Boltzmann Distribution

Question 1

Population of an energy level relative to another:

Answer 1

Ni/Nj = (gi/gj)e^-((Ei-Ej)/kT)
gi/gj is the relative degeneracies but we usually compared to the ground state so go to 1.
k is Boltzmann constant

Question 2

N in a given orbital compared to the ground state

Answer 2

Ni =Noe^-((Ei-Eo)/kT)

Question 3

Sum of all molecules

Answer 3

N =Sum between i=0 and n of: Noe^-((ΔEi)/kT), can be rearranged for ground state

Question 4

Partition function:

Answer 4

Ne^-((ΔEi)/kT/(Sum between i=0 and n of: e^-((ΔEi)/kT))

Question 5

What happens to the partition function when ΔEi»kT

Answer 5

Only lowest energy level will be occupied as very few molecules will have the required energy to jump to the next level

Question 6

What happens to the partition function when ΔEi«kT

Answer 6

All energy levels will be equally and evenly filled with electrons as energy difference is so small all molecules will have required energy to jump between energy levels

Question 7

What type of energy levels will it apply to

Answer 7

All,
Electronic, vibrational, translational and rotational (with a slight modification for translational), however they will all be occupied to a different extent

Question 8

Extent of occupancy for each mode of motion

Answer 8

Rotational and translational are fully accessible so fully occupied at room temperature, vibrational will be inaccessible at room temperature (air wont glow)

Question 9

Why does rotational motion have a slight variation to the normal distribution function

Answer 9

It’s degeneracy is equal to 2J+1, extra molecules present at each energy level so N will be multiplied by (2J+1)

Question 10

What happens for large kT so Energy level is negligibly small to the denominator of the partition function?

Answer 10

It becomes an integral with respect to J between 0 and infinity