**2. Metastable-equilibrium adsorption inequality**

Suppose the adsorption of a pure solute A in a pure solvent onto a solid surface can be schematically represented by the equation (1, 2)

$$\text{C}\_{1} + \bigotimes \text{H}\_{2}\text{O}\_{2} = \bigotimes \text{H}\_{3}\text{H}\_{3} + \text{H}\_{2}\text{O}$$

where A stands for solute in solution, for adsorbed solvent, for adsorbed A, and H2O for solvent in solution.

Since the Gibbs free energy is a state function and its change depends only on the initial and final states of the system, we can replace the real adsorption process of [1], which is generally thermodynamically irreversible, with two ideal reversible processes that lead to the same final state, in order to calculate the Gibbs free energy change,

$$\begin{array}{llll} \text{A} & + \underset{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{red}}{\stackrel{\text{red}}{\text{red}}}}}}}} \text{H}\_{2}\text{O} & \underbrace{\underset{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{red}}{\stackrel{\text{red}}{\text{red}}}}}}} \text{H}\_{2}\text{O}} + \underset{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{red}}{\stackrel{\text{red}}{\text{red}}}}}}} \text{H}\_{2}\text{O}} + \underset{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{red}}{\stackrel{\text{red}}{\text{red}}}}}} \text{H}\_{3}\text{O}} \end{array} \text{H}\_{2}\text{O} + \underset{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{\textquotedblleft}}{\stackrel{\text{red}}{\stackrel{\text{red}}{\text{red}}}}}} \text{H}\_{3}\text{O}} \text{O} \tag{2}$$

where " " indicates that the real adsorption process can be irreversible. Step 1 represents an imagined reversible adsorption process where the final concentration of *A* on the solid surface is the same as that in the real irreversible process [1]. represents an ideal equilibrium stable state of adsorbed *A*, and represents a real metastable-equilibrium adsorption state. and represent different thermodynamic states of adsorbed *A*, although they have the same value of adsorption density. Δ*G*1 and *K*eq are the change in Gibbs free energy and equilibrium constant of step 1, respectively. Δ*G*2 of step 2 is the difference in Gibbs free energy between the reaction products of the real irreversible process [1] and the ideal reversible process (step 1) . *K*me is the equilibrium constant of step 2. Thus,

$$
\Delta G\_{real} = \Delta G\_1 + \Delta G\_2 \tag{3}
$$

since

520 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

(MEA) structures. These studies represent advances on how microscopic surface molecule structures affect the macroscopic relationships in surface adsorption thermodynamics. Surface microstructures greatly affect the local chemical properties, long-range interaction, surface reactivity, and bioavailability of pollutants in the environment. Both experimental techniques and thermodynamic theoretical development on interfacial processes are

It has been a basic concept in traditional thermodynamic adsorption theories that adsorption density ( , mol/m2 ) is a state variable (a function that is only determined by the state and not affected by the path), so that the equilibrium adsorption constants defined by the ratio of equilibrium adsorption density on solid surfaces to the concentration in solution should be constant that is the reflection of the unique equilibrium characteristic of the reaction.1 Over the last century, the macroscopic methodology (e.g. surface complexation models) of equilibrium adsorption constants and adsorption isotherms are widely used to describe the equilibrium limits of adsorption reactions and predict the theoretical yield in many fields.2, 3 These relationships were deemed to obey the basic properties of chemical thermodynamics, i.e. the equilibrium constant should be constant and be independent of kinetics or initial

However, an abnormal phenomenon called particle/adsorbent concentration effect (*C*<sup>p</sup> effect), i.e. the dependence of adsorption isotherms on one of the reactant concentrations *C*p, has caused great confusion over the last three decades because it cannot be interpreted by the existing thermodynamic theories.2, 4-8 Several hundreds of papers have been published on this issue but the underlined theoretical reason, which is far more important than the *C*<sup>p</sup> effect itself, still remains not clear to most researchers. Most studies so far attribute *C*p effect to various experimental artifacts.9, 10 However, after these artifacts are excluded from the experiments, *C*p effect may disappear in some systems,9 but still exist in other systems.3, 11 Thus, the problem becomes rather confused based on empirical or experimental analysis

