Fugacity & Fugacity Coefficient - libvolume2.xyz

1 1 Chemical Engineering Thermodynamics II (0905323)02 - The Molar Gibbs Free Energy & Fugacity of a Pure ComponentDr. Ali Khalaf Al-matarChemical Engineering DepartmentUniversity of II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component2 The product of pressure times exponential of the Gibbs free energy difference divided by RTis termed the Fugacity The Fugacity divided by pressure is the Fugacity Coefficient Fugacity & Fugacity Coefficient0(,) (,)1expexpPIGgT P g T PRTfPPvdPRTRTP == 0(,) (,)1expexpPIGfgTPgTPRTvdPPRT RTP === 2 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component3 Fugacity has units of pressure. Fugacity Coefficient is dimensionless Fugacity is very useful in phase equilibria of Fugacity and its Coefficient0limPfP =00lim lim 1 PPfP ==Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component4 Equilibrium Criterion Using Fugacity : Derivation Derive an expression for the equilibrium criterion based on Fugacity .

2 }Start with the three equilibrium conditions. Note that Tand Pare fixed and the Gibbs free energy must be equal for the coexisting phases(,) (,)(,)(,)(,) ln(,) lnIIIIIIIGIGgTP g TPfTPf TPgTP RTgTP RTPP=+=+3 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component5 The ideal gas molar Gibbs free energy has the same value at the same Tand P At equilibrium:}the Fugacity must be equal in the coexisting phases.}also, Fugacity coefficients must be equal.(,)(,)lnlnIIIfTPf TPPP=Equilibrium Criterion Using Fugacity : Final Form(,) (,)(,) (,)IIIIIIfTP f TPTPTP ==Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component6 Fugacity and Phase Equilibria Since Fugacity is related to EOS. It provides an approach to solving phase equilibria problems The problem with using the definition of Fugacity directly is that the integration requires the EOS to be in the form v = f(T,P).

3 The EOS so far use P= f(T, v). With the nonlinear forms of EOS it is difficult to transform them to the form required by fugacity4 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component7 Transformation of Integration Variable Convert the integration from volume explicit to pressure explicit using the following transformation Using this transformation the expression for the Fugacity Coefficient becomes1()PP PdPd PvdvdZdvvvZv= = (,) 1ln lnln ( 1)vvfTPRTPdv Z ZPRT v = == + Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component8 Effect of Tand Pon the Fugacity The effect of temperature and pressure upon the Fugacity can be derived and are given by:ln ( , )TTfTPgRTvPP == []2ln ( , )/(,) (,)IGPfTP PhT P h T PTRT = 5 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component9 Fugacity of a Pure gaseous Species For gases, use an EOS combined with the definition of the Fugacity .

4 }The EOS is used to provide the dependence of pressure on volume.}Frequently, you may need to numerically integrate certain numerical data instead of using an EOST hermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component10 Fugacity of a Gas Using the PR EOS/(,) 1ln lnln(1)VvZRTPVVVvfTPRTPdv Z ZPRTv == == + ()(1 2 )ln(1) lnln22(1 2)VVVVVfAZBZZBPBZB ++= + 2()aPARTbPBRT==6 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component11 Representative EOS22()()()()( ) ( )() 1 TPvb vvb bvbRTaTTPTTTRTbP = ++ = =+ =+ =22()()()( ) ( )() 1 TPvb vvbRTaTTPTTTRTbP = += =+ =+ = soave -Redlich-Kwong (SRK)Peng-Robinson (PR)Thermo II.

5 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component12 Cubic EOS Coefficients320ZZ Z +++= AB + B2 + B3 AB AB 2B 3B2 1 + B B2A 1 1 B SRKvdW2()aPART=bPBRT=Initial Guess for solutionVapor (Vapor like): ideal gas (Z= 1).Liquid: Reduced covolume (Z= B).7 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component13 Solution Methodology for PR (Tc;Pc; ). bin PR ain PR .. a(T). reduced parameters Aand the cubic constants in the Solve the cubic for the roots and determine if they fall in the subcooled liquid, superheated vapor or the two phase coexistence () 1 1()( ) ( )()cccccRTbPTTTRTaTTPaPARTbPBRT ==+ =+ ===32223(1 ) ( 2 3 )()0 ZBZA B B ZAB BB+ + + + + + =Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component14 Fugacity Using Corresponding States Principle (CSP) The principle of corresponding states may be used to obtain the Fugacity utilizing the reduced temperature and pressure as parameters.

6 }A third factor may be used to enhance the accuracy. This third factor may be Pitzer s acentric factor, or the critical II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component15 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component16 You can use an EOS to get the Fugacity of a liquid in a manner similar to that for a gas , Peng-Robinson EOS yields Alternatively, one can derive an expression for Fugacity from the equilibrium condition of the equality of Fugacity to obtainFugacity of a Pure Liquid()(1 2 )ln( 1) lnln22(1 2)LLLLLfAZBZZBPBZB ++= + (),1(,)/expvapPLvapsat TPfTP P f PvdPRT = 9 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component17 Poynting Correction(),,(,)expLsatvapLvapsat TvPPffTP PPRT = Pvapis the vapor pressure at the specified T, and the (f/P)satis the Fugacity Coefficient of the fluid (liquid = vapor) at saturation.

7 The exponential term is referred to as the Poyntingpressure correction. It is important}At high pressures}At low temperatures (cryogenic systems) Assuming that liquids are incompressibleThermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component18 Solids may undergo several phase transitions. Usually dealing with solids that undergo sublimation; which is what the superscript satrefers to in this case. Similar to the derivation of liquids assuming that}Solids sublimation pressures are very small ( =1).}Solids are incompressibleFugacity of a Pure Solid (Phase)11,1(,) ()expJJPSsatJJsat TPffTP P TvdPPRT+= = ()()(,) ()expSsatSsatvPP TfTP P TRT = 10 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component19 We know that fLand fVcan be obtained via an EOS using the liquid and vapor compressibility respectively.

8 For the PR EOS At equilibriumVapor Pressure from EOS(, ) (, )LvapVvapfTPfTP=()(1 2 )ln( 1) lnln22(1 2)LLLLLfAZBZZBPBZB ++= + ()(1 2 )ln(1) lnln22(1 2)VVVVVfAZBZZBPBZB ++= + Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component20 Algorithm for vapor Pressure Using EOS Consequently, solve for the vapor pressure using any EOS. This solution is iterative in nature:}Assume a pressure}Solve the cubic for ZLand ZV}Evaluate fLand fV}If the two fugacities are equal to within a certain convergence criteria then the pressure assumed is the vapor pressure Always check for trivial solutions (too low or too high pressures leading to single phase).11 Thermo II: 02-The Molar Gibbs Free Energy & Fugacity of a Pure Component21