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Chapter 10: Problem 5

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.

Short Answer

Expert verified
The value of \(J\) for which \(\bar{F}\) is maximum can be found by solving the cubic equation \(\frac{d \bar{F}}{dJ} = 0\), and choosing the realistic solution. The exact value depends on the result of this calculation.

Step by step solution

01

Write down the given values and the equation representing \(\bar{F}\)

The given values in the problem are: \(\bar{B}=10.4400\ \mathrm{cm}^{-1}\) and \(\bar{D}=0.0004\ \mathrm{cm}^{-1}\). The equation we will work with is: \(\bar{F}=(2\bar{B}J(J+1)) - \bar{D}(2\bar{B}J(J+1))^2\). The problem asks to find the value of \(J\) for which \(\bar{F}\) is maximum.
02

Find the derivative of \(\bar{F}\) with respect to \(J\)

To find the maximum of \(\bar{F}\), we need to find where its derivative with respect to \(J\) is equal to zero. Hence, compute the derivative using the given expression: \(\frac{d \bar{F}}{dJ} = 0\).
03

Solve the resulting cubic equation

Solve the cubic equation \(\frac{d \bar{F}}{dJ} = 0\) for \(J\). Since this is a cubic equation, it will have three solutions. However, only one of those solutions will be realistic in the given context.
04

Find the realistic solution

Check all solutions from step 3. Only solutions where \(J\) is a non-negative integer are realistic, because the angular momentum \(J\) can't be a negative number or a fraction in this context.

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Most popular questions from this chapter

A diatomic molecule is found to have the following vibrational and rotational spectroscopic constants (all \\[ \left.\operatorname{in} \mathrm{cm}^{-1}\right): \tilde{v}=1525.25, \tilde{v} x_{\mathrm{e}}=21.74, \bar{B}_{\mathrm{e}}=8.295, \tilde{\alpha}_{\mathrm{e}}=0.186 \\] \(\bar{D}=0.325 ;\) the rotational constant depends on vibrational level as \(\tilde{B}_{v}=\tilde{B}_{e}-\left(\nu+\frac{1}{2}\right) \tilde{\alpha}_{e^{*}}\) If the diatomic molecule is initially in the state \((\nu=0, J=1),\) compute the wavenumbers of the \(R\) -and \(P\) -branch lines associated with the fundamental vibrational transition.

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

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.)

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

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.
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