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Chapter 10: Problem 13

\( \mathrm{An}\) 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 \(\omega x_{e^{-}}\).

Short Answer

Expert verified
The exact results could vary based on the calculations, but the process involves applying the formula \(E_{v}= \omega(v+ \frac{1}{2}) - \omega x_{e^{-}}(v+ \frac{1}{2})^{2}\) to calculate for \(\omega\) and \(\omega x_{e^{-}}\) constants.

Step by step solution

01

Identify the positions of spectral lines

Given the wavenumbers for the lowest four transitions from the \(v=0\) state as 2091, 4128, 6111,8043 cm\(^{-1}\). It is important to note that the spaces between these transitions are not symmetrical and hence forms an arithmetic progression.
02

Applications of the formula

Apply the formula \(E_{v}= \omega(v+ \frac{1}{2}) - \omega x_{e^{-}}(v+ \frac{1}{2})^{2}\) where \(E_{v}\) is the energy and \(v\) is the vibrational quantum number of the molecule. For values \(v=0\) to \(v=3\), calculate \(E_v\) using the given wavenumbers and setup equations to solve for \(\omega\) and \(\omega x_{e^{-}}\).
03

Solving for \(\omega\) and \(\omega x_{e\^{-}}\)

After setting up the equations, solve for \(\omega\) and \(\omega x_{e\^{-}}\). Ensure you end up with numerical values with understandable physical meaning.
04

Interpretation of results

From the solved values of \(\omega\) and \(\omega x_{e\^{-}}\), interpret their physical significance in relation to the vibrational energies of the molecule.

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Most popular questions from this chapter

\( \mathrm{What}\) 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?

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)?

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 \(\psi(x) \psi(y) \propto \mathrm{e}^{\mathrm{jp}}\)

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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.)
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