9780030839931, Chapter 20, Problem 1P

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Chapter 20, Problem 1P Problem A measure of the importance of quantum effects in the noble gases is the de Boer parameter. We calculated the energy per atom u(r) of a noble gas (Eq. (20.5)) under the assumption that it was entirely potential energy. In a quantum theory, however, there will zero-point vibrations even at T=0, leading to a correction to Eq. (20.5) proportional to h. (a) Show, on purely dimensional grounds, that if the correction is strictly linear in h then, the correction to the energy must have the form Au = (20.29) where if depends on the particular noble gas in question only through the ratio and = h (20.30) The quantity known as the de Boer parameter, is listed in Table 20.7. Since h/o is the uncertainty in the momentum of a particle localized to within a distance σ, is roughly the ratio of the kinetic energy of zero-point motion of an atom to the magnitude of the attractive interaction. The size of ^ is thus a measure of the importance of quantum effects (and a glance at Table 20.7 immediately indicates why our purely classical discussion cannot hope to cope with solid helium). Table 20.7 THE DE BOER PARAMETER FOR THE NOBLE GASES, INCLUDING THE TWO HELIUM ISOTOPES ³He Ne Ar Kr Xe 3.1 2.6 0.59 0.19 0.10 0.064 (b) Let rc be the equilibrium interparticle distance computed by minimizing the classical energy (20.5) and rc + be the value obtained by minimizing the classical energy quantum correction (20.29). Show, under the assumption rc, that the ratio of the values of Ar/rc for any two noble gases is equal to the ratio of their de Boer parameters. (c) Show that the result of (b) also holds for the fractional changes in internal energy and bulk moduli, due to quantum corrections. These conclusions are compared with the data for neon and argon in Table 20.8. (In the cases of krypton and xenon the deviations from the classical values are too small to be reliably extracted from the data; in the case of the helium isotopes the de Boer parameter is too large for this analysis to be reliable.) Chapter 25 describes how the effects of zero-point vibrations can be more accurately taken into account. Table 20.8 COMPARATIVE SIZE OF QUANTUM CORRECTIONS TO THE EQUILIBRIUM PROPERTIES OF NEON AND ARGON X A 0.59 0.19 3.1 0.047 0.011 4.3 0.26 0.10 2.6 AB/B 0.39 0.15 2.6 Step-by-step solution There is no solution to this problem yet. Get help from a Chegg subject expert. Ask an expert