Exponential Decay

Examples of Exponential Decay in the following topics:

• Rate of Radioactive Decay

  • Radioactive decay rate is exponential and is characterized by constants, such as half-life, as well the activity and number of particles.
  • Radioactivity is one very frequent example of exponential decay.
  • Particular radionuclides decay at different rates, so each has its own decay constant, λ.
  • A quantity undergoing exponential decay.
  • Apply the equation Nt=N0e−λt in the calculation of decay rates and decay constants

• The Arrhenius Equation

  • First, note that this is another form of the exponential decay law.
  • What is "decaying" here is not the concentration of a reactant as a function of time, but the magnitude of the rate constant as a function of the exponent –Ea /RT.
  • The Arrhenius equation can be written in a non-exponential form, which is often more convenient to use and to interpret graphically.
  • Taking the natural logarithms of both sides and separating the exponential and pre-exponential terms yields: $ln(k)=ln(A)-\frac{E_{a}}{RT}$
  • Let's look at the pre-exponential factor A in the Arrhenius equation.

• The Integrated Rate Law

  • However, the integrated first-order rate law is usually written in the form of the exponential decay equation.

• Half-Life of Radioactive Decay

  • The half-life is a parameter for the rate of decay that is related to the decay constant by: ${t}_{\frac{1}{2}}=\frac{ln2}{\lambda}$ .
  • Radioactive decay is a random process at the single-atom level; is impossible to predict exactly when a particular atom will decay.
  • However, the chance that a given atom will decay is constant over time.
  • The equation indicates that the decay constant λ has units of t-1.
  • The half-life is related to the decay constant.

• Dating Using Radioactive Decay

  • It is based on a comparison between the observed abundance of a naturally occurring radioactive isotope and its decay products, using known decay rates.
  • After one half-life has elapsed, one half of the atoms of the nuclide in question will have decayed into a "daughter" nuclide, or decay product.
  • A 100 g sample of Cs-137 is allowed to decay.
  • Each parent nuclide spontaneously decays into a daughter nuclide (the decay product) via an α decay or a β decay.
  • The final decay product, lead-208 (208Pb), is stable and can no longer undergo spontaneous radioactive decay.

• Modes of Radioactive Decay

  • Radioactive decay occurs when an unstable atomic nucleus emits particles or light waves.
  • Alpha decay is seen only in heavier elements greater than atomic number 52, tellurium.
  • The other two types of decay are seen in all of the elements.
  • Alpha decay occurs because the nucleus of a radioisotope has too many protons.
  • Examples of this can be seen in the decay of americium (Am) to neptunium (Np).

• Indoor Pollution: Radon

  • Radon gas, the result of radium's radioactive decay, can severely compromise indoor air quality.
  • Radon is a dense, colorless, odorless noble gas that occurs naturally in the soil as the product of the radioactive decay of radium; it is a decay product of uranium and thorium, which occur naturally in the Earth's crust.
  • Radon decays to form daughters, or decay products, which include radioactive polonium, lead, and bismuth.
  • Radon is a gas, but these decay products are solids that can attach to dust and enter the lungs.
  • Radon and its daughters continue to decay in the lungs, releasing alpha and beta particles that can damage cellular DNA and result in lung cancer.

• Nuclear Stability

  • However, if neutron count surpasses an ideal ratio, a nucleus becomes unstable and can undergo radioactive decay.
  • Only 90 isotopes in this region are believed to be perfectly stable, while 163 more are understood to be theoretically unstable but have never been observed to decay.
  • Technetium and promethium, as well as elements of number 83 and above, have only isotopes that will decay over time.

• Properties of Carbon

  • It has a very low natural abundance (0.0000000001%), and decays to 14N through beta decay.
  • In total, there are 15 known isotopes of carbon and the shortest-lived of these is 8C, which decays through proton emission and alpha decay, and has a half-life of 1.98739 x 10−21 seconds.

• Isotopes of Hydrogen

  • It is radioactive, decaying into helium-3 through beta-decay accompanied by a release of 18.6 keV of energy.
  • It decays through neutron emission with a half-life of 1.39 ×10−22 seconds.
  • It decays through double neutron emission and has a half-life of at least 9.1 × 10−22 seconds.
  • 6H decays through triple neutron emission and has a half-life of 2.90×10−22 seconds.