Question

Use the Boltzmann equation to calculate the excited state to ground state population ratio (to 1 significant figure) for Cl2: N(ν=1)/N(ν=0) at room temperature (25°C). The wavenumber for the fundamental vibrational frequency of Cl2 is 550 cm-1. Assume that g1 ≈ g0

Answer 1

Using Boltzmann's equation,

Nj/Ni=gj/gi(exp(-(Ej-Ei)/kT))

where Nj=Population of excited state,v=1

Ni=Population of ground state,v=0

Ej-Ei=E=energy difference between ground and excited state (fundamental vib. frequency)=550 cm-1

E=hc*v, v=frequency (cm-1)=(6.626*10^-34 Js)*(2.998*10^8 m/s)*(550 cm-1) *(100 cm/1m)=1.092*10^-20 J

T=temperature=25+273=298K

k=Boltzmann's constant=1.381*10^-23 J/K

g1=go=degeneracy of states

So,Nj/Ni=gj/gi(exp(-(Ej-Ei)/kT))=exp(-(E)/kT)=exp(-1.092*10^-20 J/(1.381*10^-23 J/K)*298K)=0.07

population ratio=0.07