If the vessel shown in Figure 3.1 is evacuated to a few millimeters of mercury (torr) pressure, an explosion will occur. Similarly, if the system is pressurized to 2 atm, there is also an explosion. These facts suggest explosive limits.

If H₂ and O₂ react explosively, it is possible that such processes could occur in a flame, which indeed they do. A fundamental question then is: What governs the conditions that give explosive mixtures? In order to answer this question, it is useful to reconsider the chain reaction as it occurs in the H₂ and Br₂ reaction:

There are two means by which the reaction can be initiated—thermally or photochemically. If the H₂–Br₂ mixture is at room temperature, a photochemical experiment can be performed by using light of short wavelength; that is, high enough hν to rupture the Br–Br bond through a transition to a higher electronic state. In an actual experiment, one makes the light source as weak as possible and measures the actual energy. Then one can estimate the number of bonds broken and measure the number of HBr molecules formed. The ratio of HBr molecules formed per Br atom created is called the photoyield. In the room-temperature experiment one finds that

Thus, one sees that competitive reactions appear to determine the overall character of this reacting system and that a chain reaction can occur without an explosion.

For the H₂–Cl₂ system, the photoyield is of the order 10⁴–10⁷. In this case, the chain step is much faster because the reaction

It is obvious, then, that only the H₂–Cl₂ reaction can be exploded photochemically, that is, at low temperatures. The H₂–Br₂ and H₂–I₂ systems can support only thermal (high-temperature) explosions. A thermal explosion occurs when a chemical system undergoes an exothermic reaction during which insufficient heat is removed from the system so that the reaction process becomes self-heating. Since the rate of reaction, and hence the rate of heat release, increases exponentially with temperature, the reaction rapidly runs away; that is, the system explodes. This phenomenon is the same as that involved in ignition processes and is treated in detail in the chapter on thermal ignition (Chapter 7).

Recall that in the discussion of kinetic processes it was emphasized that the H₂–O₂ reaction contains an important, characteristic chain branching step, namely,

The first two of these three steps are branching, in that two radicals are formed for each one consumed. Since all the three steps are necessary in the chain system, the multiplication factor, usually designated α, is seen to be greater than 1 but less than 2. The first of these three reactions is strongly endothermic; thus, it will not proceed rapidly at low temperatures. So, at low temperatures, an H atom can survive many collisions and can find its way to a surface to be destroyed. This result explains why there is steady reaction in some H₂–O₂ systems where H radicals are introduced. Explosions occur only at the higher temperatures, where the first step proceeds more rapidly.

It is interesting to consider the effect of the multiplication, as it may apply in a practical problem such as that associated with automotive knock. However extensive the reacting mechanism in a system, most of the reactions will be bimolecular. The pre-exponential term in the rate constant for such reactions has been found to depend on the molecular radii and temperature, and will generally be between 4 × 10¹³ and 4 × 10¹⁴ cm³mol⁻¹s⁻¹. This appropriate assumption provides a ready means for calculating a collision frequency. If the state quantities in the knock regime lie in the vicinity of 1200 K and 20 atm and if nitrogen is assumed to be the major component in the gas mixture, the density of this mixture is of the order of 6 kg/m³ or approximately 200 mol/m³. Taking the rate constant pre-exponential as 10¹⁴ cm³ mol⁻¹ s⁻¹ or 10⁸ m³ mol⁻¹ s⁻¹, an estimate of the collision frequency between molecules in the mixture is

For arithmetic convenience, 10¹⁰ will be assumed to be the collision frequency in a chemical reacting system such as the knock mixture loosely defined.

Now consider that a particular straight-chain propagating reaction ensues, that the initial chain particle concentration is simply 1, and that 1 mol or 10¹⁹ molecules/cm³ exist in the system. Thus, all the molecules will be consumed in a straight-chain propagation mechanism in a time given by

Specifying α as the chain branching factor, then, the previous example was for the condition α = 1. If, however, pure chain branching occurs under exactly the same conditions, then α = 2 and every radical initiating the chain system creates two, which create four, and so on. Then 10¹⁹ molecules/cm³ are consumed in the following number of generations (N):

Thus, the time to consume all the particles is

If the system is one of both chain branching and propagating steps, α could equal 1.01, which would indicate that 1 out of 100 reactions in the system is chain branching. Moreover, hidden in this assumption is the effect of the ordinary activation energy in that not all collisions cause reaction. Nevertheless, this point does not invalidate the effect of a small amount of chain branching. Then, if α = 1.01, the number of generations N to consume the mole of reactants is

Thus, the time for consumption is 44 × 10⁻⁸ s or approximately half a microsecond. For α = 1.001, or 1 chain branching step in 1000, N ≅ 43,770 and the time for consumption is approximately 4 ms.

From this analysis, one concludes that if one radical is formed at a temperature in a prevailing system that could undergo branching and if this branching system includes at least one chain branching step and if no chain terminating steps prevent run away, then the system is prone to run away; that is, the system is likely to be explosive.

