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Chapter 13: Problem 14

The following data are obtained for the vibration-rotation spectrum of \(\mathrm{H}^{\text {79 }}\) Br. Determine \(\tilde{B}_{0}, \tilde{B}_{1}, \tilde{B}_{e},\) and \(\tilde{\alpha},\) from these data. \begin{tabular}{lc} Line & Frequency/cm \(^{-1}\) \\ \hline\(R(0)\) & 2642.60 \\ \(R(1)\) & 2658.36 \\ \(P(1)\) & 2609.67 \\ \(P(2)\) & 2592.51 \end{tabular}

Short Answer

Expert verified
\(\tilde{B}_0 = 8.93\), \(\tilde{B}_1 = 7.53\), \(\tilde{B}_e = 8.58\), \(\tilde{\alpha} = 0.35\)

Step by step solution

01

Understanding the Problem

We are given vibrational-rotational spectrum lines for a diatomic molecule, \(\mathrm{H}^{79}\mathrm{Br}\), and need to calculate the rotation constants \(\tilde{B}_0\), \(\tilde{B}_1\), \(\tilde{B}_e\), and the centrifugal distortion constant \(\tilde{\alpha}\). These constants relate to molecular rotational energy levels in the vibrational states.
02

Identify Necessary Equations

The transition frequencies for R and P branches in the vibration-rotation spectrum are given by the approximations: for \(R(J)\), \(u = u_0 + (2J+1)\tilde{B}_1 - (4J+6)\tilde{\alpha}\), and for \(P(J)\), \(u = u_0 - (2J-2)\tilde{B}_0 - (4J+2)\tilde{\alpha}\). The parameters \(\tilde{B}_0\), \(\tilde{B}_1\), and \(\tilde{\alpha}\) can be derived from these relations.
03

Calculate \(\tilde{B}_0\)

Using the P branch data: \(\Delta u(P) = \tilde{B}_0 - \tilde{\alpha}\), where \(\Delta u(P(1)-P(2)) = u_{P(1)} - u_{P(2)}\). Substitute the given frequencies: \[\Delta u(P) = 2609.67 - 2592.51 = 17.16\] \[\tilde{B}_0 - \tilde{\alpha} = \frac{17.16}{2} = 8.58\]
04

Calculate \(\tilde{B}_1\)

Using the R branch data: \(\Delta u(R) = \tilde{B}_1 + \tilde{\alpha}\), where \(\Delta u(R(0)-R(1)) = u_{R(1)} - u_{R(0)}\). Substitute the given frequencies: \[\Delta u(R) = 2658.36 - 2642.60 = 15.76\] \[\tilde{B}_1 + \tilde{\alpha} = \frac{15.76}{2} = 7.88\]
05

Solve for \(\tilde{\alpha}\)

With the two equations \(\tilde{B}_0 - \tilde{\alpha} = 8.58\) and \(\tilde{B}_1 + \tilde{\alpha} = 7.88\), solve for \(\tilde{\alpha}\). Subtract the second equation from the first: \[2\tilde{\alpha} = 8.58 - 7.88 = 0.70\] \[\tilde{\alpha} = 0.35\]
06

Find \(\tilde{B}_0\) and \(\tilde{B}_1\)

Now, substitute \(\tilde{\alpha} = 0.35\) back into the equations to find \(\tilde{B}_0\) and \(\tilde{B}_1\): \[\tilde{B}_0 = 8.58 + 0.35 = 8.93\] \[\tilde{B}_1 = 7.88 - 0.35 = 7.53\]
07

Calculate \(\tilde{B}_e\)

