International Chemistry Olympiad · 2014

Problems

Problem 1Particles in a box: polyenes. In quantum mechanics, the movement of π electrons along a neutral chain of conjugated carbon atoms may be modeled using the 'particle in a box' method. The energy of the π electrons is $E_n = \dfrac{n^2 h^2}{8 m L^2}$ , where $n$ is the quantum number ( $n = 1, 2, 3, \dots$ ), $h$ is Planck's constant, $m$ is the mass of the electron, and $L$ is the length of the box, approximated by $L = (k + 2) \times 1.40\ \text{Å}$ ( $k$ being the number of conjugated double bonds along the carbon chain). A photon of wavelength $\lambda$ can promote a π electron from the HOMO to the LUMO. A semi-empirical formula relates $\lambda$ to $k$ : $\lambda = B \times \dfrac{(k+2)^2}{2k+1}$ (Equation 1). (a) Using Equation 1 with $B = 65.01$ nm, calculate $\lambda$ for octatetraene, $\ce{CH2=CH-CH=CH-CH=CH-CH=CH2}$ . (b) Derive Equation 1 from the particle-in-a-box expression and calculate the theoretical value $B_{\text{calc}}$ . (c) Find the number of conjugated double bonds $k$ and give the structure of the polyene whose HOMO–LUMO excitation requires $\lambda = 600$ nm. (d) For that polyene, calculate the HOMO–LUMO energy difference $\Delta E$ in kJ mol $^{-1}$ .Solutions: 1
Problem 2Dissociating gas cycle. Dinitrogen tetroxide forms an equilibrium mixture with nitrogen dioxide: $\ce{N2O4(g) <=> 2 NO2(g)}$ . 1.00 mol of $\ce{N2O4}$ was placed in an empty vessel of fixed volume 24.44 dm $^3$ . The equilibrium gas pressure at 298 K was 1.190 bar; when heated to 348 K the equilibrium pressure rose to 1.886 bar. (a) Calculate $\Delta G^{\circ}$ of the reaction at 298 K, assuming ideal gases. (b) Calculate $\Delta H^{\circ}$ and $\Delta S^{\circ}$ , assuming they do not vary significantly with temperature. The reversible dissociation of $\ce{N2O4}$ can be exploited in power cycles: in step $3 \to 4$ the hot gas mixture expands reversibly and adiabatically through a turbine. (c) Give the equation for the work done by 1 mol of an inert gas (air) during the reversible adiabatic expansion $3 \to 4$ , assuming constant $C_{V,m}$ and a temperature drop from $T_3$ to $T_4$ .Solutions: 1