Reactions and Reactivity: Kinetics

Definition of Rate and the Rate Equation

Rate is a term used frequently in physical science, especially in physics, where rate of displacement or velocity and rate of change of velocity or acceleration are well recognized. The term kinetics is used to describe the study of rates of chemical reactions.

The term “rate” implies how one quantity changes with respect to time. In velocity, it is a displacement in meters or kilometers per unit time of seconds or hours. In acceleration, the velocity changes per unit time, having units of  meter/second/second. In chemical reactions the quantity which changes per unit time is the concentration of a reactant or product. The concentration of a reactant will decrease as it is consumed over time; the concentration of a product will increase as it is produced over time. The units of rates of reaction are therefore moles/liter/second or M/s, where M represents moles per liter. Brackets, such as [Cl], are sometimes used to show units of molar concentration.

The diagram below shows the decomposition reaction of gaseous N₂O₅ into its elements as represented by the equation

From the slope of the reactant line on the diagram, one can see that the rate of change of the reactant is not constant. The slope is steepest at the beginning of the reaction and then levels off. The rate is fastest when the concentration of the reactant is the highest. The rate of the reaction can be defined then as the change in molar concentration of reactant [R] per unit time. Or .

However, the concentration is decreasing and rates are always considered positive, so a negative sign is used to show that the concentration of the reactant has decreased.

Thus in the reaction of , the rate of disappearance of Y₂ would be one-half the rate of disappearance of X₂. The rate of formation of product would be equal to the rate of disappearance of X₂, but twice that of the rate of disappearance of Y₂. A quantitative expression called the rate equation can be written to show that the rate is proportional to the concentration of the reactants and products.

Determining the Rate: the Mathematics of Kinetics

There are two types of rate laws:

• The differential rate law, or simply the rate law, shows how the rate depends on the concentration of each reactant.
• The integrated rate law shows how the concentration of a substance depends upon time.

Consider the example showing the decomposition of N₂O₅. Imagine the following rate data was obtained for different initial concentrations of N₂O₅.

These results show that when the concentration of N₂O₅ is halved, the rate is also halved. This means that the reaction is first order with respect to the N₂O₅ and the rate equation can be written as

or simply

Let’s look at an example of how to determine the rate equation when given the experimental data.

Sample Problem

The following initial rates of reaction were measured for the concentrations of reactants shown in the table for the reaction of . Determine the rate equation.

The general form of the equation would be . The data will be used to determine the value of the exponents. Considering experiments 1 and 2, you can see that keeping the concentration of A constant, and doubling the concentration of B, causes the rate to increase by a factor of two (the same as the increase in the concentration of B). Thus, the order of the reaction with respect to B is first order and the value of m would be 1.

If you consider the experiments 2 and 3 where the concentration of B is held constant, and the concentration of A is doubled, the rate is increased by a factor of 4;. This means that the rate depends on the square of the concentration of A, and the exponent n has a value of 2.

The rate equation can be written as . The reaction is second order with respect to reactant A and first order with respect to reactant B; the overall order of the reaction is third order which is the sum of the exponents.

We can also use this data to determine the value of the rate constant k. We can take the experimental data for any of the experiments and substitute it into the rate equation. In this case:

k = 0.0027 L²/mol²s

Use of Equations

To use the integrated form of the rate equation, one must plot the concentration of the reactant of interest as a function of time. Unless the reaction is zero-order, the graph will be a curve. The rate of the reaction can be determined by the slope of the line, but to determine the rate equation more calculations must be done. Using calculus, it can be shown that the equation relating the concentration as a function of time for a first order reaction is:

Calculating Half-life

The half-life of any reaction is the time it takes for the concentration of the reactant to become 50% of the original amount. In other words, the ratio of [A] to [A]₀ is equal to 0.50.

Substituting the values of [A] = 0.5[A]₀ in the integrated rate equation, , and rearranging gives:
where t = the half-life, t_½.

Because ln0.50 = 0.693, for any first order reaction< ; the half-life is independent of the concentration of starting material.

For second order reactions, the integrated rate law takes the form of . A plot of as a function of time will produce a straight-line with a slope equal to k. Calculations show that the equation for the half-life is .

The table below shows the characteristics of three different orders of reactions for the integrated form.

Order	Rate expression	Equation	Half-life	Linear plot
0	Rate = k			[A] vs. t
1	Rate =k[A]			ln[A] vs t
2	Rate = k[A]2			vs. t

Sample Problem

The following data were obtained for the decomposition of gaseous HI. Determine the order of the reaction, the rate equation, the value of k, and the half-life of the reaction.

Three curves must be plotted according to the information in the summary chart above. The plot that produces a straight line will show the order of the reaction.

The value of k is equal to the slope of the line:

Collision Theory

In order for a reaction to occur, the reactant particles must collide with the proper energy and proper orientation for the formation of the transition state complex. The minimum energy required is called the activation energy, E_a. There are three factors which affect the rate constant:

• A steric factor, p, takes into account that only certain orientations of the species involved will result in effective collisions.
• The frequency factor, Z, gives the number of collisions which occur in a given period of time.
• An energy factor, f, accounts for the number of particles with the minimum energy necessary for reaction, where .

