CHEM 217 Problems Chapter 6, Fall 2011

The boiling point of hexane at 1 atm is 68.7 °C. What is the boiling point at 1 bar? Given: The vapor pressure of hexane at 49.6 °C is 53.32 k Pa. 6.1 SOLUTION ~vapH(T2 -TI) P2 In PI = RT1T2 RT1T2 (P2 ) t.vapH = (T2-:-rI) In PI (8.3145)(322.8)(3419) I (101.325 k Pa ) = 30850 J mnl-l = 19.1n 53.32kPa , ( 101.325) -~ (J:-.- -~ ) In- -8.3145 Tl 341.9 1 100 = 341.5 K TI = 8.314~ 1 ( 101.325 ) ~ 30850 n + 341.9 100 Thus the boiling point is reduced 0.4 °C to 68.3 °C. What is the boiling point of water 2 miles above sea level? Assume the atmosphere follows the barometric formula (equation 1.54) with M = 0.0289 kg mol-l and T = 300 K. Assume the enthalpy of vaporization of water is 44.0 kJ mol-l independent of temperature. SOLUTION 6.2 h = (2 miles)(5280 ft/mile)(12 in/ft)(2.54 cm/in)(0.01 m/cm) = 3219 m p = Po exp(- gMh/R7) = (1.01325 b r) [ -(9.8 m s-2)(0.0289 kg mol-l)(3219 ml- a exp (8.314 J K-l mol-l)(300 K) (P2) In Pl = Llvap1l(T2 -Tl)/RT2T2 In ( 1.01325 ) -(44 000)(373 -Tl) 0.703 -(8.314)(373)Tl = 0.703 bar Chapter 6/Phase Equilibrium 96 TI = 364 K = 90.6 °C The barometric equation 1.44 and the Clausius-Clapeyron equation 6.8 can be used to estimate the boiling point of a liquid at a higher altitude. Use these equations to derive a single equation to make this calculation. Use this equation to solve problem 6.2. 6.3 SOLUTION The two equations are In(PJP) = gMhlRTatm In(PJP) = L\vapH(To -T)/RTTo where p ° is the atmospheric pressure at sea level and p is the atmospheric pressure at a higher altitude. T ° is the boiling point at sea level, and T is the boiling point at the higher altitude. Setting the right-hand sides 'of these equations equal to each other yields !-1-+ gMh T -T ° T atm~vapll Substituting the values from problem 6.2 yields ! --1- (9.8 m s-2)(0.0289 kg mol-l)(3.219 x 103 m) T- 373.15 + (298.15 K)(44,000 J mol-l) T = 363.58 K = 90.43 °C Liquid mercury has a density of 13.690 9 cm-3, and solid mercury has a density of 14.193 9 cm-3, both being measured at the melting point, -38.87 °C, at 1 bar pressure. The heat of fusion is 9.75 J g-l. Calculate the mel ting points of mercury under a pressure of (a) 10 bar and (b) 3540 bar. The observed melting point under 3540 bar is -19.9 °C. 6.4 SOLUTION ( 1 1 ) 3 -1 ~T T(\I] -~ ) (234.3 K) 13.690 -14.193 cm 9 ] -= 1 s= 1 (10-2mcm-l)3 L\P ~fusH 9.75 J g- = 6.22 x 10-8 K Pa-l (a) ~T= (6.22 x 10-8 K Pa-l)(9 x 100,000 Pa) = 0.056 K t = -38.87 + 0.06 = -38.81 °C (b) ~T= (6.22 x 10-8 K Pa-l)(3539 x 100,000 Pa) = 22.0 K t = -38.87 + 22.0 = -16.9 °C From the L\tG° of Bf2(g) at 25 °C, calculate the vapor pressure of Br2(1). The pure liquid at 1 bar and 25 °C is taken as the standard state. 6.5 = [~ 97 Chapter 6/Phase Equilibrium illL UTlOO PBr2 K=-p;- Br2(1) = Br2(g) L\GO = -RT In ( ~ ) 3110 J mol-l PBr2 [ - ] -p;-= exp -(8.314 J K-l mol-l)(298.15 K) PEr = 0.285 bar 2 Calculate l1Go for the vaporization of water at O °C using data in Appendix C.2 and assuming that MlO for the vaporization is independent of temperature. Use l1Go to calculate the vapor pressure of water at O °C. 6.6 SQLUTION H20(1) = H20(g) ~GO(298.15 K) = -228.572 + 237.129 = 8.577 kJ mol-l MlO(298.15 K) = -241.818 + 285.830 = 44.012 kJ mol-l Using the equation in problem 4.10(a) l1G1T2 ( T ) l1G2=T!+AH 1- i l1G(O °C) = 8.577(~) + 44.012( 1 -~) = 7.858 + 3.690 = 11.548 kJ mol-l = -RTln (~) -11.548 p = Po exp(S.3145 x 10-3 x 273.15) = 6.190 x 10-3 bar The change in Gibbs energy for the conversion of aragonite to calcite at 25 °C is -1046 J mol-l. The density of aragonite is 2.93 9 cm-3 at 15 °C and the density of calcite is 2.71 9 cm-3. At what pressure at 25 °C would these two forms of CaCO3 be in equilibrium? 