\(\tilde{B}_e\) is related to \(\tilde{B}_0\) and \(\tilde{B}_1\) by the equation \(\tilde{B}_e = \tilde{B}_0 - \tilde{\alpha}\). Therefore: \[\tilde{B}_e = 8.93 - 0.35 = 8.58\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vibration-Rotation Spectrum
In molecular spectroscopy, the vibration-rotation spectrum provides insight into the molecular structure and behavior of diatomic molecules. This spectrum results from the simultaneous changes in vibrational and rotational energy states of a molecule when it absorbs or emits electromagnetic radiation.
The vibration-rotation spectrum is characterized by two main branches:
  • The **R branch**, which corresponds to an increase in rotational quantum number.
  • The **P branch**, which corresponds to a decrease in the rotational quantum number.
These changes in energy levels give rise to spectral lines that help us understand molecular interactions and constants, like the rotational constants and distortion factors. Each line's frequency reflects transitions in rotational and vibrational states, revealing crucial molecular information.
Diatomic Molecule
A diatomic molecule consists of only two atoms, which may be either of the same element or different elements. In the case of ",
Rotational Energy Levels
Rotational energy levels are the quantized energy states associated with the rotation of a molecule. For a diatomic molecule, the moment of inertia, which depends on the internuclear distance and masses of the atoms involved, influences these energy levels.
The energy levels can be defined by:
  • **Suppressed centrifugal distortion**: At low energy states, distortion due to centrifugal force is minimized, and the system can be approximated using rigid rotor model.
  • **Inclusion of centrifugal distortion**: At higher energy states, distortion becomes significant. The correction factor, where centrifugal distortion constant , accounts for deviation from the rigid rotor model.
Rotational constants, such as , are extracted from the analysis of rotational energy levels and are indicative of the molecule's moment of inertia. Available spectral data helps in deriving these constants.
Spectral Lines Analysis
Spectral line analysis is essential in molecular spectroscopy as it involves examining lines formed due to transitions in molecular energy states. Each spectral line represents an energy change within the molecule and is determined by the frequency of light absorbed or emitted.
For diatomic molecules:
  • **Calculating constants**: Differences in spectral line frequencies help determine molecular constants (e.g., , rotational constants).
  • **Identifying transitions**: The frequency differences between spectral lines relate to specific transitions in vibrational or rotational states.
  • **Understanding molecular dynamics**: Such analysis provides insight into internal molecular dynamics, influencing vibrational and rotational spectra.
With accurate spectral line analysis, one retrieves valuable data on the physical properties and behavior of molecules under study.

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Most popular questions from this chapter

The following spectroscopic constants were determined for pure samples of \(^{74} \mathrm{G}^{32}\) S and \\[ \begin{array}{ccccc} ^{72} \mathrm{Ge}^{32} \mathrm{S}: & & & & \\ & \text { Molecule } & B_{e} / \mathrm{MHz} & \alpha_{e} / \mathrm{MHz} & D / \mathrm{kHz} & R_{e}(v=0) / \mathrm{pm} \\ \hline^{74} \mathrm{Ge}^{32} \mathrm{S} & 5593.08 & 22.44 & 2.349 & 0.20120 \\\ & ^{72} \mathrm{Ge}^{32} \mathrm{S} & 5640.06 & 22.74 & 2.388 & 0.20120 \end{array} \\] Determine the frequency of the \(J=0\) to \(J=1\) transition for \(^{74} \mathrm{Ge}^{32} \mathrm{S}\) and \(^{72} \mathrm{Ge}^{32} \mathrm{S}\) in their ground vibrational states. The width of a microwave absorption line is on the order of 1 KHz. Could you distinguish a pure sample of \(^{74} \mathrm{Ge}^{32} \mathrm{S}\) from a \(50 / 50\) mixture of \(^{74} \mathrm{Ge}^{32} \mathrm{S}\) and \(^{72} \mathrm{Ge}^{32} \mathrm{S}\) using microwave spectroscopy?

Determine which of the following molecules will exhibit a microwave rotational absorption spectrum: \(\mathrm{H}_{2}, \mathrm{HCl}, \mathrm{CH}_{4}, \mathrm{CH}_{3} \mathrm{I}, \mathrm{H}_{2} \mathrm{O},\) and \(\mathrm{SF}_{6}\).

The following lines were observed in the microwave absorption spectrum of \(\mathrm{H}^{127}\) I and \(\mathrm{D}^{127} \mathrm{I}\) between \(60 \mathrm{cm}^{-1}\) and \(90 \mathrm{cm}^{-1}\) \\[ \begin{array}{llllll} & \multicolumn{5}{c} {\tilde{v} / \mathrm{cm}^{-1}} \\ \mathrm{H}^{127} \mathrm{I} & 64.275 & 77.130 & 89.985 & \\ \mathrm{D}^{127} \mathrm{I} & 65.070 & 71.577 & 78.084 & 84.591 \end{array} \\] Use the rigid-rotator approximation to determine the values of \(\bar{B}, I,\) and \(R_{e}\) for each molecule. Do your results for the bond length agree with what you would expect based upon the Born-Oppenheimer approximation? Take the mass of \(^{127}\) 1 to be 126.904 amu and the mass of \(D\) to be 2.014 amu.

Assuming the rotation of a diatomic molecule in the \(J=10\) state may be approximated by classical mechanics, calculate how many revolutions per second \(^{23} \mathrm{Na}^{35} \mathrm{Cl}\) makes in the \(J=10\) rotational state. The rotational constant of \(^{23} \mathrm{Na}^{35} \mathrm{Cl}\) is \(6500 \mathrm{MHz}\).

The spacing between the lines in the microwave spectrum of \(\mathrm{H}^{35} \mathrm{Cl}\) is \(6.350 \times 10^{11} \mathrm{~Hz}\). Calculate the bond length of \(\mathrm{H}^{35} \mathrm{Cl}\).
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