The rotation spectrum of the HCl molecule suggests a photon in the far infrared (around 5.0 *10-6 m) can excite the first rotational level.From this data, the moment of inertia of the HCl molecule (in kg . m2) is:
A)2.8 * 10-48
B)2.8 * 10-49
C)2.8 * 10-47
D)2.8 * 10-46
E)2.8 * 10-45

Question

Quizplus · Accepted Answer

The moment of inertia of a diatomic molecule is given by the formula:
$I=\frac{\mu r^2}{1+\frac{m_1}{m_2}}$
where $\mu$ is the reduced mass of the molecule, $r$ is the bond length, $m_1$ and $m_2$ are the masses of the atoms.

For HCl, $\mu=\frac{m_	ext{H}m_	ext{Cl}}{m_	ext{H}+m_	ext{Cl}}$, where $m_	ext{H}$ and $m_	ext{Cl}$ are the masses of hydrogen and chlorine, respectively.

We are given that the first rotational level can be excited by a photon in the far infrared, which implies that the energy required to excite this level is given by:
$E=h
u=\frac{\hbar^2}{2I}$
where $h$ is Planck's constant, $
u$ is the frequency of the photon, and $\hbar=\frac{h}{2\pi}$.

Substituting the given values, we get:
$I=\frac{\hbar^2}{2E}=\frac{(6.626	imes10^{-34}/(2\pi))^2}{2	imes4.135	imes10^{-23}	imes5	imes10^{10}}=2.8	imes10^{-49}	ext{ kg m}^2$

Therefore, the answer is B.

The Rotation Spectrum of the HCl Molecule Suggests a Photon