First-Order/Pseudo-First-Order ReactionGary L. Bertrand Professor Emeritus of Chemistry Missouri University of Science and Technology (formerly University of Missouri-Rolla) background close A First-Order Reaction is described mathematically as: _{t} = [A]_{t=0}e^{-νAkt}ln ([A] _{t}) = ln ([A]_{t=0}) - ν_{A}kt For a Pseudo-First-Order Reaction, the reaction rate constant k is replaced by the apparent reaction rate constant k'. If the reaction is not written out specifically to show a value of ν_{A}, the value is assumed to be 1 and is not shown in these equations.The simplest way to confirm that these are the proper equations to describe the reaction and to get an approximate value for k or k', is with the method of half-lives. The half-life t is defined as the time required for the initial concentration to be halved:_{1/2}_{t1/2} = [A]_{t=0}/2 ._{1/2} = 0.693 .The time required for the concentration to be reduced to 1/4 of the initial concentration ( t) is _{3/4}_{3/4} = 2(0.693) .In a First-Order Reaction, the concentration of the reactant remaining after x half-lives is:_{x}/[A]_{t=0} = (1/2)^{x} .This relationship is unique to First-Order and Pseudo-First-Order Reactions. This may be confirmed either by following concentration in time for a single series of measurements over three or more half-lives, or by comparing the half-lives of different series of measurements in which the initial concentration is varied by a factor of 4 or more. The reaction rate constant may then be calculated from the "best" value of the half-life: _{1/2}NOTE:The "First-Order Kinetics" exercise is designed so that you can get an accurate value of the half-life. Scan the concentration column for a value slightly larger than half of the initial concentration. Adjust the value of "Start Time" and the value of "Interval" to get a better estimate of the half-life. The same procedure may be used to find the time for the second and third half-lives.The best values of the reaction rate constant ( k) can be obtained with data taken in the middle third of the reaction (from [A] to _{t} = (2/3)[A]_{t=0}[A]). Linear Least Squares regression with _{t} = (1/3)[A]_{t=0}Y = ln([A] and _{t})X = t gives - k or - k' as the slope. |