Problem Set #10 Assigned November 8, 2013 Due Friday, November 15, 2013 Please show all work for credit. To Hand in

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1 Problem Set #10 Assigned November 8, 013 Due Friday, November 15, 013 Please show all work or credit To Hand in 1. 1

2 . A least squares it o ln P versus 1/T gives the result 3. Hvaporization = 5.8 kj mol 1. Assuming constant pressure and temperature, and that the surace area o the protein is reduced by 5% due to the hydrophobic interaction: G 4 3 r V M N 3 9 r.5 10 m G 0.5 N 4 A A r 0.73mL/ g 60000g / mol ( N 4 r 0.070N / m / mol (4 ) (.5 10 m) 865kJ / mol A Convert to per mole, determine size per molecule We think this is a reasonable approach, but the value seems high 3 )

3 4. The vapor pressure o an unknown solid is approximately given by ln(p/torr) = (K/T), and the vapor pressure o the liquid phase o the same substance is approximately given by ln(p/torr) = (K/T). a. Calculate H vaporization and H sublimation. b. Calculate H usion. c. Calculate the triple point temperature and pressure. a) Calculate Hvaporization and H sublimation. From Equation (8.16) dln P H sublimation dt RT d ln P d ln P dt d ln P H T 1 dt 1 dt R d d T T For this speciic case H R sublimation sublimation 035 H J mol sublimation Following the same proedure as above, H vaporization R H J mol b. Calculate H usion. vaporization 3 1 H usion H sublimation Hvaporization J mol J mol J mol c. Calculate the triple point temperature and pressure. At the triple point, the vapor pressures o the solid and liquid are equal. Thereore, K K T T K T T tp 73.6 K tp tp Ptp 035 ln Torr P Torr tp tp 3

4 ,, 5. The UV absorbance o a solution o a double-stranded DNA is monitored at 60 nm as a unction o temperature. Data appear in the ollowing table. From the data determine the melting temperature. Temperature (K) Absorbance (60 nm) Plotting the relative absorbance versus temperature yields: Relative Absorbance T m C T [degree Celsius] The plot indicates a melting temperature o approximately C. 6. For the ormation o a sel-complementary duplex DNA rom single strands H = 177. kj mol 1, and T m = 311 K or strand concentrations o M. Calculate the equilibrium constant and Gibbs energy change or duplex ormation at T = 335 K. Assume the enthalpy change or duplex ormation is constant between T = 311 K and T = 335 K. irst calculate equilibrium constant at 311 K, the melting temperature, where = 0.5: K, 0.5 K311 K (1- ) -4 C M Then we can calculate the equilibrium constant at 335 K: K H 1 1 ln K R T T 1 1 ( ) ( ) C ds ( 4 C ds ) C ds 4 M (which means ) C ds ( 4 C ds ) C ds 5 M (which means ) And inally we can calculate the Gibbs energy at 335 K: ( ) ( ( ) 4

5 7. At 47 C, the vapor pressure o ethyl bromide is 10.0Torr and that o ethyl chloride is 40.0 Torr. Assume that the solution is ideal. Assume there is only a trace o liquid present and the mole raction o ethyl chloride in the vapor is 0.80 and then answer these questions: a. What is the total pressure and the mole raction o ethyl chloride in the liquid? b. I there are 5.00 mol o liquid and 3.00 mol o vapor present at the same pressure as in part (a), what is the overall composition o the system? (a) what is the overall composition o the system? ( ) b) We use the lever rule. tot n Z x n y Z tot liq B B vap B B Z x Z x x Z B B A A A A y z y z Z y B B A A A A Thereore, n y Z n Z x tot liq EC EC tot vap EC EC 5 3 we know that x EC = 0.50 and y EC = Z Z 0.50 Z Z EC EB EC Z EC EC 5

8. 1.053 g o bee heart myoglobin dissolved in 50.0 ml o water at T = 98 K generates suicient osmotic pressure to support a column o solution o height d. I the molar mass o myoglobin is 16.

