Half Lives (2024)

Half Lives

We use integrated rate laws, and rate constants torelate concentrations and time. The rate law to use depends on theoverall order of the reaction.

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

Equationsfor Half Lives

For a zero order reactionAproducts , rate = k:

t_½ = [A_o] /2k

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

t_½ = 0.693 / k

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

t_½ = 1 / k [A_o]

Determininga Half Life

To determine a half life, t_½, the time required forthe 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 determineit.
• In some cases, we need to know the initial concentration, [A_o]

Substitute this information into the equation for the half life of a reactionwith this order and solve for t_½. The equationsare given above.

Convertinga 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 reactionwith this order and solve for k. The equations are given above.

GraphicalRelations and Half Lives

If we plot the concentration of a reactant versustime, we can see the differences in half lives for reactions of differentorders in the graphs. We can identify a 0, 1^st, or 2^nd order reactionfrom a plot of [A] versus t by the variation in the time it takes the concentrationof a reactant to change by half.

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

For a zero order reactionAproducts , rate = k:

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

For a second order reaction2Aproducts or A + Bproducts (when [A] = [B]),rate = k[A]^2:

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