Question

Use the Boltzmann equation to calculate the excited state to ground state population ratio (to 1...

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

Homework Answers

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

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