Hydrogen chloride gas absorbs in its rovibrational band

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · Molecular structure and spectra · Molecular structure and spectra

Reading the Spectrum, Ch Two nuclei on one curve: rotation, vibration and the spectrum of a diatomic molecule

Problem

Hydrogen chloride gas absorbs in its

v=0\to1

rovibrational band, with band origin

\tilde\nu_0 = 2885.98\,\mathrm{cm^{-1}}

and rotational constant

B_0 = 10.44\,\mathrm{cm^{-1}}

in the ground vibrational state. The relative population of rotational level

J

N_J \propto (2J+1)e^{-hcB_0J(J+1)/k_{\rm B}T}

, with

k_{\rm B}T/hc = 208.5\,\mathrm{cm^{-1}}

T=300\,\mathrm{K}

and

417.0\,\mathrm{cm^{-1}}

T=600\,\mathrm{K}

. ((a)) By evaluating

N_J

for

J=2,3,4

300\,\mathrm{K}

and for

J=3,4,5

600\,\mathrm{K}

, find which

J

gives the strongest pair of lines

P(J)

R(J)

at each temperature. Does the band origin

\tilde\nu_0

itself change with temperature? ((b)) Now keep the temperature at

300\,\mathrm{K}

but replace every hydrogen atom by deuterium (mass

2.01410\,\mathrm{u}

), leaving the bond length and the chlorine isotope (mass

34.9689\,\mathrm{u}

) unchanged. Using the two-mass reduced-mass formula, find the new rotational constant

B_0(\mathrm{DCl})

and hence the strongest

J

for DCl at

300\,\mathrm{K}

. Estimate the new band origin by assuming

\tilde\nu_0 \propto \mu^{-1/2}

, the same scaling as the harmonic vibration frequency (ignore the smaller anharmonic correction). Which change, heating or isotope substitution, moves the band origin more?

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