Theoretical background

1. Introduction

This experiment is based on a computer simulation: a chain reaction between oxygen and hydrogen. At a first glance, this reaction seems strange. In some reaction conditions the reagents may react exteremely fast resulting in explosion while in other conditions they may react very slowly or not react at all. The pressures that define the separation between those extreme behaviors are called the "explosion limits".

2. Experimental Facts

We start by describing some of the experimental observations made for this reaction. It was observed that in low temperatures, the reaction is mostly heterogenic, meaning, it takes place on a solid surface and not in the gas phase. When the temperature is about 400°C, it was found that the reaction occurs in the gas phase and at a temperature above ~600°C it is almost always explosive. between those temperatures, the rate of the reaction can change with the envioronmental conditions in an interesting manner. For example, if one uses a stochometric mixture of Oxygen and Hydrogen and measures the rate of the reaction as a function of pressure, he or she will get the following diagram.

Figure 1: The variation of the rate of the Hydrogen-Oxygen reaction with pressure for a stochiometric mixture (upper line) compared with that to be expected for a non-branching exothermic reaction undergoing thermal reaction at the limit Pl.
Pl, Pu, Pt are the lower, upper and third limits, respectively.

At a very low pressure range, the reaction occurs slowly at steady state. When the pressure increases and reaches Pl (which is the first or lower limit pressure), the reaction becomes extremely fast i.e. explosion. Increasing the pressure, keeps the reaction explosive but when Pu (which is the second or upper limit pressure) is reached the reaction becomes steady again and no explosion will occur. Additional increase will keep the reaction in steady state until Pt (which is the third limit pressure) is reached and the reaction becomes explosive again. Those borders can be plotted as the function of pressure verses temperature. This plot is called "the explosion diagram". The explosion diagram for a stochiometric mixture of Oxygen and Hydrogen in a KCl-coated container is presented in figure 2.

Figure 2: The explosion limit diagram for a stoichiometric mixture of hydrogen and oxygen in a spherical KCl - coated vessel of 7.4 cm diameter.

The area to the left of the line represents the non-explosive reaction region while the area to the right of the line is the explosive region. The area between the lower and upper limits is referred to as the explosion peninsula. The lower limit is seen to decrease slowly as the temperature is raised. It is also lowered by the addition of inert gases to the reaction mixture. It is very sensitive to the nature of the surface. The influence of the inert gas on the second limit can be different, it can increase or decrease it. The third limit decreases rapidly with increasing temperature and is lowered by adding inert gases and by increasing the diameter of the vessel.

3. Theory

3.1. Chain branch reaction kinetics

In a mixture of hydrogen and oxygen, it is presumed plausible that the presence of a free radical (free valence) in the form of an OH radical or H atom should result in the following reaction cycle:

  • H+O2→OH+O
  • O+H2→OH+H
  • HO+H2→H2O+H

The first reaction(H+O2) is endothermic by 17 kcal/mole. Thus, at room temperature and even at somewhat higher temperatures a mixture of hydrogen and oxygen is very stable even if hydrogen atoms are introduced from another source. The free valence ultimately terminates at the wall through recombination process. Above some temperatures, however, the chain branch reaction becomes sufficiently frequent, when compared with the rate of removal of H atoms, to cause multiplication of free valences and possible explosion. Explosion are conveniently classified into two distinct categories: branched-chain explosion, in which the reaction rate increases without limit because of chain branching, and thermal explosions, in which there is an exponential increase in reaction rate resulting from exothermic chemical reaction, heating of reactants, and an increase in the magnitude of the specific reaction rate constants.

