If two volatile and miscible liquids are combined to form a solution, Raoult s law is not obeyed. Use the experimental data in Table 9.
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1 9.9 Real Solutons Exhbt Devatons from Raoult s Law If two volatle and mscble lquds are combned to form a soluton, Raoult s law s not obeyed. Use the expermental data n Table 9.3: Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 98
2 9.9 Real Solutons Exhbt Devatons from Raoult s Law Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
3 9.9 Real Solutons Exhbt Devatons from Raoult s Law Assumng that A and B are mscble for real soluton, we have G S V H mxng mxng mxng mxng Snce ΔV mxng and ΔH mxng can be postve or negatve, dependng on the nature of the A B nteracton n the soluton. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 00
4 Fgure 9.3 Fgure 9.3 The data n Table 9.3 are plotted versus x CS. The dashed lnes show the expected behavor f Raoult s law were obeyed. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
5 Fgure 9.4 Fgure 9.4 Devaton n the volume from the behavor expected for mol of an deal soluton are shown for the acetone-chloroform system as a functon of the mole fracton of chloroform. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
6 9.9 Real Solutons Exhbt Devatons from Raoult s Law The devaton of the volume from deal behavor can be understood by defnng the concept of partal molar quanttes. Partal molar volume The partal molar volume of a component n a soluton s defned as the volume by whch the soluton changes f mole of the component s added to such a large volume that the soluton composton can be assumed constant. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
7 9.9 Real Solutons Exhbt Devatons from Raoult s Law The partal molar volume of soluton s defned as the volume by whch the soluton changes f mol of the component s added to that soluton composton. V P, T, n, n V n P, T, n Thus the volume of a bnary soluton s gven V n P, T, n, n n V P, T, n n V, Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 04
8 9.9 Real Solutons Exhbt Devatons from Raoult s Law Fgure 9.5 The partal molar volumes of a chloroform (yellow curve) and acetone (Red curve) n a chloroform-acetone bnary soluton are shown as a functon of x chloroform. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
9 9.9 Real Solutons Exhbt Devatons from Raoult s Law support V total =n A V A +n B V B mole fracton x A n A na n B Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
10 Problem P9.8) The partal molar volumes of ethanol n a soluton wth = 0.60 at 5ºC are 7 and 57 cm 3 mol, respectvely. Calculate the volume change upon mxng suffcent ethanol wth mol of water to gve ths concentraton. The denstes of water and ethanol are and g cm 3, respectvely, at ths temperature. Soluton: V nhov HO n EtV Et V n HO 7.0 cm mol and V 57.0 cm mol nho x 3? 3? Et.00 and HO nhon Et HO mol mol n Et 0.600; n.333 Et Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 07
11 Problem P9.5) A soluton s made up of 84. g of ethanol and 08. g of H O. If the volume of the soluton s cm 3 and the partal molar volume of H O s 7.0 cm 3, what s the partal molar volume of ethanol under these condtons? Soluton: V n V n V H O H O ethanol ethanol V n V n V V ethanol H O H O ethanol ethanol V n V HO HO cm 08. g 8.0 g mol 7.0 cm mol 84. g? g mol 3 3?? n ethanol 57.8 cm mol 3? Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 08
12 Problem P9.7) The denstes of pure water and ethanol are 997 and 789 kg m 3, respectvely. The partal molar volumes of ethanol and water n a soluton wth x ethanol = 0.0 are 55. and L mol, respectvely. Calculate the change n volume relatve to the pure components when.00 L of a soluton wth x ethanol = 0.0 s prepared. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 09
13 Problem Soluton: V n V n V V n total V n total H O H O ethanol ethanol x V x V Lmol Lmol L mol H O H O ethanol ethanol?.00 L n = 39.6 mol n n Lmol total? H O ethanol x ethanol x HO n ethanol n HO M V n n ethanol deal ethanol HO ethanol ; nh O 3.7 mol nethanol 7.90 mol 4 3??????? kg mol kg mol 7.90 mol mol =.034 L?? 789 kg m 998 kg m V V V L deal M HO H O Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 0
