Direct calculation of the equilibrium composition for multi-component Langmuir isotherms in batch adsorption

Abstract

The extended Langmuir isotherm is commonly used to describe multi-component adsorption. In batch adsorption, the equilibrium composition can be computed by solving simultaneously the Langmuir equations and the material balance equations numerically. We demonstrate that for given values of the initial fluid-phase concentrations, the adsorbed phase composition of all components can be expressed in terms of the fluid-phase concentration of a reference species. As a result, the adsorbed phase composition is easily derived by combining the isotherm expression for the reference species with the corresponding material balance. We derive a general graphical representation of the underlying mathematics valid for N components along with the analytic solution for two-component systems.

1 Introduction

The Langmuir isotherm model is used extensively to describe adsorption equilibrium both in the gas phase and in the liquid phase. This model was originally developed for one-component systems [1] and describes adsorption using two parameters: the monolayer binding capacity and a temperature-dependent adsorption constant reflecting the adsorption strength [2,3,4]. The Langmuir model has been extended to multicomponent systems based on kinetics arguments [5, 6]. Although thermodynamic consistency requires that all components have the same monolayer adsorption capacity [7], in practice, the extended Langmuir isotherm is also commonly used to describe systems with unequal capacities. In this form, the model is useful to correlate adsorption equilibrium data and to predict mixture adsorption [8,9,10,11,12,13,14,15,16].

If the equilibrium fluid phase composition is known, calculating the adsorbed phase composition with the extended Langmuir isotherm is straightforward. Other cases, however, require combining the isotherm expressions with suitable material balances. An example is the case of batch adsorption where the initial composition and the amounts of adsorbent and fluid phase are known. In this case, calculating the equilibrium composition normally requires the solution of a system of equations. In this work, we provide explicit functions to predict the equilibrium composition in one and two-component systems as well as a graphical framework for the efficient calculation of equilibrium in systems with more than two components.

2 Model equations

For liquid phase adsorption, the extended Langmuir isotherm expresses the adsorbed-phase concentration $q_i$, for each component i, as a function of the fluid-phase concentrations $C_i$ as follows:

$$\begin{array}{lllll} {q_i}({C_1}, \ldots ,{C_N}) = & \frac{{{q_{m,i}}{K_i}{C_i}}}{{1 + \sum\limits_{j = 1}^N {{K_j}} {C_j}}}\\ = & \frac{{{a_i}{C_i}}}{{1 + \sum\limits_{j = 1}^N {{K_j}} {C_j}}},\;i = 1,N \end{array}$$

(1)

where $q_{m,i}$ is the monolayer adsorption capacity, $K_i$ is the Langmuir association constant, and $a_i = q_{m,i} K_i$. The adsorbed-phase concentration $q_i$ is expressed either in terms of adsorbent mass or adsorbent volume, with the two being related to each other by the density of the adsorbent material. Typical units of $q_i$ and $C_i$ are kg/$\hbox {m}^3$ or mol/$\hbox {m}^3$, while $a_i$ is dimensionless. The selectivity, $\alpha _{i,j} = a_i/a_j$, is independent of composition. In Eq. (1) the adsorbed concentration $q_i$ depends on all N fluid-phase concentrations. For example, for a binary system ($N = 2$) the adsorbed-phase composition, $q_1$ and $q_2$, can be illustrated by 3D plots such as Fig. 4.1 in [12] or Fig. 1 in [16]. For gas-phase adsorption, concentrations are typically replaced by partial pressure, but here we retain concentration units as we focus on liquid-phase adsorption.

Material balances for a batch adsorption system starting with an initially clean adsorbent yield the following equations:

$$\begin{aligned} q_i = \frac{V}{V_R} \cdot (C_{0,i} - C_i), \quad i=1,N, \end{aligned}$$

(2)

where V and $V_R$ are the fluid phase and adsorbent volumes, respectively, and $C_{0,i}$ is the initial concentration of component i. Equation 2, as well as the ensuing development, assume that both fluid phase and adsorbent volumes remain constant as a result of adsorption. This assumption is valid provided that adsorption occurs from a dilute solution in the case of liquid phase adsorption, or from an excess amount of inert gas in the case of gas phase adsorption.

A simultaneous solution of Eqs. (1) and (2) can be obtained numerically as described, for example, in [8,9,10]. In the following, we describe an analytic approach to find the equilibrium composition in batch adsorption. We obtain explicit formulas for two-component systems, and we illustrate a graphical approach, based on including the material balance as a line in the isotherm plot.

