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Wize University Chemistry Textbook > Kinetics

Half-lives for Various Orders

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Half Life for Various Orders


Another important concept in kinetics is half-life, t1/2. This is the time it takes for the concentration of a reactant to be cut in half.
Att1/2:[A]=[A]02At \hspace{3pt} t_{1/2}:[A]=\cfrac{[A]_0}{2}
  • Although it is fairly easy to derive the half-life equation from the integrated rate law we do not have to because it is given on the formula sheet and shown below.
  • If the stoichiometric coefficient of A, a, is not 1 we must include it with k in these equations just like the integrated rate laws!
t1/2=[A]02k0tht_{1/2} =\cfrac{[A]_0}{2k} \hspace{20pt} 0^{th}
t1/2=ln⁡2k1stt_{1/2} =\cfrac{\ln 2}{k} \hspace{20pt} 1^{st}
t1/2=1k[A]02ndt_{1/2} =\cfrac{1}{k[A]_0} \hspace{20pt} 2^{nd}
  • we can see that for 1st order reactions, the half-life is independent of the initial concentration
  • for radiocarbon dating, we also use first order kinetics, and can use another equation:
[At][A0]=0.5tt12\frac{\left[At\right]}{\left[A0\right]}=0.5^{\frac{t}{\frac{t1}{2}}}
[At]=concentration of substance at a particular time
[A0]=initial concentration of substance
t=how much time has passed
t1/2=half life

  • note: if you use this equation, both t and t1/2 must have the same units!
  • for 2nd order, t is inversely proportional to the initial concentration of the reactant
  • and for the n order, t1/2 directly proportional to [A0]1-n
Americium-241 is a synthetic isotope made in nuclear reactors around the world. Chances are you have been in close proximity to this radioactive isotope thousands of time in your life because it is used in household smoke detectors. For a smoke detector to be functioning properly it needs more than 22mg of Americium-241. 241Am is an alpha emitter whose decay obeys a first order rate-law (as do all nuclear decays) with a half-life of 432.6 years. If a typical smoke detector contains 25 mg of 241Am, how long will the smoke detector last assuming the batteries are replaced regularly?

Americium-241 is a synthetic isotope made in nuclear reactors around the world. Chances are you have been in close proximity to this radioactive isotope thousands of time in your life because it is used in household smoke detectors. 241Am^{241}\rm Am is an alpha emitter whose decay obeys a first order rate-law (as do all nuclear decays) with a half-life of 432.6 years. What is the rate constant of its decay? How much 241Am^{241}\rm Am would remain of a 25mg sample if it were left for 300 years.


Extra Practice
The Rate Law: Half Life
What is the half‐life of a reaction that has k = 0.254 M/s and [A]0 of 0.254 M.
The Rate Law: Half life
Given a second order reaction, derive the expression for the half life:
The integrated rate law for the second order reaction is:
1[A]t=kt +1[A]0\frac{1}{\left[A\right]_t}=kt\ +\frac{1}{\left[A\right]_0}
A reaction, A →\rightarrow is performed in a lab and the concentration of the reactant A is plotted as a function of time.
What is the half life?
A first order gas phase reaction N2O4 → 2 NO2 was carried out isothermally and the activation barrier was found to be 23.5 kJ/mol. Initially p(N2O4) = 2.6 atm and p(NO2) = 0. After 1 hour p(NO2) = 2.0 atm and p(N2O4) =1.6 atm. Assume that both NO2 and N2O4 are ideal gases.
Determine the first order rate constant k, and the half life t1/2.
The above reaction was performed at 220oC. Calculate the pre-exponential factor.
Practice: Using Integrated Rate Laws
Americium-241 is a synthetic isotope made in nuclear reactors around the world. Chances are you have been in close proximity to this radioactive isotope thousands of times in your life because it is used in household smoke detectors. 241Am is an alpha emitter whose decay obeys a first order rate-law (as do all nuclear decays) with a half-life of 432.6 years.