HalfLife
The halflife of a reaction is the amount of time it takes for the concentration of a reactant to decrease to onehalf of its initial value.
Learning Objective

Distinguish between firstorder, secondorder, and zeroorder halflife equations
Key Points
 The halflife equation for a firstorder reaction is
$t_{\frac{1}{2}}=\frac{ln(2)}{k}$ .  The halflife equation for a secondorder reaction is
$t_{\frac{1}{2}}=\frac{1}{k[A]_{0}}$ .  The halflife equation for a zeroorder reaction is
$t_{\frac{1}{2}}=\frac{[A]_{0}}{2k}$ .
Term

halflife
The time required for a quantity to fall to half its value as measured at the beginning of the time period.
Full Text
The halflife is the time required for a quantity to fall to half its initial value, as measured at the beginning of the time period. If we know the integrated rate laws, we can determine the halflives for first, second, and zeroorder reactions. For this discussion, we will focus on reactions with a single reactant.
Halflife
The halflife of a reaction is the amount of time it takes for it to become half its quantity.
HalfLife of a FirstOrder Reaction
Recall that for a firstorder reaction, the integrated rate law is given by:
This can be written another way, equivalently:
If we are interested in finding the halflife for this reaction, then we need to solve for the time at which the concentration, [A], is equal to half of what it was initially; that is,
By rearranging this equation and using the properties of logarithms, we can find that, for a first order reaction:
What is interesting about this equation is that it tells us that the halflife of a firstorder reaction does not depend on how much material we have at the start. It takes exactly the same amount of time for the reaction to proceed from all of the starting material to half of the starting material as it does to proceed from half of the starting material to onefourth of the starting material. In each case, we halve the remaining material in a time equal to the constant halflife. Keep in mind that these conclusions are only valid for firstorder reactions.
Consider, for example, a firstorder reaction that has a rate constant of 5.00 s^{1}. To find the halflife of the reaction, we would simply plug 5.00 s^{1} in for k:
HalfLife for SecondOrder Reactions
Recall our integrated rate law for a secondorder reaction:
To find the halflife, we once again plug in
Solving for t, we get:
Thus the halflife of a secondorder reaction, unlike the halflife for a firstorder reaction, does depend upon the initial concentration of A. Specifically, there is an inversely proportional relationship between
Consider, for example, a secondorder reaction with a rate constant of 3 M^{1} s^{1} in which the initial concentration of A is 0.5 M:
HalfLife for a ZeroOrder Reaction
The integrated rate law for a zeroorder reaction is given by:
Subbing in
Rearranging in terms of t, we can obtain an expression for the halflife:
Therefore, for a zeroorder reaction, halflife and initial concentration are directly proportional. As initial concentration increases, the halflife for the reaction gets longer and longer.
Key Term Reference
 Rate law
 Appears in these related concepts: ZeroOrder Reactions, Chemical Kinetics and Chemical Equilibrium, and The Rate Law
 concentration
 Appears in these related concepts: Calculating Equilibrium Concentrations , Diffusion, and Molarity
 firstorder reaction
 Appears in these related concepts: Experimental Determination of Reaction Rates, FirstOrder Reactions, and The Integrated Rate Law
 logarithm
 Appears in these related concepts: Logarithms of Powers, Changing Logarithmic Bases, and The Number e
 period
 Appears in these related concepts: Frequency of Sound Waves, Sine and Cosine as Functions, and Tangent as a Function
 reactant
 Appears in these related concepts: The Law of Conservation of Mass, Chemical Reactions and Molecules, and Writing Chemical Equations
 secondorder reaction
 Appears in this related concept: SecondOrder Reactions
 zeroorder reaction
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