There are 6 parts to this exam. Parts 1-3 deal with material covered since the midterm exam. Part 4-6 cover all course material.

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1 Chemistry 453 March 9, 03 Enter answers in a lue ook Final Examination Key There are 6 parts to this exam. Parts -3 deal with material covered since the midterm exam. Part 4-6 cover all course material. Useful Constants: Newton=N=kg-m-s - Joule=J= Nt-m= kg-m /s Svedberg= S= -3 s. Poise= P= g cm - s-=0.kg m - s - Avogadro s number=6.03x 3 mol -. Universal Gas Constant R=8.3 J K - mole - oltzmann s Constant=k=R/N A =.38x -3 J/K Planck s constant h=6.66x -34 J-sec. e Rydberg constant R= =.8x -8 J 8πε 0a0 Angstrom=Å= - m=0.nm=0pm. speed of light=c=3.00x 8 m s - Electron mass=9.x -3 kg Viscosity of water at T=98K =0.0P=cP=0.00 kg m - s - Part (8 points: Answer FOUR out of the six questions.. Give two linear transport equations that we have covered in this course. In each case what is the gradient, and what is being transported?. Place the following motions in order according to how important quantum effects are: Explain your answer: bond vibrations of a diatomic ideal gas, translation of an ideal gas in a m 3 volume, electronic motions, rotations of diatomic ideal gas molecules..3 Compare and contrast the terms equilibrium and steady state as they relate to chemical systems..4 Describe the two forces involved in rownian motion?. Give two examples of physical processes that can be described as rownian motion..5 What is the eisenberg Uncertainty Principle? Explain how this principle affects the ground state energy of a harmonic oscillator.6 What is Poiseuille s Law? Give an important consequence of this law in fluid flow.

2 Part.: (36 points Perform TREE of the following FIVE calculations. These calculations consist of solving at least one or two equations. They vary in the amount of work so chose wisely.. For the rotational temperature is θ rot = 85.3K and the vibrational temperature isθ vib = 65 K.. The ground electronic state degeneracy is g = and the classical dissociation energy is D e =457.6kJmol -. Calculate for the internal energy per mole at T=00K. State any assumptions. 5RT θvib θvib U = + R / + D θvib T e e ( 5( 8.3JK mol ( 00K 65K 65K = + ( 8.3JK mol Jmol e = 0775Jmol + 676Jmol Jmol =4063Jmol Translations and rotations are treated classically. Vibrations are quantized..the hydrodynamic radius of the enzyme ribonuclease is.57x -9 m. Assuming ribonuclease in solution at T=98K may be modeled as a sphere, calculate the frictional coefficient and the diffusion coefficient. Assume the viscosity of the solvent is 0.0 poise. 9 f = 6πηR= 6π 0.00kgm s.57 m =.96 kgs ( ( ( ( JK ( K 3 0 kt J D= = = = 4.66 f.96 kgs.96 kgs m s.3 Cyclobutadiene C 4 4 is an unstable 4-carbon cyclic molecule with 4 delocalized pi electrons. We can approximate the energies of these 4 pi electrons using the particle-in-asquare box model for which the sides of the square are 46 pm. Calculate the highest energy occupied by pi electrons in cyclobutadiene. Suppose an electron in the highest occupied molecular orbital absorbs electromagnetic radiation and is transferred to the next higher energy level. Calculate the frequency of radiation required to do this. ighest orbital is 34 3 h ( 5( 6.6 Js 3 8ma ( 8( 9. kg (.46 m 8 The next higher orbital energy is E 8.8 ( J 8 ( E = E = = = 5.8 J =. Then E = hν = J = J ( ( 8 E 8.46 J ν = = =.8 34 h 6.6 Js s 6

