is given in Eq. 9:
The term RT/b_{T} correlates the heat of adsorption. A_{T} is the equilibrium binding constant (L mg^{–}^{1}) equivalent to the maximum binding energy.
Hillde boer isotherm model: Mobile adsorption as well as lateral interaction among adsorbed molecules are well described by this HillDeboer isotherm model^{62,63}. The HillDeboer isotherm model is given in Eq. 10:
Positive value of K_{2} indicates the attraction between adsorbed species and the negative value of K_{2} is the indication of repulsion. If K_{2} is equal to zero, it indicates no interaction between adsorbed molecules and it reduces to the Volmer equation^{64}.
Fowlerguggenheim isotherm model: FowlerGuggenheim adsorption isotherm model details the lateral interaction of the adsorbed molecules. This is the simplest model allowing for the lateral interaction and it indicates that the heat of adsorption varies linearly with loading. FowlerGuggenheim adsorption isotherm model is given in Eq. 11:
Positive W indicates that the interaction between the adsorbed molecules is attractive. Negative W value is the indication of the interaction among adsorbed molecules is repulsive. If W is equal to zero, there is no interaction between adsorbed molecules and this model reduced to the Langmuir^{65,66}. This model simply relates the interaction between the adsorbed molecules and heat of adsorption. If the interaction between the adsorbed molecules is attractive it results an increase in heat of adsorption. Similarly If the interaction between the adsorbed molecules is repulsive, it results a decrease in heat of adsorption.
Floryhuggins isotherm model: FloryHuggins isotherm discusses the degree of surface coverage characteristics of the adsorbate onto the adsorbent. This model is well appropriate to find the feasibility and spontaneous nature of an adsorption process^{40,41,67,68}. FloryHuggins adsorption isotherm model is given in Eq. 12:
where, K_{FH} is used to calculate the spontaneity Gibbs free energy.
Halsey isotherm model: Halsey adsorption isotherm model is appropriate to discuss the multilayer adsorption at a relatively large distance from the surface. The fitting of the experimental data to this equation prove to the heterosporous nature of the adsorbent^{69}. Halsey adsorption isotherm model is given in Eq. 13:
Harkinjura isotherm model: HurkinJura adsorption isotherm accounts for multilayer adsorption on the surface of absorbents having heterogeneous pore distribution. The existence of a heterogeneous pore distribution can be well explained by this model^{25}. HurkinJura adsorption isotherm model is given in Eq. 14:
Jovanovic isotherm model: Jovanovic adsorption isotherm model is similar to that of Langmuir model with the approximation of monolayer localized adsorption without lateral interactions. The assumptions in this model are same in the Langmuir model in addition with the possibility of some mechanical contacts between the adsorption and desorbing molecules^{70}. Jovanovic adsorption isotherm model is given in Eq. 15:
At high concentrations of adsorbate, it becomes the Langmuir isotherm, however does not follow the Henry’s law.
Elovich isotherm model: The Elovich isotherm model assumes that the adsorption site grows exponentially with adsorption, indicating a multilayer adsorption. It is highly useful in describing chemisorption on highly heterogeneous adsorbents. This model is often valid for systems in which the adsorbing surface is heterogeneous^{71}. The Elovich adsorption isotherm model is given in Eq. 16:
Kiselev isotherm model: The Kiselev adsorption isotherm model explains the localized monomolecular layer formation of adsorbate on adsorbents^{72}. This model is valid when surface coverage is greater than 0.68. The Kiselev adsorption isotherm model is given in Eq. 17:
Three parameter models: Models containing three parameters to explain the mechanism of adsorption are discussed using sixteen models viz., Hill, RedlichPeterson, Sips, LangmuirFreundlich, FritzSchlunderIII, RadkePrausnits, Toth, Khan, KobleCorrigan, Jossens, JovanovicFreundlich, BrouersSotolongo, ViethSladek, Unilan, HollKrich and LangmuirJovanovic.
Hill isotherm model: The Hill isotherm model is derived to describe the adherence of different species onto homogeneous substrates. An assumption is made in this model is adsorption is a cooperative phenomena. It indicates that the adsorbate at one site of the adsorbent would influence the other sites on the same adsorbent^{73}. Hill adsorption isotherm model is given in Eq. 18:
If n_{H} is greater than 1, this isotherm indicates positive cooperativity in binding, n_{H} is equal to 1, it indicates noncooperative or hyperbolic binding and n_{H} is less than 1, indicating negative cooperativity in binding.
Redlichpeterson isotherm model: The RedlichPeterson isotherm model is derived with hybrid features of Langmuir and Freundlich isotherms. Consequently the mechanism of adsorption is a hybrid one and it does not follow ideal monolayer adsorption^{74}. This RedlichPeterson isotherm model is given in Eq. 19:
This model is applicable with wide concentration range, the model has a linear dependence on concentration in the numerator and an exponential function in the denominator. It is applicable for both homogeneous and heterogeneous systems. At high liquidphase concentrations of the adsorbate, RedlichPeterson isotherm model reduces to the Freundlich model. This model approaches Henrys Law model when the liquid phase concentration is low. The exponent, β_{RP}, generally ranges between 0 and 1. While β_{RP} = 1 this model approaches Langmuir model and β_{RP} = 0 this isotherm approaches to Freundlich model.
