In this problem we will use the standard deviation of

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. In this problem, we will use the standard deviation of the momentum of a particle in a box to estimate the energy of an electron confined to a region of nuclear size. We derived in class (Lecture 12) that the standard deviation of the momentum for a particle in a box, σ p is σ p = ~ a . (4) Chem 120A, Spring 2006 1
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The diameter of a typical nucleus is 10 - 14 m. Substitute a = 10 - 14 m into the expression for σ p and show that σ p is σ p & 3 × 10 - 20 kg · m · s - 1 . (5) Now, show that E = σ 2 p 2 m = 5 × 10 - 10 J 3000 MeV (6) where millions of electron volts (MeV) is the common nuclear physics unit of energy. It is observed experimentally that electrons emitted from nuclei as β radiation have energies of only a few MeV, which is far less than the energy we have calculated in this problem. Argue, then, that there can be no free electrons in nuclei because they should be ejected with much higher energies than are found experimentally. Chem 120A, Spring 2006 2
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