Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) Hmewrk 0A KEY (there will nt be a 0B) QA) Let c be the speed f sund in air. he square f the speed f sund, () f the gas with respect t the change in the density ( ) f the gas, at cnstant entrpy (sund travels adiabatically) Yu might check ut the Wikipedia n Speed f Sund. c, equals the change in the pressure A) Re-express the abve wrds int an equatin (that then will need t be slved t determine the actual speed f sund). c U B) simplify the analysis f the equatin f A: Evaluate the derivative that relates the density t the mlar vlume. (Yu will need this t simplify the expressin f A); M M W WC d M W d MW C). Frm this expressin, and the cyclic rule, relate the speed f sund t the mlecular weight and the temperature assuming an ideal diatmic gas. Using the abve tw expressins, relate c t the derivative S S d d c n n d S nm S d W S S c S C nmw S nm W C IG.. R nm W M W c C C C C C C IG.. 7 D) At rm temperature, 5 C, what is the speed f sund fr an average air atm weighing 9 g/mle. Cmpare yur answer t the actual speed f sund. Assume air behaves as an ideal gas. 7 7 c R R 45m 5 M W 5 M W sec his answer is extremely clse t the experimental speed f sund. E) what is the mean square velcity f the gas under the same cnditins as D)? Why is the rt-means-square velcity different frm the speed f sund? m v R v rms v R 506 m M W sec he speed f sund is a unidirectinal prpagatin which des arise frm the mbility f the gas mlecules. It seems t make sense that the velcity f directed infrmatin wuld be slwer than the randm mtin f the mlecules. But realize, the mlecular mtin is what causes sund t prpagate in a medium at all. 5
Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) Q) Shw that the reversible wrk W=-d cannt be an equatin f state. Use the Universal Equatin f state t shw this is nt pssible. First use the fact that is a state functin t relate beta and kappa: ln ln Nw, using that use the Euler test n the wrk: dw d d d w w 0 Q) Fr a simple A t B transitin (which can include a chemical reactin), it is instructive t realize all needed quantities fllw frm the chemical ptential. rimarily we are thinking f a transitin such as ice t water but it applies t a change between any tw states. he general expressins are given: B A Using dg Expressin d Sd d Let us assume that the chemical ptential has a simple dependence n the pressure and temperature: [his dependence is nt necessarily true fr any system, it is made up t shw hw the chemical ptential is used t derive the phase transitin -- r the equilibrium status f a chemical reactin.] A) he first item is: Find the relatin f and when the tw phases are in equilibrium. 0 B) Shw that yu can get H and Sbth frm the chemical ptential. Give explicit equatins fr H and Sthis this particular cases f
Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) S and H C) art B) gave H and S, frm the chemical ptential. Nw shw that H S equals the riginal. S H H S D) Shw, fr equilibrium (r cexistence) H S (Ntice this is nly true at equilibrium, nt all the time, but it is cnsistent with the riginal result in A). S H H S E) Derive frm the chemical ptential an expressin fr: F) Using the general frm f the chemical ptential, develp the Clausius Clapeyrn Equatin 0 d Sd d d S d d d H G) Slve the general Clausius-Clapeyrn equatin fr the specific frm f the enthalpy, H, yu fund in C) and btain a relatin between and. [It is nt suppsed t lk like anything else yu have seen.] H d H d d d d d d d 0
Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) H) Shw that the result f as a functin f is the same as yu fund in A), but nw yu went thrugh the differential frms and the expressin fr the enthalpy. Frm A gt that 0 nw we get the analgus answer that 0 0 d. 4A) Breathalyzer tests are based n Henry s law. A sample f breath taken frm a driver has a partial pressure f ethanl which is.0 0 6 atm. What is the percentage by weight f ethanl in the breath? Assume ethanl frms a Henry s Law slutin in the saliva and that ethanl in the breath is in equilibrium with ethanl in the saliva. Assume the Henry s Law cnstant fr ethanl is.0 0 atm. he vapr pressure f pure ethanl is much higher than water. Ethanl bils at 78.4C, and at 5C has a vapr pressure f 59. mmhg r trr. K H HO.00.00 6.00 We nw have the mle fractin but need the weight fractin. Assume in water f 8g/mle, ethanl is 46 g/mle. In an arbitrary vlume,, the mass fractin f ethanl is: met met net 46 f.56.560 0.6% msl mwater nwater 8 Is the persn drunk? In the breath finding 0. mg in liter f air (breath) cnverts t 0.0 g in a deciliter f bld (this 0.0% f alchl). he legal max limit is 0.08% in the bld. he density f water is gram/cc r kg/liter r 00 grams in ne deciliter. Using the abve mass fractin then there is 0.6 grams f ethanl in the persn s bld which is indeed way ver the limit 6/8 = times legal max. Q5) Imagine an ideal slutin in equilibrium with its vapr. Let the mle fractin f cmpnent i in slutin = x i. Let the mle fractin f cmpnent i in vapr = y i. he partial pressures f the gaseus cmpnents are i = y i where is the ttal pressure and i = and y i =.One way f writing Rault s Law is i = x i, where x i =. (Reading Sectin 8. in the text will be useful fr answering these questins) he pint f the exercise is simply t understand smething abut bubble pints, dew pints, and biling pints f mixtures. a. Fr a binary mixture f tw cmpnents ( and ), find an equatin fr using nly,, and x. his is the bubble pint line. If the pressure n a binary slutin is reduced frm a high value, nce the pressure crsses the bubble pint, bubbles f vapr will frm in slutin. hink f a bttle f Cke r beer. he cntents are under pressure. When yu pen the bttle, the pressure drps and bubbles frm in slutin. x y i x x x x i i i i b. Find an equatin fr y using nly,, and x. Answer : y x y x x x y x x
Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) c. Rearrange the equatin frm part (b) t find an equatin fr x using nly,, and y. x y x y y x x y x y y y d. Use the infrmatin abve t verify that: x y y x y y y x x y y his equatin relates the ttal pressure t the mle fractin f cmpnent in the vapr phase. his is the equatin fr the dew pint line. At lw pressures the system is in a vapr. When the pressure is raised t when it crsses the dew pint line, cndensatin causes drplets f liquid t frm. e. he same definitins abve can be used t understand hw slutins bil. Given that a tluene-benzene slutin bils at 00 C at.00 atm, and that = 0.74 atm and =.800 atm, find: i. the mle fractin x f tluene in the slutin (Answer: x = 0.76) x y x y.80 x 0.74.80.8 x 0.756 0.74.80 ii. the mle fractin y f tluene in the vapr (Answer: y = 0.56) x 0.756 0.74 y 0.56.0 Q6) Benzene (Bnz) and Acetne (Ace) frm an ideal slutin at 00K (i.e. the mixture beys Rault's law). and Ace 50. rr and Bnz 00. rr.[yu shuld sketch a pressure v. cmpsitin diagram. It might help in grading.] A) he pressure is reduced n the liquid (mixture) until the first vapr frms and the vapr at this pint is 75% benzene, ybnz 0.75. What is the ttal pressure n the vapr at this pint? he best way t wrk this n an exam is g in a different rder. he direct expressin is rather awkward and takes a while t develp, but as we have it abve, 0050 7.65trr y 00 0.75 50 00 B) What is the mle fractin f the acetne in the liquid?
Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) x y y 0.57. x 0.7 50 C) If the pressure is reduced further, when the last trace f liquid disappears what is the mle fractin f acetne in the vapr? y 0.7, relative cmpsitin des nt change but we mve frm a liquid t a vapr (r gas). D) If the pressure is reduced further, at what pressure des the last trace f liquid disappear? Qualitatively we will be very clse (but abve) 00 trr. Given y 0.7, we can directly cmpute the ttal pressure. x y y i.7.878 xi i i i 50 00 08..7.878 50 00 E) What is the mle fractin f acetne in the last trace f liquid? x y x y 0.7 08 y 0.7 08 x 0.05 50 It is almst pure benzene at this pint. Just as the vapr is enriched by the mre vlatile cmpnent, nw the liquid is enriched by the less vlatile cmpnent (the benzene). Additinal practice questin fr the final exam: EQ) Osmtic ressure he Clumbia river drains int the acific Ocean at a rate f 7,500 m f water/sec. he river is fresh water, and the cean is salt water, which is abut M in ins ttal (nt wrrying abut which kind, but basically NaCl). A) If there were a semi-permeable membrane that let water thrugh but nt salt, hw great wuld the smtic pressure be between the cean and the river? c R 0.08 98 5Atm slute B) One atmsphere f smtic pressure will raise water feet r 0 meters. Using the smtic pressure hw high wuld the cean rise cmpared t the fresh water? h 05 50 meters C) his membrane then culd act like a dam with the height given by part B). Hw much energy wuld be captured by a turbine run by the water falling ver a barrier such as this? [Hint: Cmpute the ptential energy difference fr the water frm the tp t the bttm f a height generated by the smtic pressure.] he mass f water is kg/liter r 0 kg/m. E mgh 0 9.850 J.4 MJ m m D) [Fllwing what yu did in part C).] What is the ptential energy utput (in Jules) fr ne cubic meter f water t fall 00 meters? E mgh 0 9.800 J.9 MJ m m
Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) 6 E) A typical pwer plant generates 600 MW r 600 0 J sec. Given that the Clumbia drains 7,500 m sec f water hw many pwer plants culd be replaced by a 00 meter high dam at the muth f the Clumbia? [he US uses abut 5000 such plants fr the entire US electrical need.] E MJ 7.50 m 50 MJ m sec sec 50 MJ E sec 5 pwer plants 600 MJ sec/ pwer plant kj vap. mle EQ) Water bils at 00C at ne atmsphere, the heat f vaprizatin is H 40 [Explain what apprximatins yu use t determine yur results?] Water is 55M, r 0.08. A) What is the partial pressure f the water abve a.0 M NaCl slutin f water at 00C? he chemical ptentials f the water is lwered, s the gas must d the same (at 00C). g g g R ln R ln g g salt 4.0 0.07 55 g 0.07 0.97Atm At 00C, expect water t be nearly biling, but nw salt is added and this reduces the partial pressure by making the liquid mre stable. B) At what temperature will the slutin frm part A) bil? Use the C.C. equatin and ntice that the pressure is.97 at 00C and want t find where =Atm H ln R 40 ln 0.97 8. 0.7 0.006 0.C mle
Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) EQ) Benzene and luene frm an ideal slutin at 00K (i.e. the mixture beys Rault's law). 0.0rr and 90.0rr. he liquid is cmpsed f mles f tluene and mles f tluene Benzene benzene. [Yu may want t sketch a pressure vs. cmpsitin diagram.] B=benzene, A=tluene a) At what pressure des the first vapr frm? x.490.60..6 0 54trr i i i ure liquid is.4 mle fractin Benzene, and.6 luene. b) What is the cmpsitin f the first trace f vapr frmed? x y x i i.490 yi 0.67 trr Benzene 54 c) If the pressure is reduced further, at what pressure des the last trace f liquid disappear? Benzene 0.4 yi.4.6 xi i 90 0 4.0 trr.4.6 90 0 d) What is the cmpsitin f the last trace f liquid? y x y y i 0.44 xi 0.8 trr Benzene i 90 his set f questins is very similar t the ne abve with Acetne and Benzene. Nw the benzene is the mre vlatile cmpnent. EQ4) he partial pressure f ethanl and water as a functin f the mle fractin, at 5C, are given. trr trr HO
Autumn 0 CHEM45B Bruce H. Rbinsn Guld Hall HW 0(A) Using this data determine Henry s cnstant fr ethanl in water and water in ethanl. As develped in class the Henry s cnstant is fund in the limit as the mle fractin f ethanl ges t zer fr the Henry s law cnstant f ethanl, where the water is taken t satisfy Rault s law (in this limit, as the slvent). K trr trr HO HO K K H lim K H 0 HO HO 0.0.05 0.08.0 = 4.800 00.7800 87.9050 79.4800 =trr=0.atm