Integrated Forms of the First-Order and Second-Order Rate Laws

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).

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

[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).

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