5.60 Spring 2007 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 2007 Lecture #33 page 2 Proposed mechanism for this reaction: k1 Initiation: CH3CHO Propagation: CH3• + CH3CHO k3 •CH3CO Termination: 2 CH3• “Side” Reactions: CH3• + CHO• k4 k2 •CH3CO + CH4 CH3• + CO C2H6 CHO• + M k5 H• +CH3CHO CO + H• + M k6 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 •] d[• CH3 CO] dt 2 = k2 [CH3 •][CH3 CHO] − k3 [• CH3 CO] 5.60 Spring 2007 Lecture #33 Assume S. S. approximation d[CH3 •] dt = d[• CH3 CO] dt ([Intermediates = small]) =0 Steady State Approximation ⇒ So… [CH3 •]SS d[CH4 ] dt ⎛ k = ⎜⎜ 1 ⎝ 2k 4 ⎞ ⎟ ⎟ ⎠ ⎛ k = k2 ⎜⎜ 1 ⎝ 2k 4 1/2 ⎞ ⎟ ⎟ ⎠ [CH3 CHO]1 / 2 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 ⎞ 3/2 k2 ⎜⎜ 1 ⎟⎟ [CH3 CHO] ⎝ 2k 4 ⎠ = k1 [CH3 CHO] = k2 2k1k 4 page 3 [CH3 CHO]1 / 2 (experimentally =300) 5.60 Spring 2007 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: RI Initiation: H2 Branching: H• + O2 O + H2 Propagation: Termination: OH• + H2 H• 2H• k1 k2 k3 k4T H• + O2 + M OH• + O OH•+ H• H• + H2O H k5T wall HO2• + M 5.60 Spring 2007 Lecture #33 page 5 Kinetic Equations d[H] = RI − k1 [H][O2 ] + k2 [O][H2 ] + k3 [OH][H2 ] − k4T [H] − k5T [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 ⇒ EXPLOSION (because not true) ** Solve for [Int.]SS [O]SS = k1 [H][O2 ] k2 [H2 ] [OH]SS = k1 [H][O2 ] + k2 [O]SS [H2 ] 2k1 [H][O2 ] = k3 [H2 ] k3 [H2 ] d[H] = RI + 2k1 [O2 ] − (k 4T [H]SS + k5T [H]SS [O2 ][M]) [H]SS = 0 dt SS { So… [H]SS = } RI k4T + k5T [O2 ][M] − 2k1 [O2 ] 5.60 Spring 2007 Lecture #33 page 6 Limiting Cases i) Low Pressure (k1[O2], k5T[O2][M]) << k4T Wall collisions dominate over branching ⇒ S.S. app. is valid No explosion 2k1[O2] ~ k4T + k5T[O2][M] ii) Medium Pressure 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 2007 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…).