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Ion Exchange Isotherms Models Thermodynamic Exchange Constant

Ion Exchange Isotherms Models Thermodynamic Exchange Constant Only ions adsorbed as outer-sphere complexes or in the diffuse ion swarm are exchangeable Exchange capacities can be determined either at the native pH of soil or

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Ion Exchange Isotherms Models Thermodynamic Exchange Constant

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  1. Ion Exchange Isotherms Models Thermodynamic Exchange Constant Only ions adsorbed as outer-sphere complexes or in the diffuse ion swarm are exchangeable Exchange capacities can be determined either at the native pH of soil or at a buffered pH (effective and total exchange capacities, respectively)

  2. Exchange Isotherms Typically developed for binary systems Plot charge fraction adsorbed against charge fraction in solution For the surface phase, Xi = Ziqi / Q = Ziqi / (z1q1 + z2q2) where Z is valance (absolute value if an anion), q is surface excess and Q is exchange capacity For the solution phase, Ei = Zi Ci / CT where C is solution concentration of charge

  3. Homovalent exchange Ca2+ - Mg2+ XCa = 2qCa / Q XMg = 2qMg / Q XCa + XMg = 1 ECa = 2CCa / CT EMg = 2CMg / CT ECa + EMg = 1 If the adsorbent had no preference for either species, it make sense that the isotherm should conform to XCa = ECa

  4. This can be shown if start with an expression for an exchange equilibrium constant Mg2+(ads) + Ca2+(aq) = Ca2+(ads) + Mg2+(aq) K = XCa(Mg) / XMg(Ca) which assumes Xi accurately models the adsorbed phase activity For non-preference, K = 1 Let’s do some substituting in the above equilibrium expression (Ca) = γCa CCa = ECa γCa CT (Mg) = EMg γMg CT = (1 – ECa) γMg CT XMg = (1 – XCa)

  5. Therefore for K = 1, 1 = [XCa (1 – ECa) γMg CT] / [(1 – XCa)(ECa γCa CT)] [(1 – XCa)(ECa γCa CT)] = XCa [(1 – ECa) γMg CT] (1 – XCa) / XCa = (1 – ECa) / ECa since γCa = γMg and XCa = ECa Example of nearly non-preference Ca – Mg exchange on 2:1 mineral. Dominant surface was Si tetrahedral sheet with diffuse charge. Use of ClO4- avoided solution complexes.

  6. The same result is obtained if instead of modeling activity of adsorbed species by X, it is modeled by mole fraction, N, on the surface where N1 = q1 / (q1 +q2) and N2 = q2 / (q1 + q2) This is obviously true for homovalent exchange since in this case Ni = Xi, however, for heterovalent exchange, i.e., Ca – Na, the expressions are different.

  7. Deviation from non-preference Ca – Mg exchange in mixed mineralogy system

  8. For the exchange reaction, 2Na+(ads) + Ca2+(aq) = Ca2+(ads) + 2Na+(aq) for which K = NCa(Na)2 / NNa2(Ca) i.e., surface phase activities modeled as mole fractions Derive XCa = F(ECa) in the case of non-preference exchange, i.e., K = 1 This is less straightforward but start with the substitutions NCa = qCa / (qCa + qNa) XCa = 2qCa / Q XNa = qNa / Q NCa = XCa / (XCa + 2XNa) = XCa / (2 - XCa) since XCa + XNa = 1 NNa = qNa / (qCa + qNa) = 2(1 - XCa) / (2 - XCa) CCa = ECaCT / 2 CNa = ENaCT = (1 – ECa)CT since ECa + ENa = 1 where CT = 2CCa + CNa

  9. Substituting in terms of XCa and ECa and including γis 1 = [XCa (1 – ECa)2CT2 γNa2] / {[2(1 – XCa)2 / (2 – XCa)] (ECa CT γCa)} which rearranges to XCa2 - 2XCa + 2 / (1 - ECa)2CT2γNa2 / ECaγCa = 0 from which XCa = 1 - [β / (1 + β)]1/2 where β = (1 - ECa)2CTγNa2 / 2ECaγCa and ranges from  to 0 ECa = 0, XCa = 0 ECa = 1, XCa = 1 ECa = y, XCa > y Can show this by substitution or dXCa / dECa at ECa = 0 and ECa = 1

  10. Deviation from non-preference in heterovalent exchange occurs even with 2:1 minerals dominated by Si tetrahedral surface. Non-preference isotherm not shown but would lie below (Ca – Na) or above (Na – Mg) data.

