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Chapter 10: Problem 17

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)?

Short Answer

Expert verified
The term value for the v=0 level and rotational constant \(\bar{B}\) are calculated in step 1. The wavenumbers of the next P-branch line and the first R-branch line are computed in step 2 and step 3. In step 4, it's concluded that the first P-branch line will be found at a lower wavenumber if centrifugal distortion is considered.

Step by step solution

01

Compute \(v\) and \(\bar{B}\)

Firstly, the term value for the v=0 level is computed. Based on the P-branch, this can be calculated from \(v=\Delta E_{P1}/2hcB\). Substituting the given values (3142.3 cm-1 as \(\Delta E_{P1}\), 21.2 cm-1 as B, and h and c as Planck's constant and speed of light respectively), we compute the value of v. Next, the rotational constant \(\bar{B}\) can be found by rearranging the formula and substituting the computed v and the given data. With h and c as Planck's constant and speed of light and J as angular momentum.
02

Compute next P-branch line

The next P-branch line would be two units higher in rotational energy, or \(2\bar{B}\) cm-1 higher in wavenumber. We can compute this by summing 2\bar{B} with the wavenumber of the first P-branch line which is given as the P-branch line's displacement from the Q-branch. Make sure to use the value of \(\bar{B}\) calculated in the first step.
03

Compute the R-branch line

The first R-branch line has rotational energy one unit higher than the Q-branch, or \(\bar{B}\) cm-1 higher in wavenumber. We can compute this by adding \(\bar{B}\) to the wavenumber of the Q-branch line given as 3142.3 cm-1.
04

Considering centrifugal distortion

Centrifugal distortion causes the rotational lines to be found at lower wavenumbers than that in an ideal rigid rotor. Therefore, if centrifugal distortion is considered, the first P-branch line will be found at a lower wavenumber as in part (a).

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Most popular questions from this chapter

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 \(J+1 \leftarrow J\) 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.

Consider a two-dimensional harmonic oscillator with displacements in the \(x\) - and \(y\) -directions, the force constants being the same for each direction (the two bending modes of \(\mathrm{CO}_{2}\) is an example ). Show that the state resulting from the excitation of the oscillator to its first excited state can be regarded as possessing one unit of angular momentum about the \(z\) -axis. Hint. Show that \(\psi(x) \psi(y) \propto \mathrm{e}^{\mathrm{jp}}\)

A diatomic molecule for which \(\tilde{v}=4401.2 \mathrm{cm}^{-1}\) and \(\bar{B}=121.3 \mathrm{cm}^{-1}\) is initially in the state \((\nu=1, J=2)\) In a Raman experiment utilizing \(15873.0 \mathrm{cm}^{-1}\) incident radiation, determine the wavenumber of the scattered radiation for (a) the Q-branch Stokes line, (b) the O-branch Stokes line, (c) the Q-branch anti-Stokes line. How will the wavenumber computed in part (a) change if the effects of anharmonicity are included?

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.
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