If a plot is made of the number of protons versus the number of neutrons for the stable isotopes, the curve shown in Figure 3-6 is obtained. The stable isotopes lie within a relatively narrow range, indicating that the neutron-to-proton ratio must lie within certain limits if a nucleus is to be stable. Most radioactive nuclei lie outside this range of stability. The plot also shows that the slope of the curve, which initially has a value of unity, gradually increases as the atomic number increases, thereby showing the continuously increasing ratio of neutrons to protons.
Since all nuclear forces are attractive, it may appear surprising to find unstable nuclei with an excessive number of neutrons. This apparent anomaly may be ex-plained simply in terms of the shell model of the nucleus. According to the Pauli exclusion principle, like nucleons may be grouped in pairs with each pair having all quantum numbers the same except the spin quantum number. Since nuclei with completely filled energy levels are more stable than those with unfilled inner levels, additional neutrons, in the case of nuclei with unfilled proton levels but filled neu-tron levels, result in unstable nuclei. To achieve stability, the nucleus may undergo an internal rearrangement in which the additional neutron transforms itself into a proton by emitting an electron. The new proton then pairs off with a proton in one
Figure 3-6. Nuclear stability curve. The line represents the best fit to the neutron–proton coordinates of stable isotopes.
of the unfilled proton levels. As an example of this possible mechanism, consider the consequences of the addition of a neutron to3115P. This is the stable isotope of phos-phorus that occurs naturally. According to the shell model, the 15 protons inside the nucleus may be distributed among seven pairs with one proton remaining un-paired, while the neutrons may be paired off into eight groups. If now an additional neutron is added to the nucleus to make 3215P, the additional neutron may go into another energy level. This condition, however, is unstable. The additional neutron may therefore become a proton and an electron—with the electron being ejected from the nucleus and the proton pairing off with the single proton, thereby forming stable3216S. This internal nuclear transformation is called aradioactive transformation or aradioactive decay.
SUMMARY
Although the wordatomis derived from the Greek wordatomos, which means “indi-visible,” modern science has found the atom to be a complex structure consisting of a positively charged nucleus surrounded by negatively charged electrons. The nucleus in turn is composed of two different particles—positively charged protons and electrically neutral neutrons. The protons and neutrons consist of an assembly of three smaller particles called quarks. Quarks are considered to be one of the fun-damental building blocks in nature. Strong, attractive, short-ranged, nuclear forces act between the nucleons (particles within the nucleus) to overcome the repulsive electric forces that act between the protons. In a neutral atom, the number of ex-tranuclear electrons is equal to the number of inex-tranuclear protons. The overall diameter of the atom is on the order of 10−8cm while the diameter of the nucleus measures about 10−13cm.
The Bohr atomic model resembles a miniature solar system, with the electrons revolving around the nucleus in only certain allowable radii that are described by a set of four quantum numbers. The wave mechanics atomic model pictures the atom as a central nucleus surrounded by a cloud of electrons. The distances of these electrons from the nuclei are not precisely defined, as in the Bohr model, but rather are described by a wavelike probability function. These two atomic models coincide to the extent that the most probable radii of the wave mechanical model correspond to the precisely quantized radii of the Bohr model. Although the wave mechanical model has replaced the Bohr model for highly theoretical considerations, the Bohr model is adequate to explain the phenomena that underlie most health physics measurements and applications.
The atomic number and hence the chemical properties of an element are deter-mined by the number of protons within the nucleus. Different atoms of the same element, however, may have different numbers of neutrons within their nuclei. These different forms of the same element are calledisotopes.The total number of nucle-ons within a nucleus is called itsatomic mass number. An isotope usually is specified by the name of the element and its atomic mass number. In written form, we fre-quently describe an isotope by its atomic number as a subscript to the left of its chemical symbol and its atomic mass number as a superscript to the left of its sym-bol. Thus, for uranium 238, whose atomic number is 92, we have23892U. The mass of a
nucleus is less than the sum of the masses of its constituent parts. This mass difference represents the mass equivalent of the energy that was expended in assembling the nucleus (the binding energy). When a particle is ejected from the nucleus or when an atom undergoes nuclear fission, this potential energy reappears as the kinetic energy of the ejected particle or of the fission fragments and the energy of accompa-nying radiation. A stable atom requires that the ratio of neutrons to protons within the nucleus be within certain limits. If this ratio is too great or too small then the nucleus is unstable and it spontaneously attempts to become stable by a radioactive transformation.
m Problems
3.1. What is the closest approach that a 5.3-MeV alpha particle can make to a gold nucleus?
3.2. Calculate the number of atoms per cubic centimeter of lead given that the density of lead is 11.3 g/cm3and its atomic weight is 207.21.
