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MMLU-Pro / 3997 / The contribution of molecular vibrations to the molar internal energy…

Problem

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question

The contribution of molecular vibrations to the molar internal energy UmU_{\mathrm{m}} of a gas of nonlinear NN-atom molecules is (zero-point vibrational energy not included) Um,vib=R∑s=13N−6θs/(eθs/T−1)U_{\mathrm{m}, \mathrm{vib}}=R \sum_{s=1}^{3 N-6} \theta_s /\left(e^{\theta_s / T}-1\right), where θs≡hνs/k\theta_s \equiv h \nu_s / k and νs\nu_s is the vibrational frequency of normal mode ss. Calculate the contribution to Um, vib U_{\mathrm{m}, \text { vib }} at 25∘C25^{\circ} \mathrm{C} of a normal mode with wavenumber $\widetilde{v} \equiv v_s / c$ of $900 \mathrm{~cm}^{-1}$.
Plain-text mathematical notation (without MathML)
The contribution of molecular vibrations to the molar internal energy U_(m) of a gas of nonlinear N-atom molecules is (zero-point vibrational energy not included) U_(m,vib)=R∑_(s=1)^(3N−6)θ_(s)/(e^(θ_(s)/T)−1), where θ_(s)≡hν_(s)/k and ν_(s) is the vibrational frequency of normal mode s. Calculate the contribution to U_(m, vib ) at 25^(∘)C of a normal mode with wavenumber $\widetilde{v} \equiv v_s / c$ of $900 \mathrm{~cm}^{-1}$.
Original LaTeX notation
The contribution of molecular vibrations to the molar internal energy $U_{\mathrm{m}}$ of a gas of nonlinear $N$-atom molecules is (zero-point vibrational energy not included) $U_{\mathrm{m}, \mathrm{vib}}=R \sum_{s=1}^{3 N-6} \theta_s /\left(e^{\theta_s / T}-1\right)$, where $\theta_s \equiv h \nu_s / k$ and $\nu_s$ is the vibrational frequency of normal mode $s$. Calculate the contribution to $U_{\mathrm{m}, \text { vib }}$ at $25^{\circ} \mathrm{C}$ of a normal mode with wavenumber $\widetilde{v} \equiv v_s / c$ of $900 \mathrm{~cm}^{-1}$.

category

chemistry

options

  1. 0.30 kJ/mol
  2. 0.05 kJ/mol
  3. 0.20 $\mathrm{kJ} / \mathrm{mol}$
  4. 0.11 $\mathrm{kJ} / \mathrm{mol}$
  5. 0.08 kJ/mol
  6. 0.17 $\mathrm{kJ} / \mathrm{mol}$
  7. 0.22 kJ/mol
  8. 0.18 kJ/mol
  9. 0.25 kJ/mol
  10. 0.14 $\mathrm{kJ} / \mathrm{mol}$

src

scibench-quan

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Source and history

upstream_id / Original problem ID
3997
task_id
d7894780-459b-5567-8385-2ba1522489dc
task_key
default--test--3997
task_revision_id
1
config / split
default / test

Official source

initial import