Solution thermodynamics theory—Part I Chapter 11

topics Fundamental equations for mixtures Chemical potential Properties of individual species in solution (partial properties) Mixtures of real gases Mixtures of real liquids

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?

G in a mixture (open system)

G in a mixture of k components at T and P How is this equation reduced if n =1

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:

We end up with How are dnia and dnib related at constant n?

For 2 phases, k components at equilibrium Thermal equilibrium Mechanical equilibrium Chemical equilibrium For all i = 1, 2,…k

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

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

Partial molar property Solution property Partial property Pure-species property

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 ?

Total vs. partial properties See derivation page 384

Derivation of Gibbs-Duhem equation

Gibbs-Duhem at constant T&P Useful for thermodynamic consistency tests

Binary solutions See derivation page 386

Obtain dM/dx1 from (a)

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:

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

Note curves for partial molar volumes

From Gibbs-Duhem: Divide by dx1, what do you conclude respect to the slopes?

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.

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

The ideal-gas mixture model EOS for an ideal gas Calculate the partial molar volume for an ideal gas in an ideal gas mixture

For an ideal gas mixture

For any partial molar property other than volume, in an ideal gas mixture:

Partial molar entropy (igm)

Partial molar Gibbs energy Chemical potential of component i in an ideal gas mixture ******************************************************************************* This is m for a pure component !!!

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:

solution n = PV/RT= 1 bar 1 m3/ [R x 278 K] DS = 204.89 J/K

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

solution Wideal = DH – Ts DS = -2484 J/mol = -7228 J/mol = -15,813 J/mol K Wideal = DH – Ts DS = -2484 J/mol

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

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

Residual Gibbs free energy Valid for species i in any phase and any condition

Since we know how to calculate residual properties… Zi from an EOS, Virial, van der Waals, etc

examples From Virial EOS From van der Waals EOS

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?

Fugacity of a pure liquid at P and T

Fugacity of a pure liquid at P and T

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)

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

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 = 0.724 x 300 bar = 217.14 bar GR/RT = ln f = -0.323

Problem Estimate the fugacity of cyclopentane at 110oC and 275 bar. At 110 oC the vapor pressure of cyclopentane is 5.267 bar. At those conditions, cyclopentane is a high P liquid

Find Tc, Pc, Zc,, Vc and acentric factor, w, Table B1, p. 680 Calculate reduced properties: Tr, Prsat Tr = 0.7486 and Prsat = 0.117 At P < Prsat we can use the virial EOS to calculate fisat

fisat = 0.9 P-correction term: Get the volume of the saturated liquid phase, Rackett equation Vsat = 107.55 cm3/mol f = 11.78 bar

Generalized correlations: fugacity coefficient Tables E13 to E16 Lee-Kessler

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, 11.19 (b), 11.21, 11.22 (a), 11.24(a), 11.25