Mechanisms and Rate Laws
7.1 SIMPLE HOMOGENEOUS REACTIONS .1 Types of Mechanisms
7.1.3 Closed-Sequence Mechanisms; Chain Reactions
In some reactions involving gases, the rate of reaction estimated by the simple collision theory in terms of the usually inferred species is much lower than observed. Examples of these reactions are the oxidation of H, and of hydrocarbons, and the formation of HCl and of HBr. These are examples of chain reactions in which very reactive species (chain carriers) are initially produced, either thermally (i.e., by collision) or photochemically (by absorption of incident radiation), and regenerated by subsequent steps, so that re- action can occur in chain-fashion relatively rapidly. In extreme cases these become “ex- plosions,” but not all chain reactions are so rapid as to be termed explosions. The chain
constitutes a closed sequence, which, if unbroken, or broken relatively infrequently, can result in a very rapid rate overall.
The experimental detection of a chain reaction can be done in a number of ways:
(1) The rate of a chain reaction is usually sensitive to the ratio of surface to vol- ume in the reactor, since the surface serves to allow chain-breaking reactions (recombination of chain carriers) to occur. Thus, if powdered glass were added to a glass vessel in which a chain reaction occurred, the rate of reaction would decrease.
(2) The rate of a chain reaction is sensitive to the addition of any substance which reacts with the chain carriers, and hence acts as a chain breaker. The addition of NO sometimes markedly decreases the rate of a chain reaction.
Chain carriers are usually very reactive molecular fragments. Atomic species such as Ho and Cl’, which are electrically neutral, are in fact the simplest examples of “free radicals,” which are characterized by having an unpaired electron, in addition to being electrically neutral. More complex examples are the methyl and ethyl radicals, CHj and C,H;, respectively.
Evidence for the existence of free-radical chains as a mechanism in chemical reac- tions was developed about 1930. If lead tetraethyl is passed through a heated glass tube, a metallic mirror of lead is formed on the glass. This is evidently caused by de- composition according to Pb(C,H,), -+ Pb + 4qHt, for if the ensuing gas passes over a previously deposited mirror, the mirror disappears by the reverse recombination:
4C,H; + Pb -+ Pb(C,Hs),. The connection with chemical reactions was made when it was demonstrated that the same mirror-removal action occurred in the thermal de- composition of a number of substances such as ethane and acetone, thus indicating the presence of free radicals during the decomposition. More recently, spectroscopic techniques using laser probes have made possible the in-situ detection of small concen- trations of transient intermediates.
We may use the reaction mechanism for the formation of ethylene from ethane (GH, + C,H, + HZ), Section 6.1.2, to illustrate various types of steps in a typical chain reaction:
chain initiation: C,H, -+ 2CHj (1)
chain transfer: CH; + C,H, + CH, + C2H; (2)
chain propagation: C,H; + C,H, + Ho (3)
Ho + C,H, + H, + C,H; (4)
chain breaking or termination: Ho + C,H; + C2H, (5) In the first step, CHT radicals are formed by the rupture of the C-C bond in GH,.
However, CHj is not postulated as a chain carrier, and so the second step is a chain- transfer step, from CHT to GHt, one of the two chain carriers. The third and fourth steps constitute the chain cycle in which C,HS is first used up to produce one of the products (C,H,) and another chain carrier (HO), and then is reproduced, to continue the cycle, along with the other product (HZ). The last (fifth) step interrupts a chain by removing two chain carriers by recombination. For a rapid reaction overall, the chain propagation steps occur much more frequently than the others. An indication of this is given by the average chain length, CL:
cL = number of (reactant) molecules reacting number of (reactant) molecules activated
= rate of overall reaction/rate of initiation (7.1-2)
7.1 Simple Homogeneous Reactions 159 Chain mechanisms may be classified as linear-chain mechanisms or branched-chain mechanisms. In a linear chain, one chain carrier is produced for each chain carrier re- acted in the propagation steps, as in steps (3) and (4) above. In a branched chain, more than one carrier is produced. It is the latter that is involved in one type of explosion (a thermal explosion is the other type). We treat these types of chain mechanisms in turn in the next two sections.
7.1.3.1 Linear-Chain Mechanisms
We use the following two examples to illustrate the derivation of a rate law from a linear-chain mechanism.
