This question looks at the rotation spectrum for carbon monoxide, CO. The molecule is first treated as a rigid rod with a moment of inertia I ~ 1.5 × 10??? kgm² about axes perpendicular to its length, and zero parallel to its length. a. What is the energy E? and degeneracy D? of a general eigenstate with angular momentum quantum number J? b. What is the spacing between the levels in terms of J? Sketch the position of the levels with J = 0, 1 and 2, labeling their degeneracy and spacing. c. Explain in one or two sentences the origin of the selection rule ?J = ±1 for absorption of electromagnetic radiation (in the dipole approximation). The thermal occupation of the energy levels is proportional their degeneracy D? multiplied by the Boltzmann weight e???/???. d. Sketch the absorption spectrum as a function of energy for rotational excitations of CO. For CO the ratio of the energies of the 6th absorption to the 1st absorption line is found experimentally to be 5.9988. The difference from 6 is attributed to the stretching of the molecule. e. From this difference, estimate the vibration frequency of the CO bond to a precision of one significant figure. [? ~ 1.0 × 10?³? Js?¹]
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Step 1: The energy EJ of a general eigenstate with angular momentum quantum number J is given by the formula: \[E_J = \frac{{\hbar^2 J (J + 1)}}{{2I}}\] where I is the moment of inertia of the molecule. Show more…
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The WKB Approximation. It can be achallenge to solve the Schrödinger equation for the bound-state energy levels of an arbitrary potential well. An alternative approach that can yield good approximate results for the energy levels is the $W K B$ approximation (named for the physicists Gregor Wentzel, Hendrik Kramers, and Léon Brillouin, who pioneered its application to quantum mechanics). The WKB approximation begins from three physical statements: (i) According to de Broglie, the magnitude of momentum $p$ of a quantum-mechanical particle is $p=h / \lambda$ (ii) The magnitude of momentum is related to the kinetic energy $K$ by the relationship $K=p^{2} / 2 m$ (iii) If there are no nonconservative forces, then in Newtonian mechanics the energy $E$ for a particle is constant and equal at each point to the sum of the kinetic and potential energies at that point: $E=K+U(x),$ where $x$ is the coordinate. (a) Combine these three relationships to show that the wavelength of the particle at a coordinate $x$ can be written as $$ \lambda(x)=\frac{h}{\sqrt{2 m[E-U(x)]}} $$ Thus we envision a quantum-mechanical particle in a potential well $U(x)$ as being like a free particle, but with a wavelength $\lambda(x)$ that is a function of position. (b) When the particle moves into a region of increasing potential energy, what happens to its wavelength? (c) At a point where $E=U(x),$ Newtonian mechanics says that the particle has zero kinetic energy and must be instantaneously at rest. Such a point is called a classical turning point, since this is where a Newtonian particle must stop its motion and reverse direction. As an example, an object oscillating in simple harmonic motion with amphtude $A$ moves back and forth between the points $x=-A$ and $x=+A ;$ each of these is a classical turning point, since there the potential energy $\frac{1}{2} k^{\prime} x^{2}$ equals the total energy $\frac{1}{2} k^{\prime} A^{2}$ . In the WKB expression for $\lambda(x),$ what is the wavelength at a classical turning point? $d$ ) For a particle in a box with length $L,$ the walls of the box are classical turning points (see Fig. 40.1$)$ . Furthermore, the number of wavelengths that fit within the box must be a half-integer (see Fig, $40.3 ),$ so that $L=(n / 2) \lambda$ and hence $L / \lambda=n / 2,$ where $n=1,2,3, \ldots .$ [Note that this is a restatement of Eq. $(40.7) . ]$ The WKB scheme for finding the allowed bound-state energy levels of an arbitrary potential well is an extension of these observations. It demands that for an allowed energy $E,$ there must be a half-integer number of wavelengths between the classical turning points for that energy. Since the wavelength in the WKB approximation is not a constant but depends on $x$ , the number of wavelengths between the classical turning points $a$ and $b$ for a given value of the energy is the integral of 1$/ \lambda(x)$ between those points: $$ \int_{a}^{b} \frac{d x}{\lambda(x)}=\frac{n}{2} \quad(n=1,2,3, \dots) $$ Using the expression for $\lambda(x)$ you found in part (a), show that the $W K B$ condition for an allowed bound-state energy can be written as $$ \int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2} \quad(n=1,2,3, \dots) $$ (e) As a check on the expression in part (d), apply it to a particle in a box with walls at $x=0$ and $x=L$ . Evaluate the integral and show that the allowed energy levels according to the WKB approximation are the same as those given by Eq. (40.9). (Hint since the walls of the box are infinitely high, the points $x=0$ and $x=L$ are classical turning points for any energy $E$ . Inside the box, the potential energy is zero. (f) For the finite square well shown in Fig. 40.6 , show that the WKB expression given in part (d) predicts the same bound-state energies as for an infinite square well of the same width. (Hint: Assume $E<U_{0}$ . Then the classical turning points are at $x=0$ and $x=L . )$ This shows that the WKB approximation does a poor job when the potential energy function changes discontinuously, as for a finite potential well. In the next two problems we consider situations in which the potential-energy function changes gradually and the WKB approximation is much more useful.