Consider a 1 cm3 container that contains initially one chain molecule, that is, one free radical per centimeter. Assume that the number density in the container is 1019 molecules/cm3 (it is close to 0.5 atm at normal conditions) and the average collision rate is 108 collisions/sec. If the reaction in the volume is a chain-carrying reaction [i.e., one free radical can generate another free radical in the reaction (α=1.0)], then the time required for all of the molecules to react (i.e., 1019 collisions) will be:

t =   1019 molecules  ~ 1011sec ~ 30,000years
108 molecules/sec

Such a slow process cannot be called combustion. If the reaction in the volume is a chain-branching reaction [i.e., one free radical or chain particle can generate two chain particles in the reaction (α = 2.0)], then the time required for all of the molecules to react can be estimated as follows:

1+2122+23+...+2N =  2N+1-1  = 1019 molecules
2-1

given that N = 64 generations solves this equation and that where N that the average collision rate is 108 collisions/sec, one can calculate that all the molecules will react in 64·10-8sec ~ 1µsec. This is certainly an extremely rapid combustion process. In an actual combustion process, not all reactions are chain-branching reactions. However, the reaction rate can still be very fast even for a very small portion of chain-branching reactions. The coefficient α, which represents the mean number of free radicals produced in one reaction cycle, can be in this case less than 2 and rather close to 1. Preparing to quiz try to calculate what will be the time required for all molecules in the volume to react in a combustion process, in which just 1% (α=1.01) of the elementary reactions are of the chain-branching type.

3.2. Reaction between hydrogen and oxygen

The chain-branching explosion between hydrogen and oxygen is the reaction which was studied in the most details. The chemical equation of the reaction looks rather simple:

2H2+O2→2H2O ΔH0298=-115 kcal/mole

Actually, the reaction mechanism includes more than 50 elementary reactions. However, in order to understand the main features of the process we will limit ourselves by the five following reactions:

(1) H+O2→OH+O
(2) O+H2→OH+H
(3) HO+H2→H2O+H
(4) H→wall
(5) H+O2+M→HO2+M

Reactions (1) and (2) are the reactions of branching, since in these reactions one free valence produces three valences. Thus, the number of free valences increases as a result of these reactions. Since the reaction (1) has higher activation energy than reaction (2), the rate of this reaction is slower and it usually can be consider as a rate determining step. Reaction (3) is the reaction of the chain propagation, since it does not change the number of valences. In reactions (4) and (5) the number of valences decreases, and these reactions are called chain termination or chain breaking reactions. Note, that the reaction (5) actually looks as a chain propagation reaction, but radical HO2 is rather non-active radical and therefore it will much faster decay on the wall than react with other molecules in the gas phase. Thus, for the simplicity the reaction (5) can be considered as the termination reaction and HO2 as a final product of the total reaction. Let us consider the behavior of the reactions (4) and (5) in more details:

3.2.1. Reactions of the Surface Termination

Chain termination as a result of the interaction of free radicals with the walls of the reaction cell plays very important role in many gas phase chain reactions, especially at low pressures. Destruction of radicals on the wall is a result of two consecutive processes: diffusion of the radicals to the wall and the interaction of the radical with the surface. If the first process is the rate limiting step (diffusion is slower) the reaction is said to take place in the diffusion region. In opposite case, it takes place in the kinetic region Effectively, the reaction occurs as a first order reaction with rate constant κ:

(6)
1  =  1  +  1
κ κd κs

where κd is so called diffusion rate constant which for the cell with characteristic size d is equal to

(7)
κd =  A·D  [sec-1]
d2

This reaction can be easily understood when compared with the Einstein diffusion equation x2=Dt. The coefficient A depends on the shape of the cell, for example A=39.6 for a sphere, and A=23.3 for a very long cylinder. The diffusion coefficient D depends on the temperature and pressure as

(8)
D = D0  (T/T0)½
P/P0
where D0 is the diffusion coefficient at normal conditions, T0 and P0. Thus, the rate constant κd is
(9)
κd ∝ D0  T
Pd2
The rate constant of the reaction on the surface can be found from the gas kinetic theory:

(10)
κs =  ε·v·S  Exp(—  E )
4·V R·T

Here ε is the chain termination efficiency in the radical collision with wall, v is mean radical velocity, s is the surface and V is the volume of the cell. The coefficient ε strongly depends on the quality and the type of the wall surface, as well as on the type of the radical. Usually ε ≅ 10-5-10-2 for glass and quartz and ε ≅ 10-2-1 for metals.

