1、Chapter 5 Solution Thermodynamics 熱力學(xué)基礎(chǔ) Solution 溶液 Gas solution，liquid solution， solid solution solvent 溶劑 solute 溶質(zhì)The review of fundamental relations of thermodynamicsThe 8 state variables: P, T, V, U, H, S, A and G. f=c-p+2=2 for 1 mole pure component For n moles of pure component:5.1 Property r

2、elations for open systems and chemical potential 開系的熱力學(xué)性質(zhì)關(guān)系與化學(xué)位For an open homogenous system containing N components ：Where represent the sum of all the components in the system. The subscript nj means all the spices except for component i. 式中：-i 組分的化學(xué)位 chemical potentialJ.W. Gibbs，18391903Physical

3、Chemist, USAChemical potential (化學(xué)位) I ：at constant T and P, put a infinitive small amount of component i into a solution, and the mixture is still homogenous, the change in Gibbs energy divided by the amount of the added component, is the chemical potential of the component in the solution. Chemica

4、l potential plays a similar role as T and P. T gives the trend of heat transfer, P decides the moving trend of a body and governs the trends of a chemical reaction or a mass transfer between phases. is intensive. Definition:Similarly，from the other 3 thermodynamic relations we get ：Ex.5.15.2 Partial

5、 Properties 偏摩爾性質(zhì)Of particular use-the partial Gibbs energy of component i.If M represent any thermodynamic property, the partial property is :Here M maybe V, U, H, S, A and G. The partial property and molar property are alike in forms in the thermodynamic relations： Ex.5.3.The constant pressure hea

6、t capacity is defined:，try to prove: Find the its 1st derivative with respect to ni at constant T, P and nj ：Solution: Change the order of the derivation：done !Calculation of Partial Properties 偏摩爾性質(zhì)的計(jì)算 For a solution of multi-component mixture with each constituent moles of n1，n2，ni , nN at constan

7、t T and P，an extensive property is a function of the constituent moles：If we double the moles of each constituent, the intensive properties will not change, but the extensive properties will be doubled. Here we introduce an ordered homogeneous function:When m=0, the homogeneous function of order 0 (

8、0階齊次函數(shù)), the intensive property ；when m=1, the homogeneous function of order 1 (1階齊次函數(shù)), the extensive property Euler 定理： For intensive property, m 0 For extensive property, m 1 orFor a pure component, the partial property becomes molar property: Which is the eqn for mollal property calculation for

9、a mixture. For real solution, the partial property functions just as the molar property in an ideal solution. The reversed calculation ？At constant T and P，M is a function of N一1 mole fractions： The 1st term is 0 and the 2nd is 1，then: Where i is the element being considered, k refers to other eleme

10、nt except for element i , and j refers to other element except for element i and k.The above eqn is a general eqn to get partial property from the molar property of a solution. M can be V，U，H and G.For a binary system : ororreMmM22xMmx21xMmx1M2M12x2221xMmxMmM-=212xMmxMmM+=2211MxMxMm+=121xMmxMmM+=112

11、xMmxMmM-=reWe may get M values from experimental data, correlate them as a function of xi, and from the correlations calculate the partial properties.Ex.5.3The enthalpy of a binary mixture at 298K and 1.0133X105Pa can be correlated as： Hm=100 xl+150 x2+x1x2(10 xl+5x2) Try to find(a) The expression o

12、f partial H1 and partial H2 in terms of x1； (b) The pure component enthalpies Hm1 and Hm2；(c) The partial enthalpies for each component at extremely dilute solutions. solution (a)(c) The partial enthalpies for each component at extremely dilute solutions are obtained when x10 and x1=1 . (b) Pure com

13、ponent Hl and H2 can be obtained when x11 and x1=0 Gibbs-Duhem Equation 吉布斯一杜亥姆方程The partial properties are related with each otherorGibbs-Duhem EqnFor intensive properties:、 For extensive properties: M=G, H, V, A Also valid for excess properties M=G, H, V, A Applications of GibbsDuhem eqn： to check

14、 the consistence of experimental data with generalized theories； to calculate the partial properties for one component from those of the others. Ex5.4A binary system, at const T, P. Solution :Gibbs-Duhem equationdone ! Where are they from?5.3 Fugacity and Fugacity Coefficient 逸度和逸度系數(shù)Phase equilibriu