  11. 5. Given the below exchange data for solution, mNa and mMg, and adsorbed, qNa and qMg, phases, graph the exchange isotherm, examine applicability of the non-preference isotherm and compare it with a fitted isotherm. mNa mMg qNa qMg ------ mol / kg ------ - mol / kg - 0.04950 0.00117 0.53 0.28 ENa = mNa / (mNa + 2mMg) = mNa / CT 0.04740 0.00234 0.30 0.45 0.04400 0.00700 0.22 0.70 EMg = 2mMg / CT 0.03830 0.00940 0.23 0.86 0.03400 0.01240 0.10 0.74 XNa = qNa / (qNa + 2qMg) = qNa / Q 0.02910 0.01490 0.08 0.74 0.02370 0.01740 0.06 0.78 XMg = 2qMg / Q 0.01850 0.01970 0.06 0.95

  12. ENa EMg XNa XMg γNa γMg XMg-NP SQErr1 SQErr2 R2 0.9549 0.0451 0.6543 0.3457 0.8206 0.4534 0.2254 0.1788 0.0000 0.9101 0.0899 0.4000 0.6000 0.8206 0.4534 0.3625 0.0284 0.0110 0.7586 0.2414 0.2391 0.7609 0.8206 0.4534 0.6122 0.0001 0.0021 0.6708 0.3292 0.2110 0.7890 0.8206 0.4534 0.6965 0.0004 0.0001 0.5782 0.4218 0.1190 0.8810 0.8206 0.4534 0.7642 0.0126 0.0025 0.4941 0.5059 0.0976 0.9024 0.8206 0.4534 0.8140 0.0179 0.0012 0.4051 0.5949 0.0714 0.9286 0.8206 0.4534 0.8583 0.0256 0.0009 0.3195 0.6805 0.0594 0.9406 0.8206 0.4534 0.8950 0.0296 0.0002 0.7685 0.2935 0.0180 0.94

  13. The form of conditional exchange constant used mole fraction to model surface phase activities. Vanselow, KV.

  14. Exchange Models These equilibrium expressions are referred to as selectivity coefficients. Largely differ based on how surface phase activities are approximated. Either as a function of equivalent fraction or mole fraction on the adsorbent, (AZ+ads) = XAF(Z) or (AZ+ads) = NA exp(F(NA, NB)), where B is the other cation in the binary exchange. The objective in modeling surface phase activities is to best describe the exchange equilibria across the full range of surface phase compositions using a single value, a constant. This value would, therefore, approximate the thermodynamic exchange constant.

  15. (AZ+ads) = XAF(Z) Gaines-Thomas Model surface phase activities as equivalent fractions directly, for example, K = XCa(Na)2 / XNa2(Ca) where XCa = 2qCa / (2qCa + qNa) = 2qCa / Q XNa = qNa / (2qCa + qNa) = qNa / Q and XCa + XNa = 1 In this case, (AZ+ads) = XAF(Z) = XA, i.e., F(Z) = 1 Notice that this form is very close to the exchange isotherm data and if isotherm data were used to calculate k at each point, k = KGaines-Thomas(2 / CT) (γCa / γNa2), for this heterovalent exange and k = KGaines-Thomas for homovalent exchange.

  16. The form of conditional exchange constant used charge fraction to model surface phase activities. Gaines-Thomas, KGT.

  17. Gapon The exchange reaction may be written ½ Ca2+(aq) + Na+(ads) = ½ Ca2+(ads) + Na+(aq) for which K(XCa)1/2 = XCa (Na+) / XNa (Ca2+)1/2 = KGapon If (AZ+ads) = XAZ , then (Caads) = XCa2 and (Naads) = XNa which gives KGapon = XCa (Na+) / XNa (Ca2+)1/2 Thus, KGapon = XCa1/2 (KGaines-Thomas)1/2

  18. (AZ+ads) = NA exp(F(NA, NB)) Vanselow Simplest among such models with F(NA, NB) = 0, thus, surface phase activities are modeled directly as mole fractions 2Na+(ads) + Ca2+(aq) = Ca2+(ads) + 2Na+(aq) K = NCa(Na)2 / NNa2(Ca) = 1 where NCa = qCa / (qCa + qNa) NNa = qNa / (qCa + qNa) and NCa + NNa = 1 This form of a conditional exchange constant (selectivity coefficient) can be manipulated in such way as to give a thermodynamic exchange constant based on exchange isotherm data. May furthermore calculate ΔGo, ΔHo and ΔSo for the exchange reaction. See handout.

  19. 1 ln K =  ln KV dXB 0 Extrapolated to XMg = 1 with KV = 1.821

  20. ln KV = -0.278 + 1.097XMg R2 = 0.56 1 ln K =  ln KV dXB = 0.270 and K = 1.310 0

  21. Since ΔGo = ΔHo – TΔSo = -RT ln K R ln K = - ΔHo / T + ΔSo if exchange experiment done at two temperatures, R ln KT2 – R ln KT1 = -ΔHo / T1 + ΔHo / T2 = ΔHo (1 / T2 – 1 / T1) ΔHo = R ln (KT2 / KT1) x T1T2 / (T1 – T2) ΔSo = R ln K + ΔHo / T

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