An Introduction to Modern Astrophysics · Problem 12.7

Main-Sequence and Post-Main-Sequence Stellar Evolution — Carroll & Ostlie

Physics — Astrophysics · molecular-rotational-energy

Problem

The rotational kinetic energy of a molecule is given by

E_{\mathrm{rot}} = \frac{1}{2}I\omega^2 = \frac{L^2}{2I},

where

L

is the molecule's angular momentum and

I

is its moment of inertia. The angular momentum is restricted by quantum mechanics to the discrete values

L = \sqrt{\ell(\ell+1)}\hbar

where

\ell = 0, 1, 2, \ldots

(a) For a diatomic molecule,

I = m_1 r_1^2 + m_2 r_2^2,

where

m_1

and

m_2

are the masses of the individual atoms and

r_1

and

r_2

are their separations from the center of mass of the molecule. Using the ideas developed in Section 2.3, show that

I

may be written as

I = \mu r^2,

where

\mu

is the reduced mass and

r

is the separation between the atoms in the molecule. (b) The separation between the carbon and oxygen atoms in CO is approximately 0.12 nm, and the atomic masses of

^{12}

^{13}

C, and

^{16}

O are 12.000 u, 13.003 u, and 15.995 u, respectively. Calculate the moments of inertia for

^{12}

CO and

^{13}

CO. (c) What is the wavelength of the photon that is emitted by

^{12}

CO during a transition between the rotational angular momentum states

\ell = 3

and

\ell = 2

? To which part of the electromagnetic spectrum does this correspond? (d) Repeat part (c) for

^{13}

CO. How do astronomers distinguish among different isotopes in the interstellar medium?

Related theory

Work And Kinetic Energy Work Energy Theorem

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