Half Lives

• Equations for half lives
• Determining a half life
• Converting a half life to a rate constant
• Graphical relations and half lives

Equations for Half Lives

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

t_½ =  [A_o] / 2k

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

t_½ = 0.693 / k

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

t_½ = 1 / k [A_o]

Determining a Half Life

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:

• The order of the reaction or enough information to determine it.
• The rate constant, k, for the reaction or enough information to determine it.
• In some cases, we need to know the initial concentration, [A_o]

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:

• The half life of the reaction, t_½.
• The order of the reaction or enough information to determine it.
• In some cases, we need to know the initial concentration, [A_o]

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 (Half life decreases with decreasing concentration.)
• For a 1st order reaction (Half life is constant.)
• For a second order reaction (Half life increases with decreasing concentration.)

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:

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