CH2351 Chemcal Engneerng Thermodynamcs II Unt I, II www.msubbu.n Soluton Thermodynamcs www.msubbu.n Dr. M. Subramanan Assocate Professor Department of Chemcal Engneerng Sr Svasubramanya Nadar College of Engneerng Kalavakkam 603 110, Kanchpuram (Dst) Taml Nadu, Inda msubbu.n[at]gmal.com Jan-2012
Contents UNIT I: PROPERTIES OF SOLUTIONS Partal molar propertes, deal and non-deal solutons, standard states defnton and choce, Gbbs-Duhem equaton, excess propertes of mxtures. www.msubbu.n UNIT II: PHASE EQUILIBRIA Crtera for equlbrum between phases n mult component nonreactng systems n terms of chemcal potental and fugacty Jan-2012 M Subramanan
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Introducton Most of the materals of the real world are not pure substances wth all atoms or molecules dentcal but rather are mxtures of one type or another. The pure substances from whch a soluton may be prepared are called components, or consttuents, of the soluton. www.msubbu.n Solutons are not lmted to lquds: for example ar, a mxture of predomnantly N 2 and O 2, forms a vapor soluton. Sold solutons such as the sold phase n the S-Ge system are also common Jan-2012 M Subramanan
Multcomponent Systems Basc Relatons Sngle component system: Intensve propertes: depends on Pressure, Temperature Extensve propertes: depends on Pressure, Temperature, and amount Multcomponent system: www.msubbu.n Intensve propertes: depends on Pressure, Temperature, and composton Extensve propertes: depends on Pressure, Temperature, amount of each component Jan-2012 M Subramanan
Composton Mole fracton For bnary soluton www.msubbu.n In dealng wth dlute solutons t s convenent to speak of the component present n the largest amount as the solvent, whle the dluted component s called the solute. Jan-2012 M Subramanan
Other Measures of Composton Mass fracton preferable where the defnton of molecular weght s ambguous (eg. Polymer molecules) Molarty moles per ltre of soluton www.msubbu.n Molalty moles per klogram of solvent. The molalty s usually preferred, snce t does not depend on temperature or pressure, whereas any concentraton unt s so dependent. Volume fracton Mole rato or volume rato (for bnary systems) Jan-2012 M Subramanan
Propertes of Solutons The propertes of solutons are, n general, not addtve propertes of the pure components. The actual contrbuton to any extensve property s desgnated as ts partal property. The term partal property s used to desgnate the property of a component when t s n admxture wth one or more other components www.msubbu.n Because most chemcal, bologcal, and geologcal processes occur at constant temperature and pressure, t s convenent to provde a specal name for the partal dervatves of all thermodynamc propertes wth respect to mole number at constant pressure and temperature. They are called partal molar propertes Jan-2012 M Subramanan
Ethanol-Water System at 20 o C Molar volumes: Water: 18 ml/mol Ethanol: 58 ml/mol www.msubbu.n Partal molar volumes (at 50 mole% of Ethanol): Water: 16.9 ml/mol Ethanol: 57.4 ml/mol Volume before mxng = (1 mole) (18.0 ml/mole) + (1 mole) (58.0 ml/mole) = 76 ml Volume after mxng = (1 mole) (16.9 ml/mole) + (1 mole) (57.4 ml/mole) = 74.3 ml
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www.msubbu.n 1 lter of ethanol and 1 lter of water are mxed at constant temperature and pressure. What s the expected volume of the resultant mxture?
