Chapter 8 Thermodynamic Properties Of Mixtures

Chapter 8Thermodynamic Propertiesof Mixtures2012/3/291

Abstract The thermodynamic description ofmixtures, extended from pure fluids.The equations of change, i.e., energyand entropy balance, for mixtures aredeveloped.The criteria for phase and chemicalequilibrium in mixtures2012/3/292

8.1 THE THERMODYNAMICDESCRIPTION OF MIXTURESThermodynamic property for pure fluids,θ θ ( T , P , N )where N is the number of moles.θ θ (T , P )where the number of mole equals to 1.Thermodynamic property for mixtures,θ θ (T , P, N1 , N 2 ,L, N c )θ θ (T , P, x1 , x2 ,L, xc )where N i is the number of moles of the ith component.where xi is the mole fraction of the ith component.For exampleU U (T , P, N1 , N 2 ,L, N c ) orV V (T , P, N1 , N 2 ,L, N c ) or2012/3/29U U (T , P, x1 , x2 ,L , xc )V V (T , P, x1 , x2 ,L, xc )3

Summation of the properties of pure fluids (before mixing at T and P)CU (T , P, x1 , x2 ,L, xc 1 ) xi U i (T , P )(8.1-1)i 1where U is the molar internal energy, U i is the internal energyof the pure i-th component at T and P.CˆUˆ (T , P, w1 , w2 ,L, wc 1 ) wUi i (T , P )(8.1-2)i 1where wi is the mass fraction of component i.2012/3/294

At the same T and P25 cc H2OH2O25 ccA50 cc25 cc 25 cc48 ccB- 2 ccAttractive2012/3/29or52 cc 2 ccRepulsive5

Property change upon mixing (at constant T and P)CΔ mix θ θ (T , P, xi ) xi θ i (T , P )i 1Volume change upon mixingCΔ mix V (T , P ) V (T , P, xi ) xi V i (T , P )i 1Enthalpy change upon mixingCΔ mix H (T , P ) H (T , P, xi ) xi H i (T , P )i 12012/3/296

Experimental data :properties changes upon mixing (H and V)

Figure 8.1-1 Enthalpy-concentrationdiagram for aqueous sulfuric acid at0.1 MPa. The surfuric acid percentageIs by weight. Reference states:The Enthalpies of pure liquids at 0oCAnd their vapor pressures are zero.2012/3/298

Figure 8.1-2 (a) Volume change on mixing at 298.15 K: o methyl formate methanol, zmethy formate ethanol. (b) Enthalpy change on mixing at298.15 K for mixtures of benzene (C6H6) and aromatic fluorocarbons(C6F5Y), with Y H, F, Cl, Br, and I.2012/3/299

Equations relating molar and partial molarpropertiesN θ φ (T , P, N1 , N 2 ,L, N c )At constant T and P,N θ φ ( N1 , N 2 ,L, N c )Total differential of N θ is: ( Nθ ) ( Nθ ) ( Nθ ) d ( Nθ ) dN1 dN 2 L dN c N N N12c T , P , N j 1 T , P , N j 2 T , P , N j cc ( Nθ ) dN i θ i (T , P, x )dN i i 1 N ii 1 T , P , N j icDefinition of the partial molar property ( Nθ ) Ni T , P , Nθ i ( T , P, x ) (8.1-12)j icd ( N θ ) θ i dN ii 12012/3/2910

At a constant T and Pθ50 cc 1 mol of Aθ ΔθA BΔθ is a partial molar property (atConstant T, P, and NB)2012/3/2911

Total property and partial molar propertycd ( N θ ) θ i dN ii 1d ( N θ ) Nd θ θ dN(a)dN i d ( Nxi ) Ndxi xi dN(b)ccNd θ θ dN d ( N θ ) θ i dN i θ i ( Ndxi xi dN )(a)(b)i 1cci 1i 1i 1Nd θ θ dN N θ i dxi dN θ i xicc θxdθθdxNθi i dN 0i i i 1i 1 The N and dN are the number of moles in system and changing in numberof moles, respectively. Thus, N and dN are arbitrary and not equal to zero.2012/3/2912

cdθ θ i dxi 0;i 1ccdθ θ i dxiresulted from 1st term of LHS(c)i 1cθ θ i xi 0;θ θ i (T , P, x ) xi resulted from 2nd term of LHS (8.1-13)i 1i 1Euler Theorem defined by (8.1-13) means θ can be calculated from θ i (T , P, x ) .cθ θ i xiSince:i 1cci 1i 1dθ θ i dxi 6)& ) ( N& S )( NSCwherekii 1i kS&gen σ& s dVσ& s 2012/3/29λT2( ΔT )2μ1 M C φ ν ij Gi X& jTT j 1 i 12(8.4-7)45

