Lecture 28 summary

Introduction to chemical kinetics. Chemical kinetics is the study of rates of chemical reactions. At the outset, we will define what is meant by reaction rate

• What is the definition of reaction rate?
• What factors influence the rate of a reaction?
• What is the relationship between reaction rate and chemical equilibrium?
• What is a rate law?
• What is the order of a reaction?

The nature of the reactants determines whether a reaction can or will tend to occur. Once we specify a particular reacting system, the factors that influence the rate of a reaction are (1) temperature, (2) concentrations of reactants, (3) physical state of reactants, (4) presence of a catalyst. We will here be mainly concerned with the effect of concentration of reactants on rate, although we will develop a conceptual approach that will allow us to relate the rate of a reaction at a fixed temperature and reactant concentration to a quantity called activation energy.

When encountering situations in chemistry where not only can a chemical reaction happen, but there are very many reactions possible, it becomes of paramount importance to know about the relative rates of these reactions.

Reaction rates: Average and instananeous

Reaction rate is defined in terms of change in concentration of a reactant or product with time.

In the simple example shown here, over the first 20 seconds, mol A drops from 1.00 mol to 0.54 mol. Let the volume be 1.00 L. Thus

Δ[A] = [A]_final – [A]_initial = (0.54 – 1.00) M = – 0.46 M

The average rate of change the concentration of A over this time period is

Δ[A]/Δt  =  (– 0.46 M) / 20 s  =  –2.3 x 10^–2 M s^–1

Similarly, the concentration of B rises from 0 mol to 0.46 mol over the first 20 seconds, and the average rate of change of [B] is equal in magnitude to that for [A] over this time period, but positive in sign.

Δ[B] = [B]_final – [B]_initial = (0.46 – 0.00) M =  + 0.46 M

Δ[B]/Δt  =  (0.46 M) / 20 s  =  2.3 x 10^–2 M s^–1

The relationship Δ[B]/Δt  =  – Δ[A]/Δt must hold for every time period we choose to look at, as we should realize from the stoichiometry of the reaction. We define the average rate of reaction in this case as

average rate  =  V_ave =  – Δ[A]/Δt   =  Δ[B]/Δt

The graph below is a representation of the pictorial representation of the three time points of the recation shown above.

For the second 20 seconds, the average rate computes to

V_ave =  – Δ[A]/Δt   =  Δ[B]/Δt  =  (0.70 – 0.46)M / 20 s  =  1.2 x 10^–2 M s^–1

The reaction slows over the next 20 seconds, and in general the rates we observe are maximal at the start of a reaction, and decrease as the reaction progresses. This is because the rates we observe as change in concentration during a time interval are net rates. Initially, there are no products present, so no reverse reaction can occur. As the reaction progresses, the increasing amounts of product will lead to increasing rate of the reverse reaction. The net reaction rate should decrease with time, until at equilibrium we see no net reaction rate - the rates of the forward and reverse reactions are the same. The next example illustrates this by showing the complete course of a reaction from the start (reactants only present) until equilibrium is reached (unchanging amounts of reactants and products). We also represent amounts of reactants and products at many closely spaced points in time, so that the concentration vs. time graph becomes a smooth curve. This is referred to here as a progress curve for the reaction.

Here we give the expressions for equivalent rate quantities for the general reaction aA + bB = cC + dD:

V_ave =  – (1/a)(Δ[A]/Δt)   =  – (1/b)(Δ[B]/Δt)  =  (1/c)(Δ[C]/Δt)  =  (1/d)(Δ[D]/Δt)

for some finite interval Δt. For instantaneous rates,

V_inst =  – (1/a)(d[A]/dt)   =  – (1/b)(d[B]/dt)  =  (1/c)(d[C]/dt)  =  (1/d)(d[D]/dt)

Note the slope of the tangent to the curves is steepest at time t = 0, and the slopes decrease in magnitude until the curves become flat (slope = 0), and the reaction has reached equilibrium. We will shortly take up the dependence of the instantaneous rate at time t = 0, the initial instantaneous rate, on initial concentration of reactant. This will lead us to the formulation of rate laws for reactions.

