Chapter 10

\( \mathrm{What}\) is the moment of inertia of (a) a solid disc of mass \(m,\) radius \(R,\) about its axis, (b) a solid sphere of mass \(m,\) radius \(R,\) about its centre?

Find expressions for the moments of inertia of an \(\mathrm{AB}_{3}\) molecule that is (a) planar, (b) trigonal pyramidal.

Show that the rotational energy levels of a square planar \(\mathrm{AB}_{4}\) molecule may be expressed solely in terms of the rotational constant \(\bar{B}\).

At very high values of the angular momentum \(J\), the rotational wavenumbers of a linear rotor can go through a maximum due to the presence of centrifugal distortion. Find the value of \(J\) for \(\mathrm{HCl}\left(\bar{B}=10.4400 \mathrm{cm}^{-1} \text {and } \bar{D}=\right.\) \(0.0004 \mathrm{cm}^{-1}\) ) where \(\bar{F}\) is a maximum. Hint: You will need to find a root of a cubic equation.

The \(J+1 \leftarrow J\) rotational transitions of \(^{16} \mathrm{O}^{12} \mathrm{C}^{32} \mathrm{S}\) and \(^{16} \mathrm{O}^{12} \mathrm{C}^{34} \mathrm{S}\) occur at the following frequencies \((v / \mathrm{GHz})\) $$\begin{array}{lllll} \hline J & 1 & 2 & 3 & 4 \\ \hline^{16} \mathrm{O}^{12} \mathrm{C}^{32} \mathrm{S} & 24.32592 & 36.48882 & 48.65164 & 60.81408 \\ ^{16} \mathrm{O}^{12} \mathrm{C}^{34} \mathrm{S} & 23.73223 & & 47.46240 & \\ \hline \end{array}$$ Find (a) the rotational constants, (b) the moments of inertia, and (c) the CS and CO bond lengths. Hint. Begin by finding expressions for the moment of inertia \(I\) through \(I=m_{\Lambda} R_{A}^{2}+m_{B} R_{B}^{2}+m_{C} R_{C}^{2},\) where \(R_{X}\) is the distance of atom \(\mathrm{X}\) from the centre of mass. The easiest procedure is to use the result established in Exercise \(10.6,\) which leads to \(I=\left(m_{\mathrm{A}} m_{\mathrm{C}} / m\right)\left(R_{\mathrm{AB}}+R_{\mathrm{BC}}\right)^{2}+\left(m_{\mathrm{B}} / m\right)\left(m_{\mathrm{A}} R_{\mathrm{AB}}^{2}+m_{\mathrm{C}} R_{\mathrm{BC}}^{2}\right)\) The lengths \(R_{\text {AB }}\) and \(R_{\mathrm{BC}}\) may be found only if two values of \(I\) are known. Assume the bond lengths are the same in isotopomeric molecules.

In \(\mathrm{PCl}_{3}\) the bond length is \(204.3 \mathrm{pm}\) and the CIPCl angle is \(100.1^{\circ} .\) Predict the form of (a) its microwave spectrum, (b) its rotational Raman spectrum, including the general structure of the line intensities. Ignore the effects of nuclear spin statistics. Hint. Establish that \\[ I_{\perp}=m_{\mathrm{B}} R^{2}(1-\cos \theta)+\left(m_{\mathrm{A}} m_{\mathrm{B}} / m\right) R^{2}(1+2 \cos \theta) \text { for } \mathrm{AB}_{3}, \text { with } \\] \(m=m_{A}+3 m_{\mathrm{in}},\) and \(I_{\|}=2 m_{\mathrm{n}} R^{2}(1-\cos \theta) .\) Suppose that the intensities are governed predominantly by the Boltzmann distribution.

The ethyne molecule \((\mathrm{HC} \equiv \mathrm{CH})\) consists of two fermions \(\left(^{1} \mathrm{H}\right)\) and two bosons \(\left(^{12} \mathrm{C}\right) .\) What are the implications for the statistical weights of the levels of various \(J\) ? What are the implications of replacing (a) one \(^{12} \mathrm{C}\) by \(^{13} \mathrm{C},\) (b) both \(^{12} \mathrm{Cby}^{13} \mathrm{C}\). (The \(^{13} \mathrm{C}\) nucleus is a fermion, \(\left.I=\frac{1}{2} .\right)\)

\( \mathrm{An}\) infrared spectrum of gaseous DCl revealed lines at the following wavenumbers for the lowest four transitions from the \(v=0\) state: 2091,4128,6111,8043 Determine the spectroscopic constants \(\omega\) and \(\omega x_{e^{-}}\).

The rotational constant of \(^{1} \mathrm{H}^{35} \mathrm{Cl}\) is \(10.4400 \mathrm{cm}^{-1}\) in the ground vibrational state and \(10.1366 \mathrm{cm}^{-1}\) in the state \(v=1 .\) Plot the wavenumbers of the \(\mathrm{P}\) -, \(\mathrm{Q}\) -, and \(\mathrm{R}\) -branches against \(J\) as a representation of the structure of the \(1-0\) transition. Take \(\tilde{v}=2990.95 \mathrm{cm}^{-1}\) and neglect anharmonicity. (The Q-branch is not observed.)

The \(Q\) -branch line of the fundamental transition of a diatomic molecule lies at \(3142.3 \mathrm{cm}^{-1}\). The first line in the P-branch (that is, the P-branch line closest to the Q-branch) is displaced in magnitude by \(21.2 \mathrm{cm}^{-1}\) from the Q-branch. (a) Neglecting the effects of anharmonicity and centrifugal distortion, compute \(v\) and \(\bar{B}\). Assume that the rotational constant is independent of the vibrational level. (b) Predict the wavenumber of the next P-branch line. (c) Predict the wavenumber of the \(R\) -branch line closest to the Q-branch. (d) If centrifugal distortion is considered, will the first \(\mathrm{P}\) branch line be found at a higher, lower, or the same wavenumber as in part (a)?