Using Integrated Rate Laws

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.

• The common integrated rate laws
• How long does it take to react?
• What concentration remains after a given time?
• What was the initial concentration?

The Common Integrated Rate Laws

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

The integrated rate law is   [A] = -kt + [A_o]

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

The integrated rate law is   ln [A] = -kt + ln [A_o]

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

 The integrated rate law is  1/[A] = kt + 1/[A_o]

How Long Does it Take?

To determine t, the time required for the initial concentration of a reactant to be reduced to some final value, we need to know:

• The initial concentration, [A_o].
• The final concentration, [A].
• The order of the reaction or enough information to determine it.
• The rate constant, k, for the reaction or enough information to determine it.

Substitute this information into the integrated rate law for a reaction with this order and solve for t. The integrated rate laws are given above.

How Much Remains After a Given Time?

To determine [A], the concentration of a reactant remaining after some time, t, we need to know:

• The initial concentration, [A_o].
• The length of time the reaction ran, t.
• The order of the reaction or enough information to determine it.
• The rate constant, k, for the reaction or enough information to determine it.

Substitute this information into the integrated rate law for a reaction with this order and solve for [A]. The integrated rate laws are given above.

What Concentration Was Present Initially?

To determine [A_o], the initial concentration of a reactant, we need to know:

• The final concentration, [A].
• The length of time, t, the reaction ran to reach the final concentration.
• The order of the reaction or enough information to determine it.
• The rate constant, k, for the reaction or enough information to determine it.

Substitute this information into the integrated rate law for a reaction with this order and solve the equation for [A_o]. The integrated rate laws are given above.