Ion Exchange

Isotherms

Models

Thermodynamic Exchange Constant

Only ions adsorbed as outer-sphere complexes or in the diffuse ionswarm 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)

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

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

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)

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 nearlynon-preference Ca – Mgexchange on 2:1 mineral.Dominant surface was Si tetrahedral sheet with diffuse charge. Use of ClO4

- avoided solutioncomplexes.

The same result is obtained if instead of modeling activity of adsorbedspecies 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 aredifferent.

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

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

Substituting in terms of XCa and ECa and including γis

1 = [XCa (1 – ECa)2CT2 γNa

2] / {[2(1 – XCa)2 / (2 – XCa)] (ECa CT γCa)}

which rearranges to

XCa2 - 2XCa + 2 / (1 - ECa)2CT2γNa

2 / 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 substitutionor dXCa / dECa at ECa = 0and ECa = 1

Deviation from non-preference in heterovalent exchange occurs even with2:1 minerals dominated by Si tetrahedral surface. Non-preference isothermnot shown but would lie below (Ca – Na) or above (Na – Mg) data.

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 / Q0.02910 0.01490 0.08 0.74 0.02370 0.01740 0.06 0.78 XMg = 2qMg / Q0.01850 0.01970 0.06 0.95

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

E Mg

0.0 0.2 0.4 0.6 0.8 1.0

X M

g

0.0

0.2

0.4

0.6

0.8

1.0

E Mg

0.0 0.2 0.4 0.6 0.8 1.0

X M

g

0.0

0.2

0.4

0.6

0.8

1.0

KNaMg XMgAlt SQErr2 R2

1.039 0.4215 0.0057  1.872 0.5679 0.0010  1.693 0.7633 0.0000  1.243 0.8186 0.0009  2.408 0.8609 0.0004  2.196 0.8910 0.0001  2.337 0.9174 0.0001  1.821 0.9390 0.0000  1.491 0.0083 0.97

The form of conditional exchange constantused mole fraction to model surface phaseactivities. Vanselow, KV.

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) = XA

F(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 theexchange equilibria across the full range of surface phase compositionsusing a single value, a constant. This value would, therefore, approximatethe thermodynamic exchange constant.

(AZ+ads) = XA

F(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) = XA

F(Z) = XA, i.e., F(Z) = 1

Notice that this form is very close to the exchange isotherm data and if isothermdata were used to calculate k at each point, k = KGaines-Thomas(2 / CT) (γCa / γNa

2),for this heterovalent exange and k = KGaines-Thomas for homovalent exchange.

E Mg

0.0 0.2 0.4 0.6 0.8 1.0

X M

g

0.0

0.2

0.4

0.6

0.8

1.0

XMgAlt SQErr1 SQErr2 R2 KNaMg

0.3843 0.1788 0.0015   2.5110.5444 0.0284 0.0031   5.3470.7403 0.0001 0.0004   5.4650.8027 0.0004 0.0002   4.1070.8466 0.0126 0.0012   8.6060.8798 0.0179 0.0005   8.0030.9092 0.0256 0.0004   8.7250.9333 0.0296 0.0001   6.877

0.2935 0.0073 0.97 4.490

The form of conditional exchange constantused charge fraction to model surface phaseactivities. Gaines-Thomas, KGT.

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) = XA

Z , then (Caads) = XCa2 and (Naads) = XNa

which gives

KGapon = XCa (Na+) / XNa (Ca2+)1/2

Thus, KGapon = XCa1/2 (KGaines-Thomas)1/2

(AZ+ads) = NA exp(F(NA, NB))

Vanselow

Simplest among such models with F(NA, NB) = 0, thus, surfacephase 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 bemanipulated in such way as to give a thermodynamic exchange constantbased on exchange isotherm data. May furthermore calculate ΔGo, ΔHo and ΔSo for the exchange reaction. See handout.

KNaMg XMg lnKVdXMg

1.039 0.3457 0.01311.872 0.6000 0.15941.693 0.7609 0.08471.243 0.7890 0.00612.408 0.8810 0.08082.196 0.9024 0.01692.337 0.9286 0.02221.821 0.9406 0.00721.491 0.0356

0.4260 SUM

1.5312 EXP(SUM) K

1

ln K = ln KV dXB 0

Extrapolated to XMg = 1with KV = 1.821

XMg

0.0 0.2 0.4 0.6 0.8 1.0

ln K

V

-1.0

-0.5

0.0

0.5

1.0

ln KV = -0.278 + 1.097XMg R2 = 0.56

1

ln K = ln KV dXB = 0.270 and K = 1.310 0

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