University Physics with Modern Physics (14th Edition)

Published by Pearson
ISBN 10: 0321973615
ISBN 13: 978-0-32197-361-0

Chapter 39 - Particles Behaving as Waves - Problems - Discussion Questions - Page 1313: Q39.14

Answer

The atom with an electron at high n. The electron at low n.

Work Step by Step

The total energy of the atom is given by equation 39.15, $E_n=-\frac{hcR}{n^2}$. The atom with the electron at high n has the greater energy. It is important to note the minus sign; the total energy is negative in each case, but the high-n atom’s energy is not as negative. The orbital speed of electrons in the Bohr model is given by equation 39.9, $v_n=\frac{1}{\epsilon_o}\frac{e^2}{2nh}$. The atom with the electron at low n has the greater speed. There is no contradiction because an atom’s total energy is both kinetic and potential. The lower-n electron is moving faster, and has more kinetic energy (which is always positive). However, being closer to the nucleus, it has a more negative potential energy, and so the atom’s total energy ends up being more negative than that of the high-n atom.
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