6.7 SOLUTION I J mol-l ] nol-l)(298.15 K) Chapter 6/Phase Equilibrium 98 1102 -1101 = 11 V(P -1) 0 + 1046 J mol-l = (2.77 x 10-6 m3 mol-l)(p -1) p = 3780 bar *6.8 n-Propyl alcohol has the following vapor pressures: t/OC 40 60 80 100 P/kPa 6.69 19.6 50.1 112.3 Plot these data so as to obtain a nearly straight line, and calculate (a) the enthalpy of vaporization, (b) the boiling point at 1 bar, and ( c ) the boiling point at 1 atm. SOLUTION: In(PJkPs.) 5 4 3 2 1 1000JT 3.2 3.4 2.6 2.8 3 Ll vapll slope = -2.34 x 103 K = (a) (2.303) (8.314 J K-l mol-l) (b) ~vapll = 44.8 kJ mol-l In ~ = ~ (! -1-- ) 100 8.3145 T- 373.15 T- 1 -8.3145 112.3 1 = 370.18 K = 97.03 °C 44 800 In 100 + 373.15 T- 1 -~ 112.3 1 = 370.51 K = 97.36 °C 44 800 In 101.325 + 3nI5 (,..\ 6.9 For uranium hex afluoride the vapor pressures (in Pa) for the solid and liquid are given by In Ps = 29.411 -5893.5/T In PI = 22.254 -3479.9/T Calculate the temper:ature and pressure of the triple point. . £..1 vapll: 103 K= I" '1{\'1\ 'Q '11A T V-l n1-1\ Chapter 6/Phase Equilibrium 99 SOLUTION The heats of vaporization and of fusion of water are 2490 J g-l and 33.5 J g-l at 0 °C. The vapor pressure of water at 0 °C is 611 Pa. Calculate the sublimation pressure of ice at -15 °C, assuming that the enthalpy changes are independent of temperature. SOLUTION /1subH = /1fusH + /1vapH L\fusH = 33.5 + 2490 = 2824 J g-l liquid f solid .J, 6V.~U~ ~ vapor 8subH (T2 -TI) In~ PI RTIT2 P2 = PI exp[~S~~ -TI) - RTIT2 [ (2824 x 18 J mol-l)(-15 K) ] = (611 Pa) exp (8.314 J K-l mol-l)(273.15 K)(258.15 K) = 166 Pa The sublimation pressures of solid CI2 are 352 Pa at -112 °C and 35 Pa at -126.5 °C. The vapor pressures of liquid Cl2 are 1590 Pa at -100 °C and 7830 Pa at -80 °C. Calculate (a) ~subll, (b) ~vapll, (c) ~fusH, and (d) the triple point. , SOLUTION 6.11 (a) !1 .u -!!J:lI1.. In f1 su~~ -T2- TI PI (8.314 J K-I moI-I)(161.15 K)(146.65 K) I 352 = .14:5 K n35 = 31.4 kJ mol-l ~ H -(8.314 J K-l mol-l)(173.15 K)(193.15 K) l ~ yap -20 K n 1590 (b) = 22.2 kJ mol-l ~fusH = ~subH -~vapH = 31.4- 22.2 = 9.2 kJ mol-l (c) £ - -1. [ J1.subll(T2 -TI) ] PI exp RTIT2 [ (2824 x 1 , (611 Pa) exp (Sl ~1LI. T 1(-1 mn1 Chapter 6/Phase Equilibrium 100 For the solid (d) 3777 r- 1107 r- 2670 T- -15.538 = 172 K 6.12 The vapor pressure of solid benzene, C6fl6, is 299 Pa at -30 °C and 3270 Pa at 0 °C, and the vapor pressure of liquid C6H6 is 6170 Pa at 10 °C and 15,800 Pa at 30 °C. From these data, calculate (a) the triple point of C6fl6, and (b) the enthalpy of fusion of C6H6. SOLUTION Calculate the enthalpy of sublimation of C6ll6. A .u -(8.314) (253.15)(223.15) 1 ~ tisu~.. -30 n 299 = 44,030 J mol-l Express sublimation pressures as a function of T. ~subll ~subS lnPsub=-~+ -y ~subS ~ 44 030 At 273.15 K, In 3270 = -RT 44 030 + 27.481 + In Psub = - RT Calculate the enthalpy of vaporization of C6lI6 A -J/ -~8.314)(283.15)(303.15) 1 ~ Uvap'A- 20 n 6170 = 33,550 J mol-l Express vapor pressures as a function of T. ilvapH ilvapS InPvap=- RT+ -y ~vapS R 33 550 At 303.15 K, In 15,800 = -8.314(303.15) + 33 550 In Pvap = -8.314 T + 22.979 At the triple point, In Psub = In Pvap -~+.27.481 = -33 5~ + 22.979 (a) RT RT For the liquid 9200 ( 1 1) In p = In 1590 + K3I4 r7m -f = 13.762 - At the triple point 29.300- ~= 13.762- J~7 ~ T= 172 K- 273.15 K = -101.2 °C Chapter 6/Phase Equilibrium 101 T= 279.99 K = 6.85 °C ( 44 030 ) p = exp -8.314 x 279.99 + 27.481 = 5249 Pa ~fusH = ~sub1l- ~vapH = 44.03 -33.55 = 10.48 kJ mol-l The surface tension of toluene at 20 °C is 0.0284 N m-l, and its density at this temperature is 0.866 9 cm-3. What is the radius of the largest capillary that will permit the liquid to rise 2 cm? 6.13 SOLUTION 1 y = 2hpgr 2y 2(0.284 N m-i) r----pgh -(0.866 x 103 kg m-3)(9.80 m s-2)(0.02 m) = 3.35xI0-2 cm Mercury does not wet a glass surface. Calculate the capillary depression if the diameter of the capillary is (a) 0.1 mm, and (b) 2 mm. The density of mercury is 13.5 9 cm-3. The surface tension of mercury at 25 °C is 0.520 N m-l. 6.14 SOLUTION (a) 2y h=-:- gpr (2)(0.520 N m-l) == 15.7 cm (9.8 m s-2)(13.5x103 kg m-3)(O.O5x10-3 m) h = (15.7 cm)(O.O5 mm)/(l mm) = 7.86 mm (b) If the surface tension of a soap solution is 0.05 N m-l, what is the difference in pressure across the film for (a) a soap bubble of 2 mrn in diameter and (b) a bubble 2 cm in diameter. 