Chemistry 112

Half Life


The half-life of a reaction is defined as the time required for the initial concentration of a reactant to be decreased by 50%.

This is not the time for half of the reaction to occur!

Half-life is denoted by t½

The half-life is related to the rate constant; thus, if we know the half-life we can also find the rate constant.

The half-life also depends upon the order of the reaction.

Zero Order Reaction

at t½ [A] = ½[A]0

[A] = –kt + [A]0

so

½[A]0 = –kt½ + [A] 0


Notice that the half-life depends upon the initial concentration of reactant in this case.


First Order Reaction



In this case the half-life is independent of the initial concentration of reactant. Half-lives for first order reactions are equivalent to knowing the rate constant.


Second Order Reaction



Again, the half-life depends upon the initial concentration of reactant.




Effect of Temperature on Reaction Rates


The orders of reaction are independent of temperature - orders change only when the reaction changes.

Thus, the entire temperature dependence of a reaction, as expressed in a rate law, is found in the rate constant, k.

Think about a typical rate law: Rate = –k[A]m

The rate depends on how much is there (the concentration term, [A] m) and how effectively what is there can react, k.

In this interpretation, k is a probability factor!


Collision Theory

In Collision Theory, three things must happen for a reaction to occur:

Reacting molecules must encounter each other (i.e., a collision must happen).

The orientation of the collision must be correct for any atom or electron transfers.

There must be enough energy present in the collision for the reaction to occur.

Each of these events has a probability associated with it. The rate constant is found from the product of each individual probability.

Number of Collisions:

Z = collision frequency

From the kinetic theory of gases the collision frequency is found to be proportional to the square root of absolute temperature:
Z T½


Orientation of collisions:

The structure of molecules influences the outcome of reactions. The correct parts of molecules must "touch" during a collision so the correct bonds can be broken and remade into the new molecule.

Consider the example:

Cl(g) + O3(g) ClO(g) + O2(g)

This reaction is important in the upper atmosphere.

The rate of reaction depends upon the orientation of the collision between the chlorine atoms and the ozone:

If the orientation is wrong, no reaction occurs.

With the correct orientation, reaction can proceed.

There is no temperature dependence of the orientation of collisions.

Energy:

For reaction to occur, collisions must have enough energy present. Not all collisions meet this criterion, so not all collisions lead to reaction.

The temperature dependence of the energy term is found to be proportional to an exponential factor:

where Ea is called the activation energy, R is the gas constant (in energy units), and T is the absolute temperature. Ea defines the amount of energy required in a collision to "activate" the reaction

The total probability of a reaction is given by:

kZfp

where

Z is the collision frequency,

f is the fraction of molecules with the activation energy, and

p is the probability that a collision occurs with the correct orientation.

This leads to the temperature dependence of the rate constant:



Transition State Theory

Transition State Theory extends Collision Theory by ascribing structural details to collisions.

When molecules collide, a new "molecule" is formed, at least temporarily. This new molecule is called an activated complex. The activated complex is a high-energy molecule and this energy can be redistributed within the molecule. Breaking bonds, changing bond angles, and making new bonds accomplishes the energy redistribution.

When the activated complex falls apart, it may return to the same geometry as reactants or it may fall apart into a new geometry: the products.

Graphically, this can be shown using a potential energy diagram:

In this example, the enthalpy of reaction is positive (i.e., the reaction is endothermic).

The activation energy is always positive.

Catalysts work by becoming part of the activated complex and lowering E a. This speeds the reaction up.