Metastable-equilibrium adsorption (MEA) theory indicates that,12-14 for a given adsorption reaction under fixed thermodynamic conditions, a polyhedral adsorbate molecule is generally ended in various MEA states with different energies and geometries rather than a unique equilibrium state when the reaction reaches to the apparent equilibrium. Unlike concentration in solutions, adsorption density (mol/m2) on solid surfaces no longer unambiguously corresponds to thermodynamic state variables, because adsorption density can only count for the mass but not the chemical potentials/energies of different microscopic MEA states that construct the real equilibrium adsorption state. When the adsorption density is not treated as a thermodynamic state variable, a theoretical equation known as "MEA inequality" is deducted from the fundamental thermodynamic laws.12

Suppose the adsorption of a pure solute A in a pure solvent onto a solid surface can be

essential for the development of molecular environmental and geological sciences.

reactant concentrations under fixed thermodynamic conditions.

**2. Metastable-equilibrium adsorption inequality** 

schematically represented by the equation (1, 2)

only.

$$
\Delta G = -RT\ln K
$$

Thus,

$$\mathbf{K}\_{real} = \mathbf{K}\_{eq} \times \mathbf{K}\_{me} \tag{4}$$

In step2

$$\Delta \mathbf{G}\_2 = \left( \mathbf{G}\_{H\_2O} + \mathbf{G}\_A^{solid} \right)\_{real} - \left( \mathbf{G}\_{H\_2O} + \mathbf{G}\_A^{solid} \right)\_{ideal} \tag{5}$$

assuming

(1)

$$\left(\mathbf{G}\_{H\_2O}\right)\_{real} = \left(\mathbf{G}\_{H\_2O}\right)\_{ideal}$$

$$\Delta \mathbf{G}\_2 = \left(\mathbf{G}\_A^{solid}\right)\_{real} - \left(\mathbf{G}\_A^{solid}\right)\_{ideal} \ge 0\tag{6}$$

Advances in Interfacial Adsorption Thermodynamics:

*HO HO T* 2 2 *<sup>s</sup>*

Because 2 <sup>1</sup> 

defined as a constant *b*,

*x A* , where *<sup>i</sup>*

Since the activity of solvent *H O*<sup>2</sup> *<sup>l</sup>*

 *A HO* 

thus,

[13] becomes

where '

Metastable-Equilibrium Adsorption (MEA) Theory 523

eq

, [11] becomes

*A*

truly independent of the kinetics of the process.

energy level the adsorbate coverage

*K K*

 

*x a* 

*HO A <sup>l</sup> <sup>s</sup>*

2

By multiplying both *<sup>A</sup> <sup>s</sup> <sup>x</sup>* and *H O*<sup>2</sup> *<sup>s</sup> <sup>x</sup>* by the total surface area *A*T, and assuming that the

molecular size of solute and solvent are similar, we have *AA T <sup>s</sup>*

*A*

*A HO <sup>s</sup> <sup>l</sup> me*

*x a*

 2 2 1 *eq me A l l eq me A*

*me A l*

*H O H O l l*

1 *me A l*

max

where max is the characteristic saturation adsorption capacity for a given reaction, which is the maximum value of the equilibrium as the equilibrium concentration of the solute increases. In dilute solution, activity *<sup>A</sup> <sup>l</sup> a* is approximately equal to concentration *CA* , so

*bK a bK a*

Practically, the adsorption amount is often expressed in terms of the adsorption density ;

'

'1

*bK C bK C*

Equations [13] and [15] are called *Langmuir-type metastable-equilibrium isotherm equations,*  since when *K*me =1, i.e., under the ideal equilibrium condition, they are reduced to the conventional Langmuir equation. Only under this ideal condition (*K*me = 1) can the isotherm be independent of the kinetic process. Generally, the equilibrium relationship between

By assuming an exponential distribution of adsorption energy, and assuming that for each

and *C*eq would be influenced by the metastability of the adsorption state.

isotherm [13] , a Freundlich-type metastable isotherm equation can be obtained,12

**2.2 Freundlich-type metastable-equilibrium adsorption isotherm** 

*me eq me eq*

max *b b* ; *b* and *b'* are constant under fixed temperature and pressure, and are

*KK a KK a a a*

2

(11)

is the fraction of the surface occupied by component *i*.

