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 first rotational level is given by:

$E=\frac{\hbar^2}{2I}J(J+1)$

where $J$ is the rotational quantum number and $I$ is the moment of inertia.

If we assume that the first rotational level is excited by a photon in the far infrared, then we can use the relation:

$E=h
u$

where $
u$ is the frequency of the photon.

Substituting the values given in the problem, we get:

$\frac{\hbar^2}{2I}=(5.0	imes10^{-11}	ext{ m})^{-1}(6.626	imes10^{-34}	ext{ J s})$

$I=\frac{(6.626	imes10^{-34}	ext{ J s})(2)}{(5.0	imes10^{-11}	ext{ m})(1.054	imes10^{-34}	ext{ kg m}^2	ext{ s}^{-1})}=2.8	imes10^{-49}	ext{ kg m}^2$

Converting to scientific notation with the proper prefix, we get:

$I=2.8	imes10^{-49}	ext{ kg m}^2=2.8	imes10^{-149}	ext{ kg}\cdot	ext{m}^2$

The closest answer choice in scientific notation with the proper prefix is:

B) 2.8 × 10−149 kg⋅m2.

The Rotation Spectrum of the HCl Molecule Suggests a Photon