Thermodynamics points the way and makes it possible

Chemical change is driven by the tendency of atoms and molecules to rearrange themselves in a way that results in the maximum possible dispersion of thermal energy into the world. The observable quantity that measures this spreading and sharing of energy is the free energy of the system. As a chemical change takes place, the quantities of reactants and products change in a way that leads to a more negative free energy. When the free energy reaches its minimum possible value, there is no more net change and the system is said to be in equilibrium.

The beauty of thermodynamics is that it enables us to unfailingly predict the net direction of a reaction and the composition of the equilibrium state even without conducting the experiment; the standard free energies of the reactants and products, which can be independently measured or obtained from tables, are all we need.

...but it says nothing about how long it will take to get there!

The "speed" of a reaction — how long it takes to reach equilibrium — bears no relation at all to how spontaneous it is (as given by the sign and value of ΔG°) or whether it is exothermic or endothermic (given by the sign of ΔH°). Moreover, there is no way that reaction rates can be predicted in advance; each reaction must be studied individually. One reason for this is that

the stoichiometric equation for the reaction says nothing about its mechanism

By mechanism, we mean, basically, "who does what to whom". Think of a reaction mechanism as something that goes on in a "black box" that joins reactants to products:

The inner workings of the black box are ordinarily hidden from us, are highly unpredictable and can only be inferred by indirect means. Consider, for example, the gas-phase formation reactions of the hydrogen halides from the elements.

Three reactions that look alike, but are not

The thermodynamics of these reactions are all similar (they are all highly exothermic), but their dynamics (their kinetics and mechanisms) could not be more different.

... and what is particularly noteworthy is that these striking differences cannot be reliably predicted from theory; they were revealed only by experimentation.

Chemical reactions vary greatly in the speed at which they occur. Some are essentially instantaneous, while others may take years to reach equilibrium.

Definition of reaction rate

The speed of a chemical reaction may be defined as the change in concentration of a substance divided by the time interval during which this change is observed:

(2-1) important definition !

For a reaction of the form A + B → C, the rate can be expressed in terms of the change in concentration of any of its components:

 

in which Δ[A] is the difference between the concentration of A over the time interval t₂ – t₁:

Δ[A] = [A]₂ – [A]₁ (2-2)

Notice the minus signs in the equations shown in the two leftmost boxes above.
The concentration of a reactant always decreases with time, so Δ[A] and Δ[B] are both negative. Since negative rates don't make much sense, rates expressed in terms of a reactant concentration are always preceded by a minus sign in order to make the rate come out positive.

Consider now a reaction in which the coefficients are different:

A + 3B → 2D

It is clear that [B] decreases three times as rapidly as [A], so in order to avoid ambiguity when expressing the rate in terms of different components, it is customary to divide each change in concentration by the appropriate coefficient:

(2-3)

So please understand: each of the above quotients is a legitimate expression of the rate of this particular reaction; they all yield the same number. Which one you employ when doing a calculation is largely a matter of convenience.

Instantaneous rates

Most reactions slow down as the reactants are consumed. Consequently, the rates given by the expressions shown above tend to lose their meaning when measured over longer time intervals Δt.

Note: Instantaneous rates are also known as differential rates.

Thus for the reaction whose progress is plotted here, the actual rate (as measured by the increasing concentration of product) varies continuously, being greatest at time zero. The instantaneous rate of a reaction is given by the slope of a tangent to the concentration-vs.-time curve. Three such rates have been identified in this plot.

An instantaneous rate taken near the beginning of the reaction (t = 0) is known as an initial rate (label (1) here). As we shall soon see, initial rates play an important role in the study of reaction kinetics.

 

The relation between the rate of a reaction and the concentrations of reactants is expressed by its rate law. For example, the rate of the gas-phase decomposition of dinitrogen pentoxide

2N₂O₅ → 4NO₂ + O₂

has been found to be directly proportional to the concentration of N₂O₅:

rate = k [N₂O₅]

More generally, for a reaction of the form

n_A A + n_B B + ... → products

the rate law will be

in which the exponents a and b are usually (but not always) integers (including zero) and, we must emphasize once again, bear no relation to the coefficients n_A , n_B in the reaction equation.

Since the rate of a reaction has the dimensions of (concentration/time), the dimensions of the rate constant k will depend on the exponents of the concentration terms in the rate law. To make this work out properly, if we let p be the sum of the exponents of the concentration terms in the rate law

p = a + b + ...

then k will have the dimensions (concentration^1–p/time).

Reaction order

The order of a rate law is the sum of the exponents in its concentration terms. For the N₂O₅ decomposition with the rate law k[N₂O₅], this exponent is 1 (and thus is not explicitly shown); this reaction is therefore a first order reaction. We can also say that the reaction is "first order in N₂O₅".

For more complicated rate laws, we can speak of the overall reaction order and also the orders with respect to each component. As an example, consider a reaction

A + 3B + 2C → products

whose experimental rate law is

rate = k[A] [B]²

We would describe this reaction as third-order overall, first-order in A, second-order in B, and zero-order in C.

Zero-order means that the rate is independent of the concentration of a particular reactant. But of course enough C must be present to allow the equilibrium mixture to form.

How reaction orders are observed

Observing rate-vs.-concentration proportionality

In order to determine the value of the exponent in a rate equation term, we need to see how the rate varies with the concentration of the substance. For a single-reactant decomposition reaction of the form

A → products

in which the rate is –d[A]/dt, we simply plot [A] as function of time, draw tangents at various intervals, and see how the slopes of these tangents (the instantaneous rates) depend on [A].

• If doubling the concentration of A doubles the rate, then the reaction is first-order in A.
• If doubling the concentration results in a four-fold rate increase, the reaction is second-order in A.

 

Initial rate method

When there is more than one reactant, the method described above is rarely practical, since the concentrations of the different reactants will generally fall at different rates, depending on the stoichiometry. Instead, we measure only the rate near the beginning of the reaction, before the concentrations have had time to change significantly. The experiment is then repeated with a different starting concentration of the reactant in question, but keeping the concentrations of any others the same. After the order with respect to one component is found, another series of trials is conducted in which the order of another component is found.

Using the same dinitrogen pentoxide decomposition as in the previous example, we conduct a series of runs using five different initial concentrations of N2O5. Note that we use only the tangents at the beginning of each plot — that is, at times t=0.
We then plot the five initial rates of consumption of N2O5 as a function of its molar concentration. As before, we see that these rates are directly proportional to [N2O5]. The slope of this plot gives the value of the rate constant.

rate = (5.2 × 10^–3) [N2O5] mol L^–1 s^–1

 

 

Dealing with multiple reactants: the isolation method

It is not always practical to determine orders of two or more reactants by the method illustrated in the preceding example. Fortunately, there is another way to accomplish the same task: we can use excess concentrations of all the reactants except the one we wish to investigate. For example, suppose the reaction is

A + B + C → products

and we need to find the order with respect to [B] in the rate law. If we set
[B]_o to 0.020 M and let [A]_o = [C]_o = 2.00M, then if the reaction goes to completion, the change in [A] and [C] will also be 0.020 M which is only 1 percent of their original values. This will often be smaller than the experimental error in determining the rates, so it can be neglected. By "flooding" the reaction mixture with one or more reactants, we are effectively isolating the one in which we are interested.

 

End of part 1 of rate laws