Combustion Physics (Day 1 Lecture) Chung K. Law Robert H. Goddard Professor Princeton University

Princeton-CEFRC-Combustion Institute Summer School on Combustion June 20-24, 2016

1 Day 1: Chemical Thermodynamics and Kinetics

1. Chemical Thermodynamics • Chemical equilibrium

• Energy conservation & adiabatic flame temp., Tad 2. Chemical Kinetics • Reaction rates and approximations • Theories of reaction rates • Straight and branched chain reactions 3. Oxidation Mechanisms of Fuels • Hydrogen, CO, hydrocarbons

2 1. Chemical Thermodynamics

3 Chemical Equilibrium (1/2)

• First and Second Laws

d E  T d S - p d V (1.2.4)*

• Thermodynamic function: 퐸 = 퐸(푆, 푉, 푁푖)

N (1.2.6) d E = T d S - p d V +  d N  ii i 1 • Criterion for chemical equilibrium N (1.2.11)  dN  0  ii i 1

* Equation numbers refer to those in Combustion Physics 4 Chemical Equilibrium (2/2)

• From element conservation for a general chemical reaction NN vvMM  e.g.: H + O ↔ OH + O i ii i 2 ii11 dN dN i j (1.2.14) we have  v v  v  v  i i j j • Consequently, chemical equilibrium: N N  dN  0  (vv  ) 0  ii   i i i (1.2.17) i 1 i 1

5 Equilibrium Constant for Reaction

(,)T po () T R o T In(/), p p o • Chemical potential: i i ii (1.2.29) N :  ( vv    ) 0 • Apply equilibrium criterion  i i i N ()vv  i 1 pKii T  (),  ip i  1 (1.2.30)  N  K( T ) e x p  ( v  v  ) o ( T ) ()RTo pi i i    (1.2.31)  i 1  o (LHS; RHS): Function of (concentrations; temperature)

o 퐾푝(푇): Tabulated for a given reaction Equilibrium composition of a (fuel, oxidizer, inert) mixture at T and p can be calculated using the 퐾푝(푇)’s for all reactions • Comment: Procedure is cumbersome: needs tabulation for each reaction; there could be 100’s to 1000’s of reactions for oxidation of a fuel 6

Equilibrium Constant for Formation

• Simplification: Relate 퐾푝(푇) to formation reaction of

each species L v  MMo    i ,, j i ji   (1.2.32) j 1

o M ij , : element in standard state o o o (1.2.33) KTTRTp, i() exp[ i ()/( )] ()vv  N ii KTKT ()()  o (1.2.34) p p, i i 1 • Comment: Number of species ~10’s to 100’s => Much reduced tabulation

7

Energy Conservation in Adiabatic System (State 1 State 2 )

N N N h T ; T o   N h T ; T o  •  i ,1 i 1  i , 2 i 2 (1.4.17) i 1 i 1

h T;; To h o T o h s T T o • i   i  i   (1.4.16) totalenthalpy heatofformation sensibleheat 0 o Heat of Formation, h i ( T ) : heat required to form one mole of reactants from its elements in their standard state, at a L reference temperature T o , v  M o  M .  j  i i , j i , j i Sensible Heat: energy required to raise the temperature of T o substance from T to T , hso(;)(). T T  c T d T (1.4.14) io p, i T Specific Heat: function of degree of excitation,

c T   h s  T . (1.4.13) p ,i    i  p

NNNN • NhT o()()(;)(;). o NhT o o  NhTT s o  NhTT s o (1.4.18) i,1 i  i , 2 i  i , 2 i 2  i ,1 i 1 i1 i  1 i  1 i  1 8

Adiabatic Flame Temperature (1/2) • For a mixture (fuel, oxidizer, inert) of given composition and

temperature T1 , determine the final, equilibrium temperature, Tad , and product composition; the process is conducted adiabatically and at given pressure p.

o Tad calculated using 퐾푝(푇), energy conservation, and element conservation o This is the single, most important parameter for a reacting mixture

• Tad is most sensitive to mixture composition ( F / O ) o Fuel/oxidizer equivalence ratio, f  . ( F / O ) st o Tad peaks slightly on the rich side of f = 1

o Excess reactants (CO, H2 for rich mixtures) serve as inert.

o Amount and cp of inert serving as a heat sink. 9 Adiabatic Flame Temperature (2/2)