Sips isotherm model: Sips adsorption isotherm model described mainly the localized adsorption without adsorbateadsorbate interactions^{75}. This model is a combined of Langmuir and Freundlich expressions developed to predict the heterogeneous adsorption systems. The limitation of increased adsorbate concentration normally associated with the Freundlich isotherm is neglected in this model. At low adsorbate concentrations, it is transformed to Freundlich isotherm. At high adsorbate concentrations, it predicts a monolayer adsorption capacity characteristic of the Langmuir isotherm^{76}. Sips adsorption isotherm model is given in Eq. 20:
When β_{S} equal to 1 this isotherm approaches Langmuir isotherm and β_{S} equal to 0, this isotherm approaches Freundlich isotherm.
Langmuirfreundlich isotherm model: LangmuirFreundlich isotherm model describes the adsorption in heterogeneous surfaces. It explains the distribution of adsorption energy of the adsorbent onto heterogeneous surface. When the adsorbate concentration is low, this model becomes the Freundlich isotherm model, contradictorily when the adsorbate concentration is high, this model becomes the Langmuir isotherm^{51}. LangmuirFreundlich isotherm model is given in Eq. 21:
where, m_{LF} is heterogeneous parameter and it lies between 0 and 1. m_{LF} would increase with a decrease in degree of surface heterogeneity. For mLF is equal to 1, this model covert to Langmuir model.
FritzschlunderIII isotherm model: FritzSchlunder three parameter isotherm models is developed to fit over an extensive range of experimental results because of huge number of coefficients in their isotherm^{77}. This expression is given in Eq. 22:
If m_{FS3} is equal to 1, the Fritz–Schlunder–III model becomes the Langmuir model but for high concentrations of adsorbate, the Fritz–Schlunder–III reduces to the Freundlich model.
Radkeprausnitz isotherm model: The RadkePrausnitz isotherm model has several important properties which makes it more preferred in most adsorption systems at low adsorbate concentration. This isotherm is applicable over a wide range of adsorbate concentration. This isotherm model reduces to a linear isotherm (Henrys Model) when the adsorbate concentration is low. This model becomes the Freundlich isotherm when the adsorbate concentration is high. When RadkePrausnitz model exponent m_{RaP3} is equal to zero, this model becomes the Langmuir isotherm^{78}. RadkePrausnitz isotherm models are given in Eq. 2325:
If the value of both m_{RaP1} and m_{RaP2} is equal to 1, the RadkePrausnitz 1, 2 models reduce to the Langmuir model but at low concentrations, the models become Henry’s law, but for high adsorbate concentration, the RadkePrausnitz 1 and 2 models becomes the Freundlich model. But the RadkePrausnitz3 equation reduces to Henry’s law while the exponent m_{RaP3} is equal to 1 and become Langmuir isotherm when the exponent m_{RaP3} is equal to 0.
Toth isotherm model: Toth adsorption isotherm model is developed to describe the heterogeneous adsorption systems which satisfy both low and high end boundary of adsorbate concentration. This model is the modified form of Langmuir isotherm with the intension of rectifying the error between experimental and predicted data^{79}. The Toth isotherm model is given Eq. 26:
It is clear that when n = 1, this equation reduces to Langmuir isotherm equation, the process approaches onto the homogeneous surface. Therefore the parameter n characterizes the heterogeneity of the adsorption system. If it deviates further away from unity, then the system is said to be heterogeneous. This isotherm model is suitable for the modelling of several multilayer and heterogeneous adsorption systems.
Khan isotherm model: The Kahn isotherm model is developed for adsorption of biadsorbate from pure dilute equations solutions. This isotherm has applicable on both limits Freundlich on one end and Langmuir isotherm on the other end^{80}. Kahn isotherm model is given in Eq. 27:
While, a_{K} is equal to 1, Toth model approaches the Langmuir isotherm model and at higher values of concentration, Toth model reduces to the Freundlich isotherm model.
Koblecorrigan isotherm model: KobleCorrigan isotherm model is the resemblance of Sips isotherm model. This model incorporates both Langmuir and Freundlich isotherm^{81}. KobleCorrigan isotherm model is given in Eq. 28:
This model reduces to Freundlich at high adsorbate concentrations. It is only valid when the constant n is greater than or equal to 1. When n is less than unity, it signifies that the model is incapable of defining the experimental data despite high concentration coefficient or low error value.
Jossens isotherm model: The Jossens isotherm model is developed on the basis of energy distribution of adsorbateadsorbent interactions at adsorption sites. An assumption is made in this model is the adsorbent has heterogeneous surface with respect to the interactions it has with the adsorbate^{82}. At low concentrations this model is reduced to Henry’s law model. Jossens isotherm model is given in Eq. 29:
where, J is corresponds to Henry’s constant at low capacities. b_{J} is Jossens isotherm constant and it is characteristic of the adsorbent irrespective of temperature and the nature of adsorbents.
Jovanovic freundlich isotherm model: JovanovicFreundlich isotherm model is developed to describe singlecomponent adsorption equilibrium on heterogeneous surfaces. An assumption is made in this model is the rate of decrease of the fraction of the surface unoccupied by the adsorbate molecules is proportional to a certain power of the partial pressure of the adsorbate. If the adsorbent surface is homogeneous, this model reduced to Jovanovic. At low pressures, the equation reduces to the Freundlich isotherm but at high pressures, monolayer coverage is achieved. As in the case of Jovanovic model, the JovanovicFreundlich model regard as the possibility of some mechanical contacts between the adsorbing and desorbing molecules. Furthermore, this isotherm was utilized for heterogeneous surfaces without lateral interactions^{83}. JovanovicFreundlich isotherm model is given in Eq. 30:
