Ion exchange isotherms in solid: electrolyte solution systems

Abstract

It is shown show that the ion exchange isotherms and the law of mass action are equivalent, the c/a versus c functions can be derived from the law of mass action (c and a: the concentration of ions in ion exchanger and solution, respectively). The equations are applied for cation exchange processes of bentonite clay (cobalt, manganese, mercury ions with calcium-bentonite; strontium ions with sodium-bentonite; cesium ions with lanthanide bentonite; lutetium ion with calcium-bentonite). The linear or non-linear shape of the isotherms does not prove the heterogeneity of the ion exchanger or the interaction among the sorbed cations.

Appendix

Homovalent exchange

In case of homovalent exchange, all exponents in Eq. 2 are equal to 1. Thus,

$$K_{\text{B,A}} = \frac{{a_{\text{A}} c_{\text{B}} }}{{a_{\text{B}} c_{\text{A}} }}$$

(3)

In the ion exchange processes, all exchange sites are always occupied by the ions, that is, the cation exchange capacity, or by the term used in the sorption isotherm, the number of exchange sites (ζ) is the sum of the concentration of the ions on the solid (a _A and a _B):

$$\zeta = a_{\text{A}} + a_{\text{B}}$$

(4)

From here,

$$a_{\text{B}} = \zeta - a_{\text{A}}$$

(5)

By substituting Eq. 5 into Eq. 3:

$$K_{\text{B,A}} = \frac{{a_{\text{A}} c_{\text{B}} }}{{\left( {\zeta - a_{\text{A}} } \right)c_{\text{A}} }}$$

(6)

$$\frac{{\left( {\zeta - a_{\text{A}} } \right)}}{{a_{\text{A}} }} = \frac{{c_{\text{B}} }}{{K_{\text{B,A}} c_{\text{A}} }} = \frac{\zeta }{{a_{\text{A}} }} - 1$$

(7)

$$\frac{\zeta }{{a_{\text{A}} }} = 1 + \frac{{c_{\text{B}} }}{{K_{\text{B,A}} c_{\text{A}} }}$$

(8)

$$\frac{{c_{\text{A}} \zeta }}{{a_{\text{A}} }} = c_{\text{A}} + \frac{{c_{\text{B}} }}{{K_{\text{B,A}} }}$$

(9)

$$\frac{{c_{\text{A}} }}{{a_{\text{A}} }} = \frac{1}{\zeta }\left( {c_{\text{A}} + \frac{{c_{\text{B}} }}{{K_{\text{B,A}} }}} \right)$$

(10)

Equation 10 is equal to the homovalent ion exchange isotherm derived [29] where

$$K_{\text{B,A}} = \frac{{K_{\text{B}} }}{{K_{\text{A}} }}$$

(11)

where K _A and K _B are the parameter characterizing the Gibbs energy of the ions.

The differences between the homovalent ion exchange isotherm equation (Eq. 10), the simple and competitive Langmuir adsorption isotherms were discussed in detail in [29].

Similar equation can be described for the B ion:

$$\frac{{c_{\text{B}} }}{{a_{\text{B}} }} = \frac{1}{\zeta }\left( {c_{\text{B}} + K_{\text{B,A}} c_{\text{A}} } \right) = \frac{1}{\zeta }\left( {c_{\text{B}} + \frac{{c_{\text{A}} }}{{K_{\text{A,B}} }}} \right)$$

(12)

Heterovalent exchange: the exchange of monovalent and bivalent ions

Firstly, let’s assume a monovalent ion exchanger and exchange the monovalent ions to bivalent ones:

$$2{\text{Me}}_{1} - {\text{S}} + {\text{Me}}_{2}^{2 + } = {\text{Me}}_{2} - {\text{S}} + 2{\text{Me}}_{1}^{ + }$$

(13)

The indexes 1 and 2 mean the valences of the ions. The equilibrium constant of the process (Eq. 13) is:

$$K_{1,2} = \frac{{a_{2} c_{1}^{2} }}{{a_{1}^{2} c_{2} }}$$

(14)

where the index 1, 2 means that monovalent ions are exchanged to bivalent ions.

The number of exchange sites (ζ) can be expressed both for the monovalent and bivalent ions. For monovalent ions (ζ _mono):

$$\zeta_{\text{mono}} = a_{1} + 2a_{2}$$

(15)

$$a_{1} = \zeta_{\text{mono}} - 2a_{2}$$

(16)

Equation 16 can be substituted into Eq. 14, we obtain:

$$K_{1,2} = \frac{{a_{2} c_{1}^{2} }}{{\left( {\zeta_{\text{mono}} - 2a_{2} } \right)^{2} c_{2} }}$$

(17)

and from here

$$1 = \frac{1}{{K_{1,2} }}{ \times }\frac{{a_{2} c_{1}^{2} }}{{\left( {\zeta_{\text{mono}} - 2a_{2} } \right)^{2} c_{2} }}$$