• Slope asymmetry of Tad (f) largely due to asymmetrical definition of f ;

lean ; rich    0  f  1; 1  f   

• Near-symmetry attained by using normalized equivalence f ratio,   ; 1  f

lean ; rich   0    0 .5; 0 .5    1

• Tad slightly increases with increasing pressure due to reduced product dissociation

10

2. Chemical Kinetics

11 Introduction

• Chemical thermodynamics: relates the initial to the final equilibrium states of a reactive mixture; does not distinguish the path and time in the process (e.g.: acceptable cycle

analysis for i.c. engines, but not NOx formation)

• Chemical kinetics describes the path and rates of individual reactions and reactants; can be extremely complex – 103 intermediates and 104 elementary reactions.

12 Law of Mass Action

NN k vv M  f • For a single-step forward reaction: i ii i , ii11 ˆ o Molar rate of change:  i  dc i / dt ˆ ˆ  j o i and j related by: i   , (2.1.3) v v  v  v  • Law of mass action: i i j j

o Reaction rate proportional to product of concentrations (ci) N o Scaled reaction rate is given by:   (2.1.4)   k T c i f    i i 1

o Proportionality constant kf(T) : reaction rate constant; only function of temperature

• Example: H + HO 2→OH + OH

dd[H ]d [HO]2 1 [O H ]  = k [H][HO ].  =  = = , f 2 d t d t2 d t

13 Reverse Reaction

• Every forward reaction has a backward reaction: NN k vvMM b  i i i i ii11     o Net reaction rate: iif ,,   ib ()()()v iif  v    b  v ii  v 

NN vv  k cii k c f i b i (2.1.6) ii11 N k f ()vv  = cKii o At equilibrium:   0   ic(2.1.7,8) k b i 1

NN vv 1   k cii K c o Implying: f i c i (2.1.9) ii11 • Preferred to (2.1.6) because Kc can be determined more accurately than kb N

v i • Irreversible reaction approximation:   kcf  i i

14

Multiple Reactions • Practical reactions involving Reactants → Products

e.g.: 2H2+O2→2H2O Rarely (never!) occurs in one step between reactants (e.g.: two H2 directly reacting with one O2)

• Reality: For H2-O2: (at least) 19 reversible reactions and 8 species (H2, O2, H, O, OH, H2O, HO2, H2O2)

• Generalized expression:

NN k kf, v M   v  M, k 1, 2 , , K , i,, k i   i k i k kb, ii11

NN vv   k ci,, k k c i k , k k,, fii k b ii11

K ().vv  i i,, k i k k (2.1.14) 15 k 1 Rational Approximations

• Approximations based on comparison of rates of certain reaction entities

o Quasi-steady-state (QSS) species approximation

o Partial equilibrium (PE) reaction approximation

16 QSS Species Approximation

• Some chain carriers are generated and consumed at rapid rates such that their concentrations remain at low values and their net change rates are very small.

dci  i =  =, ii o For dt

dci  o If  (,),ii dt  . o Then ii • Consequence: (implicit) algebraic instead of differential solution

• Note: dci/dt may not be negligible compared to other rates

17 Partial Equilibrium Approximation

• If both the forward and backward rates of a reaction k is much larger than its net reaction

rate, NN vv  k ci,, k  k c i k  0 o then set: k k,, f i k b i ii11

NN vv  o such that k ci,, k k c i k k,, f i k b i (2.1.17) ii11

o which yields an algebraic relation between ci’s.

• Note: k not necessarily small compared to  i. • Example: formulation of the transition state theory of reaction rates, to be shown later.

18 Approximation by Global and Semi- global Reactions • Successive application of QSS species and PE reactions will eventually lead to a one-step global reaction (at least theoretically!). o The process is tedious; results still depend on the individual reaction rate parameters most of which are not known. o Solution also requires iterations. • May as well just start with a one-step reaction Fuel + Oxidizer k  Products N n   kci , described by  i i 1 where ni is the reaction order, and is empirical

19

Reaction Order and Molecularity

• Molecularity, 휈푖 : number of colliding molecules in an elementary reaction

o 휈푖 is a fundamental parameter; 휈푖 = 1, 2, 3. o 휈푖 = 3 is important for recombination reactions (negative influence on progression of reaction

e.g. H+O2+M → HO2+M. • Reaction order, 푛푖 : influence of concentration of i on the reaction rate

o 푛푖 is an empirical parameter

o 푛푖 mostly < 2

o ni can be negative! o 푛푖=푛푖(푝).