3 .4 Suppose a 0000 base pair DNA is treated as a random coil polymer in solution. Assuming the separation between adjacent Watson-Crick base pairs is 3.4x - m, calculate the root mean square (rms end-to end distance r (noted in class: changed to r. Assume the coil of DNA is roughly spherical and its radius is given by the rms end-to-end distance. Calculate the friction coefficient and the coefficient of diffusion assuming T=93k and η=0.00 kg m - s -. rms 4 / 7 = = ( 3.4 ( =.08 ( π η rms ( π( ( r N m m f = 6 r = kgm s.08 m =.03 kgs 3 (.38 JK ( 93K kt D= = =.44 9 f.03 kgs 7 9 m s.5 Suppose the diatomic molecule 35 Cl in the gas phase is in its ground rotational state. It absorbs microwave radiation and makes a transition to the next higher rotational energy state. The bond length of 35 Cl is 7 pm. Calculate the change in energy E that occurs when the molecule goes from its ground rotational state to the next higher rotational energy state. What is the rotational angular momentum of the excited energy state? What is the degeneracy of the excited state? EJ = J( J + I E = E E = = I µ R J= J= 0 ( ( kg µ = kg = = ( ( Js 6 (.6 kg (.7 m J s E = = = kgm Z ( L = + = L = 0, ±,so degeneracy = 3. Part 3. Multi-step calculations (36 points Perform ONE of the two multi-step problems 3. Consider the hypothetical dissociation of diatomic hydrogen into atomic hydrogen at T=00K + For the ground electronic states degeneracy is g =. 6 kg 3 J

4 For, θvib = 65 K, θrot = 85.3 K, g =, De = 457.6kJmol a Calculate the translational partition functions of and. at T=00K V 3/ qtrans, = 3 ( π mkt h 3/ qtrans, 0.00kgmol 3 = 3 π 3 (.38 JK ( 00K V Js 6.0 mol ( 46 (.44 kg m s 3/ 30 3 = = 5.96 m Js V 0.00kgmol 3 qtrans, = π 34 3 ( 6.6 Js (.38 JK ( 00K mol qtrans, 3/ = ( qtrans, = =.69 m V b Calculate the rotational and vibrational partition functions for at T=00K. T 00K qrot = = = 5.86 σθ 85.3K q vib ( ( rot θvib /T e 3.8 e θvib / T e 6.5 e = = = = c Calculate the total partition functions for and at T=00K and the equilibrium constant K P for the dissociation of. Assume V=m q / V = q g / V = 5.96 m =.9 m trans, 3/ De / RT /8.3 3 / = ( ( ( ( trans, vib rot / = q V q q q g e V e m = m = 3.65 ( ( q ( ( (.9 / V 3 P = = ( q / 3.65 V (.38 kgm s ( 3.88 m 5.35 Pa m 3 6 K k T JK K = = m m 54 3 d Calculate the standard Gibbs energy change G o for dissociation at T=00K G = RTln K = 8.3 Jmol ln 5.35 P ( ( ( [ ] ( [ ] = 8.3 Jmol ln ln = 8.3 Jmol = 54Jmol A sedimentation velocity experiment was performed with subunits of the enzyme aldolase at T=93 K and with a rotor speed of 5,970 rpm. The sedimentation constant was found to be.3s.

5 a Calculate the molecular weight of the aldolase subunit assuming the specific volume of the solute is 0.74 ml/gram, the density of the solvent is gram/ml, and the diffusion coefficient D at T=93K is 5.94x -7 cm /sec. From the expression for the sedimentation constant we obtain for the molecular weight f s kt s m = = ( vρ D( vρ 3 RT s (8.3 J K mol (93 K(.3 sec M = = D ( v ρ (5.94 m sec ( (0.74 ml gm ( gm ml =0.8 kg/mole. b Using your result from part a, calculate the radius of unhydrated aldolase, assuming aldolase can be treated as a sphere 4 3 V MW ( 0.74 ml / g ( 0,8 g / mole 0 V = π R0 = = =.56 ml 3 3 N 6.0 A ( /3 0 / R0 =.56 ml = ( 6. cc =.8 cm =.8 m 4π c Calculate the frictional coefficient of aldolase at T=93K. Again assume aldolase may be treated as spherical in solution. Assume the solvent viscosity at 93K is 0.0 poise. kt (.38 3 / ( 93 kt J K K D= f = = = 6.80 kgs f D 5.95 m / s d Using your answer from part d (noted in class: changed to part c calculate the radius of aldolase. Compare this answer to the answer in part c (noted in class: changed to part b. Explain any difference. 8 f 6.8 g/ s 7 9 f = 6πηR R= = = 3.6 cm= 3.6 m 6πη ( 6π ( 0.0gcm s The hydrodynamic radius is larger than the radius calculated in part b because of hydration of the protein. Start of Cumulative Exam: 0 points Total Part 4. (8 points: Answer FOUR out of the six questions 4. In each case, does the heat capacity agree with the predictions of the classical Equipartition Principle? In each case describe the mechanical model upon which your answer is based. If any system deviates from classical behavior, explain the origin of the deviation:

