Lex XXIV

Chem 1200

Angel C. de Dios

Kinetics II

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For example in a reaction A --> B, one may be able to monitor either the amount of A or the amount of B present as a function of time:

These two curves are related to each other because of stoichiometry. Remember, by stoichiometry the rate at which the concentration of a product changes is directly related to the rate at which the concentration of the reactant changes. It is possible to convert the amount of A left to the the amount of B formed, and vice-versa.

Below is an example of how the rate of a reaction can be measured:

As mentioned before, the rate of a reaction can be influenced by the concentration of the reactants.  Hence, rarely do we find the concentration or amount of reactant changing linearly with time.  As the reaction progresses, the reactant gets consumed, therefore its concentration or amount is changing with time.  This can induce a change in the rate of the reaction.  Oftentimes, we see the reaction slowing down as time progresses.  Thus, it is useful to describe an instantaneous rate of reaction (one that is specified at a given point in time).  For this reason, rates are usually defined with differentials which signify infinitesimal (very small) changes.  The graphical description of an instantaneous rate is given below:

When the reaction is occurring in solution, it is sometimes more convenient to express quantities in terms of concentrations (provided that the volume of the reaction mixture remains constant).  Therefore, in these cases, rates are expressed in terms of units of concentration per units of time, as in Molarity per second.  When the reaction is occurring in the gas phase, partial pressures can also be used, so do not be surprised to see rates in units of pressure per units of time.

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In kinetics, we must first measure the rate of a reaction. And we accomplish this by monitoring the change in concentration of the reactants, how fast are these disappearing, or by measuring the change in concentration of the products, how fast are these appearing. Only after a measurement can we then examine factors that can possibly influence how fast a reaction occurs. Since reactions require an encounter between molecules, it is reasonable to suspect that the rate of a reaction depends on the concentration of the reactants. As the concentration rises so does the chance of reactant molecules or ions colliding against each other. So we need to observe the rate of a reaction at different concentrations. How a rate of a reaction experimentally depends on concentration is summarized in what we call a rate law.

This reaction is third-order overall, second-order in A, first-order in B, and zero-order in C.

Dependence of Rate of Reaction on Concentration -  The Rate Law There are generally two ways of determining the rate law:

Methods of Initial rates -  Measuring the instantaneous rate of a reaction at time=0 several times with various starting concentrations of the reactants.  Here is an example: Rate Data for the reaction of Ammonium and Nitrite Ions in Water at 298 K. 

Experiment  number	Initial NH4+ concentration, M	Initial NO2- concentration, M	Observed Initial  rate, 10-7 M/s
1	0.0100	0.200	5.4
2	0.0200	0.200	10.8
3	0.0400	0.200	21.5
4	0.0600	0.200	32.3
5	0.200	0.0202	10.8
6	0.200	0.0404	21.6
7	0.200	0.0606	32.4
8	0.200	0.0808	43.3

By comparing two experiments (for example 1 and 2) in which only one of the reactant's concentration is varied, in this particular case, the nitrite concentration is maintained while the ammonium concentration is doubled.  The result is the rate for experiment 2 is twice as fast as that of experiment 1, indicating that this reaction depends linearly on the concentration of ammonium (first order with respect to ammonium).

The exponent at which we raise the concentration of a reactant defines the order of the reaction with respect  to that reactant.  Thus, in general, if a rate law is given by:

rate = k[A]^x[B]^ythen the reaction is x order with respect to [A], y order with respect to [B], and overall x+y order.  The proportionality constant k is called the rate constant.  As we will see later, the temperature dependence of the rate of a reaction manifests in k.  In this course, we will stay within simple rate laws.  No doubt, there are much more complicated rate laws (where concentration terms are sometimes in the denominator (that is, exponents are negative) or exponents are non integers).

 

Here is another example:
 

Experiment Number	[A], M	[B], M	Intial Rate, 10-5 M/s
1	0.100	0.100	4.0
2	0.100	0.200	4.0
3	0.200	0.100	16.0

 

By comparing 1 and 2, it can be seen that the rate is not dependent on [B].  On the other hand, based on 2 and 3, doubling [A] results in quadrupling the rate.  Thus, the reaction is second order with respect to [A] and zero order with respect to [B].

Rate Law:     Rate = k [A]2 

After determining the rate law, the rate constant k is obtained by dividing each initial rate by its corresponding [A]2.

In this case, experiments 1 through 3 yield 4.0 X 10-3 M-1s-1.

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Determining the rate law by initial rates looks easy on paper, but experimentally, this could be challenging.

The above consists of five separate experiments, each one starting at a different concentration of the reactant. The rate of disappearance of the reactant N2O5 is monitored over time (red curves), and its instantaneous rate at the start is determined (black lines). 

These instantaneous rates at the beginning of the reaction are plotted against the starting concentration:

Another example:

The above table for the reaction between hydrogen and iodine also provides as a preview of how sensitive rates of reactions are to temperature.

Example Problems:

More Examples

Let us continue with more examples for the method of initial rates:

Doubling ammonium doubles the rate, and doubling nitrite also doubles the rate, so this reaction is first order with respect to ammonium, and first order with respect to nitrite. The reaction is second-order overall.

Changing [B] does not change the rate so this reaction is zero-order with respect to [B]. We can then ignore the contribution of changing [B] in experiments 2 and 3, and focus on what [A] does. When [A] is doubled, the rate quadruples so the reaction is second order with respect to [A]. It is second-order overall.

In the above, the reaction is first order with respect to hydrogen (using experiments 1 and 2) and is second order with respect to NO (using experiments 1 and 3). The reaction is third-order overall.

If the experiments are not designed as exact multiples, we can use the logarithm function to extract the order (shown in the above example). The numbers shown above are probably closer (in terms of accuracy) to what you would see in the laboratory.

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Dependence of Rate of Reaction on Concentration -  The Rate Law

Monitoring the concentration of reactants with time

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