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Table 8.4-1 continued2012/3/2947

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Table 8.4-2 continued2012/3/2949

ILLUSTRATION 8.4-1Temperature Change on Adiabatic Mixing of an Acid and WaterThree moles of water and one mole of sulfuric acid, each at 0o C, are mixed adiabatically. Usethe data in Fig. 8.1-1 and the information in Illustration 8.1-1 to estimate the final temperatureof the mixture.SOLUTIONFrom the closed-system mass balance, we haveM f M H2 O M H2SO4 3 18.015 1 98.078 152.2 kgand from the energy balance, we haveU f H f H i M f Hˆ mix M H2 O Hˆ H2 O M H2SO4 Hˆ H2SO4 3 0 1 0 0 kJThus finally we have a mixture of 64.5 wt % sulfuric acid that has an enthalpy of 0 kJ/kg, (Notethat we have used the fact that for liquids and solids at low pressure, the internal energy andenthaipy are essentially equal, U H ). From Fig. 8.1-1 we see thal a mixture containing 64.5 wt %sulfuric acid has an enthalpy of 0 kJ/kg at about 150 o C. Therefore, if water and sulfuric acid areadiabalically mixed in the ratio of 3:1, the mixture will achieve a temperature of 150o C, whichis just below the boiling point of the mixture. This large temperature rise, and the fact that themixture is just below its boiling point, makes mixing sulfuric acid and water an operation thatmust be done with extreme care.2012/3/2950

150 oCFigure 8.1-1 Enthalpy-concentrationdiagram for aqueous sulfuric acid at0.1 MPa. The sulfuric acid percentageIs by weight. Reference states:The Enthalpies of pure liquids at 0oCAnd their vapor pressures are zero.-3152012/3/2951

COMMENTIf instead of starting with refrigerated sulfuric acid and water (at 0o C), one started with thesecomponents at 21.2o C and mixed them adiabatically, the resulting 3:1 mixture would be in theliquid vapor region; that is, the mixture would boil (and splatter). Also note that because ofthe shape of the curves on the enthalpy-concentration diagram, adding sulfuric acid to wateradiabalically (i.e., moving to the right from the pure water edge of the diagram) results in amore gradual temperature rise than adding water to sulfuric acid (i.e., moving to the left fromthe pure-sulfuric acid edge). Therefore, whenever possible, sulfuric acid should be added towater, and not vice versa.2012/3/2952

ILLUSTRATION 8.4-2Mass and Energy Balances on a Nonreacting SystemA continuous-flow steam-heated mixing kettle will be used to produce a 20 wt % sulfuric acidsolution at 65.6o C from a solution of 90 wt % sulfuric acid at 0o C and pure water at 21.1o C.Estimatea. The kilograms of pure water needed per kilogram of initial sulfuric acid solution toproduce a mixture of the desired concentrationb. The amount of heat needed per kilogram of initial sulfuric acid solution to heat themixture to the desired temperaturec. The temperature of the kettle effluent if the mixing process is carried out adiabatically2012/3/2953

SOLUTIONWe choose the contents of the mixing kettle as the system. The difference form of the equationsof change will be used for a time interval in which 1 kg of concentrated sulfuric acid enters the kettle.a. Since there is no chemical reaction, and the mixing tank operates continuously, the totaland sulfuric acid mass balances reduce to30 M& k and 0 k 1 (M3k 1H 2 SO 4)kDenoting the 90 wt % acid stream by the subscript 1 and its mass flow by M& 1 , the waterstream by the subscript 2, and the dilute acid stream by subscript 3, we have, from the totalmass balance,0 M& M& M& M& ZM& M&123113M& 3 (1 Z ) M& 1where Z is equal to the number of kilograms of water used per kilogram of the 90 wt %acid. Also from the mass balance on sulfuric acid, we have0 0.90M& 0M& 0.20M& 0.90 M& 0.20 (1 Z ) M&12311Therefore,0.90 4.5 or Z 3.50.20so that 3.5 kg of water must be added to each 1 kg of 90 wt % acid solution to produce a1 Z 20 wt % solution.2012/3/2954

b. The steady-state energy balance is(& ˆ0 MHk) Q&ksince Ws 0 and PdV 0. From the mass balance of part (a).M& 2 3.5M& 1 and M& 3 4.5M& 1From the enthalpy-concentration chart, Fig. 8.1-1, we haveHˆ Hˆ 90 wt % H SO , T 0o C 183 kJ/kg1Hˆ 2Hˆ 3() Hˆ ( pure H O, T 21.1 C ) 91 kJ/kg Hˆ ( 20 wt % H SO , T 65.56 C ) 87 kJ/kg24o2o24Q ( 4.5 87 3.5 91 1 ( 183) ) ( 391.5 318.5 183) 256 kJ/kg of initial acid solutionc. For adiabatic operation, the enerey balance is(& ˆ0 MH(k)k)0 4.5 Hˆ 3 3.5 91 1 ( 183) ; 4.5 Hˆ 3 135.5 and Hˆ 3 30.1 kJ/kgReferring to the enthalpy-concentration diagram, we find that T 50 o C.2012/3/2955

