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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.

Equations for Half Lives

For a zero order reaction  AArrow products   ,    rate =  k:

t½ =  [Ao] / 2k
For a first order reaction      AArrow products  ,   rate =  k[A]:
t½ = 0.693 / k
For a second order reaction   2AArrow products   or   A + BArrow products   (when [A] = [B]),      rate = k[A]2:
t½ = 1 / k [Ao]

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Determining a Half Life

To determine a half life, t½, 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 t½. The equations are given above.

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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.

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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, 1st, or 2nd 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      AArrow products   ,    rate =  k:

A plot of concentration of reactant versus time is a straight line for a zero order reaction.  The half life is greater when the concentration is greater.

For a first order reaction      AArrow products  ,   rate =  k[A]:

The half life of a first order reaction is independent of concentration.

For a second order reaction   2AArrow products   or   A + BArrow products   (when [A] = [B]),        rate = k[A]2:

The half life of a second order reaction increases as the concentraion decreases.

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