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Chapter 10: Problem 26

Identify the conditions for the existence and locations of heads in the \(P\) - and \(R\) -branches of a diatomic molecule.

Short Answer

Expert verified
The P and R branches in a vibrational-rotational spectrum of a diatomic molecule exist due to transitions in rotational energy levels while going to a higher vibrational level and are determined by the change in the rotational quantum number. In P branch transitions, \(J\) decreases by 1 whereas in R branch transitions, \(J\) increases by 1. Spatially, P branch is seen on lower frequency (higher wavelength) side and R branch on higher frequency (lower wavelength) side in the spectrum. The heads of the branches are cut-off points where the lines of transitions converge.

Step by step solution

01

Fundamentals of Vibrational-Rotational Spectrum

In the vibrational-rotational spectrum, the terms P and R branches refer to transitions that occur when a diatomic molecule absorbs a quantum of energy, changing its rotational and vibrational energy levels. In simple terms, they are the possible paths for an energy transition in a diatomic molecule.
02

The condition for the P-branch

In the P-branch transitions, the rotational quantum number \(J\) decreases by 1, corresponding to the molecule losing a unit of angular momentum. The change in the quantum number can be represented as: \(\Delta J = J' - J = -1\), where \(J'\) and \(J\) are the final and initial quantum numbers respectively. In these transitions, the molecule goes from a higher rotational energy level to a lower rotational energy level while going to a higher vibrational level.
03

The condition for the R-branch

In the R-branch transitions, the rotational quantum number \(J\) increases by 1, corresponding to the molecule gaining a unit of angular momentum. The change in the quantum number can be represented as: \(\Delta J = J' - J = +1\), where \(J'\) and \(J\) are the final and initial quantum numbers respectively. In these transitions, the molecule goes from a lower rotational energy level to a higher rotational energy level while going to a higher vibrational level.
04

Existence and location of heads in P and R branches

The P branch appears to the lower frequency (or higher wavelength) side of the spectrum while the R branch appears to the higher frequency (or lower wavelength) side. The lines in each branch tend to get closer and closer together as we move away from the center, and the heads of the branches are where they would converge if there were enough lines. For a real molecule, the number of rotational levels is finite, and the branches cut-off. Therefore, the cut-off points determine the heads of the P and R branches.

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Most popular questions from this chapter

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.

\( \mathrm{What}\) 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?

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?

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

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, \(\left.I=\frac{1}{2} .\right)\)
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