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Unformatted text preview: Einstein heat capacity The nucleus Radioactivity Isotopes Nucleus: properties Radioactivity Radioactive decay Alpha decay Beta decay Gamma decay Decay chains Back to BoseEinstein If the occupancy of a state can be any integer (such as with photons or phonons), we use the alreadynormalized BoseEinstein distribution function (Serway Eq. 10.3): f MB ( E ) = 1 exp [ E / k B T ] − 1 (1) Here’s Fig. 241 of Sandin, Essentials of Modern Physics . Note that at E = k B T = 25 meV for room temperature, we get f MB ( E ) ∼ 1. Einstein heat capacity The nucleus Radioactivity Isotopes Nucleus: properties Radioactivity Radioactive decay Alpha decay Beta decay Gamma decay Decay chains Classical heat capacity • Consider atoms in a solid. They all have equilibrium positions. We know that solids have some elasticity, so there must be a restoring force for displacement from starting positions. • What about the electrons? We will see that a few electrons (the conduction electrons) are so easily removed that it takes very little energy, while the rest happily stay put with their atom. For those others, they again will have a restoring force towards their equilibrium position relative to the nucleus. • The lowest order expansion term for these restoring forces is F = − kx , so we can approximate the system as a collection of harmonic oscillators. • We have a thermal energy of 1 2 k B T per degree of freedom. • How many degrees of freedom are there? • 3 for an atom being able to “choose” to have its displacement be distributed over three orthogonal directions • 2 for an atom being able to “choose” its ratio of kinetic to potential energy There are 6 degrees of freedom. Einstein heat capacity The nucleus Radioactivity Isotopes Nucleus: properties Radioactivity Radioactive decay Alpha decay Beta decay Gamma decay Decay chains Classical heat capacity of a solid • Again, we have three degrees of freedom for orthogonal directions, and two degrees of freedom for kinetic versus potential energy for storing energy, and ( 1 / 2 ) k B T energy per degree of freedom. Therefore the relationship between energy and temperature per atom should be E = 3 · 2 · 1 2 k B T = 3 k B T , or E = 3 N A k B T = 3 RT per mole of atoms. • It’s tough to measure the total energy of a solid as a function of temperature starting from zero. What’s easier to measure is the heat capacity , which tells us how the temperature changes as we add heat (Serway Eq. 10.29): C = dU dT = d dT 3 RT = 3 R = 3 N A k B = 3 · ( 6 . 02 × 10 23 atoms mol ) · ( 1 . 38 × 10 − 23 J K ) = 24 . 9 J mol · K Since 1 calorie=4.184 Joules, we have C = 5 . 95 cal/(mol · K). This matches room temperature experiments; see Serway Fig. 10.9. Einstein heat capacity The nucleus Radioactivity Isotopes Nucleus: properties Radioactivity Radioactive decay Alpha decay Beta decay Gamma decay Decay chains Heat capacity: experimental data The heat capacity C of many substances approaches 3 R = 5 . 95 cal/(mol · K) at high temperatures: Einstein heat...
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 Fall '01
 Rijssenbeek
 Physics, Radioactive Decay, Energy, Photon, Heat, Nobel Prize, decay Decay chains, decay Gamma decay, Gamma decay Decay

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