Exam Thermodynamics 2 9 November 2017

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1 1 Exam Thermodynamics 2 9 November 2017 Please, hand in your answers to problems 1, 2, 3 and 4 on separate sheets. Put your name and student number on each sheet. The examination time is 08:30 until 11:30. There are 4 problems, with each 4 subproblems, a list of constants and a formulae sheet, 8 pages in total. All 16 subproblems have equal weight for the final grade. Problem 1 a) Give a definition and short description of the following concepts -Solidus -Kelvin equation -Supercritical phase -Equipartition theorem -Congruent melting b) Give the characteristic equation for the internal energy for a closed system with both volume and surface work; use it to find a Maxwell relation between the temperature and the surface tension. c) Give the meaning of all symbols in the following formula as well as a description of its use in thermodynamic problems. ( ) X X i = n i P,T,n j i d) For a battery at constant temperature and volume the change in the number of microstates W per mol of reaction progression can be written as [ ( νf E ) ] W T q = W exp. k Prove this using the Boltzmann definition of entropy together with a Maxwell relation. Do not forget to fill in the online survey to the course.

2 2 Problem 2 AgBr is a poorly soluble salt in water (s = g/l at K). We determine the change in solubility of AgBr in water at K by adding mol/kg Cr 2 (SO 4 ) 3 18H 2 O (s = 1200 g/l) to the solution. M Ag = g/mol en M Br = g/mol. A = for water at K. a) Calculate the equilibrium constant for dissolving AgBr in a saturated solution in absence of Cr 2 (SO 4 ) 3 18H 2 O using the Debye-Hückel limiting law. b) Calculate the ionic strength of the solution with Cr 2 (SO 4 ) 3 18H 2 O but without AgBr. c) Calculate the mean activity coefficient γ ± of the Ag + and Br ions in the solution with the Cr 2 (SO 4 ) 3 18H 2 O present using the Debye-Hückel limiting law. d) Calculate the solubility of AgBr in the solution with the Cr 2 (SO 4 ) 3 18H 2 O present using the result of c). Problem 3 We study the nucleation of the solid phase of cyclohexane, C 6 H 12, from an undercooled melt at P = P. Cyclohexane crystals grow roughened already at the slightest undercooling, such that the shape of the crystals is rounded off. We will therefore assume that the nuclei are perfect spheres. The melting point of cyclohexane is T fus = K. At that temperature the following data have been found: The surface tension of solid cyclohexane in its melt is γ sl = J/m 2. The density of cyclohexane in its melt phase is ρ l = g/cm 3. The density of cyclohexane in its solid phase is ρ s = g/cm 3. The melting enthalpy of cyclohexane is fus H = kj/mol. We will further assume that these values are independent of temperature in the temperature range of the undercooling. The difference in chemical potential between the solid and melt phase, µ = µ s µ l, depends on the undercooling. The molar nucleation Gibbs free energy for the formation of a nucleus with radius r is given by G m,nucl = 4π 3Ω µr3 + 4πγ sl r 2, where Ω is the molar volume. a) Calculate Ω. b) Show that the critical nucleus radius for nuclation, r c, is given by r c = 2γ slω µ. c) Estimate the undercooling, T, for which the nucleation proceeds without a barrier. d) Discuss whether the critical nucleus radius, r c, will increase or decrease when the pressure is increased.

3 3 Problem 4 A co-crystal is a crystal, which has a well-defined combination of neutral compounds in its unit cell, while the individual pure compounds also form crystals at ambient conditions. Ibuprofen (Ibu) is a a nonsteroidal anti-inflammatory chiral drug that forms a racemic compound (both (R) and (S) in the unit cell), when crystallized from a racemic solution or melt. iteracetam () is an anti-epileptic drug and is the (S)-enantiomer of etiracetam. iteracetam forms enantiospecifically a co-crystal with (S)-Ibu, implying that no co-crystal is formed with (R)-Ibu. The co-crystal melts at 72 C. T fus ((R)-Ibu) = 47 C. M((L)-Ibu) = g/mol M() = g/mol M(CH 3 CN) = g/mol ρ(ch 3 CN) = g/cm 3 at 25 C. In figure 1 the (s-l) phase diagram of a mixture of and (S)-Ibu is presented. The two compounds do not form any solid-solution T ( oc) x (S) Ibu Figure 1: (s-l) phase diagram of (, (S)-Ibu) at P = P. a) The phase diagram is also present on a separate sheet. Use this sheet to indicate the two phase regions by hatching these areas and indicate the phases present in all regions of the phase diagram. Hand in this sheet together with your other results. b) A eutectic is found at 68 C for x = Calculate the fusion enthalpy, fus H m, of the co-crystal, assuming that the melt behaves as an ideal solution and that the fusion enthalpies and entropies are independent of temperature in the relevant temperature range. Next we add acetonitrile (CH 3 CN) as a to the system. In Figure 2 the quaterny phase diagram of (, (S)-Ibu, (R)-Ibu, ) at 25 C and P = P has been drawn.

