Question

What is the formula to compute net rate change of the population of the i^th level?

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

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

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

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

The question was asked by my college professor while I was bunking the class.

Query is from Vibrational Rate Equations topic in division Properties of High Temperature Gases of Aerodynamics

Answer 1

The correct option is (a) \(\frac {dN_i}{dt}\) = Pi + 1, i ZNi + 1 + Pi – 1, i ZNi – 1 – Pi, i + 1 ZNi – Pi, i – 1 ZNi

The explanation: The rate of change of population of the molecules in i^th level is computed by adding rate of increase of Ni (population of i^th level) and rate of decrease of Ni. Thus the formula is:

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

Where, Pi + 1, i ZNi + 1 + Pi – 1, i ZNi – 1 is the rate of increase of population in i^th level due to the molecules jumping up from i – 1 and down from i + 1 levels.

Pi, i + 1 ZNi – Pi, i – 1 ZNi is the rate of decrease of population in the i^th level due to the molecules jumping from i^th level to i + 1 and i – 1 levels.