3.3. Aμ−meson has a charge of−4.8×10−10sC and a mass 207 times that of a resting electron. If a proton should capture aμ−to form a “mesic” atom, calculate (a) the radius of the first Bohr orbit and
(b) the ionization potential.
3.4. Calculate the ionization potential of a singly ionized4He atom.
3.5. Calculate the current due to the hydrogen electron in the ground state of hydro-gen.
3.6. Calculate the ratio of the velocity of a hydrogen electron in the ground state to the velocity of light.
3.7. Calculate the Rydberg constant for deuterium.
3.8. What is the uncertainty in the momentum of a proton inside a nucleus of27Al?
What is the kinetic energy of this proton?
3.9. A sodium ion is neutralized by capturing a 1-eV electron. What is the wavelength of the emitted radiation if the ionization potential of Na is 5.41 V?
3.10. (a) How much energy would be released if 1 g of deuterium were fused to form helium according to the equation2H+2H→4He+Q?
(b) How much energy is necessary to drive the two deuterium nuclei together?
3.11. The density of beryllium (atomic number 4) is 1.84 g/cm3, and the density of lead (atomic number 82) is 11.3 g/cm3. Calculate the density of a9Be and a208Pb nucleus.
3.12. Determine the electronic shell configuration for aluminum (atomic number 13).
3.13. What is the difference in mass between the hydrogen atom and the sum of the masses of a proton and an electron? Express the answer in energy equivalent (eV) of the mass difference.
3.14. If the heat of vaporization of water is 540 cal/g at atmospheric pressure, what is the binding energy of a water molecule?
3.15. The ionization potential of He is 24.5 eV. What is the
(a) minimum velocity with which an electron is moving before it can ionize an unexcited He atom?
(b) maximum wavelength of a photon in order that it ionizes the He atom?
3.16. In a certain 25-W mercury-vapor ultraviolet lamp, 0.1% of the electric energy input appears as UV radiation of wavelength 2537 ˚A. What is the UV photon emission rate per second from this lamp?
3.17. The atomic mass of tritium is 3.017005 amu. How much energy in million electron volts is required to dissociate the tritium into its component parts?
3.18. Compute the frequency, wavelength, and energy (in electron volts) for the second and third lines in the Lyman series.
3.19. Using the Bohr atomic model, calculate the velocity of the ground-state electrons in hydrogen and in helium.
3.20. The heat of combustion when H2combines with O2to form water is 60 kcal/mol water. How much energy (in electron volts) is liberated per molecule of water produced?
3.21. The atomic weights of16O,17O, and18O are 15.994915, 16.999131, and 17.999160 amu, respectively. Calculate the atomic weight of oxygen.
3.22. Calculate the molecular weight of chlorine, Cl2, using the exact atomic weights of the chlorine isotopes given in appropriate reference sources.
3.23. If 9 g of NaCl were dissolved in 1 L of water, what would be concentration, in atoms per milliliter, of each of the constituent elements in the solution?
3.24. The visual threshold of the normal human eye is about 7.3×10−15 W/cm2 for light whose wavelength is 556 nm. What is the corresponding photon flux, in photons per square centimeter per second?
3.25. What is the binding energy of the last neutron in17O?
3.26. Calculate the number of hydrogen atoms in 1-g water.
3.27. If all the mass of an electron were converted to electromagnetic energy, what would be the
(a) energy of the photon, in joules and in million electron volts?
(b) wavelength of the photon, in angstrom units?
3.28. The thermal energy content of 1 U.S. gal (3.79 L) gasoline is 36.65 kW hours.
To what weight of nuclear fuel, grams, does this amount of energy corres-pond?
3.29. The binding energy ofK electrons in copper is 8.980 keV and 0.953 keV in the Llevel. What is the wavelength of theKαcharacteristic X-ray?
3.30. The first ionization potential of aluminum is 4.2 eV. What is the maximum wave-length of light that can ionize an aluminum atom?
3.31. Two alpha particles are separated by a distance of 4×10−15m. Calculate the (a) repulsive electrical force between them.
(b) attractive gravitational force between them.
3.32. The bonding energy of a C−C bond is about 100 kcal/mol. What is the corre-sponding energy, expressed as eV/bond?
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