Book edition 1st

Author(s) Daniel V. Schroeder

Pages 356 pages

ISBN 9780201380279

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Short Answer

For a $C O$ molecule, the constant $€$ is approximately $0.00024 e V$ .(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 $C O$ molecule at room temperature $(300 K)$ , first using the exact formula 6.30 and then using the approximate formula 6.31

The rotational partition function of a heterogeneous diatomic molecule

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Rotational partition function:

The equation is

$Z_{r o t} = \sum_{j = 0}^{\infty} (2 j + 1) e x p (\frac{- j (j + 1) \in}{k T})$

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

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

$Z_{r o t} = \frac{k T}{\in}$

Substitute $8.617 \times 10^{- 5} e V / K$ for $k, 300 K$ for $T$ , and $0.00024 e V$ in the equation $Z_{r o t} = \frac{k T}{\in}$

$Z_{r o t} = \frac{(8.617 \times 10^{- 5} e V / K) (300 K)}{0.00024 e V} = 107.7$

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

The rotational partition function of a heterogeneous diatomic molecule:

The equations are

$Z_{r o t} = \sum_{j = 0}^{\infty} (2 j + 1) e x p (\frac{- j (j + 1) \in}{k T})$

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

$Z_{r o t} = 1 + 3 e x p (- \frac{2 \in}{k T}) + 5 e x p (- \frac{6 \in}{k T}) + 7 e x p (- \frac{12 \in}{k T}) + . . . 101 e x p (- \frac{2550 \in}{k T})$

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

$Z_{r o t} = 1 + 3 e x p (- \frac{2}{107.7}) + 5 e x p (- \frac{6}{107.7}) + 7 e x p - \frac{12}{107.7} + . . . 101 e x p (- \frac{2550}{107.7})$

$= 108.03$

Therefore, the exact value of rotational partition function of a $C O$ molecule is $108.03$

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