The vibrational contribution to the molar heat capacity of a gas of nonlinear molecules is given in statistical mechanics by the formula

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where u_i = hν_i/k_BT . Here νi is the frequency of the ith normal mode of vibration, of which there are 3n ˆ’ 6 if n is the number of nuclei in the molecule, h is Planck€™s constant, k_B is Boltzmann€™s constant, R is the ideal gas constant, and T is the absolute temperature. The H₂O molecule has three normal modes. The frequencies are given by

ν₁ = 4.78 × 10¹³ s^ˆ’1 ± 0.003 × 10¹³ s^ˆ’1,
ν₂ = 1.095 × 10¹⁴ s^ˆ’1 ± 0.004 × 10¹⁴ s^ˆ’1,
ν₃ = 1.126 × 10¹⁴ s^ˆ’1 ± 0.005 × 10¹⁴ s^ˆ’1.
Calculate the vibrational contribution to the heat capacity of H₂O vapor at 500.0 K and find the 95% confidence interval.Assume the temperature to be fixed without error.

Transcribed Image Text:

Зп—6 -lj uze-Hi (1 — е-и;)2" Cm (vib) = R > i=1