First Order ReactionA general unimolecular reaction
where
A is a reactant and
P is a product is called a first-order reaction.
The rate is proportional to the concentration of a single reactant raised to the first power. The decrease in the concentration of
A over time can be written as:
Equation
(2) represents the differential form of the rate law. Integration of this equation and determination of the integration constant
C produces the corresponding integrated law.
Integrating equation
(2) yields:
The constant of integration
C can be evaluated by using boundary conditions. When
t = 0,
[A] = [A]0.
[A]0 is the original concentration of
A.
Substituting into equation
(3) gives:
Therefore the value of the constant of integration is:
Substituting
(5) into
(4) leads to:
Plotting ln[A] or ln[A] / [A]0 against time
creates a straight line with slope -k. The plot
should be linear up to a conversion of about
90%.
Equation
(6) can also be written as:
This means that the concentration of
A decreases exponentially as a function of time.
The rate constant
k can also be determined from the half-life
t1/2. Half-life is the time it takes for the concentration to fall from
[A]0 to
[A]0 / 2.
According to equation
(6) is obtained:
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