Question

What is the master equation for vibrational relaxation?

(a) \(\frac {dN_i}{dt}\) = ki + 1, i ZNi + 1 + ki – 1, i ZNi – 1 – ki, i + 1 ZNi – ki, i – 1 ZNi

(b) \(\frac {dN_i}{dt}\) = ki + 1, i Ni + 1 + ki – 1, i Ni – 1 – ki, i + 1Ni – ki, i – 1 Ni

(c) \(\frac {dN_i}{dt}\) = ZNi + 1 + ZNi – 1 – ZNi, i – 1 – ZNi

(d) \(\frac {dN_i}{dt}\) = Pi + 1, i ZNi + 1 + Pi – 1, i ZNi – 1

This question was addressed to me in an international level competition.

My doubt stems from Vibrational Rate Equations in division Properties of High Temperature Gases of Aerodynamics

Answer 1

Correct choice is (b) \(\frac {dN_i}{dt}\) = ki + 1, i Ni + 1 + ki – 1, i Ni – 1 – ki, i + 1Ni – ki, i – 1 Ni

To explain: In the formula derived to obtain then net rate of change of population of the i^th level, the product of transition probability and collision frequency is expressed in the form of a new variable known as vibrational rate constant ki + 1, i = Pi + 1, i Z (this is an example of one of the transitions). Thus the formula is reduced from \(\frac {dN_i}{dt}\) = Pi + 1, i ZNi + 1 + Pi – 1, i ZNi – 1 – Pi, i + 1 ZNi – Pi, i – 1 ZNi to:

\(\frac {dN_i}{dt}\) = ki + 1, i Ni + 1 + ki – 1, i Ni – 1 – ki, i + 1Ni – ki, i – 1 Ni