(18)

We make some equivalent mathematical transformations (Eqs. 19–21):

$$\frac{{\zeta_{\text{mono}} - 2a_{2} }}{{a_{2} }} = \frac{{\zeta_{\text{mono}} }}{{a_{2} }} - 2 = \frac{1}{{K_{1,2} }}{ \times }\frac{{c_{1}^{2} }}{{\left( {\zeta_{\text{mono}} - 2a_{2} } \right)c_{2} }}$$

(19)

$$c_{2} \left( {\frac{{\zeta_{\text{mono}} }}{{a_{2} }} - 2} \right) = \frac{1}{{K_{1,2} }}\frac{{c_{1}^{2} }}{{\left( {\zeta_{\text{mono}} - 2a_{2} } \right)}}$$

(20)

$$\zeta_{\text{mono}} \frac{{c_{2} }}{{a_{2} }} = 2c_{2 + } \frac{1}{{K_{1,2} }}{ \times }\frac{{c_{1}^{2} }}{{\left( {\zeta_{\text{mono}} - 2a_{2} } \right)}}$$

(21)

Finally, we obtain a c ₂/a ₂ versus c ₂ function (Eq. 22):

$$\frac{{c_{2} }}{{a_{2} }} = \frac{1}{{\zeta_{\text{mono}} }}\left( {2c_{2} + \frac{1}{{K_{1,2} }}\frac{{c_{1}^{2} }}{{\left( {\zeta_{\text{mono}} - 2a_{2} } \right)}}} \right)$$

(22)

Formally, Eq. 22 is a sorption isotherm for the bivalent ions in a monovalent-bivalent ion exchange process.

Similar equation can be also derived for the monovalent ions. In order to do this, consider the reverse reaction of Eq. 13 when bivalent ions are exchanged to monovalent ions:

$${\text{Me}}_{2} - {\text{S}} + 2{\text{Me}}_{1}^{ + } = 2{\text{Me}}_{1} - {\text{S}} + {\text{Me}}_{2}^{2 + }$$

(23)

The equilibrium constant of Eq. 23 is:

$$K_{2,1} = \frac{{c_{2} a_{1}^{2} }}{{c_{1}^{2} a_{2} }} = \frac{1}{{K_{1,2} }}$$

(24)

Equation 24 expresses that the equilibrium constants (K _1,2 and K _2,1) are reciprocal to each other.

From Eq. 15, we can express a ₂:

$$a_{2} = \frac{{\zeta_{\text{mono}} - a_{1} }}{2}$$

(25)

By substituting Eq. 25 into Eq. 24, we obtain:

$$K_{2,1} = \frac{{c_{2} a_{1}^{2} }}{{c_{1}^{2} \frac{{\zeta_{\text{mono}} - a_{1} }}{2}}}$$

(26)

Equation 26 is transformed as Eq. 17 (details in Eqs. 18–21):

$$\frac{{c_{1} }}{{a_{1} }} = \frac{1}{{\zeta_{\text{mono}} }}\left( {c_{1} + \frac{2}{{K_{2,1} }}\frac{{c_{2} a_{1} }}{{c_{1} }}} \right)$$

(27)

As mentioned previously in this paragraph, the number of exchange sites can be expressed to bivalent ions (ζ _bi) as follows:

$$\zeta_{\text{bi}} = \frac{{a_{1} }}{2} + a_{2}$$

(28)

From here the concentration of the ion in the solid ion exchanger are:

$$a_{1} = 2\left( {\zeta_{\text{bi}} - a_{2} } \right)$$

(29)

$$a_{2} = \zeta_{\text{bi}} - \frac{{a_{1} }}{2}$$

(30)

By substituting Eqs. 29 and 30 into Eqs. 17 and 24, respectively, and after similar mathematical transformations, we obtain the c ₂/a ₂ versus c ₂ functions:

$$\frac{{c_{2} }}{{a_{2} }} = \frac{1}{{\zeta_{\text{bi}} }}\left( {c_{2} + \frac{1}{{K_{1,2} }}\frac{{c_{1}^{2} }}{{2^{2} { \times }\left( {\zeta_{\text{bi}} - a_{2} } \right)}}} \right)$$

(31)

$$\frac{{c_{1} }}{{a_{1} }} = \frac{1}{{\zeta_{\text{bi}} }}\left( {\frac{1}{2}c_{1} + \frac{1}{{K_{2,1} }}\frac{{a_{1} c_{2} }}{{c_{1} }}} \right)$$

(32)

Similarly, the reaction equations and equilibrium constants of monovalent and trivalent, bivalent and trivalent ions, respectively, can be described. The number of exchange sites can be expressed all for the mono, bi and trivalent cations; and from here the c/a versus c functions can be derived. These functions are summarized in Table 1.