A simple mass spectrometer may include an electron ionization (EI) source and a magnetic sector mass analyzer. In this type of instrument, singly charged ions are produced and accelerated through the slit to the analyzer by applying high potentials to accelerator plates. If an ion with mass 293 amu and charge z = 1 is accelerated by a potential of 7500 V, what is its kinetic energy (in J)? 1. What is its kinetic energy (in J)? Note: The following information may be useful for solving the two parts of this problem. 1 amu = 1.66 x 10^-27 kg. Electronic charge, e = 1.602 x 10^-19 C. 1 J = 1 kg m^2/s^2. 1 V = 1 J/C. 2. What is the velocity of the ion? Part B Match each statement with the term it most closely matches. A. Electrospray Ionization (ESI) B. Quadrupole C. Magnetic sector D. Hard ionization E. Molecular ion peak F. Fast atom bombardment (FAB) G. Matrix-Assisted Laser Desorption/Ionization (MALDI) H. Ion-Trap I. Soft ionization J. Time of Flight (TOF) K. Base peak ........ Feature in a mass spectrum that is due to a singly charged ion that has the same mass as the analyte. ........ General term used to describe ionization techniques that produce few ion fragments. Chemical ionization is an example of this. ........ This ionization technique is used to analyze large, non-volatile molecules. The analyte is dissolved in a liquid matrix and placed on a target where it is impacted with an atom beam. ........ Mass analyzer in which ions are separated by their velocity. Ions sequentially strike the detector in order of increasing m/z. ........ This ionization technique is used to analyze large, non-volatile molecules. Ions are formed by placing the analyte in an excess of UV-absorbing molecules, followed by irradiation with a laser. ........ Analyzer that is based upon the deflection of ions in a magnetic field. ........ Analyzer that scans DC and RF amplitudes to generate a mass spectrum. Part C Consider the following statements as they apply to mass spectrometry. Decide if each is true or false. ....... Nominal mass, isotopic mass, and average mass are three terms that have the same meaning for mass spectroscopists. ....... A GC is commonly used as a mass spec inlet system because it can be used to separate mixtures of analytes before analysis. ........ A sample must be ionized in order to be separated in a mass analyzer. ........ In order to get a mass spectrum of an analyte, it must have a significant vapor pressure. ........ A mass spectrometer is an instrument that produces ions and separates them according to their mass-to-charge ratios, m/z.
Adi S.
It can be a challenge to solve the Schrodinger equation for the bound-state energy levels of an arbitrary potential well. An alternative approach that can yield good approximate results for the energy levels is the $WKB$ $approximation$ (named for the physicists Gregor Wentzel, Hendrik Kramers, and Leon Brillouin, who pioneered its application to quantum mechanics). The WKB approximation begins from three physical statements: (i) According to de Broglie, the magnitude of momentum $p$ of a quantum-mechanical particle is $p = h/\lambda$. (ii) The magnitude of momentum is related to the kinetic energy $K$ by the relationship $K = p^2/2m$. (iii) If there are no nonconservative forces, then in Newtonian mechanics the energy $E$ for a particle is constant and equal at each point to the sum of the kinetic and potential energies at that point: $E = K + U(x)$, where $x$ is the coordinate. (a) Combine these three relationships to show that the wavelength of the particle at a coordinate $x$ can be written as $$\lambda(x) = h \sqrt{2m[E - U(x)]}$$ Thus we envision a quantum-mechanical particle in a potential well $U(x)$ as being like a free particle, but with a wavelength $\lambda(x)$ that is a function of position. (b) When the particle moves into a region of increasing potential energy, what happens to its wavelength? (c) At a point where $E = U(x)$, Newtonian mechanics says that the particle has zero kinetic energy and must be instantaneously at rest. Such a point is called a $classical$ $turning$ $point$, since this is where a Newtonian particle must stop its motion and reverse direction. As an example, an object oscillating in simple harmonic motion with amplitude $A$ moves back and forth between the points $x = -A$ and $x = +A$; each of these is a classical turning point, since there the potential energy ${1\over2} k'x^2$ equals the total energy ${1\over2} k'A^2$. In the WKB expression for l1x2, what is the wavelength at a classical turning point? (d) For a particle in a box with length $L$, the walls of the box are classical turning points (see Fig. 40.8). Furthermore, the number of wavelengths that fit within the box must be a half-integer (see Fig. 40.10), so that $L = (n/2)\lambda$ and hence $L/\lambda = n/2$, where $n$ = 1, 2, 3,c. [Note that this is a restatement of Eq. (40.29).] The WKB scheme for finding the allowed bound-state energy levels of an $arbitrary$ potential well is an extension of these observations. It demands that for an allowed energy $E$, there must be a half-integer number of wavelengths between the classical turning points for that energy. Since the wavelength in the WKB approximation is not a constant but depends on $x$, the number of wavelengths between the classical turning points a and b for a given value of the energy is the integral of 1/$\lambda(x)$ between those points: $$\int ^b _a {dx \over \lambda(x)} = {n \over 2} (n = 1, 2, 3, . . .)$$ Using the expression for l1x2 you found in part (a), show that the $WKB$ $condition$ $for$ $an$ $allowed$ $bound-state$ $energy$ can be written as $$\int ^b _a \sqrt{2m[E - U(x)]} dx = {nh \over 2} (n = 1, 2, 3, . . .)$$ (e) As a check on the expression in part (d), apply it to a particle in a box with walls at $x$ = 0 and $x = L$. Evaluate the integral and show that the allowed energy levels according to the WKB approximation are the same as those given by Eq. (40.31). ($Hint$: Since the walls of the box are infinitely high, the points $x$ = 0 and $x = L$ are classical turning points for any energy E. Inside the box, the potential energy is zero.) (f) For the finite square well shown in Fig. 40.13, show that the WKB expression given in part (d) predicts the same bound-state energies as for an infinite square well of the same width. ($Hint$: Assume $E < U_0$ . Then the classical turning points are at $x$ = 0 and $x = L$.) This shows that the WKB approximation does a poor job when the potential-energy function changes discontinuously, as for a finite potential well. In the next two problems we consider situations in which the potentialenergy function changes gradually and the WKB approximation is much more useful.
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