Reaction rate always increases with increasing temperature…

            Well, nearly always. Chemistry is a complicated subject and dependent on many variables; an immense variety of compounds, reactions beyond counting, but behind all this is a relatively small number of broad principles. However, to the fascination of people like me and the irritation of those who'd be better studying Physics, there's always the exception that tests the rule.

Reaction rates and the Arrhenius equation.

            For a reaction between compounds A and B the rate equation could be

rate = k[A]²[B]

The equation is that for a third-order reaction - not common, but the subject of this article. The rate equation would have to be found experimentally since it depends on mechanism, and is not related to the stoichiometric equation – but then everything else (including the stoichiometric equation for the reaction) has to be found experimentally as well, so perhaps this doesn't say very much. In the rate equation the temperature dependent quantity is the rate constant, k; the rate of reaction changes with a change in temperature because k changes. The quantitative effect is given by the Arrhenius equation:

k = Ae ^{– Ea/RT}

            Since the activation energy E_a is always a positive quantity, the result is that an increase in temperature always results in an increase in k. This should mean that for every reaction an increase in temperature increases the rate. Excluded from this are enzyme-mediated reactions; the rate of these will fall with increasing temperature beyond the optimum for the enzyme since the enzyme gets cooked.

An exception: the oxidation of nitrogen monoxide, NO.

            For a very small number of reactions, all involving NO, the rate of reaction falls with an increase in temperature. This implies a negative activation energy. The reason for this apparent peculiarity is that although the rate constant k does indeed increase with increasing temperature, the mechanism is such that another constant, the equilibrium constant for one of the mechanistic steps, is also involved in the rate equation. This falls with increasing temperature.

            The reaction of nitrogen monoxide (nitric oxide, nitrogen(II) oxide) with oxygen is:

2NO(g) + O₂(g) → 2NO₂(g)

This is one of only five homogeneous gas reactions known to be third order:

rate = k[NO]²[O₂]

the other four being the reactions of NO with chlorine, bromine, hydrogen and deuterium (Bodenstein, 1922 ¹).

            The probability of a termolecular reaction, where the three species collide simultaneously with the correct energy and the correct orientation in a single step, is very small. The suggested mechanism for the oxidation of NO with O₂ involves an initial dimerisation in an equilibrium reaction, followed by reaction of the dimer with the oxygen:

                        2NO(g) ⇔(NO)₂(g) . . . . . . . . . . . . . . . . . . . . . . . . . (1) 

                        (NO)₂ (g) + O₂(g) → 2NO₂(g) . . . . . . . . . . . . . . . ..... . . . . .(2)

The following discussion depends on two things; firstly that the rate of attainment of the equilibrium (1) is very fast compared with reaction (2), and secondly that reaction (2) is the rate limiting step.

Consider the equilibrium reaction (1). From usual equilibrium considerations

                        K_c = [(NO)₂]/[NO]²  . . . . . . . . . . . . . . . . . . . . . . . . . . . (3).

For the rate limiting step (2) we can write

                        rate = k'[(NO)₂][O₂];

but from (3)

                        [(NO)₂]  =  K_c[NO]²

so                     rate = k'K_c[NO]²[O₂]

This is the rate equation quoted with k = k'K_c.
 

The temperature dependence of the reaction rate.

            In the rate equation

                        rate = k [NO]²[O₂] = k'K_c [NO]²[O₂]

both k' and K_c are temperature dependent. The constant k' increases with increasing temperature; the variation of K_c with changing temperature depends on the thermicity of the equilibrium producing (NO)₂. The reaction

2NO → (NO)₂

involves bond formation and is therefore exothermic. For an equilibrium where the reaction from left to right is exothermic, K_c decreases with an increase in temperature. So; k' increases in value with temperature increase, but K_c falls. In this reaction k' increases less than the fall in K_c, so that the overall value k'K_c also falls with an increase in temperature. So, then, does the reaction rate. The activation energy is only apparently negative; for the rate-limiting step it is, as is usual, positive.

The reaction and other useful ideas in kinetics can be found in Atkins ^2.

References, and a caution.
 

	1.   M. Bodenstein,  Z. Physik. Chem.  100, 118, 1922.
	2.   P.W.Atkins, Physical Chemistry, 4th edition, p 802: OUP 1990.

Take care with reading early papers or books on kinetics; the term 'molecularity' was often used to refer to what we would call overall order. The molecularity of a reaction is the number of particles that collide in the rate-determining step; the overall order is the number of particles involved up to and including the rate-determining step. The oxidation of NO is therefore a third-order, bimolecular reaction, but is often referred to in earlier works as termolecular. It is an example of the evolution of nomenclature.