Carbon disulfide,

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

Action and Symmetry, Ch Small oscillations: stability, degeneracy, symmetry and spectra

Problem

Carbon disulfide, S=C=S\mathrm{S=C=S}, is a linear symmetric triatomic molecule with carbon mass mC=12.01 um_{\rm C} = 12.01\,\mathrm u and sulfur mass mS=32.07 um_{\rm S} = 32.07\,\mathrm u, where 1 u=1.6605×10−27 kg1\,\mathrm u = 1.6605\times10^{-27}\ \mathrm{kg}. Model each C=S\mathrm{C=S} bond as a spring of stiffness κ=700 N m−1\kappa = 700\ \mathrm{N\,m^{-1}} along the molecular axis, exactly as for carbon dioxide. ((a)) Give the symmetric-stretch frequency ωsym=κ/mS\omega_{\rm sym} = \sqrt{\kappa/m_{\rm S}} and the antisymmetric-stretch frequency ωasym=κ(1/mS+2/mC)\omega_{\rm asym} = \sqrt{\kappa(1/m_{\rm S} + 2/m_{\rm C})}, in rad s−1\mathrm{rad\,s^{-1}}. ((b)) Convert each to a spectroscopic wavenumber ν~=ω/(2πc)\tilde\nu = \omega/(2\pi c) in cm−1\mathrm{cm^{-1}}, using c=2.998×108 m s−1c = 2.998\times10^8\ \mathrm{m\,s^{-1}}. ((c)) The observed fundamentals of carbon disulfide are 658 cm−1658\ \mathrm{cm^{-1}} (symmetric stretch) and 1535 cm−11535\ \mathrm{cm^{-1}} (antisymmetric stretch). Comment briefly on the agreement.

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