For the generic reaction “A ⟶ products”, the integrated rate law:
for when the reaction is first order with respect to [A], that is when m = 1, is:
This integrated rate law for a first-order reaction can be alternatively expressed as:
It is easier to use this form of the equation when trying to calculate the time required for a reaction to proceed to a certain extent.
On the other hand, if you raise e (the base of the natural logarithm system) to the power of each side of the equation, it gives:
It is easier to use this form of the equation when trying to determine the concentration of reactant remaining after a certain period of time.
The integrated rate law for a first-order reaction can be rearranged to have a standard linear equation format:
Hence, if a reaction is first order in [A], a plot of “ln[A]t vs. t” must give a straight line. The slope of such a plot would be −k and the y-intercept would correspond to ln[A]0. If the plot is not a straight line, the reaction is not first order with respect to [A].