15、m is the fundamental theory for many separation processes and is an important part of chemical engineering. The concept of fugacity is of great significant in discussing phase equilibrium problems.Definition 逸度和逸度系數(shù)的定義The thermodynamic principles are rigid, from which many mathematical relations can

16、 be obtained. Rigid, but not very convenient. By ideal gas law:(const T)Not valid for most real cases.For ideal gas：Gilbert Newton Lewis suggest a similar relation for real fluids:where fi ，the fugacity of pure component i , in the unit of pressure.Fugacitysubatract withWith the definition of residu

17、al property：Where fi/P is dimensionless, and is named as the fugacity coefficient, noted byiWe get:Where:We have:reFugacityActivity（Gilbert Newton Lewis，a scientist in physical chemistry,18751946）fthe calibrated pressure the effective pressure no matter how much the actual pressure is, it serves as

18、the effective pressure with a value of “f”. in phase equilibrium, it measures the ability of escaping.For real gas :For ideal gas:5.3.1 Fugacity for Pure Gases 純氣體的逸度 1)To calculate fugacity and fugacity coefficient with enthalpy and entropy(const T)Integrated from a reference state (denoted by *) t

19、o P:re If the reference pressure is low enough so that the gas con be considered as ideal gas， fi*=p*，then： Which can be used to calculate fugacity and fugacity coefficient with enthalpy and entropy.2）. From P-V-T data (const T)Used to calculate f and from PVT data (numerical integration or graphica

20、l integration) .reSuppl. Ex.P(MPa) Vm (cm3/mol)P(MPa)Vm (cm3/mol)2.02661865.520.266107.46.0798570.825.33374.1810.13331030.39959.615.200176.640.53247.68Calculate fugacity for NH3 at T473.15K、P10.1MPa and 40.5MPa。The P-V data： Solution:P(MPa) VmiR (cm3/mol)P(MPa)VmiR(cm3/mol)2.0266-75.5020.266-86.716.

21、0798-76.2225.333-81.1210.133-78.2030.399-69.8215.200-82.1040.532-49.38Similarly, at P40.5MPa, Take R-K eqn as ex.Since:Substitute R-K eqn into 3)Calculate f and by EOS從狀態(tài)方程計(jì)算逸度系數(shù) Then: 4） Calculate f and with the theorem of coresponding state 從對應(yīng)態(tài)原理計(jì)算逸度系數(shù) In the form of reduced pressure:re， We set:a

22、). 2 parameter method 兩參數(shù)法: 0.452re0.810.3080.844b). 3 parameter method 三參數(shù)法More precisely , use table 7 in the appendix (p325).Since:c). Generalized Virial coefficient method 普遍化維里系數(shù)法If Virial eqn is applicable,Ex. 5.5Propane, 10.203MPa, 407K. f=?Solution:a). Take propane vapor as ideal gas: f=P=10

23、.203MPab). By R-K eqn: Numerically.c). With generalized 2-parameter correlation35d). With generalized 3-parameter correlation=0.152,Pr=2.403,Tr=1.101 From table G in appendix p325,329Comparison of the results by deferent method:method deviation Ideal gas lawR-K eqn2 parameter correlation3 parameter

24、correlation-102.6%-0.95%+8.39%0.081% The ideal gas law is not available. R-K eqn and 3 parameter correlation give quite satisfactory results. 3 parameter correlation is better than 2 parameter correlation. 5.3.2 fugacity calculation for liquid 液體的逸度For liquid and solid, the definition of fugacity is

25、 the same as gas:For pure component:In liquid state:The 1st term is for saturate vapor. And the 2nd is the calibration of liquid at P with respect to Pis, the Poynting Factor. If the system is at vapor-liquid equilibrium:If the liquid state is far from its critical point, it can be referred as non-c

26、ompressive. Then ：Poynting Factor ：Poynting factor effects significantly at very high pressures. Ex. 5.6Liquid C3H8, 38.8oC(Ps=1.312MPa)，6.890MPa, VLm=2.06cm3/g, f=? Solution:a). With generalized 2-parameter correlationFor fv :35At equilibrium:Pressure effect:b). With acentric factor( Lee-Kesler Tab