(13)

(14)

(15)

follows the Langmuir-type metastable-equilibrium

(12)

*<sup>a</sup>* can be considered constant, *K*eq/ *H O*<sup>2</sup> *<sup>l</sup>*

*x A* , and

*a* may be

where ''='' represents a reversible process and ''>'' corresponds to an irreversible process. Equation [6] indicates that if the process is not thermodynamically reversible, then for the same amount of adsorbed *A*, the real state of , which is of metastable equilibrium in nature, will have a higher Gibbs free energy *solid <sup>A</sup> real G* than the ideal equilibrium state

 *solid <sup>A</sup> ideal G* . Step 2 is therefore not a thermodynamically spontaneous process. Since 2 me *G RT <sup>K</sup> e* and 2 *<sup>G</sup>* <sup>0</sup> ,

$$0 < K\_{mc} \le 1 \text{ (} \nleq \text{for irreversible process} \text{ = for reversible process)}.\tag{7}$$

We call *K*me the metastable-equilibrium coefficient. It measures the deviation of a metastable-equilibrium state from the ideal equilibrium state. Combining [4] and [7], we get the *MEA inequality*:

$$K\_{\text{real}} \le K\_{\text{eq}} \text{ (} \Leftarrow \text{for irreversible process} \text{ = for reversible process)}.\tag{8}$$

*K*eq is the ideal equilibrium constant for an ideal reversible process which has a unique value under constant temperature, pressure, and composition of the solution. *K*real is the experimentally measured equilibrium constant for a real adsorption process and is not necessarily constant under fixed temperature, pressure, and composition of solution, but decreases as *K*me decreases.

MEA inequality indicates that equilibrium constants or adsorption isotherms are fundamentally affected by the kinetic factor of thermodynamic irreversibility (including both mass and energetic irreversibility for a forward-backward reaction), because when the surface reaction is processed through different irreversible kinetic pathways it may reach to different MEA states under the same thermodynamic conditions. By using the MEA inequality to reformulate the existing equilibrium adsorption theories, it is possible to modify some of the existing isotherm equations into metastable-equilibrium equations.

#### **2.1 Langmuir-type metastable equilibrium adsorption isotherm**

The equilibrium constant for the adsorption process [1] is

$$K\_{\text{real}} = \frac{\left(\mathbf{a}\_A\right)\_s \times \left(\mathbf{a}\_{H\_2O}\right)\_l}{\left(a\_{H\_2O}\right)\_s \times \left(a\_A\right)\_l} = \frac{\left(f\_A\right)\_s \times \left(\mathbf{x}\_A\right)\_s \times \left(a\_{H\_2O}\right)\_l}{\left(f\_{H\_2O}\right)\_s \times \left(\mathbf{x}\_{H\_2O}\right)\_s \times \left(a\_A\right)\_l} \tag{9}$$

where *a*i stands for the activity of a given component in [1], the subscripts *s* and *l* refer to surface and bulk values, respectively, ( *fi* )s is the surface activity coefficient, and (*xi* )s is the mole fraction surface concentration.

In dilute solution, the surface activity coefficient in the solid may be set equal to unity (1), so that

$$K\_{\text{real}} = \frac{\left(\mathbf{x}\_A\right)\_s \times \left(a\_{H\_2O}\right)\_l}{\left(\mathbf{x}\_{H\_2O}\right)\_s \times \left(a\_A\right)\_l} \tag{10}$$

According to Eq. [4], we have

where ''='' represents a reversible process and ''>'' corresponds to an irreversible process. Equation [6] indicates that if the process is not thermodynamically reversible, then for the same amount of adsorbed *A*, the real state of , which is of metastable equilibrium in

We call *K*me the metastable-equilibrium coefficient. It measures the deviation of a metastable-equilibrium state from the ideal equilibrium state. Combining [4] and [7], we get

*K*eq is the ideal equilibrium constant for an ideal reversible process which has a unique value under constant temperature, pressure, and composition of the solution. *K*real is the experimentally measured equilibrium constant for a real adsorption process and is not necessarily constant under fixed temperature, pressure, and composition of solution, but

MEA inequality indicates that equilibrium constants or adsorption isotherms are fundamentally affected by the kinetic factor of thermodynamic irreversibility (including both mass and energetic irreversibility for a forward-backward reaction), because when the surface reaction is processed through different irreversible kinetic pathways it may reach to different MEA states under the same thermodynamic conditions. By using the MEA inequality to reformulate the existing equilibrium adsorption theories, it is possible to modify some of the existing isotherm equations into metastable-equilibrium equations.