20 The Arrhenius Law

• Prescribes the dependence of the reaction rate constant on temperature:

d ln k ( T ) E  a , d T Ro T 2

o  ERTa / o For constant 퐸푎: k( T ) = A e

o Modified form: AA(T)BT = = 

21 The Activation Energy

• 퐸푎,푓: minimum energy needed to initiate a reaction

o Large 퐸푎 leads to temperature sensitivity o Some reactions (e.g. 3-body termination) have 퐸푎 ≡0.

22 The Arrhenius Number, Ar

ET Ar  aa o RTTm a x m a x

e x p (Ta / TT )  m a x = e x p Ar  1  e x p ( Ta / TTm a x )  • Reaction is temperature sensitive for Ar≫1 T/Tmax • Combustion systems: Ar≫1 • Ar≫1 => localizes reaction regions (spatial or temporal)

o dc  ERT/ B c e a dt

23 Collision Theory of Reaction Rate (1/3)

• Assumptions: o Equilibrium Maxwell velocity distribution o Two-body hard-sphere collision o Reaction occurs if collision (translational) energy exceeds activation energy

24 Collision Theory of Reaction Rate (2/3)

m m m/() m m • Reduced mass: i, j i j i j

 = (  +  )/ 2 • Collision diameter: i, j i j 1 / 2 8 kTo • Collision velocity: V ij,  ,  m 1 /2 i o 2 8 kT Z =  n n  . • Collision frequency per volume: i,, j i j i j   m ij, *  n o • Boltzmann velocity distribution: = e  ERT*/ . n • Collision frequency with energy in excess of

(Ei+Ej=Ea) 1 /2 o  o dn * 2 * * 8 kT  ERT/ dni j a Z =  i,, j n n Z i j e   =  i, j i j  m d t d t ij, 25 Collision Theory of Reaction Rate (3/3)

0 • Relating cii n /A

1 /2 o  oo dci o 2 8  kT ERTERT//  =  = A  c c eaa A ( T ) c c e . i, j i j i j d t m ij,  • Comparing: 1 / 2 o o 2 8 kT ATA( )  ij,  , 1 / 2 m ij, • Deviation from theory accounted by steric factor

AZ 

26 Transition State Theory of Reaction Rate (1/3) • Overall reaction consists of two steps:

N k1, f vR    R ‡ o Activation:  ii   k i 1 1,b N ‡ k RP , 2 v  o Product formation:  ii i 1

‡ • Reaction rates for Ri and R

N dcR v  i j  v k c  v k c ‡ i1, f j i 1, b R dt j 1

N dc ‡  R v j  k c  k c‡‡  k c . 1,f j 1, b RR 2 dt j 1

27 Transition State Theory of Reaction Rate (2/3)

• Assumptions: o Partial equilibrium for activation step N  ‡ v j c‡  k c d c/0 d t  R cj R i j 1 ‡ o Steady-state for activated complex, R

dc dc ‡ R i R    v k c ‡ . i 2, R d t d t

dc ‡ dcR R  i yields d t d t

N dc ‡ R v  i j  v k c‡   v k K c , i2R i 2 , c j dt j 1

N dcR v  ‡ i v k c j • Compared with ij  yields k k2 K c dt j 1

28 Transition State Theory of Reaction Rate (3/3)

• To estimate k2: o Kinetic energy = Vibrational energy 1 2 (ko T ) h 0 2 o Assume products form during one vibration k   2 • Therefore:

kTo kK  ‡ . o c h

29 Theory of Unimolecular Reactions

A unimolecular reaction

RP , k  is really the high pressure limit of a second order reaction R + M P + M where M is a collision partner

k  k constant as p   dc R  First-order reaction    kcR dt k  k00 p~ k cR a s p  0  Second-order reaction

30 Straight Chain Reactions: Halogen-Hydrogen System (1/3)

• Straight chain: The consumption of one radical produces another radical

• Hydrogen-halogen system (X2: I2, Br2, Cl2, F2)

k1, f X 2 + M   X + X + M Chain initiation (X1f )

k 2,f X + H2    HX + H Chain carrying (X2f )

k 3,f H + X2    HX + X Chain carrying (X3f )

k X + X + M 1,b X2 + M Chain termination (X1f )

k H + HX 2,b  X+ H2 Chain carrying (X2 f )