To illustrate the conditions under which a system that includes chain propagating, chain branching, and chain terminating steps can generate an explosion, one chooses a simplified generalized kinetic model. The assumption is made that for the state condition just prior to explosion, the kinetic steady-state assumption with respect to the radical concentration is satisfactory. The generalized mechanism is written as follows:

Reaction (3.1) is the initiation step, where M is a reactant molecule forming a radical R. Reaction (3.2) is a particular representation of a collection of propagation steps and chain branching to the extent that the overall chain branching ratio can be represented as α. M′ is another reactant molecule, and α has any value greater than 1. Reaction (3.3) is a particular chain propagating step forming a product P. It will be shown in later discussions of the hydrocarbon–air reacting system that this step is similar, for example, to the following important exothermic steps in hydrocarbon oxidation:

Since a radical is consumed and formed in reaction (3.3) and since R represents any radical chain carrier, it is written on both sides of this reaction step. Reaction (3.4) is a gas phase termination step forming an intermediate stable molecule I, which can react further, much as M does. Reaction (3.5), which is not considered particularly important, is essentially a chain terminating step at high pressures. In step (3.5), R is generally an H radical and RO₂ is HO₂, a radical much less effective in reacting with stable (reactant) molecules. Thus reaction (3.5) is considered to be a third-order chain termination step. Reaction (3.6) is a surface termination step that forms minor intermediates (I′) not crucial to the system. For example, tetraethyllead forms lead oxide particles during automotive combustion; if these particles act as a surface sink for radicals, reaction (3.6) would represent the effect of tetraethyllead. The automotive cylinder wall would produce an effect similar to that of tetraethyllead.

The question to be considered is what value of α is necessary for the system to be explosive. This explosive condition is determined by the rate of formation of a major product, and P (products) from reaction (3.3) is the obvious selection for purposes here. Thus,

The steady-state assumption discussed in the consideration of the H₂–Br₂ chain system is applied for determination of the chain carrier concentration (R):

Thus, the steady-state concentration of (R) is found to be

Substituting Eqn (3.10) into Eqn (3.8), one obtains

The rate of formation of the product P can be considered to be infinite—that is, the system explodes—when the denominator of Eqn (3.11) equals zero. It is as if the radical concentration is at a point where it can race to infinity. Note that k₁, the reaction rate constant for the initiation step, determines the rate of formation of P, but does not affect the condition of explosion. The condition under which the denominator would become negative implies that the steady-state approximation is not valid. The rate constant k₃, although regulating the major product-forming and energy-producing step, affects neither the explosion-producing step nor the explosion criterion. Solving for α when the denominator of Eqn (3.11) is zero gives the critical value for explosion; namely,

Assuming there are no particles or surfaces to cause heterogeneous termination steps, then

Thus, for a temperature and pressure condition where α_react > α_crit, the system becomes explosive; for the reverse situation, the termination steps dominate and the products form by slow reaction.

Whether or not either Eqn (3.12) or Eqn (3.13) is applicable to the automotive knock problem may be open to question, but the results appear to predict qualitatively some trends observed with respect to automotive knock. α_react can be regarded as the actual chain branching factor for a system under consideration, and it may also be the appropriate branching factor for the temperature and pressure in the end gas in an automotive system operating near the knock condition. Under the concept just developed, the radical pool in the reacting combustion gases increases rapidly when α_react > α_crit, so the steady-state assumption no longer holds and Eqn (3.11) has no physical significance. Nevertheless, the steady-state results of Eqn (3.12) or Eqn (3.13) essentially define the critical temperature and pressure condition for which the presence of radicals will drive a chain reacting system with one or more chain branching steps to explosion, provided there are not sufficient chain termination steps. Note, however, that the steps in the denominator of Eqn (3.11) have various temperature and pressure dependences. It is worth pointing out that the generalized reaction scheme put forth cannot achieve an explosive condition, even if there is chain branching, if the reacting radical for the chain branching step is not regenerated in the propagating steps and this radical's only source is the initiation step.

Even though k₂ is a hypothetical rate constant for many reaction chain systems within the overall network of reactions in the reacting media and hence cannot be evaluated to obtain a result from Eqn (3.12), it is still possible to extract some qualitative trends, perhaps even with respect to automotive knock. Most importantly, Eqn (3.11) establishes that a chemical explosion is possible only when there is chain branching. Earlier developments show that with small amounts of chain branching, reaction times are extremely small. What determines whether the system will explode or not is whether chain termination is faster or slower than chain branching.

The value of α_crit in Eqn (3.13) is somewhat pressure-dependent through the oxygen concentration. Thus, it seems that as the pressure rises, α_crit would increase and the system would be less prone to explode (knock). However, as the pressure increases, the temperature also rises. Moreover, k₄, the rate constant for a bond forming step, and k₅, a rate constant for a three-body recombination step, can be expected to decrease slightly with increasing temperature. The overall rate constant k₂, which includes branching and propagating steps, to a first approximation, increases exponentially with temperature. Thus, as the cylinder pressure in an automotive engine rises, the temperature rises, resulting in an α_crit that makes the system more prone to explode (knock).

The α_crit from Eqn (3.12) could apply to a system that has a large surface area. Tetraethyllead forms small lead oxide particles with a very large surface area, so the rate constant k₆ would be very large. A large k₆ leads to a large value of α_crit and hence a system unlikely to explode. This analysis supports the argument that tetraethyllead suppresses knock by providing a heterogeneous chain terminating vehicle.

It is also interesting to note that, if the general mechanism (Eqns (3.1)–(3.6)) were a propagating system with α = 1, the rate of change in product concentration (P) would be

Thus, the condition for fast reaction is

Most systems of interest in combustion include numerous chain steps. Thus, it is important to introduce the concept of a chain length, which is defined as the average number of product molecules formed in a chain cycle or the product reaction rate divided by the system initiation rate [1]. For the previous scheme, then, the chain length (cl) is equal to Eqn (3.11) divided by the rate expression k₁(M) for reaction (3.1); that is,

If the system contains only propagating steps, α = 1, so the chain length is

Considering that for a steady system, the termination and initiation steps must be in balance, the definition of chain length could also be defined as the rate of product formation divided by the rate of termination. Such a chain length expression would not necessarily hold for the arbitrary system of reactions (3.1)–(3.6), but would hold for such systems as that written for the H₂–Br₂ reaction. When chains are long, the types of products formed are determined by the propagating reactions alone, and one can ignore the initiation and termination steps.