• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! 6.26

Expert-verified Found in: Page 236 ### An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279 # For a $CO$ molecule, the constant $€$is approximately $0.00024eV$.(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 temperature $\left(300K\right)$, first using the exact formula 6.30 and then using the approximate formula 6.31

The rotational partition function of a heterogeneous diatomic molecule

See the step by step solution

## Rotational partition function:

The equation is

${Z}_{rot}=\sum _{j=0}^{\infty }\left(2j+1\right)exp\left(\frac{-j\left(j+1\right)\in }{kT}\right)$

Here, $\in$ is the rotational constant, $k$ is the Boltzmann constant, and $T$ is the absolute temperature.

At higher temperatures, for $kT>>\in$, the rotational partition function becomes as follows:

${Z}_{rot}=\frac{kT}{\in }$

Substitute $8.617×{10}^{-5}eV/K$ for $k,300K$ for $T$, and $0.00024eV$ in the equation ${Z}_{rot}=\frac{kT}{\in }$

${Z}_{rot}=\frac{\left(8.617×{10}^{-5}eV/K\right)\left(300K\right)}{0.00024eV}\phantom{\rule{0ex}{0ex}}=107.7$

Therefore, the rotational partition function of a $CO$ molecule is $107.7$

## The rotational partition function of a heterogeneous diatomic molecule:

The equations are

${Z}_{rot}=\sum _{j=0}^{\infty }\left(2j+1\right)exp\left(\frac{-j\left(j+1\right)\in }{kT}\right)$

Expand the above summation from $j=0$ to $j=50$:

${Z}_{rot}=1+3exp\left(-\frac{2\in }{kT}\right)+5exp\left(-\frac{6\in }{kT}\right)+7exp\left(-\frac{12\in }{kT}\right)+...101exp\left(-\frac{2550\in }{kT}\right)$

Substitute $107.7$ for $\frac{kT}{\in }$ in the above equation.

${Z}_{rot}=1+3exp\left(-\frac{2}{107.7}\right)+5exp\left(-\frac{6}{107.7}\right)+7exp-\frac{12}{107.7}+...101exp\left(-\frac{2550}{107.7}\right)$

$=108.03$

Therefore, the exact value of rotational partition function of a $CO$ molecule is $108.03$ ### Want to see more solutions like these? 