COMPLEXITY OF REAL REACTIONS .1 h ydROGen i Odide

Chapter 2 Introduction to Kinetic Processes in Materials 41 Chapter 3 Second-Order and Multistep Reactions 61 Chapter 4 Temperature Dependence of the Reaction Rate Constant 95

3.6 COMPLEXITY OF REAL REACTIONS .1 h ydROGen i Odide

d B

dt k A k B

[ ]

₌ ¹

[ ]

^{− 2}

[ ]

⁼⁰ (3.44)

this leads immediately to Equations 3.42 and 3.43 and is, of course, tantamount to the assump- tion that k2 ≫ k1, for which, as Equation 3.42 shows, [B] does not vary rapidly with time because it depends on e^{−k t1} and k1 is small. Of course, Equations 3.42 and 3.44 are mathematically inconsistent, but the result is a reasonable approximation if k2 ≫ k1.

3.5.3.3 Example: Nuclear Decay

In a nuclear reactor, the roughly ²³⁵⁹²U enriched to 4% or 5% of the uranium present undergoes fission generating energy and several neutrons. Some of these neutrons are captured by the 96% ²³⁸⁹²U to give

23992U, which decays to ²³⁹⁹³Np and then to ²³⁹⁹⁴Pu by the following reactions:

23992 23 5 23993

23993 2 35 23994

1 28

U Np MeV

Np ^d P

. min

.

→ + + .

→

β−

uu MeV

Pu ^yr U He MeV

+ +

→ + +

β− 0 23 5 23

23994 24 000 23592

24

.

. .

,

(3.45)

If a reactor is operating at a constant power level for a long period of time, typically several years, then the amount of ²³⁹⁹²U will stay at some small time-independent value as will the amount of ²³⁹⁹³Np because of their relatively short half-lives. On the other hand, the amount of ²³⁹⁹⁴Pu in the reactor core will continually increase. However, it too will reach a steady-state concentration because ²³⁹⁹⁴Pu is also fissionable and can be used as fuel in nuclear power reactors, as it is in France. Nevertheless, it would be expected that the relative steady-state concentrations of ²³⁹⁹²U and²³⁹⁹³Np, which from Equations 3.44 and 3.45, would be ²³⁹⁹³Np / ²³⁹⁹²U =2 35 23 5. d/ . min=720.

Of course, the concentration in the atmosphere of radioactive ¹⁴⁶C used for radiocarbon dating is a perfect example of a steady-state concentration between its formation by cosmic rays and its own nuclear decay.

3.6 COMPLEXITY OF REAL REACTIONS

mol/cm³. EA = Q, the activation energy in this case is EA = 171 kJ/mol.^* However, the reality even for most gas-phase reactions is that they are not simple single-step reactions but consist of many serial and parallel steps.

3.6.2 c

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Similar to the reaction for the production of HI, the formation of HCl can be written as H g2( )+Cl g2( )2HCl g( ).

However, for this reaction the kinetics are more complex (Kee et al. 2003). In fact, the kinetics of this reaction represent a chain reaction. In a chain reaction, there is an initiation step, a chain- propagating step, and a termination step. For this reaction

Initiation: Cl2Clⁱ+Clⁱ

Propagation: Clⁱ+H2HCl H+ ⁱ and Cl2+HⁱHCl Cl+ ⁱ Termination: Clⁱ+ClⁱCl2 and Clⁱ+HⁱHCl.

Clearly, the propagation step leads to a continuation of the reaction by producing hydrogen and chlo- rine radicals (the species with the “dot”) to react with Cl2 and H2, respectively, to continue the reac- tion to HCl.^† In principle, the differential equations for the concentrations of all of these species can be developed and, because the reaction rate constants for the reactions exist in the NIST Kinetics database, the equations for the concentrations of the various species could be solved as a function of time. However, rather than solving the equations analytically, it is easier to solve these simultaneous differential equations numerically. In fact, commercial software programs exist to perform complex gas-phase kinetic calculations. Such a program was developed at the Sandia Livermore Laboratory and is now available commercially (Kee et al. 2003; CHEMKIN). A similar series of reactions are involved in the formation of HBr by the reaction (Chang 2000)

H g2( )+Br g2( )2HBr g( ).