31 Straight Chain Reactions: Halogen-Hydrogen System (2/3) • Reaction rates: • Apply steady- state assumption d [H]2 = -kk2 , f [X][H2 ] + 2 ,b [H][HX] dt for H and X: d [X] 2  -k [X ][M] - k [H][X ] + k [X]2 [M] 1 , f22 3 , f 1 ,b dt d [H] d [H]  k2 , f [X][H22 ] - k 3 , f [H][X ] - k 2 ,b [H][HX] dt  0 dt d [X] d [X] = 2k1 , f [X2 ][M]- k 2 , f [X][H 2 ] + k 3 , f [H][X 2 ] dt  0 dt 2 + kk2 ,b [H][HX] - 2 1 ,b [X] [M] d [HX]  k2 , f[X][H22 ] + k 3 , f [H][X ] - k 2 ,b [H][HX] dt

32 Straight Chain Reactions: Halogen-Hydrogen System (3/3)

• Detailed analysis yields: 1 /2 1 /2 d [HX] 2k2 ,f ( k 1, f / k 1, b ) [H 2 ][X 2 ] = . (2.4.8) d t1+( k / k )[HX]/[X ] 2 ,b 3, f 2 • One-step reaction yields: k H + X 0 2HX (X0) 22

d [HX] = 2k 0 [H 2 ][X 2 ], dt (2.4.1)

• Comparing, detailed analysis shows o Complex instead of linear dependence on [X2] o Inhibiting effect of [HX] 33

Branched Chain Reactions:

H2-O2 System • The consumption of one radical generates more than one radical

H + O2 → OH + O Chain branching (H1)

O + H2 → OH + H Chain branching (H2)

OH + H2 → H2O + H Chain carrying (H3) • The net of (H1) to (H3) yields 2H per cycle:

3H2 + O2 → 2H2O +2H • Different radicals have different reactivities => Chain carrying steps can be weakening

H + O2 +M→ HO2 +M (HO2 less reactive than H)

CH4 + H → CH3+H2 (CH3 less reactive than H) 34 Branched Chain Reactions: Pressure Effect (1/2)

k n R 1 C In itia tio n

k R C  2 a C  P Chain branching cycle

k CRRP   g Gas termination

k CP w Wall termination

d [C ] n 2 = k12 [R] + ( a 1) k [R][C]  kgw [R][C]  k [C] dt

n k1 [R] + k 2 [R]( a a c )[C]

2 kkgw[R ] + a c = 1 + . k 2 [R]

35 Branched Chain Reactions: Pressure Effect (2/2)

d [C ] n • = k1 [R] + k 2 [R]( a a c )[C], (2.4.15) dt

o blows up for a > ac

o delays for a < ac

2 kkgw[R ] + • a c = 1 + k 2 [R]

k 1 + w  a s p  0

k 2 [R]

k g [R ] 1 +  a s p  

k 2 36 3. Oxidation Mechanisms of Fuels

37 Oxidation of Hydrogen (1/2)

38 Oxidation of Hydrogen (2/2) • Initiation: q  104 kcal/mole H2 + M → H+ H+ M p

O2 + M → O + O + M 118kcal/mole

H2 + O2 → HO2 + H 55kcal/mole; preferred • First-limit chemistry: o Wall termination of H as p  • Second-limit chemistry o Branching & carrying:

H + O2 → O + OH

O + H2 → H + OH

OH + H2 → H + H2O o Termination:

H + O2 + M→ HO2 + M (3-body reaction promoted as p↑)

HO2 → wall termination 39

Second-limit Chemistry

d [H] = -k1 [H][O2 ] + k 2 [O][H 2 ] + k 3 [OH][H 2 ] - k 9 [H][O 2 ][M] dt d [O] = kk12 [H][O22 ] - [O][H ] dt d [OH] = k1 [H][O2 ] + k 2 [O][H 2 ] - k 3 [OH][H 2 ]. dt • Assume steady state for O and OH:

d [H]  (2kk19 - [M])[H][O2 ] = 2 1 - 9 dt (3.2.4,5)

(2kk - [M])>0 o System explodes if 19 2 k p1 R T . o Second limit: 2kk19 = [M ] k 9

40 Third-Limit Chemistry

• [HO2] increases with increasing pressure, leading to:

HO2 + H2 → H2O2 + H

H2O2 + M → OH + OH + M • At high temperatures and pressures, more radicals are produced, leading to radical-radical reactions

