Half Lives
We use integrated rate laws, and rate constants to relate concentrations and time. The rate law to use depends on the overall order of the reaction.
For a zero order reaction A products , rate = k:
t_{½} = [A_{o}] / 2kFor a first order reaction A products , rate = k[A]:
t_{½} = 0.693 / kFor a second order reaction 2A products or A + B products (when [A] = [B]), rate = k[A]^{2}:
t_{½} = 1 / k [A_{o}]
To determine a half life, t_{½}, the time required for the initial concentration of a reactant to be reduced to one-half its initial value, we need to know:
Substitute this information into the equation for the half life of a reaction with this order and solve for t_{½}. The equations are given above.
Converting a Half Life to a Rate Constant
To convert a half life to a rate constant we need to know:
Substitute this information into the equation for the half life of a reaction with this order and solve for k. The equations are given above.
Graphical Relations and Half Lives
If we plot the concentration of a reactant versus time, we can see the differences in half lives for reactions of different orders in the graphs. We can identify a 0, 1^{st}, or 2^{nd} order reaction from a plot of [A] versus t by the variation in the time it takes the concentration of a reactant to change by half.
For a zero order reaction A products , rate = k:
For a first order reaction A products , rate = k[A]:
For a second order reaction 2A products or A + B products (when [A] = [B]), rate = k[A]^{2:}