Complete the following problems and submit your work as a single PDF document. Graphs must be included in the PDF file and must be clearly labeled (graphs must be generated electronically and cannot be sketched by hand). Hand-written problems must be clearly photographed and processed with a document scanning app. All work must be neat and clearly labeled. 1. We discussed in class how a system close to an energy minimum can be approximated by a harmonic oscillator (i.e. a spring) with potential energy U(x) = 1/2 mw² (x - xo)² where x is the position along the energy surface, xo is the equilibrium position, m is the mass of the system, w is the angular frequency along x. Given that the total energy of the system is given as H(x,p) = p²/2m+ U(x) (where p is the momentum along x) calculate the following: a. Probability that the system is located at xo b. The partition function for the whole system (hint: your answer should be divided by Planck's constant to ensure correct units, also fe¯a(x-b)² dx = √π/a). c. The average energy of the system. 2. An unstructured, unfolded protein of N amino acids in length are covalently modified such that residues i and j are attached (forming a loop). From this, estimate the change in chain entropy associated with this covalent modification. Assume the following: 。 the unmodified protein's conformational space can be approximated as a sphere whose radius is equal to the root mean square distance of a 3D random walk of length N o the modified protein's conformational space can be approximated as two conjoined spheres: one whose radius is equal to the root mean square distance of a 3D random walk of length N (i) and one whose radius is equal to "loop How would the entropy change if i and j were adjacent to each other as opposed to being at the very beginning and end of the sequence? What would you expect to be the size of loop relative to √j-il (where is the distance between adjacent amino acid alpha carbons)? 3. Calculate a fundamental thermodynamic equation for enthalpy. Start with dH = d(E + pV) to derive your answer. Use this to give expressions for the temperature and volume of a system at constant entropy and pressure. 1 4. Given that the time independent wave function for the particle in a one-dimensional box is: √2 sin(MEX) y(x)=. L L a. Determine if this wave function is an eigenfunction of the momentum operator. What is the expectation value of this operator? 10 b. Determine if this wave function is an eigenfunction of the momentum operator squared. What is the expectation value of this operator? (42)² = -ħ² c. Could you measure both quantities at the same time (i.e., measure one without upsetting the other)? 5. The two peptides pictured below are both 10 amino acids long and completely hydrophobic. They can form stable dimers via hydrophobic interactions. Peptide 1 Peptide 2 Rank the following peptide complexes in terms of their net dipole moment least to greatest. Explain your answer. A. Peptide 1: Peptide_1 arranged parallel B. Peptide 1: Peptide_1 arranged anti- parallel C. Peptide 1: Peptide_2 arranged anti- parallel D. Peptide_2: Peptide_2 arranged parallel 2 6. Given the 3 level photophysical-system with a singlet ground state, singlet excited state and triplet excited state and respective rate constants: A. determine the fluorescence emission rate, F, dependence on the other parameters in a single equation B. Using Mathematica, graph the fluorescence emission rate dependence on the excitation intensity, I (i.e., fluorescence emission rate, F, on the Y-axis and excitation intensity, I, on the X- axis) C. C. Determine the maximal fluorescence emission rate when the excitation intensity approaches infinity (i.e., F =? when I∞) D. Determine the saturation intensity, Is k = σl ex ns nG KISC пт 3 7. In commercially available fluorescence quantum nanocrystals (quantum dots), the fluorescence is caused from electrons going from a semiconductor conduction band (singlet excited state) with n = 2 de- exciting by releasing a photon and by the electron going into a valance band (ground state) with n = 1 with the electrons in both bands being confined to a BOX LENGTH of L = 0.6911 nm with each band having an energy level corresponding to the values of the particle in the box: En h²n² 8mL² Given the relationship that E = h c / λ = E¡ – Ef, with initial energy E¡ to final state energy Ef with wavelength λ, determine the emission wavelength of the photon emitted from the quantum dot. 8. For di-chloro-ethene, C₂H₂Cl₂, determine all possible symmetry elements and point groups for the major configurations of the molecule. Using the character tables, determine which of the combinations of A->B and A->E transitions are electronically allowed for all of the configurations. 1? Molecule Linear? N Select C, with highest n; then is nC₂LC? Two or more σ? C n>2? no?N C IN Linear groups i? Cubic groups ph nd D nh nσ C San S? 20 C₁ C C₁ A 半 E 1 CS E στ A' 1 1 x, y, R₂ x², y², z², xy A" 1 -1 z, Rx, Ry yz, xz C₁ E i Ag 1 1 Rx, Ry, R₂ x², y², z² xy, xz, yz Au 1 -1 x, y, z A В 81B E C₂ 1 1 Z 1 -1 x, y R₂ Rx, Ry C3 E C3 A 1 1 1 Z R₂ 1 E €* E x, y Rx, Ry 1 €* € x, y, z, xy yz, xz € = exp(2πi/3) x² + y², z² (x²- y², xy)(yz, xz) C4 E C4 C₂ A 1 1 1 1 Z R₁₂ 2+ y², z B 1 -1 1 -1 x² - y², xy i E x, y Rx, Ry (yz, xz) Cs E C5 € = exp(2πi/5) A 1 1 1 1 1 Z R, x² + y², z² E €" 2* €" E₁ €* €* _2* €2 x, y Rx, Ry E (yz, xz) €2 €* € €2* E2 2% E (x² - y², xy) G-AB E C6 E C6 C3 C₂ € = exp(2πi/6) 1 1 1 1 Z R₂ x² + y², z² 1 -1 1 -1 E * 1 -E €* x, y (xz, yz) F* -E 1 -€* Rx, Ry F* 1 -€* -e E2 (x² - y², xy) 5