Problem Set #6 soluton, Chem 340, Fall 2013 Due Frday, Oct 11, 2013 Please show all work for credt To hand n: Atkns Chap 3 Exercses: 3.3(b), 3.8(b), 3.13(b), 3.15(b) Problems: 3.1, 3.12, 3.36, 3.43 Engel Chap. 5: P5.14, P5.19, P5.20, P5.33 Extras : Atkns Chap 3 Exercses: 3.4(b), 3.7 (b), 3.14(b) Problems: 3.3, 3.7, 3.10, 3.11, 3.42, 3.45 To hand n : Atkns Exercses: 3.3(b) Calculate ΔS (for the system) when the state of 2.00 mol datomc perfect gas molecules, for whch C p,m = 7/2 R, s changed from 25 C and 1.50 atm to 135 C and 7.00 atm. How do you ratonalze the sgn of ΔS?
3.8(b) Calculate the standard entropy at 298 K of (a) Zn(s) + Cu 2+ (aq) Zn 2+ (aq) + Cu(s) (b) C 12 H 22 O 11 (s) + 12 O 2 (g) 12 CO 2 (g) + 11 H 2 O(l) 3.13(b) Calculate the change n the entropes of the system and the surroundngs, and the total change n entropy, when the volume of a sample of argon gas of mass 21 g at 298 K and 1.50 bar ncreases from 1.20 dm 3 to 4.60 dm 3 n (a) an sothermal reversble expanson, (b) an sothermal rreversble expanson aganst p ex = 0, and (c) an adabatc reversble expanson. 3.15(b) A certan heat engne operates between 1000 K and 500 K. (a) What s the maxmum effcency of the engne? (b) Calculate the maxmum work that can be done by for each 1.0 kj of heat suppled by the hot source. (c) How much heat s dscharged nto the cold snk n a reversble process for each 1.0 kj suppled by the hot source? (a) η= 1- T c /T h [eqn 3.10] = 1-500K/1000K = 0.500 (b) Maxmum work = η q h = 0.500 x 1.0 kj = 0.5 kj (c) ηmax =ηrev and wmax = q h - q c,mn q c,mn = q h - wmax = 1.0 kj 0.5 kj = 0.5 kj
Problems: 3.1 Calculate the dfference n molar entropy (a) between lqud water and ce at 5 C, (b) between lqud water and ts vapour at 95 C and 1.00 atm. The dfferences n heat capactes on meltng and on vaporzaton are 37.3 J K 1 mol 1 and 41.9 J K 1 mol 1, respectvely. Dstngush between the entropy changes of the sample, the surroundngs, and the total system, and dscuss the spontanety of the transtons at the two temperatures.confusng snce flp T f /TT/T f,later flp T
3.12 From standard enthalpes of formaton, standard entropes, and standard heat capactes avalable from tables n the Data secton, calculate the standard enthalpes and entropes at 298 K and 398 K for the CO 2 (g) + H 2 (g) CO(g) + H 2 O(g). Assume that the heat capactes are constant over the temperature range nvolved.
3.36 The proten lysozyme unfolds at a transton temperature of 75.5 C and the standard enthalpy of transton s 509 kj mol 1. Calculate the entropy of unfoldng of lysozyme at 25.0 C, gven that the dfference n the constant-pressure heat capactes upon unfoldng s 6.28 kj K 1 mol 1 and can be assumed to be ndependent of temperature. Hnt. Imagne that the transton at 25.0 C occurs n three steps: () heatng of the folded proten from 25.0 C to the transton temperature, () unfoldng at the transton temperature, and () coolng of the unfolded proten to 25.0 C. Because the entropy s a state functon, the entropy change at 25.0 C s equal to the sum of the entropy changes of the steps.
3.43 The cycle nvolved n the operaton of an nternal combuston engne s called the Otto cycle. Ar can be consdered to be the workng substance and can be assumed to be a perfect gas. The cycle conssts of the followng steps: (1) reversble adabatc compresson from A to B, (2) reversble constant-volume pressure ncrease from B to C due to the combuston of a small amount of fuel, (3) reversble adabatc expanson from C to D, and (4) reversble and constant-volume pressure decrease back to state A. Determne the change n entropy (of the system and of the surroundngs) for each step of the cycle and determne an expresson for the effcency of the cycle, assumng that the heat s suppled n Step 2. Evaluate the effcency for a compresson rato of 10:1. Assume that n state A, V = 4.00 dm 3, p = 1.00 atm, and T = 300 K, that V A = 10V B, p C /p B = 5, and that C p,m = 7/2 R.