6 For hydrogen ( gas, C V =is 0.75 J K - mole - For argon (Ar gas, C V =.5 J K - mole - For solid silver (Ag, C V = 5.5 J K - mole - For solid iron (Fe, C V = 4.8 J K - mole Explain the Principle of A Priori Probabilities and the Ergodic Principle. ow are these principles relevant to statistical mechanics?. 4.3 State the Correspondence Principle. Give an example of how a quantum system displays this principle. Include a sketch to illustrate your example. 4.4 Explain the velocity sedimentation experiment in terms of the three forces that determine the sedimentation profile. 4.5 Define the rotational temperature θ rot. Put the following diatomic molecules in the order of increasing rotational temperature: I, Cl,, N. Explain your ordering. In which system will quantum effects be most prominent in rotational motions? Explain. 4.6 Explain Planck s model for black body radiation. What fundamental assumptions did Planck make in constructing this model? In particular, describeplanck s quantization hypothesis. ow does Planck s model for atomic oscillations compare to Schroedinger s wave equation-based treatment of oscillations? Part 5 (36 points Perform TREE of the following FIVE calculations. These calculations consist of solving at least one or two equations. They vary in the amount of work so chose wisely. 5. Calculate the fractional helicity f for a ragg-zimm trimer (i.e. a peptide composed of three monomers if σ=0.0 and s=0. 3 Q= + 3σs+ σs + σ s + σs s Q s 3σ + 4σs+ σ s+ 3σs f = = 3 NQ s N+ 3σs+ σs + σ s + σs 3 3σs+ 4σs + σ s + 3σs = = = = N + 3σ s+ σs + σ s + σs Cl has a vibrational temperature of θ vib = 808K. Calculate the number of molecules in the vibrational ground state. Assume T=00K. Assume one mole of Cl molecules total.

7 e e P e e E0 / kt hν / kt hν/ kt hν/ kt 0 = = / ( = hν kt qvib e θvib / T 808/ = e = e = e = N = N P = = A 0 ( ( The molecule r rotates as a three dimensional rigid rotor. The bond length of 79 r is 0.4 nm. Calculate the energy, the angular momentum L, and the z component of the angular momentum L Z if the wave function is Ψ ( θ, ϕ =Θ ( θ Φ ( ϕ. You can express L and L Z in units of. J= and m=. Therefore: E = E = ( + = ( + I µ R m ( 0.00 ( mr MM kgmol kgmol r µ = = = m + mr M + M r N A 0.00kgmol kgmol 6.03 mol = kg =.66 kg (.05 Js ( ( + E = ( + = =.0 J µ R kg 0.4 m L = = 6 and LZ ( ( ( Assume a protein with two oxygen (i.e. X binding sites obeys a concerted, (i.e. allosteric binding model, where the unbound R and T forms of the protein have the [ T ] equilibrium constant L = = ; the equilibrium constant for R binding oxygen is R [ RX ] [ ][ ] [ ] [ RX ] [ ][ ] [ ] [ ] TX TX KR = = = and KT = = = 0.0. Calculate the R X RX X [ T][ X] [ TX][ X] average number of sites bound when[x]=.0.

