Chemical Kinetics - Reaction Rates

Chemical kinetics is the branch of chemistry which addresses the question: "how fast do reactions go?" Chemistry can be thought of, at the simplest level, as the science that concerns itself with making new substances from other substances. Or, one could say, chemistry is taking molecules apart and putting the atoms and fragments back together to form new molecules. (OK, so once in a while one uses atoms or gets atoms, but that doesn't change the argument.) All of this is to say that chemical reactions are the core of chemistry.

If Chemistry is making new substances out of old substances (i.e., chemical reactions), then there are two basic questions that must be answered:

1. Does the reaction want to go? This is the subject of chemical thermodynamics.

2. If the reaction wants to go, how fast will it go? This is the subject of chemical kinetics.

Here are some examples. Consider the reaction,
2 H2(g) + O2(g) → 2 H2O(l).
We can calculate ΔrGo for this reaction from tables of free energies of formation (actually this one is just twice the free energy of formation of liquid water). We find that ΔrGo for this reaction is very large and negative, which means that the reaction wants to go very strongly. A more scientific way to say this would be to say that the equilibrium constant for this reaction is very very large.

However, we can mix hydrogen gas and oxygen gas together in a bulb or other container, even in their correct stoichiometric proportions, and they will stay there for centuries, perhaps even forever, without reacting. (If we drop in a catalyst - say a tiny piece of platinum - or introduce a spark, or even illuminate the mixture with sufficiently high frequency uv light, or compress and heat the mixture, the mixture will explode.) The problem is not that the reactants do not want to form the products, they do, but they cannot find a "pathway" to get from reactants to products.

Another example: consider the reaction,

C(diamond) → C(graphite).
If you calculate ΔrGo for this reaction from data in the tables of thermodynamic properties you will find once again that it is negative (not very large, but still negative). This result tells us that diamonds are thermodynamically unstable. Yet diamonds are highly regarded as gem stones ("diamonds are forever") and are considered by some financial advisors as a good long-term investment hedge against inflation. On the other hand, if you were to vaporize a diamond in a furnace, under an inert atmosphere, and then condense the vapor, the carbon would come back as graphite and not as diamond.

How can all these things be?

The answer is that thermodynamics is not the whole story in chemistry. Not only do we have to know whether a reaction is thermodynamically favored, we also have to know whether the reaction can or will proceed at a finite rate. The study of the rate of reactions is called chemical kinetics.

The study of chemical kinetics requires new definitions, new types of experimental data, and new theories and equations to organize the data. We begin with the definition of reaction rate.
 

Reaction Rates

Consider the reaction,

2 NO(g) + O2(g) → 2 NO2(g).
We can specify the rate of this reaction by telling the rate of change of the partial pressures of one the gases. However, it is convenient to convert these pressures into concentrations, so we will write our rates and rate equations in terms of concentrations, where square brackets, [ ], mean concentration in mol/L.

We might try to write the rate variously as,

or as
but these are not the same because each molecule of O2 gives two molecules of NO2. To arrive at an unambiguous definition of reaction rate we define the "reaction velocity," v, as
                (1)
This is unambiguous. The negative sign tells us that that species is being consumed and the fractions take care of the stoichiometry.  Any one of the three derivatives can be used to define the rate of the reaction.

For a general reaction,

aA + bB → cC + dD,                (2)
the reaction velocity can be written in a number of different but equivalent ways,
                (3)
As in our previous example, the negative signs account for material that is being consumed in the reaction and the positive signs account for material that is being formed in the reaction. The stoichiometry is preserved by dividing the rate of change of concentration of each substance by its stoichiometric coefficient.
 

Rate Laws

A rate law is an equation that tells us how fast the reaction proceeds and how the reaction rate depends on the concentrations of the chemical species involved. A rate law is an equation of the form,

                (4)
Equation 4 is gives us a first order differential equation in t because the reaction velocity is related to a time-derivative of one of the concentrations (as in Equation 3).

The rate law may contain substances which are not in the balanced reaction and may not contain some things that are in the balanced equation (even on the reactant side).

Usually rate laws take the form,

                (5)
where x, y, z, are small whole numbers or simple fractions and k is called the "rate constant." The sum of x + y + z + . . . is called the "order" of the reaction.
 

Common types of rate laws:

1. First Order Reactions

In a first order reaction the rate is proportional to the concentration of one of the reactants. That is,

v = rate = k[B],                (6)
where B is a reactant. If we have a reaction which is known to be first order in B, such as
B + other reactants → products,
we would write the rate law as,
                (7)
The constant, k, in this rate equation is the first order rate constant.
 

2. Second Order Reactions

In a second order reaction the rate is proportional to concentration squared. For example, possible second order rate laws might be written as

Rate = k[B]2                (8)
or as
Rate = k[A][B].                (9)
That is, the rate might be proportional to the square of the concentration of one of the reactants, or it might be proportional to the product of two different concentrations.
 

3. Third Order Reactions

There are several different ways to write a rate law for a third order reaction. One might have cases where

Rate = k[A]3,           (10)
or
Rate = k[A]2[B],           (11)
or
Rate = k[A][B][C],                (12)
and so on.

We will see later that there are other, more "interesting" rate laws in nature, but a large fraction of rate laws will fit in one of the above categories.
 

Integrated forms of rate laws

In order to understand how the concentrations of the species in a chemical reaction change with time it is necessary to integrate the rate law (which is given as the time-derivative of one of the concentrations) to find out how the concentrations change over time.

1. First Order Reactions

Suppose we have a first order reaction of the form,

B + . . . . → products.                (13)
Then we can write the rate law and integrate it as follows (recall that the derivative is negative because the concentration of the reactant, B, is decreasing):
                (14a, b, c, d)
The first order rate law is a very important rate law, radioactive decay and many chemical reactions follow this rate law and some of the language of kinetics comes from this law. The form of Equation 14d is called an "exponential decay." This form appears in many places in nature. One of its consequences is that it gives rise to a concept called "half-life."
 

Half-life

The half-life, usually symbolized by t1/2, is the time required for [B] to drop from its initial value [B]o to [B]o/2.

Using the integrated form of the first order rate law we find that

                (15a, b)
Taking the logarithm of both sides gives,
                (16a, b)
or
                (17)
(You can also write
                (18)
which may actually give a little more insight into what is meant by half-life. This equation demonstrates clearly that the concentration drops by a factor of two for every t1/2 increment in time.)

For first order processes it is common to define a "relaxation time." τ , by

                (19)
so that one can write the integrated form of the rate law as
                 (20)
τ is the time required for [B] to drop from [B]o to [B]o/e. Sometimes τ is called the "one over e" time. Although the half-life is almost always used to describe the decay rate of radioactive elements, it is common for chemists to talk about the rate of first order processes in chemistry in terms of the relaxation time.

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Copyright 2004, W. R. Salzman
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Last updated 4 Nov 04
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