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Solution thermodynamics theory
Chapter 11
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topics Fundamental equations for mixtures Chemical potential
Properties of individual species in solution (partial properties) Mixtures of real gases Mixtures of real liquids
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A few equations For a closed system
Total differential form, what are (nV) and (nS) Which are the main variables for G?? What are the main variables for G in an open system of k components?
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G in a mixture (open system)
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G in a mixture of k components at T and P
How is this equation reduced if n =1
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2 phases (each at T and P) in a closed system
Apply this equation to each phase Sum the equations for each phase, take into account that In a closed system:
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We end up with How are dnia and dnib related at constant n?
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For 2 phases, k components at equilibrium
Thermal equilibrium Mechanical equilibrium Chemical equilibrium For all i = 1, 2,…k
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In order to solve the PE problem
Need models for mi in each phase Examples of models of mi in the vapor phase Examples of models of mi in the liquid phase
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Now we are going to learn about
Partial molar properties Because the chemical potential is a partial molar property At the end of this section think about this What is the chemical potential in physical terms What are the units of the chemical potential How do we use the chemical potential to solve a PE (phase equilibrium) problem
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Partial molar property
Solution property Partial property Pure-species property
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example Open beaker: ethanol + water, equimolar Total volume nV
T and P Add a drop of pure water, Dnw Mix, allow for heat exchange, until temp T Change in volume ?
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Total vs. partial properties
See derivation page 384
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Derivation of Gibbs-Duhem equation
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Gibbs-Duhem at constant T&P
Useful for thermodynamic consistency tests
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Binary solutions See derivation page 386
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Obtain dM/dx1 from (a)
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Example 11.3 We need 2,000 cm3 of antifreeze solution: 30 mol% methanol in water. What volumes of methanol and water (at 25oC) need to be mixed to obtain 2,000 cm3 of antifreeze solution at 25oC Data:
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solution Calculate total molar volume
We know the total volume, calculate the number of moles required, n Calculate n1 and n2 Calculate the volume of each pure species
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Note curves for partial molar volumes
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From Gibbs-Duhem: Divide by dx1, what do you conclude respect to the slopes?
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Read and work example 11.4 Given H=400x1+600x2+x1x2(40x1+20x2) determine partial molar enthalpies as functions of x1, numerical values for pure-species enthalpies, and numerical values for partial enthalpies at infinite dilution Also show that the expressions for the partial molar enthalpies satisfy Gibbs-Duhem equation, and they result in the same expression given for total H.
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Now we are going to start looking at models for the chemical potential of a given component in a mixture The first model is the ideal gas mixture The second model is the ideal solution As you study this, think about the differences, not only mathematical but also the physical differences of these models
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The ideal-gas mixture model
EOS for an ideal gas Calculate the partial molar volume for an ideal gas in an ideal gas mixture
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For an ideal gas mixture
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For any partial molar property other than volume, in an ideal gas mixture:
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Partial molar entropy (igm)
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Partial molar Gibbs energy
Chemical potential of component i in an ideal gas mixture ******************************************************************************* This is m for a pure component !!!
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Problem What is the change in entropy when 0.7 m3 of CO2 and 0.3 m3 of N2, each at 1 bar and 25oC blend to form a gas mixture at the same conditions? Assume ideal gases. We showed that:
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solution n = PV/RT= 1 bar 1 m3/ [R x 278 K] DS = J/K
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Problem What is the ideal work for the separation of an equimolar mixture of methane and ethane at 175oC and 3 bar in a steady-flow process into product streams of the pure gases at 35oC and 1 bar if the surroundings temperature Ts = 300K? Read section 5.8 (calculation of ideal work) Think about the process: separation of gases and change of state First calculate DH and DS for methane and for ethane changing their state from P1, T1, to P2T2 Second, calculate DH for de-mixing and DS for de-mixing from a mixture of ideal gases
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solution Wideal = DH – Ts DS = -2484 J/mol = -7228 J/mol
= -15,813 J/mol K Wideal = DH – Ts DS = J/mol
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Now we introduce a new concept: fugacity
When we try to model “real” systems, the expression for the chemical potential that we used for ideal systems is no longer valid We introduce the concept of fugacity that for a pure component is the analogous (but is not equal) to the pressure
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We showed that: Pure component i, ideal gas Component i in a mixture
of ideal gases Let’s define: For a real fluid, we define Fugacity of pure species i
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Residual Gibbs free energy
Valid for species i in any phase and any condition
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Since we know how to calculate residual properties…
Zi from an EOS, Virial, van der Waals, etc
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examples From Virial EOS From van der Waals EOS
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Fugacities of a 2-phase system
One component, two phases: saturated liquid and saturated vapor at Pisat and Tisat What are the equilibrium conditions for a pure component?
