Handout #2. Reading Thompson (1967) Researches in Geochem, v.2.,

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Thomas Malone
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1 1.480 Handout # Non-ideal solutions Reading Thompson (1967) Researches in Geochem, v.., Solutions for which ΔHmix > 0 Depending on the contribution from ΔHmix, we can have one of two general cases. G G ex G soln I. Solution is stable over entire composition range. G ideal X II. Contribution from ΔH is such that solution is stable only near the end members and we have phase separation for intermediate compositions. G niss G G ex G soln G ideal X In general, we will separate out an excess free energy of mixing term. Gex 1

2 then G soln = X i G i - TΔS mix + G ex i and we allow G ex = E ex - PV ex - TS ex Another way of rationalizing Gex is that it is the contribution to free energy left over after the ideal and standard state free energy contributions are subtracted out. Representations of G ex There are many ways of representing Gex, and we'll talk about one representation here and about some other examples later. We will use Margules' power series expansion in terms of the solution components. The choice of a power series as the mathematical representation is a matter of convenience. Some of the criteria are: 1) The method be useful in prediction. ) Should account for variations in data using the minimum number of fitting parameters, 3) The model should be consistent with mineral site symmetry constraints or molecular constraints. The Margules' expression represents Gex as the third power of composition. To obtain values for the coefficients of this power series we need to define another expression: G niss G niss = G soln + TΔS mix = X i G i + G ex = A + BX + CX + D X 3 G 1 * and G * are the hypothetical excess free energies implied by Henry s law behavior. G * is obtained by extrapolating the behavior of component (the solute) in a solvent consisting dominantly of component 1 to a hypothetical state of behavior where the concentration is pure component. G 1 and G are the standard state free energies for phases consisting of pure component 1 or, respectively. We will derive an expression for G ex that expresses the deviation from ideal behavior as the difference between these two end member standard states (G * - G). First, we'll look at the values of these coefficients in the limits as X 0

3 G ex + X 1 G 1 + X G = A + BX + CX + DX 3 as X 0, G ex 0 so G ex + G 1 = A and A = G 1 3

4 as X 1, G ex 0 and G 1 + B + C + D = G G * Now take the derivative of G si G G niss / X = B + CX + 3DX So, well evaluate these in the limits of infinite dilution This is the well known Henry s Law behavior exhibited by solutions. G * 1 W G1 G niss G ex W G G 0 as X 0 G si / X = B = (G * - G 1 ) G X 1 as X 1 G si / X = B + C + 3D = (G - G 1* ) Therefore, (G * - G 1 ) + C +3D = (G - G 1* ) (expression for G niss / X ) G 1 + (G * - G 1 ) + C +D = G (Expression for G niss ) multiply second expression by and subtract -G 1 - G * + D = -G - G * 1 D = (G * - G ) - (G * 1 - G 1 ) plug back and solve for C C = G - G * - (G * - G ) +(G * 1 - G 1 ) 4

5 C = (G 1 * - G 1 ) - (G * - G ) plug back into G niss and collect terms G si = G 1 + (G * - G 1 )X + [(G 1 * - G 1 ) - (G * - G ) ]X + [(G * - G ) - (G 1 * - G 1 ) ]X 3 if we define W G1 = (G 1 * - G 1 ) W G = [(G * - G ) C = W G1 - W G D = W G - W G1 (error in JBT) No we are ready to plug in for G niss and simplify G si = X 1 G 1 + X G + W G X + [W G1 - W G ]X +[ W G - W G1 ]X 3 G si = X i G i + W G X (1-X + X ) + W G1 X (1-X ) G si = X i G i + W G X (X 1 ) + W G1 X (X 1 ) So, G ex = W G X (X 1 ) + W G1 X (X 1 ) and the total free energy of the solution can be expressed by plugging G niss in the ideal contribution to the free energy from the existence of a solution. This gives us G soln. Since μ i = G i we can write G soln in the following way G soln = X 1 μ 1 + X μ + nrt(x 1 ln X 1 + X lnx ) + (W G1 X +W G X 1 )X 1 X recall μ 1 = G soln + X G soln / X and μ = G soln + (1-X ) G soln / X P,T = constant P,T = constant 5

6 G soln / X = (μ - μ 1 ) + RTln X /X 1 + W G1 X (3X 1-1) - W G X 1 (3X -1) then plug in & simplify: μ 1 = μ 1 + nrtlnx 1 + X (W G1 + (W G -W G1 )X 1 ) The observed departure of a solution from ideal behavior for component 1 is given by the term: X (W G1 + (W G -W G1 )X 1 ) This term can be thought of as μ 1 ex and μ 1 ex = nrt ln γ 1 The γ term is a way of expressing the deviation of a solution from ideal behavior through the activity of a component in a phase. a i = γ i X i and μ 1 = μ nrt lna 1 RT lna 1 = RTln X 1 + RTlnγ 1 Therefore, nrtlnγ 1 = X (W G1 + (W G - W G1 )X 1 ) and γ 1 = exp[x (W G1 + (W G - W G1 )X 1 )/nrt] Symmetric solutions We call solutions obeying this power series relation asymmetric solutions. There is a special case called a symmetric solution where W G = W G1 and our expressions reduce to: 6

7 G ex = W G X 1 X G soln = X 1 μ 1 o + X μ o + nrt (X 1 lnx 1 + X lnx ) + X 1 X W G And μ 1 = μ 1 + RTlnX 1 + (1-X) 1 (W G ) The G 1 * and G * in our expressions are related to the Henry's law chemical potential. They are the chemical potential of component 1 in a physically unattainable state - that is in pure component. We used Margules (1895) power series and we truncated the power series and obtained a parameter fit. Originally this expression was used in gas-liq systems to describe vapor pressure of a component in a binary solution. Other expressions for excess free energy (l) Van laar equations log γ 1 = A 1 /[1+(A 1 X 1 /A 1 X )] () Redlich-Kister or Guggenheim equation ΔG ex /RT = X 1 (1-X 1 ) B + C(X 1-1) + D(X 1-1) +... an infinite series - again truncated. 7

8 (3) Wilson Equation ΔG ex /RT = ΣX i ln (ΣXΛ ij ) i j 8

Chapter 5. Simple Mixtures Fall Semester Physical Chemistry 1 (CHM2201)

Chapter 5. Simple Mixtures Fall Semester Physical Chemistry 1 (CHM2201) Chapter 5. Simple Mixtures 2011 Fall Semester Physical Chemistry 1 (CHM2201) Contents The thermodynamic description of mixtures 5.1 Partial molar quantities 5.2 The thermodynamic of Mixing 5.3 The chemical