PH253: HW9 hints

Copious hints on HW9 below the fold ...


1) If you neglect the inner electrons, the nucleus has Z=3, i.e., a +3 charge. If you assume perfect screening, then Z=1, i.e., a +1 charge. Use the generalized Bohr formula for the energies.

2) The dissociation energy is the Coulomb repulsion (ke^2/r) plus the ionization energy minus the electron affinity. You know everything but r. The answer is of order 3nm.

(dissociation energy) = (ke^2/r) + (electron affinity) - (ionization energy)

3) a. Differentiate. b. Use your formula for b from the first part in the expression for U(r).

4) a. Differentiate. Dissociation occurs when r tends to infinity (the atoms are totally unbound). b. Force is -dU/dr, which you found in the first part. Use a Taylor series to approximate the exponential terms as a power series up to 1st order terms in r-r_o = d. Collecting terms you should get something like F = -(const)d, which looks just like a spring. The (const) part must then be the spring constant. Check the units to verify.

5) The wikipedia article on the Lennard-Jones potential is useful. a. Similar to #2, to break up the molecule the energy you need is the difference between the ionization energy and the electron affinity. It should be positive, since it takes work to break up the molecule. b. Differentiate to find the equilibrium spacing r_o,  and thus sigma. The value of U(r_o) is the negative of the dissociation energy, which will allow you to find epsilon. c. Force is -dU/dr. The force goes through a maximum as a function of separation distance; if you apply this force or greater, you will be able to break the molecule. Find the r that corresponds to maximum force via differentiation, and plug it back in to F(r). d. Use a Taylor series to expand the powers of r in the expression for U(r+d). Keep terms up to second order, and you should get something like U(r) = U(r_o) + (const)d^2. The (const) part must be your spring constant k.

6) Write the potential energy in terms of an infinite sum, which should end up being the alternating harmonic series times a constant.

7. a. We'll deal with this in class. b. See example 6.4 in your textbook.

8. We'll deal with this in class.

9. The number of electrons per unit volume is n=(density)(moles/gram)(Avogadro #)(electrons per atom). Drift velocity is (current)/(n*e*cross-sectional area of wire).