Chemistry 2720 Fall 2000 Assignment 6

Due: Tuesday, Nov. 7, 9:25a.m.

1.

Suppose that you want to initiate a reaction involving Cl₂ by breaking the Cl-Cl bond photochemically. The dissociation energy of Cl₂ is 239.2kJ/mol. What is the minimum frequency of photons capable of achieving this task? To what spectral region do these photons belong? [5 marks]

2.

Suppose that you want to measure the speed of some neutrons ( $m = 1.674\,927\times 10^{-27}\,\mathrm{kg}$) with a maximum uncertainty of 1m/s. What is the minimum uncertainty in their positions? [4 marks]

3.

The kinetic energy gained by a charge q accelerated through a voltage V is |qV|. The resolution of an electron microscope (the minimum size of objects which can be resolved) is given by the formula

$$\Delta r\approx 0.7C_s^{1/4}\lambda^{3/4}$$

where C_s is a constant which depends on the microscope design and $\lambda$ is the wavelength of the electrons. Suppose that $C_s = 2\,\mathrm{mm}$ and that we want a a resolution of 3Å.

(a)

Calculate the momentum of electrons with the appropriate wavelength. [4 marks]

(b)

The electrons in this microscope have relativistic speeds. In other words, if you calculated their speed from classical mechanics, you would obtain a value close to c, and classical mechanics is invalid in this range. Instead, use the relativistic energy equation to compute the total energy of one of these electrons. [4 marks]

(c)

The kinetic energy is the difference between the total energy and m₀c² (the energy associated with the rest mass). Calculate the voltage used to accelerate the electrons to the required speed. [4 marks]

4.

In a cubic lattice, the distances between the crystallographic planes is given by

$$d_{hkl} = \frac{a}{(h^2+k^2+l^2)^{1/2}},$$

where a is the length of one of the sides of the cube which defines the unit cell (the repeating unit of the crystal), and h, k and l are non-negative integers (including zero) called Miller indices. The Miller indices identify the orientation of a plane of atoms with respect to the edges of the unit cell. Potassium chloride crystallizes in a cubic symmetry. When a diffraction pattern is obtained for KCl with X-rays of wavelength 70.8pm, reflections are observed at the following angles: 6.6, 9.2, 11.4, 13.1 and $14.7^\circ$.

(a)

Calculate the interplanar distances d corresponding to each of the measured angles. [6 marks]

Note: Using the Miller indexing system, all reflections can be treated as first-order reflections.

(b)

Find a set of Miller indices for each of the reflections and calculate a. [10 marks]

Hints and notes: The smallest angles correspond to the largest distances which in turn correspond to the smallest sum h²+k²+l². Thus the first reflection is probably due to the (100) orientation (or, equivalently, to the (010) or (001) orientations, all of which are identical in a cubic symmetry). Once you have labeled the reflections, you will be able to calculate a from each. Average these values to get an accurate value for a.¹

Footnotes

....¹

An even better method would be to get a by regression from the slope of a suitably constructed plot. This allows all the data to be used at once to estimate a and tends to negate the effects of certain kinds of systematic measurement errors. One bonus mark may be awarded if you can obtain the solution to this problem by a correct regression method.