Supplementary Notes for Chapter 9 Mxture Thermodynamcs Key ponts Nne major topcs of Chapter 9 are revewed below: 1. Notaton and operatonal equatons for mxtures 2. PVTN EOSs for mxtures 3. General effects of mxng on heat and work nteractons and state property changes 4. Gbbs-Duhem relatonshp and thermodynamc consstency 5. Mxng functons 6. Ideal gas mxtures and deal solutons 7. Fugacty and actvty concepts 8. Fugacty coeffcents from PVTN EOS property relatonshps 9. Actvty coeffcents from G property relatonshps 10. Ideal reversble work effects n mxng or separatng components 1. Notaton and operatonal equatons for mxtures Partal molar propertes are extensve propertes that determne how derved propertes change as a functon of mole number or composton. For a general property, B, whch could be U, H, S, V, or A, the partal molar B s defned as: B B = (9-8) N TPN,, j [] There are a few mportant relatonshps n Chapter 9 that allow you to calculate partal molar propertes gven the extensve form B of the property or the ntensve form B. An mportant one of these s: B B = B x j (9-53) j 1 x j TPxj,,, Note that for a bnary, two component system that Eq.(9-53) quckly yelds the tangentntercept method of evaluatng partal molar propertes from a graph of B versus x j at constant T and P. For example, see Fgure 9.1 and the dscusson n the text. 2. PVTN EOSs for mxtures Volume and pressure explct equatons of state are commonly used to represent the volumetrc propertes of flud mxtures of both gases and lquds. The key feature that dstngushes a mxture EOS from ts pure component counterpart s the presence of compostonal dependence. Ths dependence expresses tself n the form of so-called mxng rules that ncorporate pure component EOS parameters and weght them [ ] Modfed: 10/16/2003 1
proportonally to the concentraton of each component followng a specfc mathematcal recpe. Wth the excepton of the vral equaton of state no rgorous theoretcal approach exsts to specfy a mxng rule recpe. The most common approach s to use some varaton of the Lorentz-Bertlelot rules used by van der Waals and others over a century ago (see Eqs (9-17 and 9-18). In mxtures that exhbt consderable non-dealty, a bnary or hgher order nteracton parameter δ j s ntroduced to capture specfc nteractons between molecules of type and j. Each mxture PVTN EOS wll have some prescrbed recpe for ts mxng rules. For example see the conventons followed for the RK and PR EOS on pages 324-326. In stuatons where hgh level non-deal effects are present often more complex forms for mxng rules are ntroduced. The Wong-Sandler and Chueh-Prausntz rules dscussed on pages 327-328 are examples of ths type of mxng rule. 3. General effects of mxng on heat and work nteractons and state property changes There are a few typcal classes of problems that you should be able to solve. These nclude: Calculatng the change n enthalpy as a result of mxng to determne a heat nteracton for mantanng a constant temperature or to follow some prescrbed recpe for temperature, for example, see Problem 9.2. Calculatng a change n volume as a result of mxng to determne a change n densty or the magntude of a work nteracton at constant pressure. 4. Gbbs-Duhem relatonshp and thermodynamc consstency For mxture data and correlatons for any property B, the partal molar quanttes are nterrelated through the use of the general Gbbs-Duhem relatonshp: n B B xdb = dt dp = 1 T + Px, P Tx, Note that B s a functon of T, P, and x for = 1,, n. The Gbbs-Duhem relatonshp can be appled to any partal molar property, such as: H,V, µ, ln ˆφ, ln ˆf,G, ln a, ln γ,etc. For a bnary system, the Gbbs-Duhem relatonshp s frequently used to check thermodynamc consstency of thermodynamc data; for example, actvty coeffcents. Addtonally, for a bnary mxture, f you have a measurement of the partal molar property of one component as a functon of composton you can determne the property for the other component. Modfed: 10/16/2003 2
5. Mxng functons n + + + + ( 1 ) ( 1 ) B B T,P,x, =,...,n x B T,P,x, =,...,n (9-68) mx = 1 whch s n ntensve form. A smlar expresson results for B mx n extensve form wth x replaced by N. You should be famlar wth the concept of a reference state (+). Strctly speakng, reference states are arbtrary, but n practce several common forms appear for x +, T +, and P +, for example: (1) pure at T and P of the mxture (2) x + 0 at T and P of the mxture that s n an nfnte dluton state where Henry s law behavor s followed (3) a fxed composton of 1 molal or 1 molar concentraton at T and P of the mxture that behaves n some deal manner commonly used for electrolytes (see Chapter 12) 6. Ideal gas mxtures and deal solutons In ths secton of the text, a set of defntons were used to characterze non-deal solutons n terms of a devaton from deal behavor. g deal gas mxture: G =µ = RT ln yp +λ( T ) ID deal soluton: G =µ = RT ln x +Λ ( T,P ) where λ and Λ are constants specfc to component. The key ponts to remember are that partal pressure y P s the deal gas mxture compostonally dependent varable whle for deal condensed phase soluton t s the mole fracton x. 7. Fugacty and actvty concepts For a real mxture or soluton we ntroduce the followng models: real flud mxture: G = µ = RT ln ˆf +λ + + + + real soluton: G =µ = RT ln a +µ ( T,P,x ) whch lead to the followng defntons of the fugacty coeffcent and actvty coeffcent ˆφ and γ, respectvely: and ˆ ˆf φ ---- represents a devaton from deal gas/flud mxture behavor yp ˆf f ˆ γ = ---- represents a devaton from deal soluton behavor a ˆ f + x Modfed: 10/16/2003 3
