6. Coping with Non-Ideality SVNA 10.3 Up until now, we have considered only ideal mixtures that (unfortunately) do not represent many cases experienced in practice. Non-ideality can take two forms: Deviations in pure component behaviour e.g. pure gases at high pressure Deviations in mixture behaviour e.g. V   xi Vi So far, our treatment of non-ideality has involved: the development of a method for describing non-ideal, single-component, gas behaviour the extension of this treatment to the description of pure liquids What remains is to revise our treatment of perfect gas mixtures and ideal solutions to account for non-ideal mixing effects. CHEE 311 J.S. Parent

Partial Properties: Thought Experiment Suppose we add a drop of water to pure acetone. What change in volume would result? If the resulting water/acetone mixture is ideal (recall definition of ideal) the volume increase is simply that of the volume of the water droplet. If the mixture behaves non-ideally, the volume increase will not equal the volume of the water droplet. The effect may, in fact, be quite different. Non-ideality in mixtures results from complex intermolecular interactions that we cannot predict. We still have to solve engineering problems (separations, property calculations, …) using these non-ideal systems. na moles Acetone nw moles H2O CHEE 311 J.S. Parent

Partial Properties: Thought Experiment If the volume change of the acetone-water mixture does not equal the volume of the water droplet, then the properties of pure water are irrelevant. We like to assign values or “contributions” to each component in non-ideal mixtures to account for the variation of a property with respect to composition. This leads us to define partial molar properties, which in our thought experiment gives us the partial molar volume for water in an acetone-water solution. This quantity represents change in solution volume as the number of moles of water is varied at a given P,T, and nacetone. CHEE 311 J.S. Parent

Partial Molar Quantities We prefer to think of mixtures in terms of their components: Overall property has a contribution from each component in the mixture. In non-ideal systems, the properties of the pure components have little meaning, forcing us to find an alternate way of defining molar quantities. If nM represents the total thermodynamic property of interest: (10.7) where is a partial molar property, also a function of (T,P, nj) A partial molar property is specific to the P,T and composition from which it is derived by equation 10.7 It is difficult to predict, but can be measured experimentally. CHEE 311 J.S. Parent

Total Properties of Non-Ideal Mixtures Ideal mixtures result from a lack of molecular interactions (ideal gas) or equivalent molecular interactions (ideal solution). In these cases, a total thermodynamic property (nM) for a mixture is: nM =  ni Mi where Mi represents the pure component property of i. Non-ideal systems do not obey this simple formula, as cross-component molecular interactions differ from pure component interactions. nM =  ni where represents the partial molar property of component i. In terms of mole fractions (dividing by n): M =  xi (10.11) If we know the partial properties of the components of the mixture (from experimental data) we can derive its total property. This is summability relation, which is opposite to 10.7 that defines a partial property CHEE 311 J.S. Parent

The Gibbs-Duhem Equation An important question we need to answer is: how do the partial molar properties of a mixture relate? Start with the definition of total Gibbs Energy at T,P: nG =  ni Gi =  ni i If we change the composition of the system at constant T,P, the Gibbs energy responds accordingly: dnG =  d(ni i) =  ni di +  i dni But we know that the total change in Gibbs energy is defined by: d(nG) = nV dP - nS dT +  i dni (10.3) which at constant P,T (dP=0, dT=0), is: d(nG) =  i dni CHEE 311 J.S. Parent

The Gibbs-Duhem Equation d(nG) provided by these two relations must be equal. Therefore, d(nG) =  ni di +  i dni =  i dni For this to be true,  ni di = 0 (10.14) This is the Gibbs-Duhem equation applied to chemical potential Why is it useful? It states that partial molar properties cannot change independently, if one partial property increases, others must decrease Estimates of partial molar properties (from experimental data, correlations…) can be checked for consistency CHEE 311 J.S. Parent

Calculating Partial Molar Props. - Binary Systems Partial molar properties (Mi) can always be derived from an equation for the solution property (M) as a function of composition: (10.7) Comparatively simple relations exist for binary systems: (10.11) therefore The Gibbs-Duhem equation tells us that: (10.14) Leaving us with: CHEE 311 J.S. Parent

Calculating Partial Molar Props. - Binary Systems We left off with: Since x1+x2=1, dx1 = - dx2 and which we can write as: Substituting the total solution property: (10.15, 10.16) What we need to calculate a partial molar property is an expression for the total molar property (M) as a function of composition. CHEE 311 J.S. Parent

Notation for the Course Our superscripts and subscripts are propagating rapidly, so let’s revisit our definitions: S = entropy of one mole of the mixture Si = entropy of one mole of pure i Si = entropy of one mole of pure i in the mixture = partial molar entropy Suggested Readings Examples 10.1, 10.2, 10.3, 10.4 - Partial Molar Properties CHEE 311 J.S. Parent

6. Non-Ideal Mixtures In our attempt to describe the Gibbs energy of real gas and liquid mixtures, we examine two “sources” of non-ideal behaviour: Pure component non-ideality concept of fugacity Non-ideality in mixtures partial molar properties mixture fugacity and residual properties We will begin our treatment of non-ideality in mixtures by considering gas behaviour. Start with the perfect gas mixture model derived earlier. Modify this expression for cases where pure component non-ideality is observed. Further modify this expression for cases in which non-ideal mixing effects occur. CHEE 311 J.S. Parent

Perfect Gas Mixtures We examined perfect gas mixtures in lecture 9. The assumptions made in developing an expression for the chemical potential of species i in a perfect gas mixture were: all molecules have negligible volume interactions between molecules of any type are negligible. Based on this model, the chemical potential of any component in a perfect gas mixture is: where the reference state, Giig(T,P) is the pure component Gibbs energy at the given P,T. We can choose a more convenient reference pressure that is standard for all fluids, that is P=unit pressure (1 bar,1 psi,etc) In this case the pure component Gibbs energy becomes: CHEE 311 J.S. Parent

Perfect Gas Mixtures Substituting for our new reference state yields: (10.28) which is the chemical potential of component i in a perfect gas mixture at T,P. The total Gibbs energy of the perfect gas mixture is provided by the summability relation: (10.11) (10.29) CHEE 311 J.S. Parent

Ideal Mixtures of Real Gases One source of mixture non-ideality resides within the pure components. Consider an ideal solution that is composed of real gases. In this case, we acknowledge that molecules have finite volume and interact, but assume these interactions are equivalent between components The appropriate model is that of an ideal solution: where Gi(T,P) is the Gibbs energy of the real pure gas: (10.30) Our ideal solution model applied to real gases is therefore: CHEE 311 J.S. Parent

Non-Ideal Mixtures of Real Gases In cases where molecular interactions differ between the components (polar/non-polar mixtures) the ideal solution model does not apply Our knowledge of pure component fugacity is of little use in predicting the mixture properties We require experimental data or correlations pertaining to the specific mixture of interest To cope with highly non-ideal gas mixtures, we define a solution fugacity: (10.42) where fi is the fugacity of species i in solution, which replaces the product yiP in the perfect gas model, and yifi of the ideal solution model. CHEE 311 J.S. Parent

Non-Ideal Mixtures of Real Gases To describe non-ideal gas mixtures, we define the solution fugacity: and the fugacity coefficient for species i in solution: (10.47) In terms of the solution fugacity coefficient: Notation: fi, i - fugacity and fugacity coefficient for pure species i fi, i - fugacity and fugacity coefficient for species i in solution CHEE 311 J.S. Parent

Calculating iv from Compressibility Data Consider a two-component vapour of known composition at a given pressure and temperature If we wish to know the chemical potential of each component, we must calculate their respective fugacity coefficients In the laboratory, we could prepare mixtures of various composition and perform PVT experiments on each. For each mixture, the compressibility (Z) of the gas can be measured from zero pressure to the given pressure. For each mixture, an overall fugacity coefficient can be derived at the given P,T: How do we use this overall fugacity coefficient to derive the fugacity coefficients of each component in the mixture? CHEE 311 J.S. Parent

Calculating iv from Compressibility Data It can be shown that mixture fugacity coefficients are partial molar properties of the residual Gibbs energy, and hence partial molar properties of the overall fugacity coefficient: In terms of our measured compressiblity: CHEE 311 J.S. Parent

Calculating iv from the Virial EOS We have used the virial equation of state to calculate the fugacity and fugacity coefficient of pure, non-polar gases at moderate pressures. Under these conditions, it represents non-ideal PVT behaviour of pure gases quite accurately The virial equation can be generalized to describe the calculation of mixture properties. The truncated virial equation is the simplest alternative: (3.31) where B is a function of temperature and composition according to: (10.65) Bij characterizes binary interactions between i and j; Bij=Bji CHEE 311 J.S. Parent