Engel P5.14) The standard entropy of Pb(s) at 298.15 K s 64.80 J K 1 mol 1. Assume that the heat capacty of Pb(s) s gven by C P,m Pb,s J mol 1 K 1 22.13 0.01172 T K 1.00 105 T The meltng pont s 327.4 C and the heat of fuson under these condtons s 4770. J mol 1. Assume that the heat capacty of Pb(l) s gven by a) C P,m Pb,l J K 1 mol 1 32.51 0.00301T K a. Calculate the standard entropy of Pb(l) at 500 C. b.calculate H for the transformaton Pb(s,25 C) Pb(l,500 C). 600.55 K 773.15 K p,m ΔHfuson p,m SmPb,,773 K SmPb,s,298.15 K dt/k dt/k 298.15 K T/K T fuson 600.55 K T/K 600.55 K 298.15 K 773.15 K 600.55 K 22.13 0.01172 T/K 1.00 10 T /K 4770 J mol dt/k T/K 600.55 K T/K 32.51 0.00301 T/K d T/K C 2 K 2 5 2 2 1 1 1 1 1 1 1 1 1 64.80 J mol K 20.40 J mol K 7.94 J mol K 7.69 J mol K b) 100.8 J mol K 600.55K 1 1 lqud T/K ΔHfuson Cp,m dt/k sold total Cp,m d 298.15K 600.55K 1 1 1 8918 J mol 4770 J mol 5254 J mol H 18.94 10 3 J mol 1 773.15K C
P5.19) Under anaerobc condtons, glucose s broken down n muscle tssue to form lactc acd accordng to the C 6 H 12 O 6 2CH 3 CHOHCOOH. Thermodynamc data at T = 298 K for glucose and lactc acd are gven here: H f kj mol 1 C p J K 1 mol 1 S (J K 1 mol 1 ) Glucose 1273.1 219.2 209.2 Lactc acd 673.6 127.6 192.1 Calculate the entropy of the system, the surroundngs, and the unverse at T = 310. K. Assume the heat capactes are constant between T = 298 K and T = 330. K. The standard entropy and enthalpy and heat capacty for ths are: note: here C P o rxn = C p o ΔS ΔH (298 K) (298 K) ν ν S f, H f, 2 1 1 192.1J K mol 1 209.2 J K mol 175.0 J K mol 1 1 2 673.6 kj mol 1 1273.1kJ mol 74.1kJ mol 1 1 2 127.6 J K mol 1 219.2 J K mol 36.0 J K Cp, (298 K) mol At 310 K the entropy and enthalpy for ths are then: ΔS (310 K) ΔS (298 K) C p, Tf ln T 310 K 175.0 J K mol 36.0 J K mol ln 176.4 J K mol ΔH (310 K) ΔH (298 K) C 298.15 K 74.1kJ mol 36.0 J K mol 310 K 298.15 K 73.7 kj mol And the entropes for the surroundngs and unverse: ΔS surroundngs - dq T H T T 73.7 kj mol 310 K f T 237.7 J mol K 1 ΔS unverse ΔS ΔS surroundn gs 1 1 176.4 J K mol 237.7 J mol K 414.1J mol K
P5.20) Consder the formaton of glucose from carbon doxde and water, that s, the of the followng photosynthetc process: 6CO 2 (g) + 6H 2 O(l) C 6 H 12 O 6 (s) + 6O 2 (g). The followng table of nformaton wll be useful n workng ths problem: T = 298 K CO 2 (g) H 2 O (l) C 6 H 12 O 6 (s) O 2 (g) H f kj mol 1 393.5 285.8 1273.1 0.0 S J mol 1 K 1 213.8 70.0 209.2 205.2 C P,m J mol 1 K 1 37.1 75.3 219.2 29.4 Calculate the entropy and enthalpy changes for ths chemcal system at T = 298 K and T = 330. K. Calculate also the entropy of the surroundng and the unverse at both temperatures. The standard entropy and enthalpy and heat capacty for ths are: ΔS (298 K) ν S f, 1 1 1 209.2 J K mol 6 205.2 J K mol 1 1 6 70.0 J K mol 6 213.8 J K mol 262.4 J K mol (298 K) ν H f, 1 1 1 1273.1 kj mol 6 0 kj mol 1 1 6 285.8 kj mol 6 393.5 kj mol ΔH C 2802.7 kj mol 1 1 p, (298 K) 1 219.2 J K mol 6 29.4 J K mol 1 1 6 75.3 J K mol 6 37.1J K mol 278.8 J K mol At 310 K the entropy and enthalpy for ths are then: ΔS (310 K) ΔS (298 K) C p, Tf ln T 330 K 262.4 J K mol 278.8 J K mol ln 290.7 J K mol ΔH (310 K) ΔH (298 K) C T T 298.15 K 2802.7 kj mol 278.8 J K mol 330 K 298.15 K 2793.8 kj mol And the entropes for the surroundngs and unverse: ΔS -dq H surroundngs 2793.8 kj mol T T 330 K unverse surroundngs 3 1 f 8.47 10 J mol K 3 1 3 1 ΔS ΔS ΔS 290.47 J K mol 8.47 10 J mol K 8.76 10 J mol K