Brouerssotolongo isotherm model: This isotherm is designed in the form of deformed exponential function for adsorption onto the heterogeneous surface mainly because of Langmuir who has recommended the extension of the simple Langmuir isotherm to nonuniform adsorbent surfaces. The assumption made in this isotherm is the surface of adsorbent consists of a fixed number of patches of active sites of equal energy^{84}. BrouersSotolongo model is given in Eq. 31:
The parameter a_{BS} is related with distribution of adsorption energy and the energy of heterogeneity of the adsorbent surfaces at the given temperature^{85}.
Viethsladek isotherm model: This model incorporates two distinct sections to calculate the diffusion rates in solid adsorbents from transient adsorption. The first one is defined by a linear section (Henry’s law) and second one is nonlinear section (Langmuir isotherm). The linear section clarifies the physisorption of gas molecules onto the amorphous adsorbent surfaces and the nonlinear section explains the adherence of gas molecules to sites on the porous adsorbent surfaces^{86}. ViethSladek isotherm model is given in Eq. 32:
Unilan isotherm model: Unilan isotherm model is presumed the application of the local Langmuir isotherm and uniform energy distribution. This equation is restricted to Henry’s law, thus it is valid at extremely low adsorbate concentrations. It is frequently used for adsorption of gas phase onto a heterogeneous adsorbent surface^{76}. Unilan isotherm model is given in Eq. 33:
The higher the model exponent β_{U}, the system is more heterogeneous. If β_{U} is equal to 0, the Unilan isotherm model becomes the classical Langmuir model as the range of energy distribution is zero in this limit^{82,87,88}.
Hollkrich isotherm model: HollKrich Isotherm Model is a modified form of Langmuir isotherm^{89}. This model becomes the Freundlich isotherm at low concentrations. The capacity reaches a finite capacity more leisurely than the Langmuir isotherm at high concentrations^{55}. HollKrich Isotherm Model is given in Eq. 34:
Langmuirjovanovic isotherm model: This empirical model is the combined form of both Langmuir and Jovanovic isotherm^{90}. The LangmuirJovanovic model is given in Eq. 35:
Four parameter models: The four parameter models discussed in this study are FritzSchlunderIV, Baudu, Webervan Vliet and MarczewskiJaroniec models.
FritzschlunderIV isotherm model: FritzSchlunder IV model is another model comprised of fourparameter with combine features of LangmuirFreundlich isotherm^{77}. The model is given in Eq. 36:
This isotherm is valid when the values of α_{FS5} and β_{FS5} are less than or equal to 1. At high adsorbate concentration, FritzSchlunderIV isotherm becomes Freundlich equation. Conversely if the value of both a_{FS5} and β_{FS5} equal to 1, this isotherm reduces to Langmuir isotherm. At high concentrations of the adsorbate in the liquidphase this isotherm model becomes the Freundlich.
Baudu isotherm model: Baudu isotherm model has been developed mainly due to the arise of discrepancy in calculating Langmuir constant and coefficient from slope an tangent over a broad range of concentrations^{91}. Baudu isotherm model is the transformed form of the Langmuir isotherm. It is given in Eq. 37:
This model is only applicable in the range of (1+x+y)<1 and (1+x)<1. For lower surface coverage, Baudu model reduces to the Freundlich equation^{92}. It is given in Eq. 38:
Webervan vliet isotherm model: Weber and van Vliet isotherm model is to describe equilibrium adsorption data with four parameters^{93,94}. The model is given in Eq. 39:
The isotherm parameters P_{1}, P_{2}, P_{3} and P_{4} can be defined by multiple nonlinear curve fitting techniques which is predicated on the minimization of sum of square of residual.
Marczewskijaroniec isotherm model: The MarczewskiJaroniec isotherm model is the resemblance of Langmuir isotherm model. It is developed on the basis of the supposition of local Langmuir isotherm and adsorption energies distribution in the active sites on adsorbent^{89,94}. The MarczewskiJaroniec isotherm model is given in Eq. 40:
where, K_{MJ} describes the spreading of distribution in the path of higher adsorption energy. n_{MJ} describes the spreading in the path of lesser adsorption energies. The isotherm reduces to Langmuir isotherm when n_{MJ} and K_{MJ} are equal to unity. The isotherm reduces to LangmuirFreundlich model when n_{MJ} equal to KMJ.
Five parameter model: Accounting the high parameter models certainly provides clear information on mechanism of adsorption under equilibrium condition. In this section, only one, five parameter model i.e., FritzSchlunderV isotherm model is applied.
FritzschlunderV isotherm model: FritzSchlunder adsorption isotherm model is developed with the aim of simulating the model variations more precisely for application over a wide range of equilibrium data^{77}. FritzSchlunder adsorption isotherm model is given in Eq. 41:
RESULTS AND DISCUSSION
Experimental results show that the maximum adsorption of Cr (VI) by BGMA can be achieved at pH of 5 and at 2 g of BGMA biomass loading. The maximum adsorption capacity of BGMA is found to be 37.426 mg g^{–}^{1}. The optimum agitation rate 120 rpm is maintained during the continuous 24 h of contact time.
To validate the prediction of maximum adsorption capacity of BGMA, the finding is compared with several authors who have reported their results of the investigation on adsorption of Cr (VI) using different sorbents. The maximum adsorption capacity of activated charcoal (prepared from wood apple shell) used by Doke and Khan^{95} for removal of Cr (VI) from aqueous solution of concentration 1250 mg g^{–}^{1} is reported to be 151.51 mg g^{–}^{1} at a pH 1.8. Though the q_{max} is relatively very higher than the present investigation, their sorbent are found to be more effective at a low pH which is highly acidic.

