Reference no: EM132755293
Question 1
Answer all parts.
(a) Consider a free electron.
(i) With reference to this electron explain the fundamental concept of de Broglie equation.
(ii) Calculate the wavelength of this electron if its velocity is 10% of the speed of light.
(b) A fluorescent dye is characterized by a fluorescence lifetime, . The resulting fluorescence spectral line is characterized by a Full Width at Half Maximum, .
(i) With reference to this fluorescent dye, explain the energy-time uncertainty principle.
(ii) The dye has a fluorescence lifetime of , calculate the minimum Full Width at Half Maximum, , of the resulting fluorescence spectral line (in joules).
(c) Identify all of the orbitals within the L shell () of hydrogen. For each one state the values of the quantum numbers , and .
(d) Consider the 3p4 electron configuration.
(i) In a table, like the one below (using arrows to represent electron spin) arrange the 4 electrons to give first the maximum possible value of the total spin magnetic quantum number, and then the maximum possible value of the total orbital magnetic quantum number,.
(ii) Identify the atomic term (using the appropriate term symbol) to which this electronic microstate belongs.
(iii) Calculate the degeneracy of this term.
(e) Consider the vibrations of a homo-atomic molecule.
(i) Sketch the harmonic and Morse potentials describing this vibrational motion. Highlight the key features and the differences between the two.
(ii) Prove mathematically that the energy levels of a harmonic oscillator are evenly spaced energetically.
(f) For the hypothetical cyclic square hydrogen molecular di-cation (H42+), list all electrostatic pairwise interactions including their signs and contributions to stabilising and de-stabilising the molecular structure.
(g) Provide the mathematical condition (analytical expression) for a wave function to be normalised. Justify your answer.
(h) For the hexatriene molecule:
(i) Draw a fully labelled orbital energy level diagram for the Π-molecular orbitals, clearly indicating which orbitals are occupied.
(ii) Sketch the unoccupied Π-molecular orbitals and their respective symmetries, clearly indicating the location of any nodes.
(i) The general solution for the energies (E) of the molecular orbitals of an extended linear conjugated system when overlap is neglected is:
E = α + 2βcos(Πk/n+1)
n : is the total number of atoms in the conjugated chain
k : is a quantum number, identifying the MO ( = 1, 2, ...., n )
Explain what this equation indicates about how the optical absorption characteristics of conjugated molecules are changed by increasing the chain length. Clearly indicate which aspects of the equation are responsible for the changes in optical absorption. Justify your answer.
Question 2
Answer all parts.
Consider an electron confined to a 1-dimensional quantum well of length L. Unlike in previous examples in the lecture notes this quantum well is centred on the origin and we have where for and otherwise.
(a) By considering the boundary conditions of the system derive an appropriate trial solution.
(b) The electron will cause fluorescence when it transitions from the first excited state to the ground state. How long should the quantum well be if you want an emission line at 600 nm?
(c) What similarities and/or differences would one expect in the full secular determinants for the Π-bonding in the 1,3,5,7-octatetraene molecule (C8H10) with respect to that for the cyclooctatetraene molecule (C8H8). Sketch both secular determinants, clearly indicating all similarities and/or differences and the expected differences in chemical bonding and molecular structure.
(d) The MOs of the ethene (C2H4) molecule form from the constituent AOs with inclusion of overlap. Show that the anti-bonding π-orbital is increased in energy more than the bonding π-orbital is lowered in energy. Your solution should include an analytical expression that isolates the coulombic and bonding terms.