8. Fugacty coeffcent relatonshps from PVTN EOS property models There are two basc approaches one nvolvng pressure explct EOSs lke the PR or RKS, and the other for volume explct EOSs, such as Vral or Correspondng States formulatons, ncludng compressblty expansons n densty or molar volume or smlarly structured equatons. Equatons (9-142) and (9-143) provde convenent forms for pressure explct EOS models for mxture and pure components, respectvely, whle Eqs(9-129) and (9-127) work for volume explct EOS models. For example, for component n a mxture: or RT V ln ˆ P RT φ = dv RT ln Z N V T,V,Nj [] RT ln ˆφ =+ P 0 RT V P dp At ths pont you should know how to calculate the fugacty or fugacty coeffcent for a pure component usng a pressure- or volume-explct EOS or for a component n a bnary mxture usng a sutable PVTN EOS that has been properly formulated wth mxng rules for ts constants (for example, the a mx and b mx constants n the RKS EOS) n terms of pure component propertes and a bnary nteracton parameter, eg δ j. Beng able to do ths provdes a powerful tool for calculatons requred later n the course, for example to estmate the vapor pressure of a pure component you would equate ˆφ for the lqud and vapor phases at a gven temperature by estmatng the P and usng the EOS to calculate lqud and vapor volumes (denstes) untl the fugacty coeffcents for each phase were equal. 9. Actvty coeffcents from G property relatonshps The actvty coeffcent s defned n terms of the partal molar excess Gbbs free energy of mxng and can be drectly related to the fugacty usng the defnton of actvty: G RT ln γ = G = N [] T,P,Nj + where γ a ˆ ˆ x = f f x To evaluate the partal dervatve we need an expresson for G : ID G Gmx G = f T, P, N ( = 1,..., n) whch represents the dfference between the actual enthalpy of mxng and the enthalpy of mxng for an deal soluton at the same T, P and composton as the actual mxture. Modfed: 10/16/2003 4
Typcally, one has access to a model that gves G = G / N as a functon of T, P, and x, and wth Eq. (9-53) one can easly obtan γ for each component, e.g. G RT ln γ = G x j x j T,P,x j, In addton, the Gbbs-Duhem equaton can be appled to calculate the other actvty coeffcent gven a set of data for one component (e.g. f γ solvent s known as a functon of composton then γ solute can be estmated by ntegratng the Gbbs-Duhem relatonshp). Conversely, f both actvty coeffcents of a bnary mxture are known then the Gbbs- Duhem equaton can be used to check the thermodynamc consstency of a gven set of data. For example, the slope and area tests have been developed specfcally for ths purpose (see pages 357-358 and Fg. 9.3). A key ssue here s how to deal wth the standard or reference state condton (+), as that wll have a drect effect on the magntude and behavor of γ. If a symmetrc reference state s used then ˆ f + = f( pure ) and the Lews and Randall rule s followed as x goes to 1.0 wth γ approachng 1.0. Alternatvely, a unsymmetrcal reference state can be used where the nfnte dluton behavor as defned by Henry s Law determnes that γ ** approaches 1.0 as x goes to 0. Ths s frequently called the McMllan-Mayer reference condton. Another popular alternatve commonly employed for electrolytes s to use a 1 molal soluton at T and P of the mxture where the mxture follows Henry s Law n dlute solutons as x or m 0 (see Fg. 9.9 for example). The dscusson n the text from pages 360 to 365 should be carefully revewed to see how the actvty coeffcent s related to fugacty behavor for dfferent reference state condtons. Partcular attenton should be pad to Fgures 9.6 9.9. 10. Ideal reversble work assocated wth mxng or separatng components Under sothermal condtons the reversble work s equal to the net change n Gbbs free energy assocated wth the process, more generally the reversble work can be related drectly to the change n avalablty usng the methods ntroduced n Chapter 14. Examples 9.8 and 9.9 should be revewed to see how the concepts are employed. The approach s qute straghtforward f you are dealng wth a steady state process. Thngs are a bt more complcated f the system or process condtons are changng wth tme, but the general concept remans unchanged. [ ] Modfed: 10/16/2003 5