8 Q = [ R] + [ RX ] + [ RX ] + [ T ] + [ TX ] + [ TX ] = [ R] (( + KR[ X] + L( + KT [ X] [ X ] Q KR( + KR[ X] + LKT ( + KT [ X] ν = = [ X ] Q [ X] ( + KR[ X] + L( + KT [ X] (( + (( + ( ( ( + ( ( + ( ( ( ( + (( + ( ( + ( 0.0( 4+ ( ( = = = At low concentration a protein is sphere with radius Å. At high concentration the protein forms a hexameric aggregate, which velocity sedimentation determines has a friction coefficient of f = 6.3 kgs. If the aggregate has a volume exactly six times the monomer volume, is the aggregate spherical or a straight chain of six protein monomers? Assume the solvent viscosity is 0.00 kg m - s -. Calculate friction coefficients for both geometries to support your answer. Solution. From the homework a spherical aggregate of 6 monomers has a radius /3 9 R = 6 R =.8 m. So if the aggregate is a sphere: 6 ( π ( ( fsphere = 6πηR6 = kgm s.8 m = 3.40 kgs If it forms a chain of six ( ( 3πηRN 3π 0.00kgm s ( 9 m( 6 monomers... fchain = = = 6.3 ln N ln 6 It is a chain. 9 kgs Part 6. (36 points Perform ONE of the TWO multi-step calculations given below. 6. The vibration of a diatomic gas molecule may be modeled as the motion of a quantized simple harmonic oscillation. Consider 4 N 6 O, for which the vibrational temperature is θ = 79K. vib a Calculate the single particle vibrational partition function for 4 N 6 O and the vibrational internal energy of one mole of 4 N 6 O molecules. Assume T=00K. θvib /T.79/ e e 0.57 qvib = = = = 0.75 θvib / T.79 e e θvib θvib 79K 79K Uvib = R + / ( 8.3JK mol θ vib T = +.79 e e = 8.3JK mol 360K + 9K = 560Jmol ( (

9 b Calculate the contribution of the bond vibrational motion to the entropy of 4 N 6 O. Assume the partition function Q is related to the single particle partition function q by N Q= q. Assume also T=00K U 560Jmol S = + Rln q= + 8.3JK mol ln 0.75 T 00K =.560JK mol.73jk mol =9.7JK mol ( ( c Calculate the contribution of the bond vibrational motion of 4 N 6 O to the elmholtz energy. Assume T=00K ( ( A = U TS = 560Jmol 00K 9.7JK mol = 560Jmol + 970Jmol = 730Jmol d Suppose 4 N 6 O adsorbs onto a surface at T=00K with the oxygen fixed to the surface and the nitrogen free. If the spring constant is κ = 596Nm calculate the new hν vibrational temperature θ vib = the new vibrational partition function. k 0.04kgmol 6 µ = =.33 kg mol 596Nm 3 ν = = 4.7 s 6 π.33 kg ( 34 ( 3 hν 6.6 Js 4.7 s θvib = = = 55K 3 k.38 JK.55/ e 0.34 qvib = = = e For hemoglobin at T=98K the diffusion coefficient is D=6.9x - m s -, the specific volume isv = 0.75 ml g -, and the molecular weight is kg mol - a Calculate the root mean square displacement of a hemoglobin molecule that is left to diffuse for 30 ms. ( ( ( 4 rrms = r = 6Dt = m s 800s = 8.63 m b Assume hemoglobin is a hydrated sphere in solution. Calculate its coefficient of friction, and its hydrodynamic radius. Let the viscosity η=0.89x -3 kg m - s -.

10 3 (.38 JK ( 98K kt f = = D 6.9 m s = 5.96 f 5.96 kgs R= = 6πη 6π 0.89 = ( kgm s c Calculate the radius of unhydrated hemoglobin assuming the unhydrated protein is also spherical. ( / m kgs ( 6.45 gmol ( 0.75cm g / π 3 MV 3 MV 3 R0 = R0 = = 3 3 NA 4π NA 4π 6.0 mol = 9. =.68 cm =.68 d Using you answers to parts b and c, calculate the number of water molecules that hydrate a single hemoglobin molecule. Assume for water V =.0mLg V =.3 =.68.3V δ = = + Vδ V (.3( 0.75mLg V mlg 64500gmol protein = 0.99g water per g protein = gmol m /3

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