Reaction coordinate diagrams

To help us understand the reason why an energetically favorable reaction (i.e. products are favored, and K_eq large) may nonetheless take place at only a very slow rate, we introduce a representation of a reaction called a reaction coordinate diagram. Such diagrams are used frequently in chemistry, being useful in illustrating many features of chemical reactions. Here we look at a hypothetical reaction coordinate diagram for the simple reaction A = B, which will help illuminate the relationship between chemical kinetics and chemical equilibrium.

The reaction coordinate diagram suggests an explanation. In order for the reaction to take place, the reactant molecules have to possess enough energy to reach an unstable configuration called the transition state (symbolized by the double dagger, ‡) before they can "fall" to the potential energy minimum of coversion to B. We can think of this activation energy, E_a, for the forward reaction, as the energetic cost required to advance the bond breaking in A to the point where formation of the new bond in B can begin to provide an energetic payback.

A key point to note is that the change in chemical potential energy is related to the equilibrium constant for the reaction, whereas the magnitude of the activation energy is related to the rate of the reaction. The greater the loss in chemical potential energy in going from A to B, the larger the equilibrium constant. However, the greater the activation energy, the slower the rate of conversion of A to B will be. Recall the Boltzmann distribution of molecular energies, which can tell us how many molecules of A at given temperature will have an energy of E_a or greater, and thus the fraction of A molecules capable of reacting. The greater the value of E_a, the fewer molecules of A will have the energy to react. This helps us understand why a mixture of hydrogen and oxygen can sit peacefully at room temperature, with no detectible production of water. The activation energy for the reaction is so high that only an infinitesimal number of molecules have the energy to possibly react. However, if a spark is provided, the molecules in the immediate vicinity acquire more energy, and a sufficient number of them react, which in turn releases much more energy (the reaction is strongly exothermic). A runaway process of reacting molecules and energy release is initiated, and an explosion immediately ensues.

The latter fact implies that the reverse reaction, conversion of B to A, will be much slower than the forward reaction when the concentrations of A and B are equal, since E_a, for the reverse reaction is much greater than that for the forward reaction. In order for this reaction to reach equilibrium, the concentration of B will have to increase well beyond that of A in order to fulfill the equilibrium condition that the rates of the forward and reverse reactions are equal.

Rate laws, rate constants, and reaction half-life

Consider a simple chemical reaction with only one reactant. Let c be the concentration of this reactant (for example, c = [N₂O₅] in the figure below), let n be its stoichiometric coefficient, and let t represent the time variable. As we have seen, measuring c as the reaction progresses, and graphing as a function of time, yields a progress curve. The average rate, over a time interval Dt, is given by Dc/Dt. In making this time interval very small, the average rate approahes the instantaneous rate at any time point within the interval. The instantaneous rate at any time t is the slope of the tangent to the curve at that value for t. The instantaneous rate when the reaction starts at time t = 0 is the slope of the tangent to the progress curve for t = 0 and is called the initial rate. We'll often use V₀ to symbolize the initial rate.

When progress curves for a reaction are determined experimentally, they reveal patterns in the dependence of rate on concentration. By consistently determining the initial rate of a reaction as initial concentration is varied, it is found that initial rates depend on starting concentrations as expressed by the equation

V₀ = kc^m

where represents the initial rate and m is some exponent (integer or in some cases, half-integer values). This equation is called the rate law. We call m the order of reaction, and most common are m = 1 and 2. If m = 1, the reaction is first-order in c, and if m = 2, it is second-order in c. It is also possible to observe zero-order reactions, in which intial rate is independent of concentration, as well as reactions with fractional order, such as 1/2 or 3/2.

We can also look at the half-life (t_1/2)of a reaction. The half-life is the time required for one-half of the initial reactant to be consumed. For a first-order reaction, t_1/2 is proportional to 1/k, and so is independent of concentration. For a second-order reaction, t_1/2 is inversely proportional to both k and c₀, the initial concentration of reactant. In a zero-order reaction, (t_1/2) is directly dependent on c₀, and inversely proportional to k.