Fig. 1: 
Experimental results of adsorption of Cr (VI) onto BGMA 
However such acidic environments are tedious, hazardous and also uneconomical to industrial scale. Similarly a new lowcost activated carbon is prepared by Gorzin and Abadi^{96} from paper mill sludge in order to remove Cr (VI) ions from aqueous solution of concentration 100 mg L^{–}^{1}. The maximum adsorption capacity of 23.18 mg g^{–}^{1} is reported at optimum pH 4.0 and contact time of 180 min. Subsequently magnetic natural zeoliteChitosan composite is used as adsorbent to remove Cr (VI) from an aqueous solution of 200 mg L^{–}^{1} concentration by Gaffer et al.^{97}. The 98% of removal efficiency is reported at pH 2 and 0.2 g of dosage. Chemically treated banana peels is used by Ali et al.^{98} as an adsorbent for removal of Cr (VI) from aqueous solution of concentration 400 mg L^{–}^{1} and the maximum adsorption capacity reported is reported to be 6.178 mg g^{–}^{1} at 120 min contact time, 4.0 g L^{–}^{1} adsorbent dosage and 3.0 pH. Potato peels is used by Mutongo et al.^{99} as adsorbent for removal of Cr (VI) from aqueous solution of concentration 100 mg L^{–}^{1} and the maximum adsorption capacity is reported as 3.28 mg g^{–}^{1} at 48 min contact time, 4.0 g L^{–}^{1} adsorbent dose and 2.5 pH. Mulani et al.^{100} studied the removal of chromium (VI) was performed using coffee polyphenolformaldehyde/acetaldehyde resins as adsorbent. The maximum adsorption capacity of coffee polyphenolformaldehyde resins is reported as 19.342 mg g^{–}^{1} at pH 2 and 150 min contact time.
Contradictorily, Guar gum–nano zinc oxide (GG/n ZnO) bio composite is used as an adsorbent by Khan et al.^{101} for enhanced removal of Cr (VI) from aqueous solution of concentration 25 mg L^{–}^{1} and the maximum adsorption capacity is reported as 55.56 mg g^{–}^{1} at 50 min contact time, 1.0 g L^{–}^{1} adsorbent dose and 7.0 pH. Though the reported q_{max} is found to be very high efficiency with an artificially made sorbent, this is comparatively inferior to naturally occurring BGMA in terms of availability, accessibility and cost.
Adsorption isotherms: The shape of the adsorption isotherms aids to classify the nature of the phenomena of adsorption of Cr (VI) onto BGMA. The experimental adsorption nature of Cr (VI) from its synthetic aqueous solutions onto BGMA is shown in Fig. 1 which is very much useful to perceive the shape of the isotherm^{102,41}. Giles et al.^{103} classified the isotherms according to their shapes and grouped as L, S, H and C. Based on the above classification, the isotherm of Cr (VI) onto BGMA shows the L curve pattern. The L shape is the indication of no strong competition between solvent and the adsorbate to occupy the adsorbent surface sites. Figure 1 shows concavity curve which indicates that the ratio between the concentration of the compound remaining in solution and adsorbed on the solid decreases when the solute concentration increases. It reveals the progressive saturation of the solid.
Limousin et al.^{46} details further, two subgroups of L shaped isotherms: (i) The curve reaches a strict asymptotic plateau (the solid has a limited sorption capacity) and (ii) The curve does not reach any plateau (the solid does not show clearly a limited sorption capacity). Figure 1 infers that BGMA has a limited sorption capacity for adsorption of Cr (VI) under the conditions employed in this investigation.
Two parameter models: The values of parameters and regression coefficient R^{2} of two parameter adsorption isotherm models for adsorption of Cr (VI) onto BGMA are given in Table 1.