3.2.2. Reaction of the Gas Phase Recombination

The gas phase recombination reaction (5) is called termolecular reaction. This reaction is not truly elementary one, but composed of two consecutive bimolecular step. To be accurate this reaction should be referred as a third order reaction since the term termolecular can only be applied to an elementary step. Let us consider the reaction of two atoms A and B which forms molecule AB:

(11) A+B+M → AB+M

The "third body" molecule, M, is needed since in the first elementary step:

(12)
ka
A+B  →  AB

the molecule AB is the hot molecule which has energy equaled to the formed bond energy plus kinetic energy of the atoms. Therefore without third molecule body M, the molecule AB will dissociate back very fast, during one molecular vibration:

(13)
kb
AB  →  A+B

The third body M can absorb excess of the energy from AB and convert it to the stable molecule AB:

(14)
kc
AB+M  →  AB+M

Applying the steady state hypothesis to the excited molecule AB which is short lived and is present in a very small concentration we obtain:

(15)
d AB  = ka·A·B -kbAB -kcAB·M
dt

Therefore in the steady state:

(16)
AB =  ka·A·B
kb+kc·M

The rate of product formation is:

(17)
d AB  = kc·AB·M =  ka·kc·A·B·M
dt kb+kc·M

In case of atoms or small radicals the lifetime of AB is very short, meaning, the rate constant kb is high. Therefore at pressures which are not very high (less than 1 atm) kb≫kc·M, and we have approximately (from Eq. 1) :

(18)
d AB  ≈  ka·kc·A·B·M
dt kb

Thus, if we consider this reaction as the reaction of third order beetween A, B, and M the third order rate constant is equal to κ(3) = κa·κcb. The units of this constant are [cm6/(mole2·sec)]. Alternatively, in this conditions, we can consider the bimolecular reaction A + B → AB with rate constant which is linearly depends on concentration M: κ(2) = (κa·κcb)·M. In this case the units are cm3/(mole·sec). For the more complex radicals the lifetime of the excited molecule AB is longer since the excess of energy can be redistributed to the different internal degrees of freedom. In this case even at some pressures κb≪κc·M. Then we have a second order reaction with rate constant κa.

Let us consider in more details the first case of the third order reaction. The rate constant of the deactivation or relaxation κc strongly depends on the type of the deactivator, and is related to its ability to absorb the excess energy. First the rate constant is higher for more light molecules, since they can convert more energy to the kinetic energy. Second, it is higher for molecules with more degrees of freedom. In this case the energy transferred from the molecule AB to the vibration or rotation of the molecule M. Usually, the total rate constant of the reaction decreases with the increase of temperature. It can be explained in the following way: the rate constant κa practically does not depend on the temperature, since it has no or minimal energy of activation. Opposite, we can expect that the rate constant κb will grow with temperature, since the temperature (kinetic energy of atoms and molecules) will increase the excess energy which has the excited molecule AB for dissociation. The reaction of relaxation with rate constant κc has low temperature dependence. Thus, because of the growth of κb with temperature, the total rate constant will decrease. It can be treated mathematically as the reaction with "negative" activation energy.

3.3. Ordinary Differential Equations to Solve the Kinetics Problem

In order to solve the kinetic problem we need to build a system of coupled differential equations, using the appropriate rate constants κ1b for reactions (1) - (5), as well as the rate of radical initiation ωi:

(19)
d H  = ωi1·H·O22·O·H23·OH·H24·H-κ5·H·O2·M
dt
(20)
d O  = κ1·H·O22·O·H2
dt
(21)
d OH  = κ1·H·O22·O·H23·OH·H2
dt
(22)
d H2  = -κ2·O·H23·OH·H2
dt
(23)
d O2  =-κ1·H·O25·H·O2·M
dt
(24)
d H2O  = κ3·OH·H2
dt

