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5.60 Thermodynamics & Kinetics
Spring 2008

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5.60 Spring 2008

Lecture #33

page 1

Complex Reactions and Mechanisms (continued)
IV) Chain Reactions Where a product from a step in the mechanism is a reactant for a previous step (i.e. the reaction feeds itself).

a) Stationary or stable chain reactions. The concentration of reactive intermediates is constant in time or slowly decreasing. Example: CH3CHO CH4 + CO Experimental observations: Small amounts of C2H6 and H2 are also produced, and the Rate of Reaction [CH3CHO]3/2. (These are signatures of a chain reaction mechanism)


5.60 Spring 2008

Lecture #33

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Proposed mechanism for this reaction:
k

Initiation:

CH3CHO

1

CH3· + CHO·
k

Propagation:

CH3· + CH3CHO ·CH3CO
k k
3

2

·CH3CO + CH4

CH3· + CO

Termination:

2 CH3·

4

C2H

6

"Side" Reactions:

CHO· + M

k

5

CO + H· + M
k
6

H· +CH3CHO

H2 + ·CH3CO

Kinetic Equations:
d[CH4 dt

]

= k2 [CH3 ·][CH3 CHO

]

d[CH3 · dt

]

= k1 [CH3 CHO ] - k2 [CH3 ·][CH3 CHO ] + k3 [· CH3 CO ] - 2k 4 [CH3 ·

]2

d[· CH3 CO dt

]

= k2 [CH3 ·][CH3 CHO] - k3 [· CH3 CO

]


5.60 Spring 2008

Lecture #33

page 3

Assume S. S. approximation
d[CH3 · dt

([Intermediates = small])

]

=

d[· CH3 CO dt

]

=0

Steady State Approximation



[CH3 ·]SS

k = 1 2k 4


1/2

[CH3

CHO

]1

/ 2


So...

d[CH4 dt

]

k = k2 1 2k 4

1/2

[CH3

CHO

]3

/2

_____________________

Chain Length: The # of propagation steps per initiation step = rate of product formation rate of initial radical formation
1/2

k k2 1 [CH3 CHO 2k 4 = k1 [CH3 CHO]
= k2 2k1k
4

]3

/2

[CH3

CHO

]1

/2

(experimentally =300)


5.60 Spring 2008

Lecture #33

page 4

b) Non-stationary or unstable chain reactions: The propagation includes a branching step, which increases the concentration of reactive intermediates.
EXPLOSION!!

Example: the combustion of hydrogen to form water

2H2 + O2 2H2O

Mechanism:
Initiation: Branching: H2
R
I

2H·
k k
1

H· + O2 O + H2

OH· + O OH·+ H·

2

Propagation: Termination:

OH· + H2 H·

k k

3

H· + H2O H
k
T 5

T 4

wall HO2· + M

H· + O2 + M


5.60 Spring 2008

Lecture #33

page 5

Kinetic Equations
d[H] T T = RI - k1 [H][O2 ] + k2 [O][H2 ] + k3 [OH][H2 ] - k4 [H] - k5 [H][O2 ][M] dt
d[O] = k1 [H][O2 ] - k2 [O][H2 ] dt

d[OH] = k1 [H][O2 ] + k2 [O][H2 ] - k3 [OH][H2 ] dt

Assume S. S. Approximation ([Intermediates]=small, d[Int.]/dt~0)

** If S.S. App. Fails Solve for [Int.]
SS



EXPLOSION (because not true) **

[O]SS =

k1 [H][O2 ]
k2 [H2 ]

k1 [H][O2 ] + k2 [O]SS [H2 ] 2k1 [H][O2 ] = k3 [H2 ] k3 [H2 ]

[OH]SS =
d[H] dt

SS

T T = RI + 2k1 [O2 ] - (k 4 [H]SS + k5 [H]SS [O2 ][M]) [H]SS = 0

{

}

So... [H]SS =

R k+
T 4 T k5

I

[O2 ][M] - 2k1 [O2 ]


5.60 Spring 2008

Lecture #33

page 6

Limiting Cases

i) Low Pressure

(k1[O2], k5T[O2][M]) << k4

T

Wall collisions dominate over branching S.S. app. is valid No explosion

ii) Medium Pressure

2k1[O2] ~ k4T + k5T[O2][M]

Branching is important, [H]SS is large! S.S. is NOT valid! EXPLOSION!!

iii) Higher Pressure

k5T[O2][M] > 2 k1[O2]

Termination dominates over branching S.S. is valid No explosion

iv) Very High Pressure HO2· + H2 H2O + OH· becomes important

This feeds OH· EXPLOSION!!


5.60 Spring 2008

Lecture #33

page 7

Stable Reaction Pressure


Explosion

Temperature

P-T phase diagram for stability of Hydrogen Combustion.

Branching chain reactions also occur in nuclear reactions, for example. In fission reactions of 235U, 3 neutrons are produced for each neutron that starts the reaction. In a nuclear reactor, control rods that absorb neutrons terminate the chain and moderate the reaction (unless the operator is reckless and forgets to insert the rods, in which case there is a meltdown of the reactor core...).