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Fugacity of a pure liquid at P and T
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Fugacity of a pure liquid at P and T
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example For water at 300oC and for P up to 10,000 kPa (100 bar) calculate values of fi and fi from data in the steam tables and plot them vs. P At low P, steam is an ideal gas => fi* =P* Get Hi* and Si* from the steam tables at 300oC and the lowest P, 1 kPa Then get values of Hi and Si at 300oC and at other pressure P and calculate fi (P)
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Problem For SO2 at 600 K and 300 bar, determine good estimates of the fugacity and of GR/RT. SO2 is a gas, what equations can we use to calculate f = f/P Find Tc, Pc, and acentric factor, w, Table B1, p. 680 Calculate reduced properties: Tr, Pr Tr=1.393 and Pr=3.805
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High P, high T, gas: use Lee-Kessler correlation
From tables E15 and E16 find f0 and f1 f0 = 0.672; f1 = 1.354 f = f0 f1w = 0.724 f = f P = x 300 bar = bar GR/RT = ln f =
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Problem Estimate the fugacity of cyclopentane at 110oC and 275 bar. At 110 oC the vapor pressure of cyclopentane is bar. At those conditions, cyclopentane is a high P liquid
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Find Tc, Pc, Zc,, Vc and acentric factor, w, Table B1, p. 680
Calculate reduced properties: Tr, Prsat Tr = and Prsat = 0.117 At P < Prsat we can use the virial EOS to calculate fisat
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fisat = 0.9 P-correction term: Get the volume of the saturated liquid phase, Rackett equation Vsat = cm3/mol f = bar
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Generalized correlations: fugacity coefficient
Tables E13 to E16 Lee-Kessler
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Now we introduce the concept of fugacity for a given component in a mixture
Fugacity of component i in a mixture plays an analogous role to the partial pressure of the same component i in an ideal mixture At low pressure, the fugacity of i in the mixture tends to be the partial pressure of i. This means that the fugacity coefficient of i in the mixture tends to 1 at low pressures
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We showed that: Pure component i, ideal gas Component i in a mixture
of ideal gases Let’s define: For a real fluid, we define Fugacity of pure species i
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Now lets consider component i in solution
Component i in a mixture of ideal gases Now lets consider component i in solution Component i in a real solution Fugacity of component i in solution We can also define a fugacity coefficient of i in solution
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We can also relate fugacity (in solution) to a residual property
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Residual partial Gibbs free energy
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How to calculate fugacity coefficients in solution
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How to calculate fugacity coefficients in solution
Valid for mixtures of gases at low and moderated pressures
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How to calculate fugacity coefficients in solution
From the Virial EOS (truncated after the second term)
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You can show that Derivation is in page 406, but try to do it by yourself first
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problem For the system methane (1)/ethane (2)/propane (3) as a gas, estimate at T = 100oC, P = 35 bar, y1 =0.21, and y2 =0.43
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solution 1) Get Tc, Pc, and Vc for each of the components
2) Calculate mixture properties: wij, TCij, PCij, Zcij and Vcij equations to 11.74 For example we will have T11, T22, T33, T12, T13, T23 From this equation get Bij that we need to calculate dik
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For example Results of d in cm3/mol d11 = 0, d22 =0, d33 =0
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results
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When we deal with mixtures of liquids or solids
We define the ideal solution model Compare it to the ideal gas mixture, analyze its similarities and differences
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The ideal solution model
Component i in a mixture of ideal gases That is obtained by using In the first term of this equation Now we define Ideal solution model
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Other thermodynamic properties for the ideal solution: partial molar volume
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partial molar entropy in the ideal solution
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partial molar enthalpy in the ideal solution
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Chemical potential ideal solution
Chemical potential component i in a Real solution Chemical potential Pure component i Subtracting: For the ideal solution
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Lewis-Randall rule Lewis-Randall rule
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When is the ideal solution valid?
Mixtures of molecules of similar size and similar chemical nature Mixtures of isomers Adjacent members of homologous series
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problem For the system methane (1)/ethane (2)/propane (3) as a gas, estimate at T = 100oC, P = 35 bar, y1 =0.21, and y2 =0.43 Assume that the mixture is an ideal solution Obtain reduced pressures, reduced temperatures, and calculate
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Results: methane (1) ethane (2) propane (3)
Virial model Ideal solution
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Now we want to define a new type of residual properties
Instead of using the ideal gas as the reference, we use the ideal solution
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Excess properties The most important excess function is the excess Gibbs free energy GE Excess entropy can be calculated from the derivative of GE with respect to T Excess volume can be calculated from the derivative of GE with respect to P And we can also define partial molar excess properties
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Definition of activity coefficient
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Summary
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Summary
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HW # 3, Due Monday, September 17
Problems 11.2, 11.5, 11.8, 11.12, 11.13 HW # 4, Due Monday, September 24 Problems 11.18, (b), 11.21, (a), 11.24(a), 11.25
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