2. Derive ideal mixing and the Flory-Huggins models from the van der Waals mixture partition function.

Size: px

Start display at page:

Download "2. Derive ideal mixing and the Flory-Huggins models from the van der Waals mixture partition function."

Nicholas Johns
5 years ago
Views:

1 Lecture #5 1 Lecture 5 Objectives: 1. Identify athermal and residual terms from the van der Waals mixture partition function.. Derive ideal mixing and the Flory-Huggins models from the van der Waals mixture partition function. 3. Define the concept of local compositions and compute them given a sample mixture. 4. Compute local compositions from the Wilson and RTL equations. 1. The van der Waals partition function and mixture models. Recall the van der Waals partition function. We are going to use this as a spring board for developing and understanding liquid mixture models. The van der Waals partition function for a mixture is given by Q vdw = { C qi,int ) } i i=1 i!λ 3 i i V f exp βu 0 /) where V f is the free volume of the mixture, and u 0 is the mean-field potential of the mixture. For a binary mixture, Q vdw = q 1,int) 1 q,int ) 1!!Λ Λ 3 V f exp βu 0 /) The Helmholtz free energy is given by A = kt lnq. The change on mixing at constant T, V is given by ) Q A mix = A A 1 A = kt ln Q 1 Q = kt ln V f + 1 f,1 V u 0 1 u 0,1 u 0, ) f, We identify the following: A atherm = kt ln V f f,1 V f, Then, A resid = 1 u 0 1 u 0,1 u 0, ) A mix = A atherm + A resid Where A atherm is athermal because it involves only free volume terms, which are due to hard core interactions, and hence independent of temperature. This is like an entropy of mixing. A resid is the residual free energy of mixing. This term has to do with the energetic effects of mixing, or enthalpic mixing terms.

2 Lecture #5. This model is the starting point for many liquid mixture activity coefficient models. One may make various approximations for the athermal and residual terms. 3. Example: Derive ideal mixing from assumption for the free volume. Assume that V f. Then, ) A atherm = kt ln 1 1 ) 1+ = kt ln x 1 ) 1 x ) ) 1+ = kt ln 1+ x 1 1 x ) = kt ln x x 1 1 x x = kt lnx x 1 1 xx ) = kt [x 1 lnx 1 )+x lnx )] = A IM Making a different assumption for V f gives the Flory-Huggins equation of state. 4. Example: Derive the Flory-Huggins equation. Assume that the free volume is proportional to the pure-component molar volume. A atherm = kt ln = kt ln V 1 + m 1 V V 1 1 V m V m ) = kt ln Φ 1 1 Φ ) ) [ ) 1 ) ] V1 V = kt ln V m V m = kt [ 1 lnφ 1 + lnφ ] = kt i x i lnφ i where Φ i = V i /V m is the volume fraction of component i based on the pure component volume. This is the Flory-Huggins equation for polymers. 5. Models for A resid or U resid. Direct models for A resid are rare. It has been common to propose a model for U resid and then compute A resid by thermodynamic integration. See the text for a discussion. There are three commonly used liquid mixture models which can be derived from this approach. The models are: a) Wilson Equation b) RTL Equation on-random two-liquid) c) UIQUAC Universal Quasi-Chemical) equation

3 Lecture #5 3 These models are all based on the concept of local compositions. This is a useful concept not only for these models, but for many others as well. Local Compositions: Show Figure 9. from the book. The local mole fraction, x ij is the number of moleculesof type i around acentral moleculeof type j, divided by the total number of molecules around the central type j molecule. We restrict our view to the first solvation shell. Obviously, if we included the whole fluid we would get the bulk mole fractions, rather than the local mole fractions. For Figure 9. we see that the total number of molecules in the first shell around a type 1 molecule large sphere) is 6. The number of type molecules is 4, so the local composition x 1 = 4/6 = /3, and x 11 = /6 = 1/3. Likewise, we see from the same figure that x 1 = 4/6 = /3, and x = /6 = 1/3. x ji = ji k ki where ji is the number of molecules of type j around a central molecule of type i. From this we see that x ji = j j ji k ki = 1 Thought question: What effects could give rise to the phenomenon of nonlocal compositions? a) Unlike pair interactions in the pair potential that are stronger or weaker than the like-like interactions. b) Specific interactions, e.g., hydrogen bonding, etc. c) Polar interactions. d) Ionic interactions. The local composition phenomenon is real, and can be observed experimentally as solvent clustering around a solute, etc. The local compositions can be computed in terms of a nonrandom factor, G ji The equations are x ii = x ji = x i x i +x j G ji x j G ji x i +x j G ji If G ji > 1 then x ii < x i and there is a depletion of type i around type i, i.e., the unlike interactions are stronger more favorable) than the like-like interactions. Likewise, x ji > x j. On the other hand, if G ji < 1, then x ii > x i and x ji < x j. I.e., the like-like interactions are more energetically favorable than the unlike interactions. Activity: Open a web-browser to Select the Mixture button. Change the density to 0.8, the temperature to, and epsilon1 to 4. Press the Start button and run the simulation for a few minutes to equilibrate the mixture. Then select molecules at random and compute the local compositions. If we assume

