Kinetics :  Integrated Forms of the First-Order 
                 and Second-Order Rate Laws
Integrated Form of the First-Order Rate Law

The original first-order rate law equation is:

The integrated form of the first-order rate law equation is:

Where X is the concentration of a reactant at any moment in time, (X)o is the initial concentration of this reactant, k is the constant for the reaction, and t is the time since the reaction started.
This equation is useful in calculating how much of a substance remains after a certain amount of time has passed, or to calculate how long it takes until the concentration is at a certain point.
First-Order Reaction

[Image]

  If the rate law of a reaction is first order
with respect to [A], then the graph of ln[A] 
versus time (t) creates a straight line with a
negative slope.  The value of the slope of the
line is equal to the negative value of the 
rate constant (k).
Second-Order Reaction

[Image]

  If the rate law for a reaction is second order
with respect to [A], a graph of 1/[A] versus 
time (t) creates a straight line with a positive
slope.  The value of the slope of the line is
equal to the value of the rate constant (k).