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For a CO molecule, the constant ? is approximately 0.00024
Chapter 6, Problem 23P(choose chapter or problem)
For a CO molecule, the constant \(\epsilon\) is approximately 0.00024 eV. (This number is measured using microwave spectroscopy, that is, by measuring the microwave frequencies needed to excite the molecules into higher rotational states.) Calculate the rotational partition function for a CO molecule at room temparature (300 K), first using the exact formula 6.30 and then using the approximate formula 6.31.
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QUESTION:
For a CO molecule, the constant \(\epsilon\) is approximately 0.00024 eV. (This number is measured using microwave spectroscopy, that is, by measuring the microwave frequencies needed to excite the molecules into higher rotational states.) Calculate the rotational partition function for a CO molecule at room temparature (300 K), first using the exact formula 6.30 and then using the approximate formula 6.31.
ANSWER:Step 1 of 2
Consider a CO molecule with an energy constant of \(\epsilon=0.00024 \mathrm{eV}\). The approximated rotational partition function is given by:
\(Z_{\mathrm{rot}} \approx \frac{k T}{\epsilon}\)
substitute with the givens at temperature of \(T=300 \mathrm{~K}\), (note that the Boltzmann constant in \(\mathrm{eV}\) is \(k=8.617 \times 10^{-5} \mathrm{eV} / \mathrm{K}\)) to get:
\(\begin{array}{l}
Z_{\mathrm{rot}} \approx \frac{\left(8.617 \times 10^{-5} \mathrm{eV} / \mathrm{K}\right)(300 \mathrm{~K})}{0.00024 \mathrm{eV}} \\
\approx 107.71 \\
Z_{\mathrm{rot}} \approx 107.71
\end{array}\)
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