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1 Answer each questin in the space prvided; use back f page if extra space is needed. Answer questins s the grader can READILY understand yur wrk; nly wrk n the exam sheet will be cnsidered. Write answers, where apprpriate, with reasnable numbers f significant figures. Yu may use nly the "Student Handbk," a calculatr, and a straight edge. 1. (20 Pints) Cmplete the sentence in the left clumn using the answers prvided in the right clumn. Where requested, explicitly write ut True r False s as t avid ambiguity. DO NOT WRITE IN THIS SPACE 1. Fr a multi-cmpnent, multi-regin system that allws the transfer f the different cmpnents between regins f the system, the f f an individual species is i thrughut the entire system at equilibrium. 2. The varius Fundamental Equatins f Thermdynamics are unrelated and prvide different ways f cnsidering the same infrmatin. True r False FALSE 3. One measure f the nn-ideality f a pure vapr r a mixture f vaprs is the b. 4. Fr gas-phase reactins f species described by the fllwing equatin f state: PV = nrt, the equilibrium cnstant accunts fr the e cntributins t the Gibbs Free Energy f reactin arising frm g f the individual gases. a. equal t b. fugacity cefficient c. spatially-dependent d. Cefficient f nnideality, C NI p. 1 /20 p. 2 /20 p. 3 /20 p. 4 /20 p. 5 /20 ============= p. 6 /15 (Extra credit) ============= TOTAL PTS / Fr ideal, dilute slutins, the Henry s Law standard state represents a real, stable state f a pure fluid at the relevant system cnditins f temperature, pressure, and cmpsitin. True r False FALSE e. entrpic f. chemical ptentials g. mixing h. less than i. the same and timeindependent

2 NAME: 2 2. (20 Pints) Cnsider the fllwing reactin at T = K and P = 1 bar: H 2 (gas) + CO 2 (gas) = CO (gas) + H 2 O (gas) Starting with a mles f H 2 (gas) and b mles f CO 2 (gas), hw many mles f each f the fur cmpnents are present at equilibrium. Assume ideal gas behavir fr each species and standard pressure f 1 bar. Slutin: H 2 CO 2 H 2 O CO Initial mles a b 0 0 equilibrium Mle equilibrium a-n b- n n n a n a + b b n a + b n a + b n a + b At equilibrium, ΔG rxn = RT ln K P The standard mlar Gibbs free energy f reactin is given by: ΔG rxn =ν H2 ΔG frmatin,h 2 = ΔG frmatin,h 2 ΔG frmatin,co2 +ν CO2 ΔG frmatin,co 2 +ν CO ΔG frmatin,co +ν H2O ΔG frmatin,h 2O + ΔG frmatin,co = 0 ( kcal ml ) + ( 32.81kcal kcal ) + ( ml ml ) = 6.81 kcal kj = ml ml Thus, the dimensinless equilibrium cnstant is: J ml = RT ln K P + ΔG frmatin,h 2O K P = e J ml RT = e =1.0x10 5 At equilibrium (nte all expnents are unity, s they are nt explicitly included here), K P =1.0x10 5 = p CO p H 2 O p H 2 p CO2 = x x CO H 2O n 2 = x H 2 x CO2 (a n)(b n) Scre fr Page

3 NAME: 3 K P =1.0x10 5 = p CO p H 2 O p H 2 p CO2 n 2 = K P n 2 (a + b)n + ab 0 = (K P 1)n 2 K P (a + b)n + K P ab = x x CO H 2O n 2 = x H 2 x CO2 (a n)(b n) = K P n 2 K P (a + b)n + K P ab n = K P (a + b) ± K 2 p(a + b) 2 4abK P (K P 1) 2(K P 1) If a =1ml and b=2ml, n = 3K P ± 9K 2 p 8K P (K P 1) 2(K P 1) = 3( ) ± 9( )2 8( )( ) 2( ) = ml If yu frget the quadratic frmula, ne can cnsider that Kp is small and make sme apprximatins. With small Kp, n will be small. We can retain nly certain terms in the quadratic plynmial: K P =1.0x10 5 = p CO p H 2 O p H 2 p CO2 = x CO x H 2O n 2 = x H 2 x CO2 (a n)(b n) n 2 (a)(b) n 2 abk P n abk P = 2( ) = ml Thus, small Kp apprximatin leads t essentially the same final equilibrium mles and the number is quite small as expected fr self-cnsistency. Scre fr Page