HO2 + HO2 → H2O2 + O2

HO2 + H → OH + OH

HO2+ O → OH + O2

41

Role of Initiation Reaction

Homogeneous System Diffusive System with Flame o Shock tube, flow o Radical produced in high- reactor,… temperature flame o No diffusive transport o Radical back diffuses and reacts with incoming o Radicals generated from reactant original reactants o e.g.: H + O2 → OH + O o e.g.: H2 + O2 → HO2 + H o Different (lower) global activation energy

Flame H + O H ,O H H ,O  22  HO, H  22 H + O   2 2 2 2

42 Oxidation of Carbon Monoxide

• Direct oxidation rarely relevant:

CO + O2 → CO2 + O o High activation energy (48 kcal/mole) o No branching o Hence no dry CO oxidation • Dominant oxidation route:

CO + OH → CO2 + H (CO3)

o Integrated to the H2-O2 chain

o H2, H2O are catalysts for CO oxidation o Extremely sensitive to moisture content

43 General Considerations of Hydrocarbon Oxidation • Most important reactions in HC oxidation:

o Chain initiation: (H, HO2) o Chain branching: H + O2  O H + O o Heat release: C O + O H C O2 + H • HC oxidation is hierarchical:

o Large HC molecule breaks down into smaller C1,C2,C3 fragments in the low-temperature region/regime, with small heat release, which subsequently undergo massive oxidation with large heat release

• Dominant low-temperature chemistry: HO2 chemistry

• Dominant high-temperature chemistry: H2-O2 chain

44 Methane Oxidation

45 Methane Initiation Reactions

• High-temperature route:

CH4 + M → CH3 + H + M

H + O2 → OH + O

CH4 + (H, O, OH) → CH3 + (H2, OH, H2O)

o CH4+H →CH3+H2 is retarding (exchange H by CH3)

Ignition delay time increases with [CH4] ! • Low-temperature route:

CH4 + O2 → CH3 + HO2

CH4 + HO2 → CH3 + H2O2

H2O2 + M → OH + OH + M

46 Methyl (CH3) Reactions

• Oxidation path:

o Continuous stripping of H eventually leads to CO2

o Oxidation of H leads to H2O • Growth path:

o CH3 + CH3 + M→C2H6 + M latched onto the ethane oxidation path

o CH3 + CH3 →C2H5 + H latched onto the ethyl oxidation path

47 Closing Remarks of Day 1 Lecture (1/2)

• Adiabatic flame temperature, Tad , and equilibrium composition, ci o Consequence of chemical equilibrium and energy conservation o Single most important property of a combustible mixture; peaks around f = 1

o Is there a Tad for nonpremixed systems?

48 Closing Remarks of Day 1 Lecture (2/2) • Oxidation mechanisms of fuels are described by a plethora of coupled, nonlinearly interacting elementary reactions o Track radicals in branching, carrying & termination reactions

o (H, HO2) are key radicals in (high, low) temperature regimes

o Increasing T favors 2-body,large Ea reactions; increasing pressure favors 3-body, termination reactions, with Ea=0

o Combustion is characterized by large values of overall Ea; i.e. the Arrhenius number, Ar, when referenced to Tad o Large Ar confines reactions to narrow spatial regions or

time intervals 49

! Daily Specials !

50 Day 1 Specials

1. Significance of normalized parametric representation (e.g.: scatters in laminar flame speed determination) 2. Growth & reduction of complexity in combustion CFD

3. Cubic description of the H2-O2 Z-curve: towards “analytical” chemistry

51 1. Significance of Normalized Parametric Representation: The Equivalence Ratio

• Assessment of premixed systems frequently are based on equivalence ratio: (퐹/푂) 휙 = ; Lean (0 < 휙 < 1); Rich (1 < 휙 < ∞) (퐹/푂)푆 • A normalized, symmetrical definition: 휙 훷 = ; Lean (0 < Φ < 0.5); Rich (0.5 < Φ < 1) 1+휙

Example: Hydrogen/Air Flame Speeds

• Hydrogen/oxygen kinetics forms the backbone of hydrocarbon chemistry • Laminar flame speed is a key component in validating the mechanism • When plotted in f, tight correlation on lean side versus wider scatter on rich side has led to concern on adequacy of rich chemistry Normalized (x, y) Representations