4 4 Figure 2: Quaternary phase diagram of (, (S)-Ibu, (R)-Ibu, ) at 25 C and P = P. Three side faces of the tetrahedron, each forming a ternary phase diagram, are also drawn schematically. c) This phase diagram is also present on the separate sheet. Use that sheet to indicate the two phase regions by hatching these areas and indicate the phases present in all regions of all three ternary phase diagrams. d) The solubility of in acetonitrile is 27 mg/ml at 25 C. At the eutectic in the ((R)- IBU,, acetonitrile) ternary phase diagram 626 mg/ml (R)-IBU and 359 mg/ml is present, all at 25 C. Determine the composition of the phases present and their relative amounts for an overall composition given by (x (R) IBU, x, x ) = (0.1, 0.2, 0.7). Hint: First determine the mole fractions for the euectic point as well as for the solubility of. List of constants Elementary charge e C Faraday s constant F Cmol 1 Boltzmann s constant k JK 1 Planck s constant h Js Bohr magneton µ B JT 1 Atomic mass constant m u kg Amadeo Avogadro di Quaregna e Ceretto s constant N A mol 1 Gas constant R JK 1 mol 1 Free fall acceleration g ms 2 Unit of energy 1 cal = J Standard pressure P 1 bar = 10 5 Nm 2 = atm = 750 Torr

5 5 Formulae Thermodynamics (1) P V = nrt = NkT U = 3 2 nrt = 3 2 NkT U = W + Q dw = P ext. dv + dw and dw max = (dg) P,T dq P = C P dt and dq V = C V dt Q 1 Q 2 = T 1 T 2 ds = dqrev dq T T ds tot = ds + ds omg 0 du = P dv + T ds + i µ i dn i H = U + P V dh = V dp + T ds + i µ i dn i A = U T S da = P dv SdT + i µ i dn i G = H T S dg = V dp SdT + i µ i dn i r G = ( ) G = r G + RT ln Q, where Q = ξ P,T i RT ln K = r G E = E RT νf ln Q and dw = Edq and E = IR and P = EI µ i = µ i + RT ln a i = µ i + RT ln P i P a νi i G P,T = i µ i n i n j dµ j = 0 j ( RT 2 T = trs H ) x B S = nr (x A ln x A + x B ln x B ) Π = [B]RT = n B V RT S = k ln W n i N = exp ɛi kt q, where q = i exp ɛ i kt and < X >= N < x >= N i x i n i N

6 6 Formulae Thermodynamics 2 ( ) V = T P,W,n i ( ) X X i = n i n j dµ j = 0 j P j = x j P j P j = y j P P,T,n j i ( S P ) T,W,n i P B = x B K B ( ) ( ) µβ µα = trs V P T P T ( ) ( ) µβ µα = trs S T P T P ( ) P = P Vm P exp RT dp dt = trss trs V = trsh T trs V d ln P dt trsh RT 2 ( ) RT 2 T = x B trs H µ = µ + RT ln a = µ + RT ln x + RT ln γ µ = µ + RT ln a = µ + RT ln b + RT ln γ b F = C P + 2 n α l α = n β l β ) solv G = z2 i e2 N A (1 1ɛr 8πɛ 0 r i γ ± = ( γ+γ p q ) 1 p+q log γ ± = z + z A I I = 1 z 2 b i i 2 b A = i F 3 4πN A ln 10 ( ρb 2ɛ 3 R 3 T 3 ) 1 2 P in = P out + 2γ r ; P = ρgh w ad = γ sg + γ lg γ sl γ sg = γ sl + γ lg cos Θ c

7 7 Use this sheet to indicate the two phase regions by hatching these areas and indicate the phases present in all regions of the binary (problem 4a) as well as the ternary phase diagrams (problem 4c). Hand in this sheet together with your other results. THE NEXT PAGE CAN BE USED AS WELL IN CASE YOU RECON- SIDER YOUR RESULT. INDICATE FINAL/DRAFT. Name: Student no.: FINAL/DRAFT (please, indicate) T ( oc) x (S) Ibu

8 8 Use this sheet to indicate the two phase regions by hatching these areas and indicate the phases present in all regions of the binary (problem 4a) as well as the ternary phase diagrams (problem 4c). Hand in this sheet together with your other results. THIS PAGE CAN BE USED AS WELL IN CASE YOU RECONSIDER YOUR RESULT. INDICATE FINAL/DRAFT. Name: Student no.: FINAL/DRAFT (please, indicate) T ( oc) x (S) Ibu

Exam Thermodynamics 12 April 2018

1 Exam Thermodynamics 12 April 2018 Please, hand in your answers to problems 1, 2, 3 and 4 on separate sheets. Put your name and student number on each sheet. The examination time is 12:30 until 15:30.