3 Direct calculation of the equilibrium composition

3.1 Explicit relations for one and two-component systems

Let

$$\begin{aligned} \gamma = V_R / V. \end{aligned}$$

(3)

Based on $\gamma$ and on the initial concentrations $C_{0,i}$, we will determine the equilibrium state $(C_i,q_i)$. First, we define the dimensionless intermediate quantity

$$\begin{aligned} u = \gamma \big / \Bigl ({1 + \sum _{j=1}^N K_j C_j} \Bigl ). \end{aligned}$$

(4)

We show how to calculate u without using $C_i$. Once the value of u is determined, we can readily obtain $C_i$ by Eq. (8) below and thus also $q_i$ by Eq. (2).

Equation 1 implies

$$\begin{aligned} \begin{aligned} \gamma q_i&= a_i C_i u, \end{aligned} \end{aligned}$$

(5)

while Eq. (2) implies

$$\begin{aligned} \gamma q_i = C_{0,i} - C_i. \end{aligned}$$

(6)

Combining Eqs. (5) and (6) and solving for u gives

$$\begin{aligned} \begin{aligned} u = \frac{1}{a_i} \Bigl (\frac{C_{0,i}}{C_i} - 1 \Bigr ), \end{aligned} \end{aligned}$$

(7)

or solving for $C_i$,

$$\begin{aligned} \begin{aligned} C_i&= \frac{C_{0,i}}{1 + a_i u}. \end{aligned} \end{aligned}$$

(8)

By combining Eqs. (4) and (8) we obtain

$$\begin{aligned} \gamma /u = 1 + \sum _{j=1}^{N} K_j \, C_{0,j} \cdot \frac{1}{1 + a_j u}. \end{aligned}$$

(9)

Define the polynomials P(u) and $P_1(u),\dots , P_N(u)$ of degree N and $N-1$, respectively, by

$$\begin{aligned} \begin{gathered} P(u) = \prod _{j=1}^{N} (1 + a_j u) \quad \text { and} \\ P_j(u) = \frac{ P(u) }{1 + a_j u} = \prod _{\begin{array}{c} k=1 \\ k\ne j \end{array}}^{N} (1 + a_k \, u). \end{gathered} \end{aligned}$$

(10)

We can rewrite Eq. (9) by using Eq. (10) as

$$\begin{aligned} \gamma /u = 1 + \sum _{j=1}^{N} K_j \, C_{0,j} \cdot \frac{P_j(u)}{P(u)} \end{aligned}$$

(11)

or rearranging,

$$\begin{aligned} ( \gamma - u ) \cdot P(u) = u \cdot \sum _{j=1}^{N} K_j \, C_{0,j} \cdot P_j(u), \end{aligned}$$

(12)

or in other words, $Q(u) = 0$, with the polynomial Q(u) defined by

$$\begin{aligned} Q(u) = ( u - \gamma ) \cdot P(u) + u \cdot \sum _{j=1}^{N} K_j \, C_{0,j} \cdot P_j(u). \end{aligned}$$

(13)

Thus we can determine u as a polynomial root. The polynomial Q(u) is of degree $N+1$, so for one- and two component systems ($N=1,2$) this approach gives an explicit formula for u. The appropriate root among the $N+1$ roots is to be selected by physical considerations. The selection is immediate if the other roots are negative or complex valued, as is the case with all our examples. In other cases, roots giving $C_i > C_{0,i}$ can be eliminated. From u compute $C_i$ by Eq. (8) and compute $q_i$ by Eq. (2).

The roots of Q(u) are as follows:

• For one-component adsorption ($N=1$), we have

  $$\begin{aligned} \begin{gathered} P(u) = 1 + a_1 \, u, \quad P_1(u) = 1, \end{gathered} \end{aligned}$$

  (14)

  and Q(u) is quadratic,

  $$\begin{aligned} Q(u) = a_1 u^2 + ( K_1 \, C_{0,1} - \gamma a_1 + 1 ) u - \gamma , \end{aligned}$$

  (15)

  which is easily solved to find u.

• For two-component adsorption ($N=2$), we have

  $$\begin{aligned} \begin{aligned}&P(u) = (1 + a_1 \, u) \cdot (1 + a_2 \, u), \\&P_1(u) = 1 + a_2 \, u, \quad P_2(u) = 1 + a_1 \, u. \end{aligned} \end{aligned}$$

  (16)

  Thus, the polynomial is cubic or $Q(u) = a u^3 + b u^2 + c u + d$, with

  $$\begin{aligned} \begin{aligned} a&= a_1 a_2, \\ b&= a_1 \cdot ( 1 + K_2 \, C_{0,2} ) + a_2 \cdot ( 1 + K_1 \, C_{0,1} ) - \gamma a_1 a_2, \\ c&= 1 + K_1 \, C_{0,1} + K_2 \, C_{0,2} - \gamma (a_1 + a_2), \\ d&= -\gamma . \end{aligned} \end{aligned}$$

  (17)

  Appendix A provides the explicit solution of this cubic equation.