This is a system of nonlinear differential equations which cannot be solved analytically. During the computer experiment you will solve it numerically. On the other hand, in the framework of so call quasi-steady state approximation this system can be treated analytically. within the limits of this approximation we will consider only the three first equations, since we will be interested in the early moments of the explosion when change in hydrogen and oxygen concentration is low. Since the concentration of O and OH are usually much lower than that of H, we can suggest that (dO/dt)=(dOH/dt)=0. Then:

(25)
d H  = ωi+(2κ1·O245·O2·M)·H
dt

or

(26)
d H  = ωi+φ·H
dt

where branching factor φ is the difference between the rate of the branching reactions and the termination reactions:

(27)
φ ≡ 2κ1·O245·O2·M

In case of constant φ (neglecting the change of oxygen molecule concentration) the solution of the Eq (26) is:

(28)
H =  ωi [exp(φ·t)-1]
φ

The system behavior falls into two clear-cut cases depending on whether the branching factor, φ, is greater or less than zero. When φ<0 reaction shows "normal" behavior approaching steady state (see Fig. 3). When φ>0 the variation of rate with time follows a continuously accelerating, exponential, rise as shown in Fig. 3. The sharp division between these two types of behavior is expressed by the condition, &phi=0.

Figure 3: A plot of [exp(φ·t)-1]/φ versus t for φ<0 and φ>0 showing the difference between steady and explosive reaction.

This shows that there is an explosion limit at which a very small change in conditions, such as the temperature, pressure or composition of the reactant, can change the reaction rate from a slow steady value to one which is self-accelerating and would eventually become infinitely fast, other factors remain unchanged. When the rate of reaction reaches a certain value the observable characteristics of an explosion appear such as the shock wave causing a bang to be heard, the high temperature causing a flash of light to be seen, etc. There is however a time-lag before the rate reaches this critical value, that is there is an induction period before explosion. This may be of the order of milliseconds or minutes depending on the value of φ.

Let Hcrit be the critical active center concentration at which the rate is just fast enough to produce the observable criteria of explosion. Then if ti is the induction period:

(29)
ti 1 ln(1+ φ·Hcrit )
φ ωi

The logarithmic term in this equation varies relatively slowly with φ so we get a crude approximation: ti ∼ 1/φ.

This result shows that large net branching factors result in short induction periods before explosion while net branching factors only very slightly in excess of zero lead to very long induction periods. Although this discussion of explosion limits has been restricted to linearly branched and terminated chains for simplicity, it nevertheless enables us to interpret the experimental data of a number of branched chain reactions.

Thus, first and second explosion limits can be found from the condition φ = 0:

(30)
0 = 2κ1·O245·O2·M

Let us consider mixture of the hydrogen and oxygen only (without inert gas) with mole fraction of oxygen α. At constant pressure the concentrations of species can be measured in pressure units. Therefore O2 = α·P and M=P. We can also consider the wall decay in a similar way to the diffusion region (see Eq. (7) and (8)):

(31)
1·α·P— κT  —κ5·α·P2 = 0
P·d2

where

(32)
κ A·D0·P0
T0

This is a cubic equation for P and as can be seen from the signs of the terms, two of the roots are positive and one is negative. The latter cannot correspond to any physically observable system but the two positive roots account for the lower and upper explosion limits that are observed. At the lower limit Pl, the principal termination reaction is the surface one and the third term of the equation, the gas-phase termination, is small so that we have approximately,

(33)
1·α·Pl κT  = 0
Pl·d2

and

(34)
Pl = ( κT ) ½
1·α·d2

At the upper limit Pu, gas-phase termination is more important than surface termination so that equation becomes approximately,

(35)
1·α·P —κ5·α·P2 = 0

or

(36)
Pu 2·κ1
κ5

We can replace the rate constants in equations 34 and 36 with Ahrenious equation and by that derive the relation between the upper and lower pressure limits and the activation energies.