Fig. 2: 
Comparison of experimental values of equilibrium uptake of Cr (VI) with two parameter model values 
Table 1: 
Parameter values of two parameter adsorption isotherm models for adsorption of Cr (VI) onto BGMA 

DubininRadushkevich, Hillde Boer, FowlerGuggenheim, FloryHuggins, Halsey, HarkinJura, Elovich and Kiselev are eliminated from the running discussion in the initial stage itself due to the negative Rsquare values. Jovanovic and Temkin are also removed from the discussion due to their low R^{2} value when compared to other isotherm models under study. Only two models viz. Freundlich and Langmuir are taken into consideration for further discussion. A plot of C_{eq} (mg L^{–}^{1}) against q_{eq} (mg g^{–}^{1}) for both the models along with experiential values is shown in Fig. 2. Among the two models, Freundlich isotherm model shows greater accuracy with experimental data (R^{2} = 0.9662). It indicates that the extent of heterogeneity of the BGMA surface. Due to this, the adsorptive surfaces of BGMA are expected to be made up of small heterogeneous adsorption sites that are homogeneous in themselves. Through the surface exchange mechanism, adsorption sites are activated which results in increased adsorption. The value of a_{F} and n_{F} are found to be 5.315 mg g^{–}^{1} and 2.194, respectively. Since the value of n_{F} is in the range of 110, it infers that the adsorption of Cr (VI) from its synthetic solution onto BGMA is favourable one.