27、le):When ; ; 5.3.3 Fugacity for species in mixture 混合物中組分的逸度Most practical streams are mixtures. Thermodynamic properties are complex, and it is impossible to get all data experimentally.It is useful to obtain thermodynamic data from limit experimental data through theoretical or semi-theoretical ap

28、proaches.Here we will discuss the calculation of fugacity for component in mixtures. For ideal gas:Similar expression is given for a component in a mixture :where is the fugacity of component i in a mixture . The definition of fugacity for a component in a homogeneous mixture is similar to that for

29、pure component.Const T, xreThe fugacity coefficientwith P,V and T:Available for both gases and liquids. reThe relations ofRelations among f, fi, andFor ideal gas mixture:For real gas mixture:Is the partial property of Is the partial property of Calculation of fugacity for a species in solution with

30、Lewis-Randall mixing rule 路易斯-蘭德爾逸度規(guī)則 For component I in mixture:For pure component:The deference of the two: , Then:The relationship of fugacities between pure component and a component in a solution. For ideal solution：Lewis-Randall fugacity Rule（2）with 2nd Virial coefficient 從第二維里系數(shù)計(jì)算 If the gas

31、mixture obeys virial eqn terminated at the 2nd coefficient: Available for gaseous mixture under low or moderate pressures. The mixing rule：Bij represents for the intermolecular forces between component i and j.Bij=Bji For a bineray system, i=1，2 and j=1，2， then:B11,B22 and B12 are the only function

32、of T.Differentiated with respect to n1 ： set reThe expression can be expanded to multi-component systems :Differentiated with respect to n1 ：kij is the interact parameter correlated from know data. In homework, if not specified, take it 0The resultant eqn is available for non or less polar gases.whe

33、reEx. 5.7C1+(n-C6), equi-molecularly mixed, 500K , 2M Pa. B=?, fugacity for each ?Solution:For methane:Similarly:取k120，re60; Therefore,For the mixture,（3）By EOS 從狀態(tài)方程計(jì)算 If a gas is around its critical point, the virial eqn is not applicable. The method with EOS can be tried.The mixing rules should b

34、e carefully selected, since they effect the results remarkably. The above eqn combined with mixing rule gives the calculation of fugacity for a component in a mixture.For binary system, with R-K eqn and Prausnitz mixing rule:Where Vm is the molar volume of the mixture and generally calculated by ite

35、rative calculations；and： reIn the lack of a and kij，aij and TCij can be obtained by: Ex. 5.8H2(1)C3H8, y1=0.208, 344.8K, 3.7972MPa,Solution: H2 is a quantum gas, the effective Tr, and Pr is given by Prausnitz:yl0.208，y210.2080.792 With R-K eqn:reSubstitute in R-K eqn:Solved numerically for VR-K eqn

36、can be rewritten in the form of:Then:The result is quite satisfactory5.3.4 Application of Microsoft Excel in the thermodynamic property calculationsRead after classInfluence of T and P on fugacity 溫度與壓力對逸度的影響5.4 The Ideal Solution and Standard State 理想溶液和標(biāo)準(zhǔn)態(tài)Model : simplify the system studied, revea

37、l laws and principles，provide a comparison standard.Solutions are always complex.The ideal solution, presents the properties approximately for real solutions, simplifies their calculations, is a fundamental concept for the analysis of real solution. LewisRandall Rule: the fugacity of a component in

38、a solution is proportional to its mole fraction. The standard state 標(biāo)準(zhǔn)態(tài) Is the standard fugacity of pure component iWhere The ideal solution modelequal interaction forces, equal in molecular volume (but no interaction force, 0 molecular volume for ideal gas )For gases, the ideal solution occupies a

39、wider range than ideal gas. When each component in a gas mixture gets a reduced pressure of less than 0.8，the mixture can be regarded as an ideal solution. The Standard State Different types of standard fugacity 1 The pure component standard (based on the Rourts law).2 the imaged standard based on H

40、enrys law.Other standard properties： When the solution is nearly ideal solution, the two lines get closer and impose to the straight line.The selected standard state must at the same T as the actual state.( the pressure maybe different )For pure component:For component i in solution:Const TAt const

41、T, P, from pure component to solution:For ideal solution:orThe partial Gibbs energy of component I in an ideal solution The mixing change in Gibbs energy from pure component I to an ideal solution(a) G Property Change in Mixing for Ideal Solutions Considering the relation between solution properties

