(c) 2000-2003 by Akira Miyoshi. All rights reserved.

[Top][section-1] [section-2] [section-3] [section-4]

Reaction Dynamics 2002 - section-1

Canonical Ensemble

- Under free and random energy exchange among molecules in the system (normal gas-state)
- The total energy of the system is defined by temperature, .

**<Boltzmann Statistics>**

Probability of finding a molecule in the state (of energy and multiplicity )

ex.) | Distribution of the ground triplet states of atomic oxygen; ( 0, = 5), ( = 158.5 , = 3), and ( = 226.5 , = 1); at 298 K (1 = 1.4388 K) | |

**<Partition Function>**

Probability of finding molecule A in the system
(e.g., A = HCN and not HNC, 1/2 H_{2} +
1/2 N_{2} + C, nor etc.) = Sum of the probability
of finding for all states of A
[partition function] (1.1.1)

ex.) | Distribution of the triplet ground states ( = + + ) and the excited () state [ = 15867.7 , = 5] of atomic oxygen at 298 K: | |

ex.) | Vibrational partition function of a harmonic oscillator (): | |

(1.1.2) |

**<Chemical Equilibrium>**

Ratio of the distributions of molecule B to A:

(When the same zero energy point was used to calculate both
and
)
or

(1.1.3)
(When and
are calculated from an each
ground sate)

**<Transition-State Theory>**

Rate coefficient (A [TS*] B) (1.1.4)

- Probability of passing through the states in the transition state obey Boltzmann distribution

- Isolated molecules (molecules after photolysis, molecules just after reaction, or etc.j
- The system is defined by energy .
- The prior distribution

**<Multiplicity>**

- Degeneracy of multiple solutions to a (time-independent) Schrodinger equation with the same eigenvalue (energy).
- Ratio of the probabilities of finding a system in the state A (multiplicity ) to that in the state B (multiplicity ; at the same energy as A) = :

ex.) | Atomic orbitals of H: = (: principal q.n.), = (: azimuthal q.n.) |

K-shell ( = 1):
1s ( = 0,
= 1) only = = 1 | |

L-shell ( = 2):
2s ( = 0,
= 1),
2p ( = 1,
= 3) = = 4 | |

M-shell ( = 3):
3s ( = 0,
= 1),
3p ( = 1,
= 3),
3d ( = 2,
= 5) = = 9 |

ex.) | Electronic states of H-atom: | total multiplicity | |

ground state: | () [ = = 2 1 = 2] | (K-shell) 2 | |

excited states: | () [ = 2 3 = 6] | (L-shell) 8 | |

() [ = 2 1 = 2] |

ex.) | Bending vibration () of
CO_{2} ... 2-D harmonic oscillator |

, ; (: vibrational q.n.) |

**<Density of states>**

- = Number of states per unit energy [states / ]
- Probability ratio between molecule A (density of states = ) and B () = :

ex.) | Photodecomposition of HCl at 248 nm (40320 ): | ||

HCl + (40320 ) | H + Cl() + (4240 ) | (a) | |

H + Cl() + (3359 ) | (b) | ||

Statistical branching ratio, : | |||

= [Cl()] (4240 ) : [Cl()] (3359 ) | |||

= |

Problem-1Calculate the statistical branching ratio between H + I() [channel-a] and H + I() [channel-b] upon the photolysis of HI at 266 nm. Use the bond dissociation energy of HI [ H + I() ] = 298 and excitation energy of I() form I() (ground state) = 0.943 eV. |