P5.33) An electrcal motor s used to operate a Carnot refrgerator wth an nteror temperature of 0.00 C. Lqud water at 0.00 C s placed nto the refrgerator and transformed to ce at 0.00 C. If the room temperature s 20 C, what mass of ce can be produced n 1 mn by a 0.25-hp motor that s runnng contnuously? Assume that the refrgerator s perfectly nsulated and operates at the maxmum theoretcal effcency. We need to fnd the amount of heat per unt tme that can be removed from the nteror of the refrgerator. q t η r w t T hot 273 K 293 K 273 K T cold T cold w t 0.25 hp 746 W hp 2546 J s The number of grams of ce that can be frozen n one mnute by ths amount of heat s: q / t H fuson t M ce 2546 J s 6008 J mol 2 60 s mn 18.02 g mol 4.510 g mn Extras : Atkns Exercses: 3.4(b) A sample consstng of 2.00 mol of datomc perfect gas molecules at 250 K s compressed reversbly and adabatcally untl ts temperature reaches 300 K. Gven that C V,m = 27.5 J K 1 mol 1, calculate q, w, ΔU, ΔH, and ΔS. note: ths s an unusual C V looks lke C P, but do problem as wrtten anyway
3.7(b) The enthalpy of vaporzaton of methanol s 35.27 kj mol 1 at ts normal bolng pont of 64.1 C. Calculate (a) the entropy of vaporzaton of methanol at ths temperature and (b) the entropy change of the surroundngs. 3.14(b) Calculate the maxmum non-expanson work per mole that may be obtaned from a fuel cell n whch the chemcal s the combuston of propane at 298 K. (for next week!) Problems: 3.3 A block of copper of mass 2.00 kg (C p,m = 24.44 J K 1 mol 1 ) and temperature 0 C s ntroduced nto an nsulated contaner n whch there s 1.00 mol H 2 O(g) at 100 C and 1.00 atm. (a) Assumng all the steam s condensed to water, what wll be the fnal temperature of the system, the heat transferred from water to copper, and the entropy change of the water, copper, and the total system? (b) In fact, some water vapor s present at equlbrum. From the vapor pressure of water at the temperature calculated n (a), and assumng that the heat capactes of both gaseous and lqud water are constant and gven by ther values at that temperature, obtan an mproved value of the fnal temperature, the heat transferred, and the varous entropes. (Hnt. You wll need to make plausble approxmatons.)
3.7 The standard molar entropy of NH 3 (g) s 192.45 J K 1 mol 1 at 298 K, and ts heat capacty s gven by eqn 2.25 wth the coeffcents gven n Table 2.2. Calculate the standard molar entropy at (a) 100 C and (b) 500 C. equaton 2.25 and Table 2.2:
3.10 A gaseous sample consstng of 1.00 mol molecules s descrbed by the equaton of state pv m = RT(1 + Bp). Intally at 373 K, t undergoes Joule Thomson expanson from 100 atm to 1.00 atm. Gven that C p,m = 5/2 R, μ = 0.21 K atm 1, B = 0.525(K/T) atm 1, and that these are constant over the temperature range nvolved, calculate ΔT and ΔS for the gas.
3.11 The molar heat capacty of lead vares wth temperature as follows: T/K 10 15 20 25 30 50 Cp,m /(J K 1 mol 1) 2.8 7.0 10.8 14.1 16.5 21.4 T/K 70 100 150 200 250 298 Cp,m /(J K 1 mol 1) 23.3 24.5 25.3 25.8 26.2 26.6 Calculate the standard Thrd-Law entropy of lead at (a) 0 C and (b) 25 C.
3.42 Suppose that an nternal combuston engne runs on octane, for whch the enthalpy of combuston s 5512 kj mol 1 and take the mass of 1 gallon of fuel as 3 kg. What s the maxmum heght, neglectng all forms of frcton, to whch a car of mass 1000 kg can be drven on 1.00 gallon of fuel gven that the engne cylnder temperature s 2000 C and the ext temperature s 800 C? 3.45 The expressons that apply to the treatment of refrgerators also descrbe the behavor of heat pumps, where warmth s obtaned from the back of a refrgerator whle ts front s beng used to cool the outsde world. Heat pumps are popular home heatng devces because they are very effcent. Compare heatng of a room at 295 K by each of two methods: (a) drect converson of 1.00 kj of electrcal energy n an electrcal heater, and (b) use of 1.00 kj of electrcal energy to run a reversble heat pump wth the outsde at 260 K. Dscuss the orgn of the dfference n the energy delvered to the nteror of the house by the two methods.