Why the He , Ne , and Ar atoms are chemically inert.

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Chapter 29, Problem 32GP

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Why the He, Ne, and Ar atoms are chemically inert.

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3. Consider a monatomic linear with equilibrium separation a. Suppose the outer electrons (of mass m) in a given atom move with a displacement different from that of the corresponding ion core (of mass M). Let the displacement of the ion core s be: Us = uei(Ksa-wt) and the displacement of the center of mass of the outer electrons associated with ion s be: V = vei(Ksa-wt) Each ion core is assumed to interact only with its own outer electrons with a force proportional to the displacement of the electron distribution from the nucleus, and the force constant is C₂. However, neighboring electron distributions interact with a force constant C₁. a) Show that -w² Mus = C₂ (vs - Us) -w²mvs = C₂ (us - Vs) + C₁ (Vs+1 + Vs-1-2vs) b) Substitute for the displacements, and solve the resulting simultaneous equations. Find an expression for w². c) Take the limit as m → 0 (the mass of electrons is much smaller than that of the ion core), and show that the dispersion relation for the acoustic mode is…