The Arrhenius equation relates the factors to the rate constant . The steric factor and frequency factor are combined in the variable A. A useful form of the equation is obtained by taking the natural logarithm of both sides, which yields an equation in the form y = mx + b, which can be graphed as a straight line:

Factors Affecting Rates

The Collision Theory assumes that in order for a reaction to occur the reacting particles must collide with the proper orientation and proper energy. Any factor that enhances the ability of the particles to interact will necessarily affect the rate. The five factors normally considered are:

• Temperature
• Concentration of reactants
• Nature of reactants
• Surface area
• Presence of a catalyst

Temperature

As we have seen, temperature affects the rate of a reaction. According to the kinetic molecular theory, molecules are in constant motion. The higher the temperature, the higher the kinetic energy due to an increase in velocity of the molecules. In order for a reaction to occur, the reactant particles must collide with sufficient energy and the proper orientation for the formation of the transition state complex. Increasing the temperature increases both the number of collisions and the energy of collisions. An approximate rule of thumb (which is derived using the Arrhenius equation) is that an increase of 10^o C doubles the rate of a reaction. For example, if the rate of a reaction at 0^o C is 5.0 mol/L-min, we can expect that increasing the temperature to 10^o C would make the rate 10 mol/L-min.

Concentration

The concentration of a reactant is the relative amount of reactant per unit volume. Remember, the more collisions that occur, the greater the chance that a reaction will have the proper orientation. Thus, higher concentrations of reactant molecules result in a greater rate of collisions between reactant molecules, and a faster rate of reaction. Just how much the rate is increased for a particular reaction must be determined experimentally.

Nature of Reactants

The nature of the substances which are reacting does affect the rate. In general, ionic reactions occur faster than molecular ones in solution. The latter require the breaking of covalent bonds and the formation of new ones. For example, a weak acid such as acetic acid reacts much more slowly with zinc metal than does the strong acid, like hydrochloric acid, of the same concentration. The difference is in the ability of the hydrochloric acid to ionize more completely than the acetic acid. Another example would be the different reaction rates of metals when they react with dilute hydrochloric acid. Magnesium metal reacts much more rapidly with acids than does lead.

Surface Area

The ability of reactants to meet affects the rate of a reaction. One way to enhance the opportunity for the reactants to come into contact with each other is to increase their surface area. Consider attempting to burn a charcoal briquette using the flame from a candle. The rate of combustion is very slow, because only the charcoal particles on the outside surface of the briquette are in contact with the oxygen in the air. If the briquette is ground up and blown into the candle flame the reaction rate is very rapid, in fact it is explosive. Grain elevator explosions are another impressive example of such surface area phenomena. These explosions occur, not in full grain elevators, but in ones that are almost empty of grain, but full of grain dust. A spark from a light switch can provide enough energy to start a reaction which can bring down the elevator.

Presence of a Catalyst

A catalyst is a substance which alters the rate of a chemical reaction without itself being consumed. Since the activation energy is the minimum amount of energy required for a reaction to occur, if the particles do not have that minimum energy, no reaction occurs. Catalysts provide a different reaction pathway with a lower activation energy. Frequently the reaction mechanism for catalyzed reactions has multiple steps, each with a lower activation energy than the uncatalyzed reaction.
Catalysts may be homogeneous (same phase) or heterogeneous(different phase) with the reactants. Enzymes are examples of biological catalysts which allow reactions in the body to occur at much lower temperatures than would be required in a test tube.

Effect of a catalyst on a reaction

Other substances called inhibitors also alter the rate of reactions by slowing them. For example, some preservatives that are found in commercial food products are inhibitors. Anti-depressant medications selectively inhibit chemical reactions in the body. The action of inhibitors is complex, but inhibitors work by reducing the effectiveness of a catalyst. It is generally thought that they offer alternate reaction pathways that result in other products.

Reaction Mechanisms

A reaction mechanism is a proposed series of steps by which a reaction is thought to proceed. Each step in a mechanism is considered as an elementary step which may involve one particle (a unimolecular step), two particles (a bimolecular step), or in rare cases, three particles (a termolecular step). These steps have corresponding rate equations as shown below, in which the rate equation is equal to the constant multiplied by the molar concentration of each reactant.

A → B + C with rate = k[A]

A + B → C + D with rate = k[A][B]

A + B + C → D + E with rate = k[A][B][C]

When a mechanism is proposed, two requirements must be satisfied:

• The sum of the elementary steps must give the overall balanced equation for the reaction.
• The mechanism must agree with the experimentally determined rate law.

The slowest step in a mechanism is called the rate-determining step. The overall reaction can only proceed as fast as the slowest step. As an example, consider the formation of a compound by the reaction of nitrogen dioxide with fluorine.

with rate = k[NO₂][F₂] determined by experiment.

A proposed mechanism is slow

With a second step of fast

The first requirement is met because the algebraic sum of the two steps is the net reaction. The second requirement is also met because the rate of the overall reaction is that of the rate-determining step which is bimolecular with rate = k₁[NO₂][F₂]. The proposed mechanism is acceptable because it meets both requirements, but it is not necessarily the correct mechanism. Chemists need to perform many additional experiments to decide the most probable mechanisms.

Question

The following data were obtained for the reaction shown below at various temperatures. Plot the data (lnk vs 1/T) and calculate the activation energy.

k (s-1)	0.048	2.3	49	590
T (oC)	500	600	700	800

Reveal Answer

Plotting the lnk as a function of 1/T gives a straight line. The activation energy is equal to –R·(slope of the line). In this case the slope is -2.6 × 10⁴. So calculating the activation energy is (8.31 J/mol K)(2.6 10⁴) = 216 kJ/mol

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