Khan Chapter 2 Summary

2.1-2.3: Radioactivity, Decay Constant, & Activity

  • A radioactive nucleus with excess energy may distribute enough energy onto a nucleon for it to escape the nucleus.
    • This energy can be emitted in the form of electromagnetic or particulate radiation.The emission of this energy will result in a lower energy state of the nucleus.
  • Radioactive decay or disintegration cannot be precisely predicted for an individual atom because the phenomenon is statistical. However, this prediction becomes more possible when discussing a large number of atoms.
    • Every radioactive isotope is assigned a decay constant (λ) which is the fraction/percentage of the atoms for a population (N_0) that will decay per unit time (t). The number of decayed atoms can be determined when this constant is multiplied with the total number of atoms.
    • As this is a “loss” in the number of original radioactive atoms, the value of the number of atoms decayed will be negative: -λ * N = ΔN / Δt.
    • When expressed as a differential equation: N = N_0 * e^-λ * t.
  • Activity represents the number of decay per unit time as opposed to the change in the number of radioactive atoms. Thus, activity is a positive value and is given by the equation: A = λ * N.
    • When expressed as a differential equation: A = A_0 * e^-λ * t.
    • The SI unit of activity is Becquerel (Bq) that is defined as 1 disintegration per second (dps). The conversion to a more commonly used unit Curie (Ci) can be made by: 1 Ci = 3.7 * 10^10 Bq.

2.4: The Half-Life & The Mean Life

  • Half-life refers to the amount of time required for the number of radioactive atoms to decay to half the initial amount.
    • Substituting “0.5 for N / N_0 and A / A_0” and “t_1/2 for t” in the differential equations mentioned above and applying “ln” to both sides will give the equation T_1/2 = ln 2 / λ.
  • The mean or average life refers to the sum of the lifespans of all the radioactive atoms until decay divided by the number of atoms.
    • Conceptually speaking based on the idea of half-life, it will take an infinite amount of time for a population of radioactive atoms to completely decay as continually halving N will never yield 0. Mean life can be better understood in terms of activity where a source disintegrates at a rate of an absolute number.
    • As the half-life equation is derived from the differential activity equation, the source that decays from t = 0 to t = ∞ will produce an equal number of disintegrations in time T_a as the source that decays at an exponentially constant rate equal to the initial activity.
    • Thus, the total number of disintegrations can be found by the equation: T_a * A = N_0.
      • Substitute λ * N_0 for A:  T_a * λ * N_0 = N_0.
      • Cancel out N_0 and rearrange: T_a = 1 / λ.
      • Substitute λ = ln 2 / T_1/2 for λ: T_a = 1.44 * T_1/2.
  • How many atoms are in 1 g of Ra-226 and what is the activity of 1 g of Ra-226 (T_1/2 = 1,622 years)?
    • If 1 amu = 1.66 * 10^-24 g, then 266 amu = 3.75 * 10^-22 g.
    • Thus 1 Ra-226 atom = 3.75 * 10^-22 g, or 3.75 * 10^-22 g/atom.
    • 1 g of Ra-226 / 3.75 * 10^-22 g/atom = 2.66 * 10^21 atoms/g.
    • A = λ * N_0; A = ln 2 * N_0 / T_1/2; ln 2 * 2.66 * 10^21 atoms/g / 5.12 * 10^10 sec =3.6 * 10^10 dps/gram.
    • If 1 Ci = 3.7 * 10^10 Bq, then 0.974 Ci/g = 3.6 * 10^10 dps/g.
  • 0.974 Ci/g is the specific activity of Ra-226, i.e., the activity per unit mass. The Ci was originally defined as the decay rate of 1 g of radium, but the specific activity of the radium is slightly less than 1 Ci/g. The actual decay rate of radium has since been revised, but value of the Ci still remains.
  • What is the decay constant in units of month for Co-60 (T_1/2 = 5.26)? After 4 years, what will be the activity for a 5,000 Ci Co-60 source?
    • λ = ln 2 / 63.12 months; λ = 0.011 months^-1.
    • A = 5,000 Ci * e^-0.011 months^-1 * 48 months; A = 2,949 Ci.
  • When will 5 mCi of I-131 (T_1/2 = 8.05 days) and 2 mCi of P-32 (T_1/2 = 14.3 days) have equal activities?
    • Set differential activity equations equal to one another: A_0 * e^-λ * t = A_0 * e^-λ * t.
    • Calculate T_1/2 of each isotopes:
      • λ_I-131 = ln 2 / 8.05 days; λ_I-131 = 0.0861 days^-1.
      • λ_P-32 = ln 2 / 14.3 days; λ_P-32 = 0.0485 days^-1.
    • 5 mCi * e^-0.0861 days^-1 * t = 2 mCi * e^-0.0485 days^-1 * t.
    • ln 5 mCi – 0.0861 days^-1 * t = ln 2 mCi – 0.0485 days^-1 * t.
    • t = 24.37 days.