• For three-component adsorption ($N = 3$), the polynomial Q(u) in Eq. (13) is quartic, so root formulas exist but they are intricate and numerical root-finding is simpler.

• For adsorption with four or more components ($N \ge 4$), the polynomial Q(u) in Eq. (13) has degree five or higher, so there are no root formulas. Numerical root-finding is required.

3.2 Example

Table 1 Data for the adsorption of mixtures of antibody dimer (component 1) and monomer (component 2) on CHT Type I from [16] and calculations based on Eqs. (16) and (17)

Consider the adsorption data for mixtures of an antibody dimer (component 1) and monomer (component 2) on ceramic hydroxyapatite (CHT Type I) reported by [16]. Regression of isotherm data with the extended Langmuir isotherm, Eq. (1), gave $q_{m,1} = 110$, $q_{m,2} = 94.7$ both in mg/mL, $K_1 = 230$, $K_2 = 65.6$ both in mL/mg. Table 1 summarizes the experimental data and the calculated values based on Eqs. (16) and (17). The calculated values are exact to three significant digits for the given parameter values. The agreement between experimental and calculated values is within the reported estimated errors of the regressed parameter values.

4 Multi-component Langmuir isotherm graphs

4.1 Explicit formulas

Combining Eq. (8) for a general component j with Eq. (7) we obtain

$$\begin{aligned} \begin{aligned}&C_j(C_i) = C_{0,j} \Big / \Bigl [{1 + \frac{a_j}{a_i} \Bigl (\frac{C_{0,i}}{C_i} - 1 \Bigr )}\Bigr ]. \end{aligned} \end{aligned}$$

(18)

Substituting Eq. (18) in Eq. (1) gives the explicit isotherm formula

$$\begin{aligned} \begin{aligned}&q_i(C_i) = \frac{a_i C_i}{ 1 + \sum _{j=1}^{N} \displaystyle {K_j C_{0,j}} \Big / \Bigl [{1 + \frac{a_j}{a_i} \Bigl (\frac{C_{0,i}}{C_i} - 1 \Bigr )}\Bigr ] }. \end{aligned} \end{aligned}$$

(19)

4.2 Common abscissa for the isotherms

Substituting Eq. (18) with suitably adjusted index not only in the denominator of Eq. (1) but also in the numerator gives for a general component k and the reference component i,

$$\begin{aligned} \begin{aligned}&q_k(C_i) = \frac{\displaystyle {a_k C_{0,k}} \Big / \Bigl [{1 + \frac{a_k}{a_i} \Bigl (\frac{C_{0,i}}{C_i} - 1 \Bigr )}\Bigr ]}{ 1 + \sum _{j=1}^{N} \displaystyle {K_j C_{0,j}} \Big / \Bigl [{1 + \frac{a_j}{a_i} \Bigl (\frac{C_{0,i}}{C_i} - 1 \Bigr )}\Bigr ] }. \end{aligned} \end{aligned}$$

(20)

By Eq. (20) we can draw the N isotherms on a common abscissa $C_i$. Note that by Eq. (18) the coordinate transform $C_i \rightarrow C_k$ is not linear, so the graph of $q_k(C_i)$ is not just a scaled version of $q_k(C_k)$.

In the figures discussed below we choose the most strongly adsorbed species as the reference component and we assign to it the index $i=1$, such that $C_1$ is the common abscissa in each of the plots.

4.3 Graphical construction of equilibrium points

The equilibrium point $(C_i,q_i)$ computed in Sect. 3.1 can also be obtained by equating the isotherm equation Eq. (19) with the material balance equation Eq. (2). This computation, with $C_i$ instead of u as variable, produces a less tangible polynomial, but it allows us to illustrate the solution directly in the isotherm plot, as follows. Intersect the graph of the isotherm $q_i(C_i)$ from Eq. (19) with the straight line defined by the material balance Eq. (2). The intersection is the equilibrium point $(C_i,q_i)$, see Fig. 1 where $i=1$. The figure, for a binary mixture ($N=2$), also illustrates that if the isotherm graphs $q_1(C_i),\dots ,q_N(C_i)$ are combined in the same plot, note the common abscissa, then their intersection with a vertical line from $(C_i,q_i)$ lets us also read $q_1,\dots ,q_N$ from that same plot.

By Fig. 2 we illustrate the equilibrium composition calculation in a hypothetical three-component case, for various initial fluid-phase concentrations. We keep $C_{0,1} = C_{0,3} = 2$ fixed and vary $1 \le C_{0,2} \le 4$. The material balance line for component 1 is shown in red, for an arbitrarily chosen value of $V/V_R = 1$.

Graphical representation of phase equilibrium problems by plotting an equilibrium line, describing the thermodynamic relationship between the phase compositions, along with an operating line describing material balances, are commonplace in many fields such as distillation, absorption, liquid-liquid extraction, and crystallization, e.g., [17] as well as for single-component adsorption problems, e.g., [18]. The method presented in this section extends this approach to the case of multicomponent adsorption systems.

