|All Solutions of Type: Statistical Mechanics|
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Statistical Mechanics}Specific Heat
Both Debye and Einstein assumed that there are 3N oscillators. (In fact, one can argue that the core of condensed matter begins with the assumption that a continuum piece of matter is basically a tiny mattress---a bunch of springs laden together.) Answer is thus (B).
However, Einstein was too lazy, and he decided that all 3N oscillators have the same frequency. Debye assigned a spectrum of frequencies (phonons).
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Statistical Mechanics}Fermi Temperature
When one deals with metals, one thinks of Fermi branding---i.e., stuff like Fermi energy, Fermi velocity, Fermi temperature, etc. So, . Fermi stuff is based on the Fermi-Dirac distribution, which assumes that the particles are fermions. Fermions obey the Pauli-exclusion principle. (c.f. Bose-Einstein distribution, where the particles are bosons, who are less discriminating and inclusive than fermions. Both Bose-Einstein and Fermi-Dirac assume indistinguishable particles, but the Boltzmann distribution, which assumes the particles are distinguishable)
All the other choices are too general, since bosons can also satisfy them. (Moreover, the Born approximation is pretty much the fundamental assumption of all of QM---every single calculation you do involving the interpretation of mod square of wave functions as probability depend on the Born approx!)
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Statistical Mechanics}Diatomic Molecules
Vibrational energy of a diatomic molecule goes to 0 at low temperatures. Thus, the springy dumbbell would be approximately a rigid dumbbell at a low-enough temperature. The other choices are too specific, and thus (E), the most general, must be it.
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At high temperatures, one has .
One expands the argument of the denominator according to (for small x). The denominator becomes . Since the whole quantity of the denominator is squared, the top term is canceled.
In the numerator, one has , since .
Thus, one arrives at choice (D).
One either remembers this fact about solid or one derives it, as shown above. (Also, for low temperatures, at say around 20K for most solids, the Debye law applies, and specific heat is proportional to . One can't find this result from the Einstein formula, which is why Debye's theory is more accurate for solids.)
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Statistical Mechanics}Blackbody Radiation Formula
where is the power and the energy and the temperature.
So, initially, the blackbody radiation emits . When its temperature is doubled, it emits .
Recall that water heats according to . So, initially, the heat gain in the water is . Finally, , where is the unknown change in temperature.
Conservation of energy in each step requires that and , i.e., that . Divide the two to get . Assuming the experiment is repeated from the same initial temperature, this would bring the initial to , as in choice (C).
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Statistical Mechanics}Maximal Probability
According to statistical mechanics, maximal probability is the sate of highest entropy---it's the peak of a Gaussian curve, the average score on a normally-curved test.
Spontaneous change to lower probability thus does not occur since maximal probability is the most stable state--one of highest entropy. Boltzmann's constant never approaches 0, however in the third law of thermodynamics, one has the entropy approaching 0 for .
Eliminating choices, one has choice (D).
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