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Half Life

Radioactive isotopes breakdown into other elements in an inverse exponential relationship to time.

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Half Life

Students will learn what is meant by half life and how to solve problems involving radioactive decay.

Key Equations

\begin{align*}M = M_0 (\frac{1}{2})^{\frac{t}{t_H}}\end{align*}M=M0(12)ttH

The amount of mass \begin{align*}M\end{align*}M of the substance surviving after an original \begin{align*}\mathit{M}_0\end{align*} \begin{align*}n\end{align*} uclei decay for time \begin{align*}t\end{align*} with a half-life of \begin{align*}\mathit{t}_H\end{align*}.

\begin{align*}t = t_H \frac{\ln{\frac{M}{M_0}}}{\ln{\frac{1}{2}}} \end{align*}

The amount of time \begin{align*}t\end{align*} it takes a set of nuclei to decay to a specified amount.

\begin{align*}N = N_0 exp^{-\lambda t}\end{align*}

The number \begin{align*}N\end{align*} of nuclei surviving after an original \begin{align*}\mathit{N}_0\end{align*} \begin{align*}n\end{align*} uclei decay for time \begin{align*}t\end{align*} with a half-life of \begin{align*}\mathit{t}_H\end{align*}.

  • Some of the matter on Earth is unstable and undergoing nuclear decay.
  • Alpha decay is the emission of a helium nucleus and causes the product to have an atomic number \begin{align*}2\end{align*} lower than the original and an atomic mass number \begin{align*}4\end{align*} lower than the original.
  • Beta minus decay is the emission of an electron, causing the product to have an atomic number \begin{align*}1\end{align*} greater than the original
  • Beta plus decay is the emission of a positron, causing the product to have an atomic number \begin{align*}1\end{align*} lower than the original.
  • When an atomic nucleus decays, it does so by releasing one or more particles. The atom often (but not always) turns into a different element during the decay process. The amount of radiation given off by a certain sample of radioactive material depends on the amount of material, how quickly it decays, and the nature of the decay product. Big, rapidly decaying samples are most dangerous.
  • The measure of how quickly a nucleus decays is given by the half-life of the nucleus. One half-life is the amount of time it will take for half of the radioactive material to decay.
  • The type of atom is determined by the atomic number (i.e. the number of protons). The atomic mass of an atom is approximately the number of protons plus the number of neutrons. Typically, the atomic mass listed in a periodic table is an average, weighted by the natural abundances of different isotopes.
  • The atomic mass number in a nuclear decay process is conserved. This means that you will have the same total atomic mass number on both sides of the equation. Charge is also conserved in a nuclear process.
  • It is impossible to predict when an individual atom will decay; one can only predict the probability. However, it is possible to predict when a portion of a macroscopic sample will decay extremely accurately because the sample contains a vast number of atoms.