Fig. 1

Equilibrium composition calculation for the third data of Table 1, which is $V = 0.5$ mL, $V_R = 0.00571$ mL, $C_{0,1} = 2.54$ mg/mL, $C_{0,2} = 1.74$ mg/mL. The graphs of $q_1(C_1)$ and $q_2(C_1)$ are blue and orange, respectively. The material balance line is shown in red. The two dots indicate the predicted equilibrium values

Fig. 2

Adsorbed-phase concentrations $q_i(C_1)$ attained at equilibrium, for batch adsorption in a three-component system, with $q_{m,1} = 3$, $q_{m,2} = 2$, $q_{m,3} = 1$, $K_1 = 13$, $K_2 = 12$, $K_3 = 11$, for various initial concentrations ($C_{0,1} = C_{0,3} = 2$ fixed, $C_{0,2}$ varying). The blue, yellow, and green graphs are $q_1$, $q_2$, and $q_3$, respectively. The straight red line is the material balance line for component 1 with $V/V_R = 1$. The intersection of the red line with the blue graph gives $C_1$ and thus $q_1,q_2,q_3$

4.4 Further geometric properties

The explicit formulas in this section also allow us to compute geometric properties of the isotherm graphs. For example, from the slopes of the isotherm $q_k(C_k)$ at $C_k=0$,

$$\begin{aligned} \begin{aligned} \frac{\partial q_k}{\partial C_k} \bigg |_{C_k \rightarrow 0} = a_k, \end{aligned} \end{aligned}$$

(21)

and using Eq. (18) to compute

$$\begin{aligned} \begin{aligned} \frac{\partial C_k}{\partial C_i} \bigg |_{C_i \rightarrow 0} = \frac{a_i}{a_k} \cdot \frac{ C_{0,k} }{ C_{0,i} } \end{aligned} \end{aligned}$$

(22)

we obtain that the slopes of the isotherms $q_k(C_i)$ drawn on the common abscissa $C_i$ are by the chain rule, at $C_i=0$,

$$\begin{aligned} \begin{aligned} \frac{\partial q_k}{\partial C_i} \bigg |_{C_i \rightarrow 0} = \frac{\partial q_k}{\partial C_k} \bigg |_{C_k \rightarrow 0} \, \frac{\partial C_k}{\partial C_i} \bigg |_{C_i \rightarrow 0} = a_i \frac{ C_{0,k} }{ C_{0,i} }. \end{aligned} \end{aligned}$$

(23)

5 Conclusions

The Langmuir model is the cornerstone of adsorption theories, Its extension to multi-component adsorption is the Extended Langmuir model, which can be solved numerically to find the equilibrium composition in a batch adsorption system. Our results show how to determine the equilibrium composition directly, with an explicit formula for the case of one or two components ($N = 1,2$). From the analytic results we also derive a graphical construction and geometric properties of the isotherm graph.

Appendix A: Solution of the cubic equation

The analytical solution of cubic equations is available in a number of standard handbooks, e.g., [19]. A cubic equation $a u^3 + b u^2 + c u + d = 0$ as encountered in Sect. 3.1 has three distinct real roots if and only if the auxiliary numbers

$$\begin{aligned} \begin{aligned} p = \frac{3ac - b^2}{3a^2}, \qquad q = \frac{27a^2d - 9abc + 2b^3}{27a^3} \end{aligned} \end{aligned}$$

(24)

satisfy $4p^3 + 27q^2 < 0$ (note it implies $p<0$). In this case the three roots $u_1, u_2, u_3$ are

$$\begin{aligned} \begin{gathered} u_k = 2\,\sqrt{-\frac{p}{3}}\,\cos \biggl [ \frac{\varphi }{3} + \frac{2\pi k}{3} \biggr ] - \frac{b}{3a}, \quad k=0,1,2, \\ \text {with } \ \varphi = \arccos \biggl [\frac{3q}{2p}\sqrt{-\frac{3}{p}}\,\biggr ]. \end{gathered} \end{aligned}$$

(25)

Similar formulas are available for the case $4p^3 + 27q^2 \ge 0$ that involves multiple roots or complex roots.

To compare (25) with the version in [19], note that for $p<0$ we have $\sqrt{-p^3} = -p \cdot \sqrt{-p}$ and that

$$\begin{aligned} \cos \left[ \frac{\arccos (x)}{3} + \frac{2 \pi k_1}{3}\right] = -\cos \left[ \frac{\arccos (-x)}{3} + \frac{2 \pi k_2}{3}\right] , \end{aligned}$$

(26)

for $(k_1,k_2) = (0,1)$, (1, 0) and (2, 2).