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Partal Molar Propertes The partal molar property of a gven component n soluton s defned as the dfferental change n that property wth respect to a dfferental change n the amount of a gven component under condtons of constant pressure and temperature, and constant number of moles of all components other than the one under consderaton. www.msubbu.n where M s any thermodynamc property. The concept of partal molar quantty can be appled to any extensve state functon. Jan-2012 M Subramanan
Partal Molar Volume Benzene-Toluene:Benzene and toluene form an deal soluton. The volume of 1 mole pure benzene s 88.9 ml; the volume of 1 mole pure toluene s 106.4 ml. 88.9 ml benzene mxed wth 106.4 ml toluene results n 88.9 ml + 106.4 ml, or 195.3 ml of soluton. (deal soluton) www.msubbu.n Ethanol-Water: The volume of 1 mole pure ethanol s 58.0 ml and the volume of 1 mole pure water s 18.0 ml. However, 1 mole water mxed wth 1 mole ethanol does not result n 58.0 ml + 18.0 ml, or 76.0 ml, but rather 74.3 ml. When the mole fracton s 0.5, the partal molal volume of ethanol s 57.4 ml and the partal molal volume of water s 16.9 ml. (non-deal soluton) Jan-2012 M Subramanan
Fundamental Equatons of Soluton Thermodynamcs www.msubbu.n Jan-2012 M Subramanan
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Gbbs-Duhem Equaton Ths equaton s very useful n dervng certan relatonshps between the partal molar quantty for a solute and that for the solvent. www.msubbu.n Jan-2012 M Subramanan
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Determnaton of Partal Molar Propertes of Bnary Solutons www.msubbu.n Jan-2012 M Subramanan
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Partal molar propertes n bnary soluton For bnary system M = x + 1M 1 x2m 2 www.msubbu.n dm = x1 dm 1 + M 1 dx1 + x2 dm 2 + M 2 dx 2 Const. P and T, usng Gbbs/Duhem equaton dm = M + dm dx 1 = 1dx1 M 2dx 2 M 1 x 1 + x2 = M 2 1 M = M + 1 x 2 dm dx 1 M 2 = M x 1 dm dx 1 Jan-2012 M Subramanan
Partal Molar Quanttes Physcal Interpretaton The partal molar volume of component n a system s equal to the nfntesmal ncrease or decrease n the volume, dvded by the nfntesmal number of moles of the substance whch s added, whle mantanng T, P and quanttes of all other components constant. www.msubbu.n Another way to vsualze ths s to thnk of the change n volume on addng a small amount of component to an nfnte total volume of the system. Note: partal molar quanttes can be postve or negatve! Jan-2012 M Subramanan
Example Problem www.msubbu.n Jan-2012 M Subramanan
x1 H (J/mol) H-deal (J/mol) H1bar (J/mol) H2bar (J/mol) 0 600 600 420 600 0.1 581.98 580 419.44 600.04 0.2 563.84 560 417.92 600.32 0.3 545.46 540 415.68 601.08 www.msubbu.n 0.4 526.72 520 412.96 602.56 0.5 507.5 500 410 605 0.6 487.68 480 407.04 608.64 0.7 467.14 460 404.32 613.72 0.8 445.76 440 402.08 620.48 0.9 423.42 420 400.56 629.16 1 400 400 400 640 Jan-2012 M Subramanan
700 650 H (J/mol) 600 550 500 www.msubbu.n H-mx H-deal H1bar H2bar 450 400 350 0 0.2 0.4 0.6 0.8 1 x 1 (-) Jan-2012 M Subramanan
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Partal Molar Propertes from Expermental Data Partal molar volume: Densty data (ρ vs. x 1 ) Partal molar enthalpy: www.msubbu.n Enthalpy data (H vs. x 1 ); can be drectly used Heat of mxng (also called as enthalpy change on mxng) data ( H mx vs. x 1 ) Obtaned usng dfferental scannng calormetry Reported normally as J/mol of solute; to be converted to J/mol of soluton Jan-2012 M Subramanan
Densty data for Water (1)-Methanol(2) system at 298.15 K x 1 ρ(kg/m 3 ) avgmw V V deal V mx m 3 /kmol m 3 /kmol m 3 /kmol 0 786.846 32.042 0.040722 0.040722 0.00000000 0.1162 806.655 30.41032 0.037699 0.038088-0.00038867 0.2221 825.959 28.92327 0.035018 0.035687-0.00066954 0.2841 837.504 28.05267 0.033496 0.034282-0.00078631 0.3729 855.031 26.80574 0.031351 0.032269-0.00091829 0.4186 864.245 26.16402 0.030274 0.031233-0.00095908 www.msubbu.n 0.5266 887.222 24.64748 0.027781 0.028785-0.00100419 0.6119 905.376 23.4497 0.025901 0.026851-0.00095055 0.722 929.537 21.90368 0.023564 0.024355-0.00079116 0.8509 957.522 20.09366 0.020985 0.021433-0.00044816 0.9489 981.906 18.71755 0.019062 0.019212-0.00014922 1 997.047 18 0.018053 0.018053 0.00000000 V deal = x 1 V 1 + x 2 V 2 V mx = V - V deal Jan-2012 M Subramanan