Example 1

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Time for Practice

  1. After \begin{align*}6\end{align*} seconds, the mass of a sample of radioactive material has reduced from \begin{align*}100\end{align*} grams to \begin{align*}25\end{align*} grams. Its half-life must be
    1. \begin{align*}1 \;\mathrm{s}\end{align*}
    2. \begin{align*}2 \;\mathrm{s}\end{align*}
    3. \begin{align*}3 \;\mathrm{s}\end{align*}
    4. \begin{align*}4 \;\mathrm{s}\end{align*}
    5. \begin{align*}6 \;\mathrm{s}\end{align*}
  2. For any radioactive material, when does its half-life,
    1. First decrease and then increase?
    2. First increase and then decrease?
    3. Increase with time?
    4. Decrease with time?
    5. Stay the same?
  3. If the half-life of a substance is \begin{align*}5\end{align*} seconds, it ceases to be radioactive (i.e. it ceases emitting particles), …
    1. … after \begin{align*}5\end{align*} seconds.
    2. … after \begin{align*}10\end{align*} seconds
    3. … after \begin{align*}20\end{align*} seconds.
    4. … after a very long time.
  4. You have \begin{align*}5\end{align*} grams of radioactive substance A and \begin{align*}5\end{align*} grams of radioactive substance B. Both decay by emitting alpha-radiation, and you know that the higher the number of alpha-particles emitted in a given amount of time, the more dangerous the sample is. Substance A has a short half-life (around \begin{align*}4\end{align*} days or so) and substance B has a longer half-life (around \begin{align*}10\end{align*} months or so).
    1. Which substance is more dangerous right now? Explain.
    2. Which substance will be more dangerous in two years? Explain.
  5. A certain radioactive material has a half-life of \begin{align*}8\end{align*} minutes. Suppose you have a large sample of this material, containing \begin{align*}10^{25}\end{align*} atoms.
    1. How many atoms decay in the first \begin{align*}8\end{align*} minutes?
    2. Does this strike you as a dangerous release of radiation? Explain.
    3. How many atoms decay in the second \begin{align*}8\end{align*} minutes?
    4. What is the ratio of the number of atoms that decay in the first \begin{align*}8\end{align*} minutes to the number of atoms that decay in the second \begin{align*}8\end{align*} minutes?
    5. How long would you have to wait until the decay rate drops to \begin{align*}1\end{align*}% of its value in the first 8 minutes?
  6. Hidden in your devious secret laboratory are \begin{align*}5.0\end{align*} grams of radioactive substance A and \begin{align*}5.0\end{align*} grams of radioactive substance B. Both emit alpha-radiation. Quick tests determine that substance A has a half-life of \begin{align*}4.2\end{align*} days and substance B has a half-life of \begin{align*}310\end{align*} days.
    1. How many grams of substance A and how many grams of substance B will you have after waiting \begin{align*}30\end{align*} days?
    2. Which sample (A or B) is more dangerous at this point (i.e., after the \begin{align*}30\end{align*} days have passed)?
  7. The half-life of a certain radioactive material is \begin{align*}4\end{align*} years. After \begin{align*}24\end{align*} years, how much of a \begin{align*}75\end{align*} g sample of this material will remain?
  8. The half life of \begin{align*}^{53}\mathrm{Ti}\end{align*} is \begin{align*}33.0\end{align*} seconds. You begin with \begin{align*}1000\;\mathrm{g}\end{align*} of \begin{align*}^{53}\mathrm{Ti}\end{align*}. How much is left after \begin{align*}99.0\end{align*} seconds?
  9. You want to determine the half-life of a radioactive substance. At the moment you start your stopwatch, the radioactive substance has a mass of \begin{align*}10 \;\mathrm{g}\end{align*}. After \begin{align*}2.0\end{align*} minutes, the radioactive substance has \begin{align*}0.5\end{align*} grams left. What is its half-life?
  10. There are two equal amounts of radioactive material. One has a short half-life and the other has a very long half-life. If you measured the decay rates coming from each sample, which would you expect to have a higher decay rate? Why?

Answers to Selected Problems

  1. .
  2. .
  3. .
  4. a. Substance \begin{align*}A\end{align*} decays faster than \begin{align*}B\end{align*} b. Substance \begin{align*}B\end{align*} because there is more material left to decay.
  5. a. \begin{align*}5 \times 10^{24}\end{align*} atoms b. Decay of a lot of atoms in a short period of time c. \begin{align*}2.5 \times 10^{24}\end{align*} atoms d. \begin{align*}\frac{1}{2}\end{align*} e. \begin{align*}26.6\end{align*} minutes
  6. a. Substance \begin{align*}B = 4.6 \;\mathrm{g}\end{align*} and substance \begin{align*}A = 0.035 \;\mathrm{g}\end{align*} b. substance \begin{align*}B\end{align*}
  7. \begin{align*}1.2 \;\mathrm{g}\end{align*}
  8. \begin{align*}125 \;\mathrm{g}\end{align*}
  9. \begin{align*}0.46\end{align*} minutes
  10. The one with the short half life, because half life is the rate of decay.

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