Reactions occur at many different rates. Some reactions in geological
processes occur at imperceptably slow speeds where product change takes
place in intervals of years or even decades. Other reactions occur so rapidly
that they are measured in nanoseconds. Most reactions fall somewhere in
the middle. For example the combustion of a hydrocarbon occurs fairly rapidly

whereas the hydrolysis of the bismuth(III) ion in a chloride containing
solution does not produce the white BiOCl for several minutes in some cases
depending upon a number of factors.

**Reaction Rates**

** Instantaneous Rates** are the change in product formation
at a moment in time. If we plot the concentration of a product forming
against time we will get a curve. The tangental slope at any place on that
curve would be the instantaneous rate at that moment in time. The tangental
slope can be determined by drawing a straight line that just touches the
concentration vs time curve at the moment in time of which you wish to
determine the instantaneous rate. It we draw such a line then we
can determine the slope of the tangental line.

Rate = -[Reactant] /t or [Product] /t

To relate the rate with respect to different reactants or products, stoichiometry is used. For example, for reaction:

2 A 3 B

2[B] /t = -3[A] /t

This slope value is the instantaneous rate at the time indicated at
the point which the tangental line touches the concentration curve. The
best instantaneous rate is the rate at t = 0, or the ** initial rate**.

Reaction rates can be affected in a number of ways. The four factors are:

- Concentration
- Phase of the reactants
- Temperature
- The presence of a catalyst

aA + bB + cC... dD + eE + fF....

is usually:

rate = *k*[A]^{x }[B]* ^{y }*[C]

where *k* is the ** rate constant**, [ ] is the molarity
of the reactant, and

*Method of Initial Rates*

For a reaction:

a set of experiments are performed, with different initial conditions.
The experiments are designed to investigate the concentration effect of
one of the reactants, while keeping the others constant.

Experiment | [BrO_{3}^{-}] (M) |
[Br^{-}] (M) |
[H^{+}] (M) |
Initial Rate (M/s) |
---|---|---|---|---|

1 | 0.10 | 0.10 | 0.10 | 8.0 x 10 ^{-4} |

2 | 0.20 | 0.10 | 0.10 | 1.6 x 10 ^{-3} |

3 | 0.20 | 0.20 | 0.10 | 3.2 x 10 ^{-3} |

4 | 0.10 | 0.10 | 0.20 | 3.2 x 10 ^{-3} |

In experiment 1 and experiment 2, Br^{-} has a a concentration
of 0.10 M and H^{+} has a concentration of 0.10 M. Now look at
the concentrations of BrO_{3}^{-} and decide how it was
changed. The BrO_{3}^{-} concentration is doubled. Then
you have to compare it to the quotient of the initial rates. So:

**1.6 x 10 ^{-3} / 8.0 x 10 ^{-4} = 2
times faster**

In both experiment 2 and experiment 3, BrO_{3}^{-} has
a concentration of 0.20 M and H^{+} has a concentration of 0.10
M. Now look at the concentration of Br^{-} and decide how it was
changed and what effect it had on the reaction rate. The concentration_{1}
was doubled, and the rate doubled. This is found by taking the quotient
of the initial rates:

**3.2 x 10 ^{-3} / 1.6 x 10 ^{-3} = 2
times faster**

Finally, In both experiment 1 and experiment 4, BrO_{3}^{-}
has a concentration of 0.10 M and Br^{-} has a concentration of
0.10 M. Now look at the concentrations of H^{+} and decide how
it was changed and what effect it had on the reaction rate. The concentration_{1}
was doubled, and the rate goes up by four times. This is found by
taking the quotient of the initial rates:

**3.2 x 10^{-3} / 8.0 x 10^{-4} = 4 times
faster**

The rate law for the reaction, therefore is:

Rate = *k*[BrO_{3}^{-}]^{1} [Br^{-}]^{1}
[H^{+}]^{2} or rate = *k*[BrO_{3}^{-}]
[Br^{-}] [H^{+}]^{2}

*Integrated Form of the Rate Law*

If the rate law for a reaction is known to be of the form

rate = *k*[A]^{n}

where *n* is either zero, one or two, **and** the reaction depends
(or can be made to depend) on one species **and** if the reaction is
well behaved, the order of the recation can be determined graphically.

Order |
Rate Law |
Integrated Rate Equation |
Linear Plot |
Slope |
Units for k |

0 |
rate = k |
[A]_{0} - [A]_{t} = kt |
[A] vs time | -k | M / s |

1 |
rate = k [A] |
ln ([A]_{t}/[A]_{0}) = - kt |
ln [A] vs time | -k |
s^{-1} |

2 |
rate = k [A]^{2} |
(1/[A]_{t}) - (1/[A]_{0}) = kt |
(1/[A]) vs time | k | M^{-1} s^{-1} |

The following data were collected for a hypothetical reaction whose
rate is known to only depend on the concentration of a single reactant,
A. The order of the rate law with respect to A is unknown.

Time (s) | [A] (M) |

0 | 1.00 |

5 | 0.82 |

10 | 0.67 |

15 | 0.55 |

20 | 0.45 |

40 | 0.20 |

60 | 0.09 |

This data can be used to determine the order of the reaction. We assume that the reactions is either zeroth, first or second order.

**Phase of The Reactants**

Reactions produce products by having the reacting molecules come into
contact with one another. The more often they collide, the more likely
the chance that product will form. If the reacting molecules moving more
rapidly and in the gaseous state then product will have a more likely chance
to form. This is part of an over riding theory that forms the foundation
of all kinetics work. This theory is called the ** Collisional Theory
of Reaction Rates**. Reactions usually occur more rapidly when the
reactants are in the gaseous state. The reacting molecules dispersed in
a solution is the next most favorable way for product to form at a reasonable
speed. Reactions do occur in pure liquids or in solid form but the rates
tend to be rather slow because the reacting molecules are very restricted
in their movement among one another, and therefore, do not come into contact
as often. The relative rates are roughly in this manner:

gases > solutions > pure liquids > solids

**Temperature**

The ** activation energy** of a reaction is the amount of energy
needed to start the reaction. It represents the minimum energy

needed to form an activated complex during a collision between reactants. In slow reactions the fraction of molecules in the

system moving fast enough to form an activated complex when a collision occurs is low so that most collisions do not produce a

reaction. However, in a fast reaction the fraction is high so that most collisions produce a reaction. For a given reaction the rate

constant, k, is related to the temperature of the system by what is known as the Arrhenius equation:

*k *= Ae^{-Ea/RT}

where R is the ideal gas constant (8.314 J/mole-K), T is the temperature
in Kelvin, E_{a} is the activation energy in

joules/mole, and A is a constant called the frequency factor; which
is related to the fraction of collisions between reactants_{1}

having the proper orientation to form an activated complex. The
exponential equation can be converted to the linear form by taking the
logarithm of both sides:_{1}

ln*k* = -E_{a} /RT + lnA

Thus, a plot of ln*k* vs. 1/T is a straight line which can be used
to determine the activation energy of the reaction from the slope.
The rate constant at different temperatures can then be evaluated.

For example, in the hydrolysis of 2-chloro-2-methylpropane:

the following data was obtained:

T (K) |
303 | 298 | 293 | 288 | 283 |

k (x10^{2}) |
9.41 | 3.59 | 3.08 | 2.44 | 1.82 |

A plot gives the following results: