title: Zeldovich mechanism
text: # Zeldovich mechanism
  
  Zel'dovich mechanism is a chemical mechanism that describes the oxidation of nitrogen and $NO_x$ formation, first proposed by the Russian physicist Yakov Borisovich Zel'dovich in 1946. The reaction mechanisms read as
  
      ${\ce {{N2}+ O <->[k_1] {NO}+ {N}}}$
      ${\ce {{N}+ O2 <->[k_2] {NO}+ {O}}}$
  
  where $k_{1}$ and $k_{2}$ are the reaction rate constants in Arrhenius law. The overall global reaction is given by
  
      ${\ce {{N2}+ {O2}<->[k] 2NO}}$
  
  The overall reaction rate is mostly governed by the first reaction (i.e., rate-determining reaction), since the second reaction is much faster than the first reaction and occurs immediately following the first reaction. At fuel-rich conditions, due to lack of oxygen, reaction 2 becomes weak, hence, a third reaction is included in the mechanism, also known as extended Zel'dovich mechanism (with all three reactions),
  
      ${\ce {{N}+ {OH}<->[k_3] {NO}+ {H}}}$
  
  Assuming the initial concentration of NO is low and the reverse reactions can therefore be ignored, the forward rate constants of the reactions are given by
  
      ${\begin{aligned}k_{1f}&=1.47\times 10^{13}\,T^{0.3}\mathrm {e} ^{-75286.81/RT}\\k_{2f}&=6.40\times 10^{9}\,T\mathrm {e} ^{-6285.5/RT}\\k_{3f}&=3.80\times 10^{13}\end{aligned}}$
  
  where the pre-exponential factor is measured in units of cm, mol, s and K (these units are incorrect), temperature in kelvins, and the activation energy in cal/mol; R is the universal gas constant.
  
  ## NO formation
  
  The rate of NO concentration increase is given by
  
      ${\frac {d[\mathrm {NO} ]}{dt}}=k_{1f}[\mathrm {N} _{2}][\mathrm {O} ]+k_{2f}[\mathrm {N} ][\mathrm {O} _{2}]+k_{3f}[\mathrm {N} ][\mathrm {OH} ]-k_{1b}[\mathrm {NO} ][\mathrm {N} ]-k_{2b}[\mathrm {NO} ][\mathrm {O} ]-k_{3b}[\mathrm {NO} ][\mathrm {H} ]$
  
  ## N formation
  
  Similarly, the rate of N concentration increase is
  
      ${\frac {d[\mathrm {N} ]}{dt}}=k_{1f}[\mathrm {N} _{2}][\mathrm {O} ]-k_{2f}[\mathrm {N} ][\mathrm {O} _{2}]-k_{3f}[\mathrm {N} ][\mathrm {OH} ]-k_{1b}[\mathrm {NO} ][\mathrm {N} ]+k_{2b}[\mathrm {NO} ][\mathrm {O} ]+k_{3b}[\mathrm {NO} ][\mathrm {H} ]$
  