4 Lecture #5 4 a) The pair potentials between molecules can be replaced by a potential of mean force. b) The shell thickness is the distance to the first coordination shell. Then we end up with the Wilson equation. Other assumptions will lead to the RTL and the UIQUAC equations, respectively. See Table A9.1 from the book. ote that it is assumed that the mixing of the two pseudo fluids is ideal. All the nonideal effects are contained in A resid. Does this seem reasonable? 6. There are many empirical activity coefficient models: a) Margules b) van Laar c) Scatchard-Hildebrand d) See Table 4.4 in Walas 7. Example calculations with the Wilson Equation a) Problem 9.10 on page 351 of the text. Use ρ hexane = g, ρ cm 3 heptane = g, cm g 3 and molecular weights of and mole, respectively, to compute the volumes needed to compute G ji for Wilson s equation from Table 9. in the text. For the acetonitrile + water mixture use ρ acetonitrile = 0.78 g, ρ cm 3 water = g, and molecular cm g 3 weights of and 18.0 mole, respectively. The Wilson parameters for hexane1) and heptane) are: λ 1 /R = 59.1 K, λ 1 /R = 66. K. Use the equations in Table 9. to get G ji at T = K. Ṽ 1 = MW 1 /ρ 1 = /0.659 = cm 3 /mol Ṽ = /0.684 = cm 3 /mol G 1 = Ṽ Ṽ 1 exp λ 1 /RT) =.633 G 1 = We can now calculate the local mole fractions from Eq. 9.39) in the book. 0.5 x 11 = = 0.75 x 1 = 1 x 11 = 0.75 x = x 1 = 0.10 Given that hexane and heptane are very similar, these local compositions are very surprising. We would not expect that heptane would cluster around hexane to shift the local mole fraction from the bulk value of 0.5 to the calculated value of We conclude that the Wilson equation over estimates the local composition effects for this mixture. For acetonitrile1) + water) we have Ṽ 1 = MW 1 /ρ 1 = cm 3 /mol Ṽ = 18.0 cm 3 /mol G 1 = Ṽ Ṽ 1 exp λ 1 /RT) = 0.71 G 1 = 0.11

5 Lecture #5 5 We can now calculate the local mole fractions from Eq. 9.39) in the book. 0.5 x 11 = = x 1 = 1 x 11 = 0.13 x = x 1 = The acetonitrile + water mixture is highly nonideal, with hydrogen bonding and polar interactions contributing the the complexity of this system. The water hydrogen bonds with its self, giving very strong like-like interactions. The acetonitrile tends to disrupt this hydrogen network. The large value of x 11 reflects this phenomenon. Hence, these local mole fractions are consistent with our molecular picture of the mixture. b) Problem 9.11 on page 351 of the text. Use the RTL nonrandom two-liquid) model to repeat the calculations above. For hexane1)+heptane)wehavea 1 /R = 616.3K,anda 1 /R = 364K,withα = Using The equations for RTL in Table 9. we have τ 1 = a 1 = 616.3/ =.033 RT τ 1 = 364/ = G 1 = exp ατ 1 ) = exp ) = 0.56 G 1 = exp[ 0.316) 1.007)] = x 11 = x 1 = x = x 1 = These local compositions are closer to what one would expect to observe for the hexane + heptane mixture. The local mole fractions are much closer to the bulk mole fractions. For acetonitrile1)+water) we have a 1 /R = K, a 1 /R = K, and α = These values give τ 1 =.1603 τ 1 = G 1 = exp ατ 1 ) = exp ) = G 1 = exp ) = x 11 = x 1 = 0.91 x = x 1 = 0.39 These local compositions are in line with the local compositions computed from the Wilson equation. They predict stronger like-like interactions than unlike interactions. This fits with our molecular reasoning of this system.

CH.7 Fugacities in Liquid Mixtures: Models and Theories of Solutions

CH.7 Fugacities in Liquid Mixtures: Models and Theories of Solutions The aim of solution theory is to express the properties of liquid mixture in terms of intermolecular forces and liquid structure. The