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OlymeRizatiOn 3.6.3.1 Mechanism

There are many different mechanisms for polymerization of monomers. In fact, a central feature of polymer science is the types of polymerization processes that can take place, how they differ for different polymers, the effects of temperature and concentration, and—most important—the role of various catalysts. The intention here is not to try to summarize all of this information that is best left to polymer chemistry. However, one mechanism that fits very nicely into the cur- rent discussion, and gives an example of one of the processes, radical chain polymerization that includes steps of initiation, propagation, and termination to form polymer chains. The process is covered in many polymer textbooks (Billmeyer 1984; Saunders 1988; Young and Lovell 1991;

Elias 1997).

* “A” and “EA” are the symbols used in the NIST Database for the “k0” and “Q” terms used here.

† The “dot” superscript represents a reactive radical, an atom or molecule with an unsatisfied electron bond—called free radicals in older literature. The literature is somewhat inconsistent in the designation of a radical. Sometimes the dot is placed directly on top of the symbol (Kee et al. 2003), sometimes to the side (Silbey and Alberty 2001), and sometimes it is not used at all (Laidler 1987; Houston 2001), but most frequently it is raised and to the right (or left of the atom with the unsatisfied bond) (Cowie 1991) as used here.

Figure 3.11 shows a typical initiator material, benzoyl peroxide, that on heating, splits into two parts, gives off CO2, and provides two radicals, I^• , that can act as chain initiation for polymerization. This radical reacts with a mono- mer molecule

Initiation step: I^•+CH2=CRH⇒ −I CH2−C RH^• Propagation step: I CH C RH CH CRH

I CH CRH CH C RH etc

− − + = ⇒

− − − −

2 2

2 2

•

• , .

Termination step: ← − − + − − →

← − − − − − →

CH C RH C RH CH CH CRH CRH CH

2 2

2 2

• •

In these equation, R = H, Cl, CH3, and so on and the single arrows indicate the extension of the polymer chain in both the positive and negative directions in the termination step.

3.6.3.2 Rate of Polymerization Initiation can be represented by two steps:

I nI

I M IM

slow

fast i

 →

+  →

·

· · (3.47)

where M is the monomer molecule. Let RI be the rate of initiator formation RI=d I dt[ ]/^• and kp the rate of chain propagation: Mi M k M

p i

+  → +1 so the rate of consumption of the monomer is

−

[ ]

₌

 

[ ]

⁺ _{ }

[ ]

_{+  }

[ ]

d M

dt k Mp 1i M k Mp 2i M Mii M

−

[ ]

₌

[ ]

_{  =}

[ ]

_{ }

=

∑

∞

d M

dt k Mp Mi k M M

i

p

• •

1

(3.48)

where [M^•] is the total concentration of all the radical species. The rate of termination is given by Mi Mj k M

t i j

•+ • → + where kt is the rate of termination so

−[M]= [ ] . dt k Mt

•

2 • ² (3.49)

The 2 comes from the fact that two radical species are destroyed in a termination reaction. Steady- state conditions are assumed; that is, d I dt[ ]^• / = −( [d M dt^•] )/ so RI=2k Mt[ ^•]² or [M^•]=

(

RI^/2kt

)

^{1 2/} . Substituting this into Equation 3.48 and calling −d M dt R[ ] = p, the “rate of polymerization,” the result is

R k

k R M

p p

t

=  I

 



[ ]

2^{1 2 1 2}

1 2

/ /

/ (3.50)

which indicates that the rate of monomer consumption is proportional to the monomer concentra- tion with the proportionality consisting of the various rate constants. This is a first-order equation, at least in the specified steady state.

O O O

O C

O

2 O 2 + 2 CO₂

FIGURE 3.11 Benzoyl peroxide, a typical chain initiator for radical polymerization undergoing thermolysis— decomposition by heat—with the breaking of the oxygen–oxygen bond and release of CO2 to give two radicals, that can act as chain initiators, I^• .

3.6.4 c

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Many combustion reactions include chain branching reactions and can lead to explosive mixtures.

For the very familiar simple reaction of the combustion of hydrogen with oxygen to form water vapor

H g2( )+1 2O g2( )→H O g2 ( )

the reaction has a total of 19 identified separate steps including the following as examples (Li et al.

2004):

Initiation: H2+O2Hⁱ+HO2ⁱ

Propagation: H O H HO

OH H H O H

2 2 2

2 2

+ +

+ +

i i

i i

Branching: Hⁱ+O2Oⁱ+ⁱOH Termination: HO2i+iOHH O O2 + 2

the solution of which, of course, requires some sort of numerical solution (Kee et al. 2003) to obtain the rate of the reaction and/or the concentrations of the various possible gas species as a function of time and temperature.