6.15 SOLUTION 4(0.05 N m-l , 2r (a) M=R= 0.001 m =200 Pa (b) M = 4(0.§.~~ ~ m-i) = 20 Pa *6.16 From tables giving L\tGD, L\fJlO, and cPp for H2O(1) and H2O(g) at 298 K, calculate (a) the vapor pressure of H2O(1) at 25 °C and (b) the boiling point at 1 atm. SOLUTION (a) H20(1) = H20(g) LlGo = -228.572 -( -237.129) = 8.557 kJ mol-l l~ -2y 4(0.05 N m-i) = 200 Pa =R= 0.001 m .4/1\ I\~I\ ---1 \ Chapter 6/Phase Equilibrium 102 8557 J mol-l p --exp- -3168x10-2 po -(8.314 J K-l mol-l)(298.15 K) -. p = (3.168 x 10-2)(105 Pa) = 3.17 x 103 Pa MlO (298.15 K) = -214.818- (- 285.830) = 44.012 kJ mol-l (b) In the absence of data on the dependence of C p on T we will calculate MIO(7) from T MlO(7) = 44012 + f 298.15 = 44012 -41.714 (T- 298.15) (~ ) [~ (33.577- 75.291) dT ~0(1) 12 -- p ~GO -44012 41.714 T =-J I 44012 - (41.714)(298.15} ]dT 12 T + 12 T +41.7141nT+~!4¥29~+I where I is an integration constant to be evaluated from l1GO(298.15 K). ~-~ 417141 29815 {11.714)(298.15) I 298.15- 298.15 + .n .+ (298.15) + I= -398.191 At the boiling point l1Go = O and so we need to solve the following equation by successive approximations. O = .'!.'!:Q!1 + 41.714 In T + ~7J4¥298~ill- 398.191 T Trying T= 373 K RHS = 0.163 Trying T= 374 K RHS = -0.130 Thus, the standard boiling point calculated in this way is close to 373.5 K. What is the maximum number of phases that can be in equilibrium in one-, two-, and three-component systems? 6.17 SOLUTION F=C-p+2 0 = 1 -p + 2, p = 3 0.= 2 -p + 2, p = 4 0 = 3 -p + 2, p = 5 The vapor pressure of water at 25 °C is 23.756 mm Hg. What is the vapor pressure of water when it is. in a container with an air pressure of 100 bar, assuming the 6.18 -=- , -~~ v..~ ~~. .~ .,. - (~GO) [~] =-~ dT p l~ ~-- f [ --44012-~+! T -7'2 T =~+41.7141n T+ ( Chapter 6/Phase Equilibrium 103 dissolved gases do not affect the vapor pressure. The density of water is 0.99707 g/m3. SOLUTION - RTln (P/Po) = VL(P- Po) (8.3145 J K-l mol-l)(298.15 K) In (P/23.756) = (18.015 9 mol-l/0.99707 9 m-3)(10-2 m/cm)3(99 x 105 Pa) P/23.756 = exp 0.07289 P = 25.552 mm Hg = 3406 Pa 6.19 A binary liquid mixture of A and B is in equilibrium with its vapor at constant temperature and pressure. Prove that 'UA(g) = 'UA(l) and 'UB(g) = 'UBO) by starting with G = G(g) + GO) and the fact that dG = O when infinitesimal amounts of A and B are simultaneously transferred from the liquid to the vapor. SOLUTION ~ dO = dO(I) + dO(g) = O (I) since the transfer is made at equilibrium. dO(I) = ,LtA(I) dnA(1) + ,LtB(1) dnB(1) (2) dO(g) = ,LtA(g) dnA(g) + ,LtB(g) dnB(g) (3) since dnA(1) = -dnA(g) and dnB(1) = -dnB(g), substituting equation 2 and 3 into equation 1 yields dO = [,LtA(g) -,LtA(1)] dnA(g) + [,LtB(g) -,LtA(1)] dnB(g) = 0 (4) Since dnA(g) and dnB(g) are independently variable, both terms in equation 4 have to be zero for dO to be zero. This yields the desired expression for components A andB. 6.20 Ethanol and methanol form very nearly ideal solutions. At 20 °C, the vapor pressure of ethanol is 5.93 kPa, and that of methanol is 11.83 kPa. (a) Calculate the mole fractions of methanol and ethanol in a solution obtained by mixing 100 9 of each. (b) Calculate the partial pressures and the total vapor pressure of the solution. (c) Calculate the mole fraction of methanol in the vapor. ~OLUTION 100/32 (a) XCH3OH = lOO/46 + lOO/32 = 0.590 100/46 lOO/32 , 13OH = lOO/46 + 100732 = 0.