Fig. 3: 
Accuracy of two parameter model values of equilibrium uptake of Cr (VI) with experimental values 
The value of 1/n_{F} is calculated as 0.4558 which is closer to zero ensures that the active sites of BGMA for adsorption of Cr (VI) onto its surface are more heterogeneous.
Followed by the Freundlich, Langmuir isotherm model shows better agreement with experimental adsorption data with an R^{2} value of 0.9019. It suggests the applicability of the Langmuir adsorption isotherm and indicates the formations of monolayer coverage of the sorbate at the outer surface of the sorbent. The value of Langmuir adsorption equilibrium constant b_{L} is 0.0354 mL g^{–}^{1} which shows quantitatively the affinity between Cr (VI) and BGMA. Since the calculated value of R_{L} is equal to 0.3608, which is in between 01, it ensures that the adsorption of Cr (VI) onto BGMA is favourable. Based on Langmuir isotherm model, the maximum adsorption capacity of BGMA for Cr (VI) is found to be 50.04 mg g^{–}^{1} and it indicates the reasonable agreement of Langmuir isotherm model with experimental data of Cr (VI) adsorption by BGMA form its synthetic solution under the conditions employed in this investigation.
Since the R^{2} values of both the Freundlich and Langmuir are highly significant and gives good agreement with the experimental data mathematically, It is suppose to be arrived two different opinions: One is surface heterogeneity of adsorbent through Freundlich isotherm and the another is monolayer coverage of adsorbate onto the surface of adsorbent from Langmuir isotherm. Figure 3 ensures the same and shows the comparison and extent of concurrence of both models with experimental equilibrium uptake.
However it cannot be claimed that both the mechanisms suit for adsorption of Cr (VI) onto BGMA for the entire range of concentration employed in this investigation. From Fig. 4, it is clearly observed that the concurrence of Langmuir isotherm mechanism is lesser than Freundlich when the initial concentration of Cr (VI) synthetic solution is low. Subsequently as the initial concentration is increased to a level of 100 mg L^{–}^{1}, the mechanism shows a closer association with experimental data under equilibrium condition. This trend continuous up to the initial concentration of 120 mg L^{–}^{1}, thereafter the trend deviates from the concurrence. Again inbetween 200220 mg L^{–}^{1} initial concentration level a closer association is noticed. Up to 70 mg L^{–}^{1} and above 220 mg L^{–}^{1} of initial concentration level, though the Freundlich adsorption mechanism shows a deviation from experimental data under equilibrium condition, the magnitude of deviation is less when compared with Langmuir mechanism. Not only that, inbetween 70220 mg L^{–}^{1} initial concentration level is a closer association observed with Freundlich isotherm mechanism than Langmuir. Freundlich isotherm mechanism provides maximum satisfaction with the equilibrium experimental data under the range of experimental conditions investigated in this study.
Three parameter models: Table 2 provides the parameter values for three parameter adsorption isotherm models for adsorption of Cr (VI) onto BGMA. As in the case of two parameter models, LangmuirFreundlich, Jossens, JovanovicFreundlich, RedlichPeterson Isotherm Model, RadkePrausnits isotherm ModelII and Toth isotherm model are eliminated from the running discussion in the initial stage itself due to its poor R^{2} values.