2.5-2.6: Radioactive Series & Radioactive Equilibrium

  • The three series that all naturally occurring radioactive elements are grouped into are the uranium series (U-238), the actinium series (U-235), and the thorium series (Th-232).
  • These series end at the stable isotopes of lead that have mass numbers of 206, 207, and 208, respectively.
  • In the process of radioactive decay, the original nuclide and the product is referred to as the parent and daughter, respectively.
    • Equilibrium between the parent and the daughter can be achieved when the ratio of daughter to parent activity becomes constant. The half-life of the parent must be longer than the daughter for equilibrium to exist.
      • To better understand the concept of equilibrium, picture a fish tank that had just been punctured resulting in a outflow of water. The puncture is unable to be blocked so you place a hose by the rim of the tank to offset the water loss. Assuming the water flow from the hose is constant, the water level in the tank and the outflow of water through the puncture will become constant as well.
      • Secular equilibrium occurs when the half-life of the parents is much longer than that of the daughter. The half-life is long enough such that there is no noticeable decay.
      • Transient equilibrium occurs when the half-life of the parents is not that much longer than that of the daughter. The half-life is short enough such that a noticeable decay is evident, but the daughter-parent proportion remains constant.

2.7: Modes of Radioactive Decay

  • In alpha particle decay, an alpha particle containing 2 protons and 2 neutrons leave the nucleus.
    • Accompanied by the alpha emission is the disintegration energy (Q) which is the energy equivalent to the difference between the mass of the parent and product nuclei and will manifest as kinetic energy distributed amongst the products.
      • Reactant: radioactive nuclei.
      • Product: product nuclei (A – 4, Z -2), alpha particle, and Q.
  • Beta decay can refer to either negatron or positron emission.
    • In negatron emission, the radioactive nucleus will possess a high neutron-to-proton ratio, and a neutron transforms into a proton to achieve nuclear stability. A negatron will be emitted in the process accompanied by an antineutrino and Q.
      • Reactant: radioactive nuclei.
      • Product: product nuclei (Z + 1), negatron, antineutrino, and Q.
    • The opposite occurs in positron emission where the radioactive nucleus possess too low of a neutron-to-proton ratio, and a proton transforms into a neutron to achieve nuclear stability. A positron is emitted in the process accompanied by an neutrino and Q.
      • Reactant: radioactive nuclei.
      • Product: product nuclei (Z – 1), positron, neutrino, and Q.
  • Electron capture is an alternative to positron decay by which the nucleus with a neutron deficiency will capture a surrounding orbital electron to raise its neutron-to-proton ratio.
    • A proton will transform into a neutron, and the product nuclei is accompanied by a neutrino emission and Q.
      • Reactant: radioactive nuclei and electron.
      • Product: product nuclei (Z – 1), neutrino, and Q.
    • Electron capture will create a vacancy in the shell where the electron had previously resided. The transition of the outer shell electrons to fill the vacancy will produce characteristic x-rays and Auger electrons.
  • A nucleus that had undergone previous nuclear decay may be left in an excited state and can rid this excess energy by internal conversion.
    • A gamma ray is emitted from the nucleus that ejects an orbital electron, and as in electron capture, the resultant shell vacancy will produce characteristic x-rays and Auger electrons.
    • If the excited nucleus remains in this state for an extended period of time, this nucleus is said to exist in the metastable state and is an isomer of the final product nucleus.

2.8-2.10: Nuclear Reactions, Activation of Nuclides, & Nuclear Reactors

  • Particles are accelerated by the likes of a cyclotron to bombard a nucleus to produce radioactive nuclei.
    • When nuclei are initially activated/made unstable, the transformed population will grow exponentially, but the growth of activity will eventually taper at a maximum value.
    • This ceiling is referred to as the saturation activity where the rate of activation equals the rate of decay.
  • When neutrons are bombarded into certain high Z nuclei in a reactor, the nucleus will split into nuclei of lower Z or fission fragments. Neutrons are also a product of this reaction, and these neutrons can chain additional fission reactions. However, these neutrons are most effective in chaining at thermal energies, and thus, are slowed using a low Z material called a moderator.
  • The following are examples of bombardment: alpha, proton reaction; alpha, neutron reaction; proton bombardment; deuteron bombardment; neutron bombardment; photodisintegration; fission; and fusion.

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