**2.1 Langmuir-type metastable equilibrium adsorption isotherm** 

s

a a ( )

real

*K*

2 2 2

where *a*i stands for the activity of a given component in [1], the subscripts *s* and *l* refer to surface and bulk values, respectively, ( *fi* )s is the surface activity coefficient, and (*xi* )s is the

In dilute solution, the surface activity coefficient in the solid may be set equal to unity (1), so

2

 

*x a* 

*x a*

*A HO <sup>s</sup> <sup>l</sup> HO A <sup>l</sup> <sup>s</sup>*

2

The equilibrium constant for the adsorption process [1] is

real

*K*

mole fraction surface concentration.

According to Eq. [4], we have

that

*G* . Step 2 is therefore not a thermodynamically spontaneous process.

*<sup>A</sup> real*

0 1 *Kme* (< for irreversible process, = for reversible process). (7)

*K K* real eq (< for irreversible process, = for reversible process). (8)

 

(9)

(10)

*fxa*

2 2

*A HO A A HO <sup>s</sup> <sup>s</sup> l l HO s A <sup>l</sup> HO HO A <sup>l</sup> s s*

*a a fxa* 

*G* than the ideal equilibrium state

nature, will have a higher Gibbs free energy *solid*

and 2 *<sup>G</sup>* <sup>0</sup> ,

 *solid <sup>A</sup> ideal*

Since 2 me

the *MEA inequality*:

*G RT <sup>K</sup> e*

decreases as *K*me decreases.

$$\mathbf{K}\_{\rm eq} \times \mathbf{K}\_{m\boldsymbol{\epsilon}} = \frac{\left(\mathbf{x}\_A\right)\_s \times \left(\mathbf{a}\_{H\_2O}\right)\_l}{\left(\mathbf{x}\_{H\_2O}\right)\_s \times \left(\mathbf{a}\_A\right)\_l} \tag{11}$$

By multiplying both *<sup>A</sup> <sup>s</sup> <sup>x</sup>* and *H O*<sup>2</sup> *<sup>s</sup> <sup>x</sup>* by the total surface area *A*T, and assuming that the molecular size of solute and solvent are similar, we have *AA T <sup>s</sup> x A* , and *HO HO T* 2 2 *<sup>s</sup> x A* , where *<sup>i</sup>* is the fraction of the surface occupied by component *i*. Because 2 <sup>1</sup> *A HO* , [11] becomes

$$\theta\_A = \frac{K\_{eq} \times K\_{me} \times \left(a\_A\right)\_l}{\left(a\_{H\_2O}\right)\_l} \int \mathbf{1} + \frac{K\_{eq} \times K\_{me} \times \left(a\_A\right)\_l}{\left(a\_{H\_2O}\right)\_l} \tag{12}$$

Since the activity of solvent *H O*<sup>2</sup> *<sup>l</sup> <sup>a</sup>* can be considered constant, *K*eq/ *H O*<sup>2</sup> *<sup>l</sup> a* may be defined as a constant *b*,

$$\theta\_A = \frac{b \times K\_{m\epsilon} \times \left(a\_A\right)\_l}{1 + b \times K\_{m\epsilon} \times \left(a\_A\right)\_l} \tag{13}$$

Practically, the adsorption amount is often expressed in terms of the adsorption density ; thus,

$$\theta = \frac{\Gamma}{\Gamma\_{\text{max}}} \tag{14}$$

where max is the characteristic saturation adsorption capacity for a given reaction, which is the maximum value of the equilibrium as the equilibrium concentration of the solute increases. In dilute solution, activity *<sup>A</sup> <sup>l</sup> a* is approximately equal to concentration *CA* , so [13] becomes

$$\Gamma = \frac{\boldsymbol{b}^{\prime} \times \boldsymbol{K}\_{me} \times \mathbb{C}\_{eq}}{\mathbf{1} + \boldsymbol{b}^{\prime} \times \boldsymbol{K}\_{me} \times \mathbb{C}\_{eq}} \tag{15}$$

where ' max *b b* ; *b* and *b'* are constant under fixed temperature and pressure, and are truly independent of the kinetics of the process.

