(a) Explain how the spatial localisation of waves leads to the quantisation ofencrgy.-(b) Starting from the de Broglie relationship and the equation for kinetic energy,show that the allowed energy levels for a wave on a circle are:-E=\frac{n^{2} h^{2}}{2 m C^{2}}-where C is the circumference of the circle and n is an integer.-The R-clectrons of benzene (CHg) can be treated as waves localised on a “circle"with a circumference that equals six times the carbon-carbon bond length. In thissimplified model, the lowest energy level (n=0) is singly degenerate, and all theother levels are doubly degenerate.-Assuming a carbon-carbon bond length of 140 pm, usc the equation derived(c)in (b) to calculate the energies of the first four allowed levels for the T-clectrons of benzenc.-(d) Calculate the wavelength of the lowest energy transition in the UVabsorption spectrum of the x-clectrons of benzene.-(c) Compare the results from (c) and (d) with:-O the measured value of 180 nm for the lowest energy transition in theUV spectrum of benzene;-(i) the measured values of 9.4 eV and 12.3 eV for the ionisation potentialsfrom the occupied benzene R-orbitals.-What is the most likely explanation for any discrepancies?-\left[\text { Mass of an electron }=9.1 \times 10^{-31} \mathrm{~kg} ; 1 \mathrm{pm}=1 \times 10^{-12} \mathrm{~m} ; 1 \mathrm{~J}=6.24 \times 10^{18} \mathrm{eV}\right]

Physical Chemistry(Chemistry)

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