(a) A proposed free-radical chain mechanism for the pyrolysis of ethyl nitrate, C,HsONO, (A), to formaldehyde, CH,O (B), and methyl nitrite, CH,NO, (D), A + B + D, is as follows (Houser and Lee, 1967):
A-+C,H,O’ + NO,kl (1)
C,H,O’ % CH; + B (2)
CHj + A-D + C,H,O*k3 (3)
2CzH500&H,CH0 + C,H,OH (4)
Apply the stationary-state hypothesis to the free radicals CH; and C,HsO* to derive the rate law for this mechanism.
(b) Some of the results reported in the same investigation from experiments carried out in a CSTR at 250°C are as follows:
c,/mol mP3 0.0713 0.0759 0.0975 0.235 0.271 (- rA)/mol rnP3s-r 0.0121 0.0122 0.0134 0.0209 0.0230 Do these results support the proposed mechanism in (a)?
(c) From the result obtained in (a), relate the activation energy for the pyrolysis, EA, to the activation energies for the four steps, EA1 to EA4.
(d) Obtain an expression for the chain length CL.
SOLUTION
(a) The first step is the chain initiation forming the ethoxy free-radical chain carrier, C,H,O’, and NO,, which is otherwise unaccounted for, taking no further part in the mech- anism. The second and third steps are chain propagation steps in which a second chain car- rier, the methyl free radical, CH;, is first produced along with the product formaldehyde (B) from C,H,O’, and then reacts with ethyl nitrate (A) to form the other product, methyl nitrite (D), and regenerate C,H,O’. The fourth step is a chain-breaking step, removing C,H,O.. In a chain reaction, addition of the chain-propagation steps typically gives the overall reaction. This may be interpreted in terms of stoichiometric numbers (see Example 7-1) by the assignment of the value 1 to the stoichiometric number for each propagation step and 0 to the other steps.
To obtain the rate law, we may use (-Y*) or rn or rn. Choosing r,, we obtain, from step (21,
rB = 'bCc,H50*
We eliminate ~~.u,~. by applying the stationary-state hypothesis to C,HsO’, ~C2u5@ = 0, and also to the other chain carrier, CHj.
rCzHsO’ = kc, - hCC2H50* + hcAcCH; - 2k4c&2Hs0a = o
‘Cl-I; = k2Cc2H5v - k3CACCH; = o
Addition of these last two equations results in
CC2H50* = (WW 112 112 cA
and substitution for cc2n500 in the equation for ru gives rB = k2(k1/2k,)“2c~2
which is the rate law predicted by the mechanism. According to this, the reaction is half- order.
(b) If we calculate the vah.te of kobs = (-rA)/cA1’2 for each of the five experiments, we obtain an approximately constant value of 0.044 (mol m-3)1’2 s-t. Testing other reaction orders in similar fashion results in values of kobs that are not constant. We conclude that the experimental results support the proposed mechanism.
(c) From (b), we also conclude that
ko b s = k2(k1/2k4) 112 from which
dlnkobs _ dlnk2 1 dln k, 1 dln k4
- c - - - - - -
dT +z dT 2 dT
or, from the Arrhenius equation, 3.1-6,
EA = EA, + $A, - EAI)
(d) From equation 7.1-2, the chain length is
CL = k2(k1/2k4) 1’2ca/2/kl cA
= k2(2kl k4cA)-“2
The rate law obtained from a chain-reaction mechanism is not necessarily of the power-law form obtained in Example 7-2. The following example for the reaction of H, and Br, illustrates how a more complex form (with respect to concentrations of reactants and products) can result. This reaction is of historical importance because it helped to establish the reality of the free-radical chain mechanism. Following the ex- perimental determination of the rate law by Bodenstein and Lind (1907), the task was to construct a mechanism consistent with their results. This was solved independently by Christiansen, Herzfeld, and Polanyi in 1919-1920, as indicated in the example.
The gas-phase reaction between H, and Br, to form HBr is considered to be a chain reac- tion in which the chain is initiated by the thermal dissociation of Brz molecules. The chain
7.1 Simple Homogeneous Reactions
161
is propagated first by reaction between Br’ and H, and second by reaction of Ho (released in the previous step) with Br,. The chain is inhibited by reaction of HBr with Ho (i.e., HBr competes with Br, for Ho). Chain termination occurs by recombination of Br’ atoms.
(a) Write the steps for a chain-reaction mechanism based on the above description.
(b) Derive the rate law (for rnnr) for the mechanism in (a), stating any assumption made.
SOLUTION
(a) The overall reaction is
H, + Br, + 2HBr The reaction steps are:
initiation: Br, 5 2Br’ (1)
propagation: Br’ + H, -%HBr + H’ (2)
Ho + Br, 2 HBr + Bf (3)
inhibition (reversal of (2)): Ho + HBr%H2 + Br* (4)
termination (reversal of (1)): 2 Br’ 3 Br, (5)
(b) By constructing the expression for rnnr from steps (2), (3), and (4), and then elimi- nating cn,.. and cn. from this by means of the SSH (rBr. = rn. = 0), we obtain the rate law (see problem 7-5):
2k3(qk-,xk,lk-,)
rHBr =
(k&Z)
+ hB&3r2) (7.1-3)
This has the same form as that obtained experimentally by Bodenstein and Lind earlier.
This rate law illustrates several complexities:
(1)
The effects on the rate of temperature (through the rate constants) and concen- tration are not separable, as they are in the power-law form of equation 6.1-1.(2) Product inhibition of the rate is shown by the presence of cHBr in the denomi- nator.
(3) At a given temperature, although the rate is first-order with respect to H2 at all conditions, the order with respect to Br, and HBr varies from low conversion (kslk-, > cHnrlcnrJ, (1/2) order for Br, and zero order for HBr, to high conver- sion (k3/k2 =K cnnr/cnrz), (3/2) d f Bor er or r2 and negative first-order for HBr. It was such experimental observations that led Bodenstein and Lind to deduce the form of equation 7.1-3 (with empirical constants replacing the groupings of rate constants).
7.1.3.2
Branched-Chain Mechanisms; Runaway Reactions (Explosions)In a branched-chain mechanism, there are elementary reactions which produce more than one chain carrier for each chain carrier reacted. An example of such an elementary reaction is involved in the hydrogen-oxygen reaction:
0’ + H, --$ OH’ + Ho
Two radicals (OH’ and Ho) are produced from the reaction of one radical (0’). This allows the reaction rate to increase without limit if it is not balanced by corresponding radical-destruction processes. The result is a “runaway reaction” or explosion. This can be demonstrated by consideration of the following simplified chain mechanism for the reaction A + . . . + P.
initiation: A&R’
chain branching: R’ + A%P + nR* (FZ > 1) (If 12 = 1, this is a linear-chain step)
termination: R’k3-X The rate of production of R’ is
rR. = klcA + (n - l)k2cAcR. - k3cR.
= klcA + [(n - 1) kZcA - k3]cR.
(7.1-4)
A runaway reaction occurs if
drRJdc,.[= (n - l)k2cA - k3] > 0 or (n - l)k,c, > k3
which can only be the case if 12 > 1. In such a case, a rapid increase in cn. and in the overall rate of reaction (rp = k2cAcR.) can take place, and an explosion results.
Note that the SSH cannot be applied to the chain carrier R* in this branched-chain mechanism. If it were applied, we would obtain, setting rRo = 0 in equation 7.1-4,
c,.(SSH) = kc,
k3 - (n - l)kzCA <o if (n - 1) kZcA > k3 which is a nonsensical result.
The region of unstable explosive behavior is influenced by temperature, in addition to pressure (concentration). The radical destruction processes generally have low acti- vation energies, since they are usually recombination events, while the chain-branching reactions have high activation energies, since more species with incomplete bonding are produced. As a consequence, a system that is nonexplosive at low T becomes ex- plosive above a certain threshold T . A species Y that interferes with a radical-chain mechanism by deactivating reactive intermediates (R* + Y + Q) can be used (1) to increase the stability of a runaway system, (2) to quench a runaway system (e.g., act as a fire retardant), and (3) to slow undesirable reactions.
Another type of explosion is a thermal explosion. Instability in a reacting system can be produced if the energy of reaction is not transferred to the surroundings at a sufficient rate to prevent T from rising rapidly. A rise in T increases the reaction rate, which reinforces the rise in T . The resulting very rapid rise in reaction rate can cause an explosion. Most explosions that occur probably involve both chain-carrier and thermal instabilities.
7.1 Simple Homogeneous Reactions 163