6 g o bee heart myoglobin dissolved in 50.0 ml o water at T = 98 K generates suicient osmotic pressure to support a column o solution o height d. I the molar mass o myoglobin is 16.9 kg per mole, calculate d. We set the osmotic pressure equal to the pressure o the column: n solutesrt π g h V We can now solve or the height, h: msolutes RT n M solutesrt solutes h V g V g g J K mol 98 K g mol m 1000 kg m 9.81 m s m 9. Calculate the change in the reezing point o water i g o a protein with molecular weight g mol 1 is dissolved in 100. ml o water K m protein / Mprotein -1 ΔT K msolute K 1860 K g mol m solute g / g mol 100 g 10. 6

8 13. ( ) ( ) ( ) Assume we have 5g cholesterol dissolved in 95g Dipalmitoyl phosphatidylcholine. ( ) ( ) ( ) 8

9 Extra practice or exam, do not hand in Phase equilibrium 1. Use the vapor pressures or C N given in the ollowing table to estimate the temperature and pressure o the triple point and also the enthalpies o usion, vaporization, and sublimation. Phase T ( C) P (Torr) Solid Solid Liquid Liquid P Torr R ln J mol K ln P Torr H sublimation 3.6 kj mol T T 10.5 K 1.4 K Torr J mol K ln 400 Torr 6 Hvaporization K 5. K kj mol 5.6 kj mol H usion kj mol 1 1 To calculate the triple point temperature, take T T 1.4 K P 100 Torr solid, re solid, re 40. K P 400 Torr liquid, re liquid, re J mol J mol 3-1 Ttp J mol 1.4 K 40. K 1 T P tp tp J mol K ln 400 Torr K, Ttp 40.3 K 100 Torr J mol Torr exp 40 Torr J mol K 1.4 K 40.3 K 9

10 . Use the vapor pressures o Cl given in the ollowing table to calculate the enthalpy o vaporization using a graphical method or a least-squares itting routine. T (K) P (atm) T (K) P (atm) A least squares it o ln P versus 1/T gives the result Hvaporization = 0.3 kj mol It has been suggested that the surace melting o ice plays a role in enabling speed skaters to achieve peak perormance. Carry out the ollowing calculation to test this H usion = 6010 J mol 1, the density o ice is 90 kg m 3, and the density o liquid water is 997 kg m 3. a. What pressure is required to lower the melting temperature by 5.0 C? b. Assume that the width o the skate in contact with the ice has been reduced by sharpening to cm, and that the length o the contact area is 15 cm. I a skater o mass 85 kg is balanced on one skate, what pressure is exerted at the interace o the skate and the ice? 10

11 c. What is the melting point o ice under this pressure? a) d. I the temperature o the ice is 5.0 C, do you expect melting o the ice at the ice skate interace to occur? usion usion 1 1 dp Sm Sm.0 J mol K usion dt M M usion Vm 997 kg m 90 kg m HO, l HO, l Pa K 144 bar K kg kg 3 3 The pressure must be increased by 70 bar to lower the melting point by 5.0ºC. b) F 85 kg 9.81 ms P A m 510 m 7 =. 10 Pa =. 10 bar c) dt 1C T P dp 144 bar usion.0 10 bar 1.5 C ; T m = 1.5ºC d) No, because the lowering o the melting temperature is less than the temperature o the ice. 4. Consider the transition between two orms o solid tin, Sn(s, gray) Sn(s, white). The two phases are in equilibrium at 1 bar and 18 C. The densities or gray and white tin are 5750 and 780 kg m 3 H transition = 8.8 J K 1 mol 1. Calculate the temperature at which the two phases are in equilibrium at 00 bar. In going rom 1 atm, 18 C to 00 atm, and the unknown temperature T G V P S T gray gray gray m G V P S T white white white m At equilibrium gray white m m T gray white S S m m gray white gray white gray white G G 0 V V P S S T M V V P Sn 1 1 gray S transition kg mol 199 x10 Pa kg m 780 kg m 9.8 C J K mol T 8. C white P 11