4 NAME: 4 3. (20 Pints) Determine the enthalpy f vaprizatin f methanl using the fllwing experimental data fr temperature versus equilibrium saturatin pressure (vapr pressure) f pure methanl. State any assumptins yu make in yur slutin f this prblem. Pressure (mm Hg) T (Celsius) Slutin: Use Clausius-Clapeyrn equatin with assumptin f ideal gas and cnstant vaprizatin enthalpy (ver the given temperature range). ln P = -ΔH vap R 1 T + c Plt ln Pressure versus inverse temperature and perfrm linear regressin. Regressin Results: Number f bservatins = 10 Mean f independent variable = Mean f dependent variable = Standard dev. f ind. variable = Standard dev. f dep. variable = Crrelatin cefficient = Regressin cefficient (SLOPE) = Standard errr f cefficient = t - value fr cefficient = Regressin cnstant (INTERCEPT) = Standard errr f cnstant = t - value fr cnstant = y = * x H vap = (-R)(slpe) = 36.8 kj/ml (see figure; circles = data; squares = regressin) Scre fr Page

5 NAME: 5 4. (20 Pints) The binary system acetnitrile(1)/nitrmethane(2) is sufficiently well-described as an ideal slutin (bth vapr and cndensed phase are ideal) fr cnsidering vapr-liquid equilibrium (VLE) (see figure). Fr this prblem, species 1 is acetnitrile and species 2 is nitrmethane. One can thus use Rault s expressin relating the mle fractins f species in the vapr t the cndensed phase mle fractins and individual species saturatin vapr pressures. Using this infrmatin, alng with the fllwing Antine relatins fr the temperature dependence f the individual species saturatin pressures, determine the equilibrium liquid and vapr cmpsitins fr a temperature f 75 Celsius and X 1 =0.6. Data: Antine equatins fr the vapr pressures f pure species. Temperature in Celsius and pressure in kilpascal (kpa). ln P 1 saturatin = T ln P 2 saturatin = T Slutin: First determine the vapr pressures f the tw species at the given temperature, T=75 Celsius = K. P saturatin 1 = e T = e = kPa P saturatin T = e = e = kPa P Ttal = x 1 P saturatin 1 + x 2 P saturatin 2 = (0.6)( ) + (0.4)( ) = = saturatin x y 1 = 1 P 1 x 1 P saturatin 1 + x 2 P = saturatin = y 2 =1 y 1 = x 1 = 0.6 x 2 = 0.4 Scre fr Page

6 NAME: 6 5. (20 Pints). State which thermdynamic ptential reaches an extremum under the fllwing external cnstraints n a system; cnsider n external wrk ther than pressure-vlume wrk is pssible fr this prblem: A. Temperature, pressure, and system cmpsitin Gibbs Free Energy. Temperature, vlume, and system cmpsitin Helmhltz Free Energy C. Entrpy, vlume, and system cmpsitin Internal Energy D. Internal energy, vlume, and system cmpsitin Entrpy

7 NAME: 7 6. (15 nus Pints). It can be shwn that the partitin functin f an ideal gas f N diatmic mlecules in an external electric field, ε, is: Q = [q]n k with q = C T sinh µε N! µε k T Here, T is temperature, k is ltzmann s cnstant, µ is the diple mment f a single mlecule, and C is a cnstant independent f ε. The partitin functin, Q, relates t the Helmhltz Free Energy thrugh the fllwing equatin: A = k T ln Q = k T ln [q]n N! Using this infrmatin alng with the Fundamental Thermdynamic Relatin fr the ttal derivative f the Helmhltz Free energy: da = SdT PdV (Nµ ) dε where µ is the average diple mment f a mlecule in the directin f the external field, ε, shw that at cnstant temperature and vlume: µ = µ cth µε k T k T µε Slutin: Frm the relatin fr the ttal differential f A at cnstant vlume and temperature: da = SdT PdV (Nµ) dε = (Nµ ) dε thus A ε = Nµ µ = -1 A N ε That s it; simply a matter f ding sme differentiatin nw: Scre fr Page

8 NAME: 8 µ = -1 A N ε = k T ln Q N ε = k T N ln q N ε = -1 k T ln Q N ε k T ln N! N ε secnd term is zer since N! des nt depend n electric field µ = k T N ln q N ε = Nk T ln q N ε = k T ln q ε = k T lnε ε = k T ε ln sinh µε k T + k T ε µ + k T cth µε k T k T = µ cth µε k T k T µε Scre fr Page