- 100/46 ~H5OH = 100/46 + 100/32 = 0 Chapter 6/Phase Equilibrium 104 (b) PCH30H = XCH30H P*CH30H = (0.590)(11,830 Pa) = 6980 Pa PC2HSOH = XC2HSOH P*C2HSOH = (0.410)(5930 Pa) = 2430 Pa Ptotal = 2430 Pa + 6980 Pa = 9410 Pa 6980 Pa YCH30H,vapor= 9410 Pa= 0.742 (c) 6.21 One mole of benzene (component 1) is mixed with two moles of toluene (component 2). At 60 °C the vapor pressures of benzene and toluene are 51.3 and 18.5 kPa, respectively. (a) As the pressure is reduced, at what pressure will boiling begin ? (b) What will be the composition of the first bubble of vapor? SOLUTION (a) * * * p = p 2 + (P I -p 2 )XI = 18.5 kPa + (51.3 kPa -18.5 kPa)(0.333) = 29.4 kPa * XlPl (b) yI = * * * p 2 + (P I -p 2 )XI (0.333)(51.3 kPa) = 18.5 kPa + 32.8 kPa (0.333) = 0.581 The vapor pressures of benzene and toluene have the following values in the temperature range between their boiling points at 1 bar: 6.22 tlOC 110.0 * p C6B6/bar * PC7HS/bar 0.508 0.616 0.742 1.000 (a) Calculate the compositions of the vapor and liquid phases at each temperature and plot the boiling point diagram. (b) If a solution contaning 0.5 mole fraction benzene and 0.5 mole fraction toluene is heated, at what temperature will the first bubble of vapor appear and what will be its composition? 79.4 88 94 100 1.000 1.285 1.526 1.801 . .l\,1~ 1 Yl = * * * p ...( p -p )}"1 105 Chapter 6/Phase Equilibrium .SOL UTION At 88 °C (a) At 94 °c At 100 °C oXB(1.285) + (1 -oXB) 0.508 = 1 oXB = 0.633 YB = (0.633)(1.285)/1 = 0.814 oXB(1.526) + (1- XB) 0.616 = 1 oXB = 0.422 yB = (0.4222)(1.526)/1 = 0.644 oXB(1.801)+(1-oXB)0.742= 1 oXB = 0.244 YB= (0.244)(1.801)/1 = 0.439 tJoC 1 0.4 0.6 0.8 0.2 The liquid curve is below the vapor curve. These curves were plotted by using Mathematica TM to fit them with quadratic equations. (b) The fit of the liquid data with t = a + bXB + CXB2 yields t = 109.9 -42.72\B + 12.29XB2. At xB = 0.5, t = 91.6 °C. Fit of the vapor data yields t = 109.9- 15.13YB -15.13YB2. Setting the temperature in this equation equal to the bubble point temperature 91.6 °C yields yB = 0.72, by use of the quadratic formula. This is the composition at which the first bubble of vapor will appear . 6.23 At 1.013 bar pressure propane boils at -42.1 oC and n-butane boils at -0.5 oC; the following vapor-press'ure data are available. t 1°C -31.2 -16.3 P/kPa(propane) 160.0 298.6 P/kPa(n-butane) 26.7 53.3 Chapter 6/Phase Equilibrium 106 Assuming that these substances fonn ideal binary solutions with each other, (a) calculate the mole fractions of propane at which the solution will boil at 1.013 bar pressure at -31.2 and -16.3 °C. (b) Calculate the mole fractions of propane in the equilibrium vapor at these temperatures. (c) Plot the temperature-mole fraction diagram at 1.013 bar, using these data, and label the regions. SOLUTION (a) At -31.2 °C, Pp = xp/60.0 kPa, PB = (1- xp) 26.7 kPa p = 101.3 kPa = xp 160.0 + (1- xp) 26.7 = 26.7 + 133.3 xp xp = 0.560 At -16.3 °C, Pp = xp 298.6 kPa, PB = (1- xp) 53.3 kPa p = 101.3 kPa = xp 298.6 + (1- xp) 53.3 = 53.3 + 245.3 xp xp = 0.196 At -31.2 °C, (b) (0.560)(160.0) yP = (0.560)(i60.0) + (0.440)(26.7) = 0.884 At -16.3 °C (0. 196)(298.6) yp = (0. 