8.5 THE HEAT OF REACTION AND A CONVENTIONFOR THE THERMODYNAMICPROPERTIES OF REACTING MIXTURESIn the ideal gas-phase reactionH 2 ( g ) 12 O 2 ( g ) H 2 O ( g )it is observed that 241.82 kJ are liberated for each mole of water vapor producedwhen this reaction is run in an isothermal, constant-pressure calorimeter at 25o C and1 bar with all species in the vapor phase. Clearly, then, the enthalpies of the reactingmolecules must be related as follows:()(( vapor, T 25 C, P 1 bar ) Δ rxn H T 25o C, P 1 bar H H2O T 25o C, P 1 bar H H2o12())H O2 T 25o C, P 1 bar 241.82kJmol of H 2 O producedso that we are not free to choose the values of the enthalpy of hydrogen, oxygen, andwater vapor all arbitrarily.2012/3/2956

Enthalpy of formationThe enthalpy of formation is Δ f H H2 O and the Gibbs energy of formation is Δ f G . By definition,o()o(Δ f H H2 O T 25o C, P 1 bar H H2 O T 25o C, P 1 baro()() H H2 vapor, T 25o C, P 1 bar 12 H O2 T 25o C, P 1 bar)H H2 O Δ f H H2 O H H2 12 H O2oAppendix A.IV contains a listing of Δ f H and Δ f G for a large collection of suboostances in their normal states of aggregation at 25o C and 1 bar.Isothermal heats (enthalpies) and Gibbs energies of formation of species may besummed to compute the enthalpy change and Gibbs free energy change that wouldoccur if the molecular species at 25o C, 1 bar, and the state of aggregation listed inAppendix A.IV. We will denote these changes by Δ rxn H (25o C, 1 bar) and Δ rxn G (25o C,oo1 bar), respectively.2012/3/2957

Enthalpy of reaction and enthalpy of formationFor the gas-phase reaction 3NO 2 H 2O 2HNO3 NO, we have()()()( 25 C,1 bar )Δ rxn H o 25o C,1 bar 2 H HNO3 25o C,1 bar H NO 25o C,1 bar() 3H NO2 25o C,1 bar H H2Oo 2 Δ f H HNO3 32 H O2 12 H H2 H N2 o Δ f H NO 12 H O2 12 H N2 25o C,1 bar25o C,1 baro 3 Δ f H NO2 H O2 12 H H2 o Δ f H H2O H H2 12 H O2 o25o C,1 bar25o C,1 baroooo 2Δ f H HNO3 Δ f H NO 3Δ f H NO2 Δ f H H2O ()( 25 C,1 bar ) ν Δ H ( 25 C,1 bar )( 25 C,1 bar ) ν Δ G ( 25 C,1 bar )25o C,1 bar ν i Δ f H i 25o C,1 baroΔ rxn H oΔ rxn G o2012/3/29oioififooioo(8.5-1)(8.5-2)58

At T and 1 bar()()( 25 C,1 bar ) ν Δ G ( 25 C,1 bar )Δ rxn H o 25o C,1 bar ν i Δ f H i 25o C,1 bar(8.5-1)Δ rxn G o(8.5-2)ooioifoThe standard heat of reaction at any temperature TΔ rxn H o (T ,1 bar ) ν i Δ f H i (T ,1 bar )o(8.5-3)Δ rxn G o (T ,1 bar ) ν i Δ f G i (T ,1 bar )o(8.5-4)oH i (T , P 1 bar ) H i (To , P 1 bar ) CP,i(T , P 1 bar ) dToToTo()Δ rxn H o (T ,1 bar ) ν i Δ f H i 25o C,1 bar ν i o()TT 25 C Δ rxn H o 25o C,1 bar ν i TT 25 CoooCP,idToCP,idT(8.5-5)CPo,i is the heat capacity of species i in its sta

8.2 THE PARTIAL MOLAR GIBBS ENERGY AND THE GENERALIZED GIBBS-DUHEM EQUATION Since the Gibbs energy of a multicomponent mixture is a function of temperature, pres-sure, and each species mole number, the total differential of the Gibbs energy function can be written as G dG T