14 9.0 The Ideal Dlute Soluton We defne the dmensonless actvty, a solvent, of the solvent by Psolvent solvent * P solvent For a nondeal soluton, the actvty coeffcent defned by asolvent solvent x solvent The actvty coeffcent quantfes the degree to whch the soluton s non-deal. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
15 9.0 The Ideal Dlute Soluton For real soluton the chemcal potental of a component related to ts actvty: solvent * RT ln a For deal dlute soluton, solute and solvent are defned by the condtons, x solute 0 and x solvent. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
16 9.0 The Ideal Dlute Soluton Henry s law states that P x k H as x 0 where = deal dlute soluton k H = Henry s law constant Ideal dlute soluton s a soluton n whch the solvent s descrbed usng Raoult s law and the solute s descrbed usng Henry s law. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 3
17 9. Actvtes Are Defned wth Respect to Standard States The Henry s law standard state s a state n whch the pure solute has a vapor pressure k H,solute rather than ts actual value P* solute. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 4
18 9. Actvtes Are Defned wth Respect to Standard States Henry s law standard state chemcal potental s gven by solute * H * kh solute solute RT ln * Psolute The actvty and actvty coeffcent based on Henry s law are defned, respectvely, by a P k H and a x Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 5
19 Example 9.8 Calculate the actvty and actvty coeffcent for CS at x CS = Assume a Raoult s law standard state. Soluton: P R CS a CS * PCS R CS a x R CS CS Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 6
20 Example 9.9 Calculate the actvty and actvty coeffcent for CS at x CS = Assume a Henry s law standard state. Soluton: a H CS k P CS H, CS H CS a x H CS CS Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 7
21 P9.9) Calculate the actvty and actvty coeffcent for CS at = 0.70 usng the data n Table 9.3 for both a Raoult s law and a Henry s law standard state. Soluton: a R CS R CS P CS * CS P a R CS x CS Torr 5.3 Torr a H CS H CS k a CS HCS, H CS x P CS Torr 00 Torr Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 8
22 9. Henry s Law and the Solublty of Gases n a Solvent The deal dlute soluton model can be appled to the solublty of gases n lqud solvents. We wll see how to model the dssoluton of a gas n a lqud. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 9
23 Example 9. The average human wth a body weght of 70 kg has a blood volume of 5.00 L. The Henry s law constant for the solublty of N n H O s bar at 98 K. Assume that ths s also the value of the Henry s law constant for blood and that the densty of blood s.00 kg L -. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 0
24 Example 9. a. Calculate the number of moles of ntrogen absorbed n ths amount of blood n ar of composton 80% N at sea level, where the pressure s bar, and at a pressure of 50 bar. b. Assume that a dver accustomed to breathng compressed ar at a pressure of 50 bar s suddenly brought to sea level. What volume of N gas s released as bubbles n the dver s bloodstream? Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
25 Soluton a. We use the symbol c solute to desgnate the solute molarty, and c o to ndcate a molar concentraton. n N n H O 3 At 50 bar, n 5. 0L.00 kg L kg mol. 50 P k N N H mol N atbar bar bar total pressure mol Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
26 Soluton b. 3 V nrt L P. 0 bar The symptoms nduced by the release of ar nto the bloodstream are known to dvers as the bends. The volume of N just calculated s far more than s needed to cause the formaton of arteral blocks due to gas-bubble embolsms. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 3