Equations [13] and [15] are called *Langmuir-type metastable-equilibrium isotherm equations,*  since when *K*me =1, i.e., under the ideal equilibrium condition, they are reduced to the conventional Langmuir equation. Only under this ideal condition (*K*me = 1) can the isotherm be independent of the kinetic process. Generally, the equilibrium relationship between and *C*eq would be influenced by the metastability of the adsorption state.

#### **2.2 Freundlich-type metastable-equilibrium adsorption isotherm**

By assuming an exponential distribution of adsorption energy, and assuming that for each energy level the adsorbate coverage follows the Langmuir-type metastable-equilibrium isotherm [13] , a Freundlich-type metastable isotherm equation can be obtained,12

$$
\Gamma = \alpha \times K\_{me} \times \mathbb{C}\_{eq}^{\beta} \tag{16}
$$

Advances in Interfacial Adsorption Thermodynamics:

and 14 days, respectively.

Metastable-Equilibrium Adsorption (MEA) Theory 525

Fig. 1. Adsorption (solid lines, closed symbols) and desorption (dotted lines, open symbols) isotherms under different *C*p conditions in Zn–goethite (a) and Cd–goethite (b) systems. (a) *C*p1=0.38 g/L, *C*p2=1.53 g/L, *C*p3=2.3 g/L, pH=6.4, equilibration time for adsorption and desorption are 12 days and 10 days, respectively. a', a comparison of the sizes of hysteresis in Figure 1a, when the first points of the desorption isotherms are translationally moved to the same point. (b) pH=7.1, equilibration time for adsorption and desorption are 20 days

Fig. 2. Adsorption (a) and desorption (b) kinetic curves under different *C*p conditions in Zn– goethite system. pH=6.4. The inset chart in (a) shows the initial stage of the adsorption. The final Zn concentrations of the four experiments in (a) are (*C*eq )*C*p1=0.18 ppm, (*C*eq )*C*p2=0.17 ppm, (*C*eq )*C*p3=0.16 ppm, (*C*eq )*C*p4=0.15 ppm. In order to examine the desorption rate

effectively, only the initial stage of desorption is presented in (b).

where is a constant under isothermal conditions. Under ideal equilibrium conditions (*K*me = 1) , [16] is reduced to the conventional Freundlich equation. Equation [16] indicates that the adsorption isotherm is shifted to the lower as *K*me decreases.

### **2.3 Particle concentration (***C***p) effect isotherm equations**

According to reaction rate theory, adsorption speed should increase as particle concentration (i.e., reactant concentration) increases.15, 16 Since the reversibility for a physical adsorption process on a plain solid surface generally declines as the speed of the process increases, the adsorption reversibility could decline as the particle concentration increases. Here, we assume that changes in *C*p can affect the metastable-equilibrium adsorption state,

$$K\_{m\epsilon} = \mathcal{Y} \times \mathbb{C}\_p^{-n} \tag{17}$$

where is a constant and *n* is an empirical parameter, *n* ≥ 0. Substituting [17] into [15] , we obtain a semi-empirical Langmuir-type *C*p effect isotherm equation

$$\Gamma = \frac{k^{\prime} \times \mathbb{C}\_p^{-n} \times \mathbb{C}\_{eq}}{1 + k \times \mathbb{C}\_p^{-n} \times \mathbb{C}\_{eq}} \tag{18}$$

where ' ' k b and k b . For a given adsorption reaction, *k'* and *k* are equilibrium adsorption constants which are independent of the *C*eq and *C*p conditions.

Substituting [17] into [16] , we obtain a Freundlich-type *C*p effect isotherm equation,

$$
\Gamma = k\_{sp} \times \mathbb{C}\_p^{-\text{st}} \times \mathbb{C}\_{eq}^{\beta} \tag{19}
$$

where *sp k* . For a given adsorption reaction, *k*sp is an equilibrium adsorption constant which is independent of the *C*eq and *C*p conditions.

The *C*p effect isotherm equations [18] and [19] predict that, by affecting the metastableequilibrium adsorption state (or the adsorption reversibility ), particle concentration can fundamentally influence the equilibrium constants or adsorption isotherms.
