91Ó°ÊÓ

One of the most powerful modern techniques for studying structure is neutron diffraction. This technique involves generating a collimated beam of neutrons at a particular temperature from a high-energy neutron source and is accomplished at several accelerator facilities around the world. If the speed of a neutron is given by \(v_{\mathrm{n}}=\left(3 k_{\mathrm{B}} T / m\right)^{1 / 2}\) where \(m\) is the mass of a neutron, then what temperature is needed so that the neutrons have a de Broglie wavelength of \(50 \mathrm{pm} ?\) Take the mass of a neutron to be \(1.67 \times\) \(10^{-27} \mathrm{kg}\).

Show that a small change in the speed of a particle, \(\Delta v,\) causes a change in its de Broglie wavelength, \(\Delta \lambda,\) of \\[|\Delta \lambda|=\frac{|\Delta v| \lambda_{0}}{v_{0}}\\] where \(v_{0}\) and \(\lambda_{0}\) are its initial speed and de Broglie wavelength, respectively.

Derive the Bohr formula for \(\tilde{v}\) for a nucleus of atomic number \(Z\).

Using the Bohr theory, calculate the ionization energy (in electron volts and in \(\mathrm{kJ} \cdot \mathrm{mol}^{-1}\) ) of singly ionized helium.

If we locate an electron to within \(20 \mathrm{pm}\), then what is the uncertainty in its speed?

What is the uncertainty of the momentum of an electron if we know its position is somewhere in a \(10 \mathrm{pm}\) interval? How does the value compare to momentum of an electron in the first Bohr orbit?

There is also an uncertainty principle for energy and time: \\[\Delta E \Delta t \geq h\\] Show that both sides of this expression have the same units.

Ionizing a hydrogen atom in its electronic ground state requires \(2.179 \times 10^{-18} \mathrm{J}\) of energy. The sun's surface has a temperature of \(\approx 6000 \mathrm{K}\) and is composed, in part, of atomic hydrogen. Is the hydrogen present as \(\mathrm{H}(\mathrm{g})\) or \(\mathrm{H}^{+}(\mathrm{g}) ?\) What is the temperature required so that the maximum wavelength of the emission of a blackbody ionizes atomic hydrogen? In what region of the electromagnetic spectrum is this wavelength found?