196)(298.6) + (0.804)(53.3) = 0.577 (c) tJoC ~ 0 -10 -20 -30 -40 ...xp ) yp 0.2 0.4 0.6 0.8 1 107 Chapter 6/Phase Equilibrium The following table gives mole % acetic acid in aqueous solutions and in the equilibrium vapor at the boiling point of the solution at 1.013 bar. 6.24 100.0 104.4 102.1 113.8 107.5 118.1 B.P.,oC Mole % of acetic acid 0 50.0 30.0 100 90.0 70.0 In liquid 18.5 0 57.5 37.4 100 83.3 In vapor Calculate the minimum number of theoretical plates for the column required to produce an initial distillate of 28 mole % acetic acid from a solution of 80 mole % acetic acid. .SOLUTION mole t in vapor 100~ 80 60 40 20 mole t in liquid 100 60 80 20 40 Since there are four steps, three theoretical plates are required in the column. The distilling pot counts as one plate. If two li quids (1 and 2) are completely immiscible, the mixture will boil when the * * sum of the two partial pressures exceeds the applied pressure: p = p 1 + p 2 .In the vapor phase the ratio of the mole fractions of the two components is equal to the ratio of their vapor pressures. * p I xi mIM2 p*=X2=1n2M1 2 where m 1 and m2 are the masses of components 1 and 2 in the vapor phase, and M 1 and M2 are their molar masses. The boiling point of the immiscible liquid system naphthalene-water is 98 °C under a pressure of 97.7 kPa. The vapor pressure of 6.25 Chapter 6/Phase Equilibrium 108 water at 98 °C is 94.3 kPa. Calculate the weight percent of naphthalene in the distillate. SOLUTION 0.256 Weight percent naphthalene = 1.256 = 0.204 or 20.4% A regular binary solution is defined as one for which 6.26 Derive L\mixG, L\mixS, L\mixH, and L\mix V for the mixing of xi moles of component 1 with x2 moles of component 2. Assume that the coefficient w is independent of temperature. SOLUTION o o GO = ~ ni'U,.O = Xl'Ul + X2'U2 1 LlmixG = RT(xllml + x21m2) + wxlx2 LlmixS = -(aLlmixG/a7)p = -R(xllml + x21mv LlmixH = LlmixG + TLlmixS = wxlx2 Llmix V = (aLlmixG/aP)T = 0 From the data given in the following table construct a complete temperature- composition diagram for the system ethanol-ethyl acetate for 1.013 bar. A solution containing 0.8 mole fraction of ethanol, EtOH, is distilled completely at 1.013 bar. (a) What is the composition of the first vapor to come off? (b) That of the last drop of liquid to evaporate? (c) What would be the values of these quantities if the distillation were carried out in a cylinder provided with a piston so that none of the vapor escapes? 6.27 109 Chapter 6/Phase Equilibrium B.P.,oC B.P.,oC XEtOH YEtOH YEtOH XBtOH 72.0 0.563 0.507 77.15 0 0 72.8 0.710 0.600 76.7 0.070 0.025 74.2 0.735 0.833 75.0 0164 0.100 76.4 0.880 0.942 72.6 0.295 0.240 77.7 0.965 0.982 71.8 0.398 0.360 78.3 1.000 1.000 71.6 0.462 0.462 ~OLUTION t/oC 78 77' 7C I i 75 74 73 721 x(EtOH),y(EtOH) 1 0.6 0.8 0.2 0.4 I , c ", , The composition of the liquid is given by the lower curve, and the composition of the vapor is given by the upper curve. (a) The liquid with 0.8 mole fraction ethanol will start to boil at 73.7 °C. The vapor in equilibrium with it is 0.69 mole fraction in ethanol. (b) As the boiling continues, more and more of the EtAc will be removed. As the mole fraction of EtCH increases, the boiling point will also increase. Finally, the very last vapors will be pure EtCH. (c) The first vapor to come off will be the same as in part (a). However, when all of the mixture is in the vapor state, the mole fraction of the vapor state will be 0.800, simply because that is the total mole fraction of EtCH in the entire mixture. Thus, to find the composition of the last drop to evaporate into the cylinder, simply note that liquid <YEtOH = 0.9) is in equilibrium with vapor which has 0.800 EtCH in it. 6.28 The Henry law constants for oxygen and nitrogen in water at 0 °C are 2.54 x 104 bar and 5.45 x 104 bar, respectively. Calculate the lowering of the freezing point of water by dissolved air with 80% N2 and 20% 02 by volume at 1 bar pressure. Chapter 6/Phase Equilibrium 110 SOLUTION 0.8 bar 5.45 x 104 bar 0.2 bar 2.54 x 104 bar = 1.47 x 10-5 PN XN --2-- 2 -KN2 - Po xO2 = ~ = "- -= 7.87 x 10-6 2 The