0.045 0.04 V Volum me (m 3 /kmol) 0.035 0.03 0.025 0.02 www.msubbu.n V-deal 0.015 0.01 0 0.2 0.4 0.6 0.8 1 x1 Jan-2012 M Subramanan
0.00E+00 0 0.2 0.4 0.6 0.8 1-2.00E-04 Vmx -4.00E-04-6.00E-04 www.msubbu.n -8.00E-04-1.00E-03-1.20E-03 x1 Jan-2012 M Subramanan
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wt% H2SO4] H (kj/kg) x1 avgmw H (kj/mol) H mx (kj/mol) H mx Model (kj/mol) 0 278 0.0000 18.00 5.00 0 0 20 85 0.0439 21.51 1.83-3.3516-3.3818 40-78 0.1091 26.73-2.08-7.5264-7.4880 www.msubbu.n 60-175 0.2160 35.28-6.17-12.0446-12.0692 80-153 0.4235 51.88-7.94-14.6412-14.6637 90-60 0.6231 67.85-4.07-11.5746-11.5736 100 92 1.0000 98.00 9.02 0 0 H deal = x 1 H 1 + x 2 V 2 H mx = H - H deal Jan-2012 M Subramanan
10.00 6.00 (kj/mol) H 2.00-2.00 0.0 0.2 0.4 0.6 0.8 1.0 www.msubbu.n -6.00-10.00 x1 Jan-2012 M Subramanan
Redlch-Kster Model Also known as Guggenhem-Scatchard Equaton Fts well the data of M mx vs. x 1 www.msubbu.n Jan-2012 M Subramanan
0 0.00-2 0.20 0.40 0.60 0.80 1.00-4 -6 Hmx -8-10 www.msubbu.n -12-14 Data RK Model ft -16 x1 ao -55.9287 a1 27.0094 Redlch-Kster model fts well the data of M mx vs. x 1 Jan-2012 M Subramanan
www.msubbu.n Calculate the partal molar volume of Ethanol and Water as a functon of composton.
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Gbbs Josah Wllard Gbbs (1839 1904) Gbbs greatly extended the feld of thermodynamcs, whch orgnally comprsed only the relatons between heat and mechancal work. Gbbs was nstrumental n broadenng the feld to embrace transformatons of energy between all the forms n whch t may be manfested, be they thermal, mechancal, electrcal, chemcal, or radant. www.msubbu.n He s consdered to be the founder of chemcal thermodynamcs. He s an Amercan theoretcal physcst, chemst, and mathematcan. He devsed much of the theoretcal foundaton for chemcal thermodynamcs as well as physcal chemstry. Jan-2012 M Subramanan
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Fundamental Equaton for Closed System The basc relaton connectng the Gbbs energy to the temperature and pressure n any closed system: www.msubbu.n appled to a sngle-phase flud n a closed system wheren no chemcal reactons occur. Jan-2012 M Subramanan
Fundamental Equaton for Open System Consder a sngle-phase, open system: d ( ng ) = ( ng ) P T, n dp ( ng ) + T P, n dt + ( ng ) n P, T, n j dn Defnton of chemcal potental: www.msubbu.n ( ng ) µ n P, T, n j (The partal dervatve of G wth respect to the mole number n at constant T and P and mole numbers n j n j ) The fundamental property relaton for sngle-phase flud systems of constant or varable composton: Jan-2012 M Subramanan + d ( ng ) = ( nv ) dp ( ns ) dt µ dn
When n = 1, + dg = VdP SdT µ dx G = G P, T, x, x,..., x,...) ( 1 2 V = www.msubbu.n G P T, x S = G T P, x The Gbbs energy s expressed as a functon of ts canoncal varables. Soluton propertes, M Partal propertes, M Pure-speces propertes, M µ G Jan-2012 M Subramanan
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Chemcal potental and phase equlbra Consder a closed system consstng of two phases n equlbrum: + α α α α α β β β d ( ng ) = ( nv ) dp ( ns ) dt µ dn d ( ng ) = ( nv ) dp ( ns ) dt + www.msubbu.n µ β β dn nm = + α ( nm ) ( nm ) β d ( ng ) = ( nv ) dp ( ns ) dt α α + µ dn + µ β dn β Snce the two-phase system s closed, Mass balance: dn = α β dn Multple phases at the same T and P are n equlbrum when Jan-2012 chemcal M Subramanan potental of each speces s the same n all phases. α β µ = µ
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Partal Molar Energy Propertes and Chemcal Potentals The partal molar Gbbs free energy s chemcal potental; however, the other partal molar energy propertes such as that of nternal energy, enthalpy, and Helmholtz free energy are not chemcal potentals: because chemcal potentals are dervatves wth respect to the mole numbers wth the natural ndependent varables held constant. www.msubbu.n Jan-2012 M Subramanan