• Use of  moderates difference in scatters • Furthermore, since regime of lean flames consists of more low flame speeds, normalizing the flame speeds (y-axis) shows even larger scatter on the lean side! • As such, lean/rich chemistry are equally accurate/inaccurate! Further Thoughts

• Could the use of f instead of  have caused misinterpretation in other combustion phenomena (e.g. sensitivity of NOx formation)? • In scientific investigations, parameters representing the ratio of two processes/factors are frequently used to simplify/focus analysis & understanding; but it could skew interpretation & hinder unification • Consider the following parameters: Re, Gr, Pe, Da, Ka,… • Try a formulation of the N-S equations based on a

normalized Re, as ReN = Re/(1+Re) 2. The Unrelenting Growth of Size of Reaction Mechanism!

• Exponential growth of mechanism Lu-Law Diagram size with increasing fuel size o N ~ 103; K ~ 104 o Empirically: K ≈ 5N • Size prohibitively large: o For insight even with sensitivity analysis o For CFD: Not even possible for simple 1D flames; compete with fluid- scale resolution for turbulent flames • Is this growth forever enslaving us to the super-computer? • Need systematic reduction in all aspects of computation • Hope & goal: Chemistry-limited asymptotic size 56 Additional Concerns: Inadequate State of Mechanism Development

100

10 n-Heptane-Air

p = 1 atm 1 f = 1

0.1

0.01

Ignition Delay (sec) , DelayIgnition (sec) without low-T chemistry 0.001 with low-T chemistry

0.0001 0.5 1 1.5 2 1000/T (1/K)

Example: Various versions of • Adoption of mechanisms GRI-Mech yield qualitatively beyond applicability range different results • One generally cannot get the right answer with the wrong reactions! 57 Further Challenges in Computation

~K2 ~K 10-6 4 Shortest flow time 10 PRF Diffusion is o -o c ta n e n -h e p ta n e Chemistry -9 Ethylene, 10 iso-octane, skeletal p = 1 atm

T0 = 1000K n-heptane, skeletal

3 10 C 1 -C 3 1,3-Butadiene -12 n-Heptane, 10 DME p = 50 atm G R I3 .0

T0 = 800K Number of exp functions G R I1 .2 C h e m is try D iffu s io n 10-15 2 Shortest SpeciesShortest Scale,Sec Time 1000 2000 3000 10 1 2 3 10 10 10 Temperature, K Number of species , K Stiffness in reaction rates Crossing point: K~20 Evaluation of diffusion coefficients can overwhelm that of chemistry

58 Strategy for Facilitated Computation

Detailed mechanisms Skeletal mechanisms Reduced mechanisms

CH4: 30 species CH4: 13 species CH4: 9 species

C2H4: 70 species C2H4: 30 species C2H4: 20 species nC7H16: 500 species nC7H16: 80 species nC7H16: 60 species

DRG, DRGASA Isomer Lumping CSP QSSDG

Skeletal reduction Time scale reduction Analytical QSS solution Minimal diffusive species C2H4: 9 groups Reduced mechanism nC7H16: 20 groups • Fewer species/reactions CCR DCL • Less stiff • Faster simulation Computation cost reduction Diffusion reduction

WP Air Force Base Georgia Tech SANDIA

C2H4, VULCAN C2H4, 3D, LES CH4, 3D, DNS

CFD simulations

Time savings: factors of 10 ~ 100 3. Structure & Cubic Description of the

H2-O2 Explosion Z-Curve

• Explicit expression was derived for the 2nd limit. How about the entire Z-curve? • Shape of Z-curve suggests 3rd degree polynomial description • Explicit expression can be used to rule out unphysical situations

60 Structure of H2-O2 Z-Curve

1st and 3rd limits due to The backbone 2nd limit surface deactivation

61 Cubic Description of H2-O2 Z-Curve

• Assume steady state for O, OH and H2O2 • Detailed analysis yields:

d H H 퐼 = 퐴H,HO2 + d푡 HO2 HO2 퐼

2푘 O − 푘 O M − 푘 3푘 H 퐴 = 1 2 9 2 H 17푏 2 H,HO2 푘 O M −푘 H − 푘 9 2 17푏 2 HO2 • Neutral condition yields cubic equation in [M] ~ p: a(T)[M]3+b(T)[M]2+c(T)[M]+d(T) = 0 • Cubic equation yields explicit expressions for three limits, high & low pressure parabola, & loss of non-monotonicity 62