mole fraction of dissolved air is 2.26 x 10-5 = m 1000 m + 18.02 mair = 1.25 x 10-3 mol/IOOO 9 solvent ~Tfp= Kfp m = -(1.86)(1.25 x 10-3) =- 0.00233 K Use the Gibbs-Duhem equation to show that if one component of a binary liquid solution follows Raoult's law, the other component will too. SOLUTION ~ ~"i ~" j" 'i: ' J"'; c ;c I'i " , "'r'r "f ! i !::"'I' The following data on ethanol-chlorofonn solutions at 35 °C were obtained by G. Scatchard and c. L. Raymond [1. Am. Chem. Sac. 60,1278 (1938)]: O..jU 1.0 0.6 0.8 0.4 0 0.2 XEtOH,liq 0.4246 1.0000 0.2554 0.1864 YEtOH.vap 0.0000 0.1382 13.703 25.357 38.690 34.387 Total P/kPa 39.345 40.559 Calculate the activity coefficients of ethanol and chlorofonn based on the deviations from Raoult's law. 1 ! 1 ! r Chapter 6/Phase Equilibrium 111 .SOLUTION YEtO~ (0. 1382)(40.559) rEtOH = p* = (0.2)(13.703) XEtOH" EtOH = 0.2 ~tOH = 0.4 ~tOH -(0.1864)(38.690) -1316 -(0.4)(13.703) -. -(0.2554)(34.387) -1065 -(0.6)(13.703) -. -(0.4246)(25.357) - 0 982 -(0.8)(13.703) -. (1.OOO)(13.703) = (1.000)(13.703) = 1.000 = 0.8 ~tOH = 1.0 ~tOH ~ -YCHC13P -(1)(39.345) CHC13 -XCHC13P*CHC13 -(1)(39.345) (0.8618)( 40.559) = (0.8)(39.345) = (0.8136)(38.690) (0.6)(39.345) = (0.7446)(34.387) (0.4)(39.345) (0.5754 )(25.357) = (0.2)(39.345) =1 XCHCl3 = 1.111 = 1.333 = 1.627 = 1.854 = 0.2 XCHCl3 Show that the equations for the bubble-point line and dew-point line for nonideal solutions are given by 6.31 * P-r2P2 x = * * 1 r1PXl -r2P 2 SOLUTION * P-r2P2 X = * * 1 r1Pl -r2P 2 Chapter 6/Phase Equilibrium 112 PI - -- YI-PI+P2 * * * r2P2 +Xl(rlPl -r2P2) Substituting the expression for xi yields the desired expression for yi. A regular binary solution is defined as one for which 6.32 Derive the expressions for the activity coefficients of rl and r2in tenDS of w. SOLUTION wx22/ RT Therefore, wx22/RT y -e 1 - The expressions for the activity coefficients of the components of a regular binary solution were derived in the preceding problem. Derive the expression for r in tenns of the experimentally measured total pressure P, the vapor pressures ot the two components, and the composition of the solution for the case that the deviations from ideality are small. SOLUTION 6.33 r2 = exp(wxi/R7) 2 2 * ( WX2 ) * ( WXl ) = XlP 1 exp liT + X2P 2 exp liT XIYIPI -= * * * Chapter 6/Phase Equilibrium 113 If the exponentials are not far from unity, the exponentials may be expanded to obtain 2 2 * wx2 * WXl P=XIPl (1+RT)+x2P2 (l+RT) Thus from a measurement of the total pressure of a mixture of the two components, the activity coefficients of the components can be calculated over the entire concentration range using the first two equations given above. This is a particular example of the general situation that r 1 and r2 can be calculated from the total pressure as a function of xl. 6.34 Using the data in Problem 6.75, calculate the activity coefficients of water (I) and n- propanol (2) at 0.20, 0.40, 0.60, and 0.80 mole fraction of n-propanol, based on deviations from Henry's law and considering water to be the solvent. SOLUTION The data at the lowest mole fractions of n-propanol are used to calculate the Henry law constant for n-propanol in water. x2 K2 = P2/X2 0.02 33.5 0.05 28.8 0.10 17.6 !{2 x2 0.02 0.04 0.06 0.08 0.1 The intercept indicates that K2 = 37 kPa. 114 At I P2 r2= X2K2 x2 = 0.2 x2 = 0.4 X2 = 0.6 X2 = 0.8 rl = x2 = 0.2 -2.89 -W6)(3T'75 = 1.52 x2 = 0.4 x2 = 0.6 -2.65 -(o:4)(3:rrj = 2.09 -1.79 -(O:2R3:-m = 2.82 X2 = 0.8 .s.Q L UTI 00 68.4 --342 -= 3.59 x 10-3 X2 -68.4 IJ>..QQ 342 + 18 P*-PI ~=3.59x 10-3 -1 = x2 2.3149 PI (a) PI = 2.3149(1- 3.59 x 10-3) = 2.3066 kPa ~Tf = Kfm = 1.86(0.2) = -0.372 Tf = -0.372 °C (b) Chapter 6/Phase Equilibrium 115 6.36 The protein human plasma albumin has a molar mass of 69,000 9 mol-l, Calculate the osmotic pressure. of a solution of this protein containing 2 g per 100 cm3 at 25 °C in (a) pascals and (b) millimeters of water. The experiment is carried out using a salt solution for solvent and a membrane permeable to salt as well as water. SOLUTION -~- (20 x 10-3 kg L-l)(103 L m-3)(8.314 J K-t mol-l)(298.15 K) 1r- M -69 kg mol-l = 719 Pa 1! 