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Varaton of µ wth T and P Varaton of µ wth P: partal molar volume Varaton of µ wth T: partal molar entropy, can be expressed n terms of partal molar enthalpy www.msubbu.n Jan-2012 M Subramanan
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Entropy Change The second law of thermodynamcs For an solated system www.msubbu.n n whch the equalty refers to a system undergong a reversble change and the nequalty refers to a system undergong an rreversble change. For systems that are not solated t wll be convenent to use the crtera of reversblty and rreversblty such as n the followng equaton: Jan-2012 M Subramanan
Mxng at Constant T and P www.msubbu.n To carry out the mxng process n a reversble manner, the external pressure P on the rght pston s kept nfntesmally less than the pressure of B n the mxture; and the external pressure P on the left pston s kept nfntesmally less than the pressure of A n the mxture. Jan-2012 M Subramanan
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As the mxng process s sothermal, and the mxture s an deal gas, www.msubbu.n Jan-2012 M Subramanan
Entropy Change of Mxng Consder the process, where n A moles of deal gas A are confned n a bulb of volume V A at a pressure P and temperature T. Ths bulb s separated by a valve or stopcock from bulb B of volume V B that contans n B moles of deal gas B at the same pressure P and temperature T. When the stopcock s opened, the gas molecules mx spontaneously and rreversbly, and an ncrease n entropy S mx occurs. www.msubbu.n The entropy change can be calculated by recognzng that the gas molecules do not nteract, snce the gases are deal. S mx s then smply the sum of S A, the entropy change for the expanson of gas A from V A to (V A + V B ) and S B, the entropy change for the expanson of gas B from V B to (V A + V B ). That s, S mx = S A + S B Jan-2012 M Subramanan
For the sothermal process nvolvng deal gases, H s zero. Therefore, www.msubbu.n Jan-2012 M Subramanan
Partal Molar Entropy of Component n an deal gas mxture www.msubbu.n
Gbbs free energy change of mxng www.msubbu.n
Chemcal potental of component n an deal gas mxture www.msubbu.n
µ g G g = G g + RT ln y G g = Γ ( T ) + RT ln P µ g = Γ ( T ) + RT ln www.msubbu.n y P Jan-2012 M Subramanan
Resdual Property www.msubbu.n
Fugacty of a component www.msubbu.n
Chemcal potental of component n a soluton n terms of fugacty www.msubbu.n
Fugacty and fugacty coeffcent: speces n soluton For speces n a mxture of real gases or n a soluton of lquds: µ Γ ( T ) + RT www.msubbu.n ln fˆ Fugacty of speces n soluton (replacng the partcle pressure) Multple phases at the same T and P are n equlbrum when the fugacty of each consttuent speces s the same n all phases: f ˆ = fˆ =... = α β fˆ π Jan-2012 M Subramanan
The resdual property: M R = M M g The partal resdual property: M R = M M g G R = G G g µ Γ ( T ) + www.msubbu.n µ g µ µ = Γ ( T ) + g = RT ln RT ln RT ln fˆ y P fˆ y P R G = Jan-2012 M Subramanan 0 ˆ ˆ f φ = = 1 y P For deal gas, fˆ = y P G R = RT µ ln φˆ φˆ ( ng ) n fˆ y P P, T, n j = G The fugacty coeffcent of speces n soluton
The excess Gbbs energy and the actvty coeffcent The excess Gbbs energy s of partcular nterest: c.f. Jan-2012 M Subramanan G E G G G E = R E = RT G G G ln γ RT d G d = Γ ( T ) + RT = Γ ( T ) + RT fˆ x f ln γ fˆ x f ln ln fˆ www.msubbu.n x f The actvty coeffcent of speces n soluton. A factor ntroduced nto Raoult s law to account for lqud-phase non-dealtes. For deal soluton, E G 0, = = RT ln φˆ γ 1 =
ˆ ˆ f φ = = 1 y P y fˆ = y P www.msubbu.n γ fˆ x f G E = RT ln γ R G = 0 For deal soluton, E G 0, γ = 1 =
Fugacty of a pure lqud The fugacty of pure speces as a compressed lqud: G G sat = RT ln f f sat G G sat = P P sat V dp ( sothermal process ) ln f 1 = sat f RT P P sat V dp www.msubbu.n Snce V s a weak functon of P ln l sat f V ( P P ) = sat f RT f sat l sat sat V ( P P f = φ P exp = φ P RT sat sat sat )
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