Converting A Half Life To A Rate Constant

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
• 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:

t½ = [Ao] / 2k

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

t½ = 0.693 / k

For a second order reaction 2A products or A + B 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:

• 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, [Ao]

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:

• The half life of the reaction, t½.
• The order of the reaction or enough information to determine it.
• In some cases, we need to know the initial concentration, [Ao]

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 (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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