Fig. 4: 
Concurrence of experimental values of equilibrium uptake of Cr (VI) with two parameter model values upon initial metal ion concentration 

Fig. 5: 
Comparison of experimental values of equilibrium uptake of Cr (VI) with three parameter model values 
Though Unilan, Hill, Sips, FritzSchlunderIII, RadkePrausnitsI, Khan, BrouersSotolongo, HollKrich, LangmuirJovanovic Isotherm Models have significant R^{2} values, the qmax values obtained in each of these models are either negative or too high which are not physically realizable. Hence these models are eliminated from the running discussion.
KobleCorrigan isotherm Model shows an amazing concurrence with experimental data points. Its R^{2} value is equal to 0.9919. Since the value of KobleCorrigan’s isotherm constant (n_{KC} = 0.07927), it signifies that the model is incapable of defining the experimental data despite high concentration coefficient and low error value. Among the eighteen, three parameter models, only two models viz., ViethSladek and RadkePrausnitsIII are taken into the consideration for the running discussion. Figure 5 shows the plot of C_{eq} (mg L^{–}^{1}) against q_{eq} (mg g^{–}^{1}) for the two models along with experiential values. ViethSladek isotherm model gives R^{2} value of 0.9652. Hence it is arrived that the adsorption of Cr (VI) onto BGMA is influenced both by van der waals forces (physical adsorption) and ionic forces (chemical adsorption) for the entire range of concentration employed in this investigation. However a conflict of interest is arises upon the lower qmax value of 10.02 mg g^{–}^{1}.
Table 2: 
Parameter values of three parameter adsorption isotherm models for adsorption of Cr (VI) onto BGMA 

Finally The RadkePrausnitsIII isotherm model gives an appreciable R^{2} value of 0.9205. The value of m_{RaP3} is equal to 0.6439 indicates that both Langmuir and Freundlich mechanisms are suitable to describe the adsorption of Cr (VI) onto BGMA for the entire range of concentration employed in this investigation.

Fig. 6: 
Accuracy of three parameter model values of equilibrium uptake of Cr (VI) with experimental values 

Fig. 7: 
Concurrence of experimental values of equilibrium uptake of Cr (VI) with three parameter model values upon initial metal ion concentration 
As in the case of discussion in two parameter model, RadkePrausnitsIII isotherm model ensures this conclusion. Figure 6 shows the comparison and extent of concurrence of three models with experimental equilibrium uptake. Figure 7 is the indication of equilibrium uptake of metal by isotherm models upon the increase in initial concentration of Cr (VI) synthetic solution.
Four parameter models: FritzSchlunderIV Isotherm Model, Baudu Isotherm Model, Webervan Vliet Isotherm Model and MarczewskiJaroniec Isotherm Model are investigated to know the mechanism of adsorption of Cr(VI) form its synthetic solution by BGMA. Among the four models, Webervan Vliet Isotherm Model alone seems to be highly significant (R^{2} = 0.9811). Rest of the three models are not applicable due its poor R^{2} values and irreverent parameter values. Hence it can describe the mechanism of adsorption of Cr (VI) onto BGMA for the entire range of concentration employed in this investigation. Figure 8 shows the comparison of equilibrium Cr (VI) uptake by both experimental and Webervan Vliet Isotherm Model against equilibrium effluent concentration.

