# LaWeyl

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Published on August 8, 2007

Author: Wanderer

Source: authorstream.com

Deuterium and Hydrogen’s Split Personality:  Deuterium and Hydrogen’s Split Personality A Derivation of the Isotope Shift The Overview…:  The Overview… Deuterium is a stable isotope of Hydrogen The increased mass in the nucleus effects the energy of the atom’s orbits The difference in energy from one energy state to the next is also affected Deuterium’s spectral lines are slightly shifted This shift can (and will) be calculated! Deuterium’s Properties!:  Deuterium’s Properties! Hydrogen: - 1 proton in the nucleus - 1 electron in orbit (when neutral) Deuterium: - 1proton in the nucleus - 1 neutron in the nucleus - 1 electron in orbit (when neutral) Deuterium’s nucleus is TWO TIMES as massive as that of Hydrogen! Bohr’s Model:  Bohr’s Model From the following equations… F = (k q1 q2) / r2 (Coulomb force) Bohr’s Model:  Bohr’s Model From the following equations… F = (k q1 q2) / r2 (Coulomb force) F = (m v2) / r (Centripetal force) Bohr’s Model:  Bohr’s Model From the following equations… F = (k q1 q2) / r2 (Coulomb force) F = (m v2) / r (Centripetal force) L = mvr (Angular momentum) Bohr’s Model:  Bohr’s Model From the following equations… F = (k q1 q2) / r2 (Coulomb force) F = (m v2) / r (Centripetal force) L = mvr (Angular momentum) We can (do lots of nasty algebra to) find the energy of an electron in orbit to be… Bohr’s Model:  Bohr’s Model From the following equations… F = (k q1 q2) / r2 (Coulomb force) F = (m v2) / r (Centripetal force) L = mvr (Angular momentum) We can (do lots of nasty algebra to) find the energy of an electron in orbit to be… E = (m k2 e4) / (2 h2 n2) Spectral Lines:  Spectral Lines We know E = (h c) /  Hence 1 /  = R∞ [(1 / nf2) - (1 / ni2)] Where R∞ = (m k2 2π2 e4) / h3 (Known as the Rydberg constant) (coulomb constant, k = 9.988 x 109 N m2 / C2) Reduced Mass:  Reduced Mass Finite mass modifications: RH = R∞ / (1 + (me / mp)) RD = R∞ / (1 + (me / md)) (reduced mass = (m1 m2) / (m1 + m2)) Wavelength Shift:  Wavelength Shift After some ‘rather simple’ algebra, it is clear to see that… H - D = H [(1 - (mp / md)) / (1 + (mp / me))] Hydrogen’s Spectral Lines:  Hydrogen’s Spectral Lines Table of the wavelength shift in deuterium Visualizing the Shift:  Visualizing the Shift From the classic single slit interference formula  = d sin() We can find d to be the separation between the lines of a spectrometer’s diffraction grating Visualizing the Shift:  Visualizing the Shift From the classic single slit interference formula  = d sin() We can find d to be the separation between the lines of a spectrometer’s diffraction grating Working backwards to solve for d, 1.155 x 10-6 = d cos(sin-1(4.102 x 10-7 / d)) Visualizing the Shift:  Visualizing the Shift From the classic single slit interference formula  = d sin() We can find d to be the separation between the lines of a spectrometer’s diffraction grating Working backwards to solve for d, 1.155 x 10-6 = d cos(sin-1(4.102 x 10-7 / d)) d must be greater than 1.2257 x 10-6 m 816 lines / mm Conclusion…:  Conclusion… Deuterium’s increased mass effects the energy of its orbiting electrons Deuterium’s emitted photons have different associated energies than hydrogen’s This slight change in wavelength is known as an isotope shift And is visible with a diffraction grating of 816 lines / mm, or greater References and Acknowledgements:  References and Acknowledgements Thank you to Jess Kahn, Daniel Lawrence, and Eric Myers for their knowledge and support. http://scienceworld.wolfram.com/physics H. White, 1934, Introduction to Atomic Spectra, New York, McGraw-Hill Book Company, Inc

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