27 The actvty of a Pure Sold or lqud The standard state of a pure sold or lqud s the pure substance at pressure p o. At another pressure p, then p' * o o o p G p G * * ' ' V dp V p' p o m, m, m, p Assumed that the sold or lqud has a nearly constant volume. Now o RT ln a V * m, p p The actvty of a pure lqud or sold at pressure p s gven by a V exp * m, p p RT o m, Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
28 Example Fnd the actvty of pure lqud water at a pressure of,000 bar and a temperature of 98.5 K Soluton: * o V m, p p ln a RT x m mol 0 N m bar 000 bar J K mol 98. 5K. 456 a = e.456 =4.87 Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
29 Exercse Fnd the pressure such that the actvty of lqud water s equal to.000 at 98.5 K. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd
30 9.3 Chemcal Equlbrum n Solutons The concept of actvty can be used to express the thermodynamc equlbrum constant n terms of actvtes for real solutons. Usng Henry s law standard state for each solute wth Gbbs energy and the chemcal potental, G reacton RT eq v a j RT ln K The equlbrum constant can be derved as j ln K eq v j eq a v j c c eq v j Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 7
31 Example 9.3 a. Wrte the equlbrum constant for the reacton N (aq, m) N (g, P) n terms of actvtes at 5 C, where m s the molarty of N (aq). b. By makng sutable approxmatons, convert the equlbrum constant of part (a) nto one n terms of pressure and molarty only. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 8
32 Physcal Chemstry 9 00 Pearson Educaton South Asa Pte Ltd
33 From Alberty, 4e. Fg A.4 Fg A.4 Partal pressure of ether acetone soluton at 30. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 30
34 From Alberty 4e, Table 6.5 Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 3
35 Chap 9 μ Actvty coeffcent γ apply to nondeal soluton wth l γ μ l RT ln as x γ x here If there are postve dervatons from Raoult s law, γ s greater than unty; and f there are negatve dervatons from Raoult s law, γ s less than unty. As actvty a = P / P *, P* To calculate the actvty coeffcent of from expermental data: γ P x P* y P x P* a γ x a P γ x Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 3
36 Example A.0 Calculate actvty coeffcents n ether-acetone soluton Calculate the actvty coeffcents for ether() and acetone() n ether-acetone soluton at 30. The expermental data are gven n Table A.5 and are plotted n Fgure A.4. Ans: At 0.5 mole fracton acetone, the actvty coeffcents of the two components are gven by γ γ P x P* P x P * 5. kpa kpa.4 kpa kpa The actvty coeffcents of both components, calculated n ths way at other mole fractons, are summarzed n Table A.5. Note: as the mole fracton of ether component approaches unty, ts actvty coeffcent approaches unty, snce the vapor pressure asymptotcally approaches that gven by Raoult s law. Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 33
37 Chap 9 For 0.5 mole fracton of acetone, the actvty coeffcent of acetone based on the dervaton from Henry s law s yp γ ' K x a P γ ' K x P* P*.4 kpa kpa γ γ H k P* 0.57 H The two types of actvty coeffcents are related by: H or γ γp* k H Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 34
38 Chap 9 Snce γ H γ H P /K x as x yp K x P Apparent Henry's constant : K ' x P A plot of K ' versus x and extraplotng to x x 0 P γ H K x 0 Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 35
39 Fg A.5 Evaluaton of Henry s law constant K as K at x = 0 for acetone n ether acetone solutons at 30 Physcal Chemstry 00 Pearson Educaton South Asa Pte Ltd 36
40 Example A.9 Proof that f Henry s law holds for the solute, Raoult s law holds for the solvent. Show that f Henry s law holds for the solute (component ), Raoult s law holds for the solvent (component ). Ans: The Gbbs-Duhem equaton provdes a relatonshp between the dfferentals of the chemcal potentals of components and at constant temperature and pressure. If μ Snce x + x =, dx = dx then Physcal Chemstry μ RT ln a dμ 00 Pearson Educaton South Asa Pte Ltd x RT μ - dμ - dx x x μ If μ x RT, then μ * ln x RT μ ln x constant RT d μ ln μ * RT Kx P * μ *. Therefore 37 d x d μ RT d x x RT RT d ln x x ln P P*
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