42、 and the constituent partial properties:A molar property for a solution is different from the linear combination of its constituents. For a general property M: (Maxwell Eqn)(b) VConst T and xThe mole volume of a ideal solution equals the linear combination of its constituents.(c). S(d). HThere is no

43、n change in enthalpy for the formation of an ideal solution from its constituents. No heat effect.5.5 Excess Properties and Property Change in Mixing過量性質(zhì)與混合性質(zhì)變化 Properties of a real solution are generally deferent from the sum of its constituents. The deference is named as the property change in mix

44、ing The relationship between solution property and partial property is: The property change in mixing ：Which is the property change for the formation of 1 mole solution from its constituents under a given T and P5.5.1 Excess Properties 過量熱力學(xué)性質(zhì)The definition: at a given T ,P and composition, the defe

45、rence of a thermal property for a real solution from that of the imaged ideal solution is named as the excess property:For extensive properties: And similarly When M represents V、H、CV、CP、U, Notice! When M represents S、A and G， Hence, for these properties, the excess properties are deferent from thos

46、e of mixing change:The excess property is a measurement of the non-ideality of a solution.A summary for solution properties Ex.5.11At const T,P, binary system, Solution:Similarly:Similarly:From the definition of excess enthalpy:5.5.2 Enthalpy Change in mixing and Hx Graph 混合過程的焓變和焓濃圖 Mixing is alway

47、s accompanied by enthalpy changes.For controlled mass mixed isobarically:For controlled volume system, steady flow:The enthalpy change in mixing T & P:For binary system and based on L-R standardThe enthalpy change of mixing for C2H5OH（1）-H2O（2）system焓濃圖H-x Graph 焓-濃圖The H-x graph is one of the most

48、convenient graph for practical usage NaOHH2O系的焓濃圖 Ex. 5.12 p123An 1 effect evaporator, F1=5000kg/h, 10%NaOH50%, tf=293K, P=10.133kPa, tend=361K, Q=?Solution:evaporatorF1F2F3Material balance:Home Work5.1, 5.3, 5.4, 5.5，,5.6, 5.7, 5.8, 5.95.6 Activity and Activity Coefficient 活度與活度系數(shù)Fugacity is useful

49、 in dealing with gaseous mixtures, and can also be used in the calculations for liquid mixtures.Another method for dealing with liquid mixtures maybe is more convenient for use, the Activity.Definition 定義：Relates the abstractive concept “the chemical potential” with the tangible volume. For ideal so

50、lution:For ideal gas:For real fluid:Means how active the component is in a solution compared with its standard state. For ideal solution at given T and PThe the activity coefficient- maybe considered as an effective concentration ,describing the deviation from ideal solutionreEx. 5.13For a binary li

51、quid solution,Solution : or orThe regular solutionThe athermal solutionand , and Margules equationvan Laar equations Wilson equationNRTLThe excess Gibbs energy modelRedlich-Kister expansion:the symmetric modelMargules modelThe influence of T and P on activity coefficient 溫度和壓力對活度系數(shù)的影響 The expression

52、 defers with deferent rules :Rule 1： Rule2 ：5.6.1 Wohl Type Equation 伍爾型方程 Based on normal solution assumption, HE 0，constituent molecules defer in structure, size, interaction force and polarity. Where: Measures the interaction force between molecules I and j. Among 3 molecules Among 4 moleculesq t

53、he effective molar volume Zi the effective volume fraction The 4 rank Wohl type eqnScatchardHamer EquationSubstituting the effective volume q1 and q2 with pure component volumes V1L and V2L, gives:Margules equations (馬居斯方程）Exa. 5.15 a binary solution Solution :Margules equations:x100.10.20.30.40.50.

54、60.70.80.91.011.501.451.401.331.261.191.131.081.041.011.0021.001.001.011.031.061.111.181.291.451.672.001 2Van Laar equation 范拉爾方程ifTab.5.2 The van Laar constants for some binary mixturesComponent1-2Temperature range ABAcetaldehyde-waterAcetone-benzeneAcetone-methanolAcetone-waterAcetone-waterBenzene-isopropanolCabon disulfide-acetonCabon disulfide- Cabon tetrachlorideCabon tetrachloride- benzeneEthanol- benzeneEthanol-cyclohexaneEthanol-tolueneEthanol- waterEthyl