719 Pa h -dg -(1 9 cm-3)(10-3 kg g-1)(102 cm m-l)(9.8 m s-2) = 7.34 x 10-2 m = 73.4 mm of water *6.37 The following osmotic pressures were measured for solutions of a sample of polyisobutylene in benzene at 25 °C. c /kg m-3 5 10 15 20 7r /Pa 49.5 101 155 211 Calculate the number average molar mass from the value of 1rIc extrapolated to zero concentration of the polymer. SOLUTION c/kg m-3 5 10 15 20 (1r/c)/(Pa/kg m-3) 9.90 10.1 10.3 10.6 ~-RT c -M+Bc n h = dg = (I 9 cm-3)( Chapter 6/Phase Equilibrium 116 press/c ~ / 10.4 10.2 10 9.8 cJkg mA-3 20 15 10 5 ~314 J K-l m~l-l)(~9~ = 255 kg mol-l = 9.7 Pa m3 kg-l RT - M= 9.7 Pa/kg m-3 Calculate the osmotic pressure of a 1 mol L -1 sucrose solution in water from the fact that at 30 °C the vapor pressure of the solution is 4.1606 kPa. The vapor pressure of water at 30 °C is 4.2429 kPa. The density of pure water at this temperature (0.99564 9 cm-3) may be used to estimate VI for a dilute solution. To do this problem, Raoult's law is introduced into equation 6.89. .SQLUTION - VI1'&= -RTln XI Substituting Raoult's law Pl = X1P~ RT Pl n=--ln- Vl ~ -1802 Q mol-l Vl =-;: = -= 18.10cm3mol-l =0.01810Lmol- 0.99564 g cm-3 = ~314 L barK-l mol-1J(303.15~ln ~2429 kP~= 27.3 bar n 0.0180 L mol-l 4.1606 kPa 6.39 Calculate the solubility of p-dibromobenzene in benzene at 20 °C and 40 °C assuming ideal solutions are formed. The enthalpy of fusion of p-dibromobenzene is 13.22kJ mol-l at its melting point (86.9 °C). ~OLUTION At 20 °C ~fusH2(T- T2r) In x2 = -R1T2f (13220 J moI-l)(- 66.9 K) = ,~- : " ---, (8.314 J K-l moI-l)(360.1 K)(293.2 K) , In x2 = -Rmf 6.38 ( t Chapter 6/Phase Equilibrium 117 x2 = 0.365 At 40 °C -~ .(~~-2~O J ~oI-l)(-~L- In x2 -(8.314 J K-l moI-l)(360.1 K)(313.2 K) x2= 0.516 Calculate the solubility of naphthalene at 25 °C in any solvent in which it fonns an ideal solution. The melting point of naphthalene is 80 °C, and the enthalpy of fusion is 19.19 kJ mol-l. The measured solubility of naphthalene in benzene is xI = 0.296. SOLUTION -In xi = .1fusHI(TQ,I -1)/R1TQ,I -~~ !~~-J.moI-I~{~ In xi -(8.314 J K-I moI-I)(353 K)(298 K) xi = 0.297 The addition of a nonvolatile solute to a solvent increases the boiling point above that of the pure solvent. The elevation of the boiling point is given by 8Tb = !jJIbA)2MA~ = KbmB 8vapllAo where TbA is the boiling point of the pure solvent and MA is its molar mass. The derivation of this equation parallels that of equation 6.82 very closely, and so it is not given. What is the elevation of the boiling point when 0.1 mol of nonvolatile solute is added to 1 kg of water? The enthalpy of vaporization of water at the boiling point is 40.6 kJ mol-l. ~OLUTION = 0.513 K kg mol-l L\Tb = (0.513 K kg mol-l)(0.1 mol kg-l) = 0.0513 K 6.42 SQIJ.ITJ.QN 12 (cr) + [C6H6J = 12 in C6H6 L\Go = -RTln x 6.40 , Chapter 6/Phase Equilibrium 18 x = 0.0569 where x is the equilibrium mole fraction of 12 The mole fraction calculated from the chemical handbook is 16.46 ~ 16:46 (100)(0.87ffi = 0.0545 ~+ "--78.12- so the values are consistent. 6.43 100 70 93 50 58 47.5 37.5 20 0 Cd Wt. % 400 -- 419 -- 461 550 First break in curve, °c 295295 321 410 439 410 410 410 Constant temperature, °C 630 Construct a phase diagram, assuming that no breaks other than these actually occur in any cooling curve. Label the diagram completely and give the formula of any compound formed. How many degrees of freedom are there for each area and at each eutectic point? .s.Q L UTI O ~ Melt 700 F=2 600 Sb2Cd3 + melt F=O t/"c 500 I Cd + melt F=l "'\ F=O 400 .