Measuring and evaluating the Balmer series of lines for the Hydrogen atom (virtual experiment)

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• published: 09 Jun 2014
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by Dr. Jonathan Agger (University of Manchester) taken form the Coursera Course "Introduction to Physical Chemistry" The solution of the Schrödinger equation for the hydrogen atom, gives: E = Ry * n² [5] Here n is the principal quantum number and must be an integer. E is the electron's energy in quantum level n. Ry is called the Rydberg constant. The energy of a given level is referenced to an energy zero, with n = ∞, corresponding to a free electron and the ionised atom. When n = 1 the electron is in the 1s orbital, when n = 2 the electron is in either the 2s or 2p orbital and when n = 3 the electron is in either the 3s, 3p or 3d orbital etc... Emission of a photon occurs as an electron undergoes a transition from some energy level with a quantum number n2 down to a lower energy level with a quantum number n1 . The energy of the photon will be determined by the energy difference delta E = Ry * (n(2)^2 - n(1)^2) between the levels and may be expressed by the relationship delta E = h * nu [6] The spectrum consists of a number of series of lines. All the lines may be identified in terms of transitions from an excited state to a lower energy level. Different lines within the same series correspond to starting in different excited states but falling to the same lower state. The lowest energy line in the Balmer series corresponds to an electronic transition from n2 = 3 to n1 = 2. EXPERIMENTAL PROCEDURE This spectroscopic method involves exciting electrons within H atoms by means of electrical discharge. Electrons are promoted to excited states and then emit a photon as they fall back to lower states. When large numbers of atoms undergo such transitions, the emission is of sufficient intensity for spectral lines to be observed by eye. The wavelengths of the spectral lines are measured using a spectrometer. DATA TREATMENT - Using the spectrometer simulator, obtain the wavelengths in nanometres (nm) of four lines in the Balmer series and then convert these to eV. - Identify the transitions. In each case, you know that the lower energy level has a principal quantum number n1 = 2. The lines must therefore correspond to transitions from higher levels, with n2 = 3, 4 etc... - Plot a graph of the energy of each transition in eV against 1/ n(2)^2 using a spreadsheet and obtain the gradient and intercept. Think very carefully about the units of these quantities. - Adapt equation [5] to the Balmer series and express it in the form y = mx + c and use it to evaluate the Rydberg constant, R, separately, both from your calculated gradient and your intercept. - Use your value of R to estimate the ionisation energy for the hydrogen atom in kJ mol -1 , and cm -1 given that 1 eV = 96.487 kJ/mol or 8065.6 cm^-1 . - Compare your result with the literature value. This work is licensed under a Creative Commons Attribution‐NonCommercial‐NoDerivs 3.0 Unported License