Gas Expansion and Compression: Thermodynamic Analysis

Name______________ Chem. 450 Midterm II March 17, 2023 1 liter=0.001 m3 R=8.314 J/K*mol; R=8.314 x 10-2 L*bar/K*mol; R=8.206 x 10-2 L*atm/K*mol k=1.38 x 10-23 J/K and k = R/Na Na=6.022 x 1023/mol 1 bar = 105 Pa 1 Pa = 1 N/m2 1 J = 1 Nm pV=NkT pV=nRT U q w H U pV dwrev=-pdV dqrev=TdS w Vf Vi p ex dV dG = Vdp – SdT H Cp T p CV U T V dU=CVdT dH=CpdT Tf=Ti(Vi/Vf)1/c c=Cv,m/R pV =constant =(Cp,m/CV,m) S=qrev/T S = k ln (W) Cp,m-Cv,m=R A= U-T S df = g dx + h dy = dS-dq/T 0 (Clausius Inequality) trnsS= trnsH/Ttrans G n T,p S = nRln(Vf/Vi) S=nCp,mln(Tf/Ti) (1) (20 points) Consider a gas which expands reversibly inside a cylinder from Vi to Vf. The initial pressure of the gas is pi. (a) Draw a curve on a pressure-volume diagram depicting this process if it occurs isothermally. (b) On the same graph, starting from the same initial pressure and volume, draw a curve depicting the expansion process if it occurs adiabatically. Please label your axes and the two curves on the diagram. (c) Shade the region on your diagram denoting the difference in the amount of work done during the adiabatic expansion and the isothermal expansions. (d) In which expansion process was more work done? (e) For the isothermal expansion, derive an expression for the final pressure, pf, in term of pi, Vi, and Vf. (f) Why is the value of the final pressure, pf, different in the case of adiabatic expansion? aisothermalprocess II iii f ithasagreaterchangeinpressurebecause isothermal p authhyp V s urroundings without thepressureisreleasedtothe backtothe going system (2) (18 points) (a) Consider an isothermal, reversible compression that takes place inside a cylinder fitted with a piston. What is the sign change for the entropy in each of the following cases (goes up, goes down, stays the same)? (i) Ssys goes up (ii) Ssurr goes down (iii) Stot stays thesame (b) Consider an isothermal, completely irreversible expansion (i.e., no work is done) which takes place inside a cylinder fitted with a piston. What is the sign change for the entropy in each of the following cases (goes up, goes down, stays the same)? (i) Ssys (ii) Ssurr (iii) Stot goes up stays the goes same up (c) Was the process in (a) spontaneous? Was the process in (b) spontaneous? a wasspontaneous but b was not (3) (16 points) 3.25 moles of a perfect gas are placed into a 10.8 L container at 273 K. The sample is allowed to expand to 80.6 L, while it is simultaneously heated to 374 K. What is the change in entropy for this system if Cv,m = 31.6 J/mol K? (Hint: remember that entropy is a state function). Δs Δs Δg Δ5 0T NYT 32513,1 401 10.26 32513243401 (4) (18 points) (a) Derive an exact differential for dA with respect to volume and temperature starting from dA = dU – d(TS). (b) What is the Maxwell relationship associated with this expression? (c) Use the Maxwell relationship you found in (b) to show how the entropy of a perfect gas varies with volume at constant temperature. da do acts a dA Tds do Tds dA par sat Pdr d TS IT an H b c AA dA pdr Sdt S 3 AV a dt binv atblnV dT MR at binv dT dA do d dA Tds pdr dA Sdt Idt MT X Pdr b Inv a dv an pdV Pdr do_of TS Tds Sdt Sdt (5) (12 points) Consider an isotopically labeled crystal of methane that is made with one deuterium atom and three hydrogen atoms (CH3D). What is the residual entropy for 3 moles of this substance at absolute zero? ΔS Δ5 K In w 1.38 1023 In 4 Δ5 1.91 10 23 (6) (16 points) (a) Starting from the fundamental equation for chemical thermodynamics, dG = Vdp – SdT, derive an expression that predicts the change in Gibbs energy, G, for nearly incompressible liquids and solids upon increasing the pressure from pi to pf at constant temperature. (b) Starting from the fundamental equation of chemical thermodynamics, derive an expression that predicts the change in Gibbs energy, G, for a perfect gas upon increasing the pressure from pi to pf at constant temperature. 0 0 a H Of PV H PV dG PV SAT dG H SAT b dG Vdp Sdt Rtp ΔG nRTIn sat Sdt If 4 2.5 10 f 51 5 2.5 12.5