**Definition**: the **specific heat** of a system is the quantity of heat energy that must be supplied to raise the temperature of the system by one kelvin.

1.* Specific heat of a solid*

(a) Assume that each atom in a solid can be regarded as a three-dimensional harmonic oscillator. Assuming further the equipartition of energy, what is the specific heat of a *single atom* in a solid?

(b) From the above, derive the Dulong-Petit Law: for a solid, (atomic weight) x (specific heat per *kilogram*) = constant, and find the value of the constant.

(c) Now assume that the energy of the oscillating atom is quantized, just like Planck's electromagnetic oscillators in black body radiation. Assume all the atoms in the solid have the same frequency of vibration *f*. Adapt Planck's formula (see BBR lecture) to find the vibrational energy of the atom as a function of temperature, and sketch the function.

(d) From your result in (c), sketch how you would expect the specific heat of a solid to vary with temperature, all the way down to absolute zero.

(e) Most solids obey the Dulong-Petit Law at room temperature, but diamond doesn't. Which way would you expect it to deviate? What properties of diamond are important in this context?

2*. Specific heat of a gas*

Experimentally, the *molar* specific heat of hydrogen gas varies with temperature, being 1.5*R* at low temperatures, and 2.5*R* at room temperature. This can be explained by assuming that the angular momentum of the rotating molecule is quantized as* *(*nh*/2)p , with *n* = 0, 1, 2, … .

(a) The two hydrogen nuclei in the molecule are about 8. 10^{-11} meters apart. What are the moments of inertia of the molecule?

(b) Find the rotational kinetic energy of the lowest non-zero angular momentum state of the molecule, and estimate the temperature below which rotation is "frozen out".

(c) Explain why the specific heat goes to 2.5*R* rather than 3*R*.

(d) Give a very approximate estimate of the temperature below which rotational specific heat will be frozen out for oxygen gas.

3.* Rutherford scattering at higher energies*

Rutherford derived a formula for the percentage of particles scattered at various angles, assuming inverse-square repulsion between the gold nucleus and the alpha-particle. On increasing the energy of the incident alphas, it is found that there are deviations from the formula at ingoing energy 32 Mev. This suggests that some of these particles must be hitting the nucleus. Estimate the size of the nucleus.

4.* Innermost electrons*

Suppose that in the gold atom, one electron is orbiting closer to the nucleus that all the others, so all it sees is the nuclear charge (assuming the other electrons act like surrounding spheres of charge, with no effective fields inside). Assume the close-in electron can be taken to be in the lowest Bohr orbit.

(a) What is the size of the orbit? How does it compare with the nuclear size?

(b) If a photon comes along and knocks that innermost electron clear out of the atom, estimate the energy of the photon by ignoring the effects of all the other electrons (this gives the right order of magnitude). What kind of photon is that?

5. *Spectra of atoms in a magnetic field*

Just to get some idea of how the presence of a magnetic field might affect atomic energy levels, consider the lowest Bohr orbit in hydrogen (*n* = 1). Regard it as a loop of electric current, and find the strength of the equivalent magnetic dipole. If this atom is placed in a field of strength 2 Tesla, what is the energy difference between the states having maximum and minimum *magnetic* energy? (The magnetic energy depends on the orientation of the dipole.) What is the corresponding result for the *n* = 2 orbit? How would the presence of the magnetic field affect the observed *spectra*? Give a quantitative estimate. Faraday looked for an effect of magnetic fields on spectra, bu t saw nothing. Why?