12 5. A protein has a melting temperature o T m = 335 K. At T = 315 K, UV absorbance determines that the raction o native protein is N = At T = 345. K, N = Assuming a two-state model and assuming also that the enthalpy is constant between T = 315 and 345 K, determine the enthalpy o denaturation. Also, determine the entropy o denaturation at T = 335 K. By DSC, the enthalpy o denaturation was determined to be 51 kj mol 1. Is this denaturation accurately described by the two-state model? We irst calculate the equilibrium constants at 315 K and 345 K: K315 K K345 K D N D N N N N N K 345 K ln R ln J mol K K 315 K H 6. kj mol T T 345 K 315 K 1-1 This result deviates rom the DSC result, indicating that the denaturation process is not accurately described by a two-state model. 6. Suppose a DNA duplex is not sel-complementary in the sense that the two polynucleotide strands composing the double helix are not identical. Call these strands A and B. Call the duplex AB. Consider the association equilibrium o A and B to orm duplex AB Assume the total strand concentration is C and, initially, A and B have equal concentrations; that is, C A,0 = C B,0 = C/. Obtain an expression or the equilibrium constant at a point where the raction o the total strand concentration C that is duplex is deined as. I the strand concentration is M, calculate the equilibrium constant at the melting temperature. 1

13 We make the table o concentrations: C initial C equilibrium AB 0 C A = B C/ C/ C The equilibrium constant at the melting temperature with = 0.5 is given by: CAB C C C C C K A B C And or C = M: K C 110 M Ideal and Real Solutions 1. Predict the ideal solubility o lead in bismuth at 80 C given that its melting point is 37 C and its enthalpy o usion is 5. kj mol 1. 13

14 . The vapour pressure o -propanol is kpa at C, but it ell to 49.6 kpa when 8.69 g o an involatile organic compound was dissolved in 50 g o -propanol. Calculate the molar mass o the compound. 3. The addition o 5.00 g o a compound to 50 g o naphthalene lowered the reezing point o the solvent by K. Calculate the molar mass o the compound. 4. The osmotic pressure o an aqueous solution at 88 K is 99.0 kpa. Calculate the reezing point o the solution. 14

15 5. The molar mass o an enzyme was determined by dissolving it in water, measuring the osmotic pressure at 0 C, and extrapolating the data to zero concentration. The ollowing data were obtained: c/(mg cm 3 ) h/cm Calculate the molar mass o the enzyme. 6. a 15

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9. A and B orm an ideal solution. At a total pressure o 0.900 bar, y A = 0.450 and x A = 0.650.

550 total ybptotal x 0.63 bar x 0.650 * A B * * * a B A A 0.450P 0.650 P P P P P P A yp P P P y * B * * * A 0.450 B A 0.650 10.450 0.450 10.650 1.414 bar.7 10.

18 9. A and B orm an ideal solution. At a total pressure o bar, y A = and x A = Using this inormation, calculate the vapor pressure o pure A and o pure B. P x P y P * total A a B total P x * a A * B * A * B P bar total ybptotal x 0.63 bar x * A B * * * a B A A 0.450P P P P P P P A yp P P P y * B * * * A B A bar The heat o usion o water is J mol 1 at its normal melting point o K. Calculate the reezing point depression constant K. K K RM T solvent usion H usion J mol 1.86 K kg mol J mol K kg mol K

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Problem Set #10 Assigned November 8, 2013 Due Friday, November 15, 2013 Please show all work for credit To Hand in

Problem Set #10 Assigned November 8, 2013 Due Friday, November 15, 2013 Please show all work for credit To Hand in 1. 2. 1 3. 4. The vapor pressure of an unknown solid is approximately given by ln(p/torr)