Fig. 8: 
Comparison of experimental values of equilibrium uptake of Cr (VI) with four parameter model values 

Fig. 9: 
Accuracy of four parameter model values of equilibrium uptake of Cr (VI) with experimental values 
Upto 120 mg L^{–}^{1} of initial metal ion concentration, the concurrence of model data with experimental data is observed as significant. Thereafter the deviation of model data from the experimental value is high. It is ensured Fig. 9 which shows the experimental and model equilibrium metal uptake upon the initial metal ion concentration. Figure 10 shows the concurrence of equilibrium experimental metal uptake against the FritzSchlunderV adsorption isotherm model data. The values of parameters and regression coefficient R^{2} of four parameter adsorption isotherm models for adsorption of Cr (VI) onto BGMA are given in Table 3.
Five parameter model: Table 4 provides the parameter values of FritzSchlunderV parameter model. The R^{2} value indicates the significance of this model. Figure 11 shows the comparison of equilibrium Cr (VI) uptake by both experimental and FritzSchlunderV model against equilibrium effluent concentration. For the entire range of concentrations investigated in this study, a significant concurrence of model data with experimental data is observed (Fig. 12). Figure 13 infers more precisely the concurrence of equilibrium experimental metal uptake against the FritzSchlunderV adsorption isotherm model data.

Fig. 10: 
Concurrence of experimental values of equilibrium uptake of Cr (VI) with four parameter model values upon initial metal ion concentration 

Fig. 11: 
Comparison of experimental values of equilibrium uptake of Cr (VI) with five parameter model values 
Table 3 
Parameter values of four parameter adsorption isotherm models for adsorption of Cr (VI) onto BGMA 


Fig. 12: 
Concurrence of experimental values of equilibrium uptake of Cr (VI) with five parameter model values upon initial metal ion concentration 

Fig. 13: 
Accuracy of five parameter model values of equilibrium uptake of Cr (VI) with experimental values 
Table 4: 
Parameter values of five parameter adsorption isotherm model for adsorption of Cr (VI) onto BGMA 

CONCLUSION
The adsorption of Cr (VI) onto BGMA is found to be L shape. It indicates that no strong competition exists between solvent and the active sites of BGMA to occupy. Freundlich adsorption model is more suitable for the entire range of concentrations investigated in this study. Applicability of ViethSladek isotherm model infers that the adsorption of Cr (VI) onto BGMA is influenced by van der waals forces and ionic forces for the entire range of concentration employed in this investigation.
RECOMMENDATIONS, APPLICATIONS, LIMITATIONS AND FUTURE IMPLICATIONS OF THIS RESEARCH WORK
Blue green algae are recommended to recover the chromium from industrial effluent under the conditions employed in this investigation. The application of the blue green algae is favorable for the recovery of heavy metal ions. The only limitation is the procedure involved in recovery of the adsorbed metal ions by desorption method on industrial scale. In the future study, chemical modification of the algal biomass can be done to enhance the recovery of heavy metal. This can help to reduce the quantity of algae required.
SIGNIFICANCE STATEMENTS
This study discovers the modeling of adsorption isotherms for removal of hexavalent chromium from aqueous solutions using blue green algae with the application of various isotherm models that can be beneficial for gaining insight knowledge of sorbate/sorbent adsorption mechanism which is essential in design of real time operations in industrial scale. Isotherm modeling in adsorption studies is important in identifying or predicting the mechanism of adsorption processes. Hence in this present investigation, various two, three, four and five parameter adsorption isotherm models have been tested.
This study will help the researchers to uncover the critical areas modeling in adsorption that many researchers were not able to explore. Thus this article comprises of critical review of the models with experimental data which would help the readers in acquiring complete knowledge on adsorption mechanisms.