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 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).
The equation for the half-life of a substance is derived from this equation.
Half-life - The length of time it takes for exactly half of the nuclei of a radioactive sample to decay.

Integrated Form of the Second-Order Rate Law

The original equation for a second-order rate law with a single reactant is:

The integrated form of the second-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.
 Second-Order Reaction 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).