-'"'l Sb + Sb2Cd3 F = 1 3001 Sb2Cd3 Sb2Cd3 + Cd F = 1 200 1-1 I' -1- ~In ro Sb 0.2 0.4 0.6 0.8 Cd Weight fraction Cd 6.44 The phase diagram for magnesium-copper at constant pressure shows that two compounds are formed: MgCu2 that melts at 800 °C, and Mg2Cu that melts at 580 °C. Copper melts at 1085 °C, and Mg at 648 °C. The three eutectics are at 9.4% by weight Mg (680 °C), 34% by weight Mg (560 °C), and 65% by weight Mg (380 °C). Construct the phase diagram. How many degrees of freedom are there for each area and at each eutectic point? Chapter 6/Phase Equilibrium 19 SOLUT1ill:l 9001 7001 trc 500 O Cu MgCu2 Mg2Cu 50 100 Mg Weight % Mg F = 2 -p + 1 In the liquid region F = 2; in the two-phase regions F = 1; and at the eutectic point F = 0. 6.45 The Gibbs-Duhem equation in the form (~) dT dM) aP T:x dP -L (XidMi) = 0 PoX applies to any molar thennodynamic property M in a homogeneous phase If this is applied to GE. it may be shown that if the vapor is an ideal gas d In ("1?) d In ("2?) X]~+X2~=O (constant 7) dx] dx] dP dyl -f(Yl -xI) -Yl(l-yl) SQLUTIO~ Chapter 6/Phase Equilibrium 120 For a solution of ethanol and water at 20 °C that has 0.2 mole fraction ethanol, the partial molar volume of water is 17.9 cm3 mol-l and the partial molar volume of ethanol is 55.0 cm3 mol-l. What volumes of pure ethanol and water are required to make a liter of this solution? At 20 °C the density of ethanol is 0.789 9 cm-3 and the density of water is 0.998 9 cm-3. SOLUTION 6.46 Component 1 is water and component 2 is ethanol. V= nlVl + n2V2 = 4mVl + mV2 = m(4Vl + Vv 1000 cm3 = n2[(4)(17.9 cm3 mol-l) + 55.0 cm3 mol-l] m = 7.90 mol = W2/46.07 9 mol-l W2 = 363.9 9 of ethanol Volume of pure ethanol = 363.9 glO.789 9 cm-3 = 461 cm3 nl = 4n2 = 31.50 = wl/18.016 9 mol-l w 1 = 569.3 9 of water Volume of pure water = 569.3 glO.998 9 cm-3 = 570 cm3 Thus there is a shrinkage of 31 cm3 when these amounts of ethanol and water are mixed at 20 °C. When 1 mol of water is added to an infinitely large amount of an aqueous methanol solution having a mole fraction of methanol of 0.40, the volume of the solution is increased by 17.35 cm3 at 25 °C. When one mole of methanol is added to such a solution, the volume increases by 39.01 cm3. (a) Calculate the volume of solution containing 0.40 mol of methanol and 0.60 mol of water. (b) What is the volume of 0.6 mol of water plus the volume of 0.4 mol of methanol before they are mixed? The densities of water and methanol at 25 °C are 0.998 g cm-3 and 0.791 g cm-3, respectively. SOLUTION 6.47 v= 0.4(39.01 cm3 mol-l) + 0.6(17.35) = 26.01 cm3 mol-l (a) (b) Derive 6.48 a2.u) Cp = -r(a"72 p from equation 4.44, H= G -T(~) aT p Chapter 6/Phase Equilibrium 12 SOLUTION For one mole of substance - ( d.u ) H=.u-T - dT p The freezing point is lowered to -5 °C under the skates, and so the answer is yes. 3.611 x 103 Pa K-t, 0.05% 6.50 6.29 x 10-3 bar 2.66 x 10-6 bar 4.9K 53.0 kJ mol-l, 28.4 kJ mol-l, 158.6 J K-l mol-l 6.54 383.38 K, -0.4 °C 96°C 6.56 27.57 kJ mol-l 0.0773 Pa (a) 0.12314 bar, (b) 0.125110 bar (a) 38.1 kJ mol-l, (b) 3.78 kPa (a) 50.91 kJ mol-l, (b) 50.14 Pa K-l, 44.20 Pa K-l, (c) Pice = 361 Pa, Pliq = 390 Pa (a) -0.46 °C, (b) -0.59 °C -8.59 kJ mol-l 6.65 4 6.66 (a) I, (b) 2 6.67 166.5 Pa 6.49 6.50 6.51 6.52 6.53 6.54 6.55 6.56 6.57 6.58 6.59 6.61 6.62 6.63 6.64 6.65 6.66 6.67 6.68 122 6.71 6.70 6.72 6.73 6.75 6.76 6.80 6.79 6.82 6.81 6.83 6.84 6.88 6.86 6.89 6.90 Chapter 6/Phase Equilibrium (a) YEtBr2 = 0.802, (b) -XEtBr2 = 0.425 -XB = 0.240, YB = 0.434 (a) XCIB = 0.591; (b) YCIB = 0.731; (c) 1.081 bar 571