PHYSICAL CHEMISTRY MOLECULAR
PHYSICAL CHEMISTRY MOLECULAR CHGN 351
Colorado School of Mines
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CHGN 351 FALL 2004 PRACTICE MIDTERM 1 Principles of Quantum Mechanics True or False IL PP 90gtC7 Sh gt0 In a photoelectric experiment the work function is the sum of the incident photon energy and the electron kinetic energy Any wave on a classical string can be expressed as a linear combination of the normal modes The state function 1p can never be negative The state function 1px for a particle in a box size a must be of the form A sinn7xa where n is a positive integer and A a constant The system s state function 1p must be an eigenfunction of the system s Hamiltonian The smaller the box a particle is in the higher the kinetic energy in the ground state The ground state of a particle in a two dimensional box is degenerate A measurement of the momentum for a particle in a box ground state can only give two possible values The momentum and Hamiltonian operators commute for a free particle If we measure the property A when the system s state function is not an eigenfunction of A then we can get a result that is not an eigenvalue of A Quantum Phenomena 1 Given that the work function of chromium is 44eV calculate the kinetic energy of electrons emitted from a chromium surface that is irradiated with UV radiation of wavelength 200nm Phenomena and significance of the quantum theory 1 Write a couple of sentences explaining what deBroglie s hypothesis is some experimental veri cations of the hypothesis and some significant consequences Postulates and formalism of quantum mechanics 1 What are the five postulates of quantum mechanics as discussed in class and in the textbook 2 Consider a mass on a vertical spring under the in uence of gravity Give an equation and draw a curve for the potential energy Write down the Hamiltonian operator for this system For the remaining questions consider a particle in a box where the walls are located at x a and xa The ground state wavefunction for this system is 1px c cos2 a 1 3 Normalize the ground state wavefunction to show that c F a 4 Without calculation what is the expectation value of x 5 Write down the explicit expression for the momentum operator 6 What is the expectation value of p2 the momentum squared You can give your answer as an integral you don t need to evaluate it Absorption spectrum of benzene Benzene has 6 pi electrons which can be considered to be delocalized over the molecule As a crude model of the electronic motion we can consider these 6 electrons to be moving freely in a two dimensional square box 3 angstroms on a side 5 v Q v D v What is the ground state lowest energy state of the molecule What energy does the ground state counting all 6 electrons have What is the first excited state next highest energy state of the molecule Draw a diagram indicating which orbitals the electrons are in and label the states with their quantum numbers What is the energy gap between these two states Justify why we expect benzene to be primarily found in the ground state when the temperature is room temperature Calculate the wavelength of light absorbed by benzene in making the transition from its ground state to the next highest state Is it possible to determine the momentum of a benzene electron to arbitrary precision If not give an estimate for the maximum possible precision CHGN 351 FALL 2004 PRACTICE MIDTERM 2 QM principles of vibration rotation and the hydrogen atom True or False gt105 gt0 For a 1D harmonic oscillator HO every eigenstate with an odd quantum number is actually an even function The zero point Vibrational energy of HCl is higher than that of DCl In the Morse potential the energy level spacing gets larger and larger as the Vibrational quantum number increases In the harmonic oscillator model for molecular Vibrations a photon with exactly 2km of energy can be absorbed to change the molecular Vibrational state The reduced mass of a diatomic is always less than the mass of either atom The spacing between lines in the rotational spectrum in HCl is higher that that of DCl The Q branch in the IR spectrum of HCl shows no absorption because of the Vibrational selection rule Av 1 The 5pZ orbital is an eigenfunction of the operators 2 and 1 The radial part of the wavefunction for the hydrogen atom is the same for the 5s and the 5p states For the H atom all orbitals with the same quantum numbers 1 and m have the same number of angular nodes Relation between spectroscopy and molecular parameters 1 The force constant of 79Br79Br is 240 Nm l Calculate the fundamental Vibrational frequency and zero point energy of 79Br79Br 2 The J20 to J1 transition for carbon monoxide IZCIGO occurs at 1153 x 105MHz Calculate the value of the bond length of carbon monoxide 3 Muonium Mu is a variant of an H atom where the nucleus instead of being a proton is a muon A muon is a particle with the same charge as a proton but has 19 h the mass What is the reduced mass 4 of muonium and what fraction is it of the reduced mass of normal hydrogen 4 Sketch the rotation Vibration spectrum of HCl Label the axes indicating the direction of increasing wavenumber On the spectrum indicate which peaks correspond to the following transitions a J20 to J21 b J1 to J22 c J1 to J20 Nodal character of H atom orbitals Answer the following questions for the 3pZ orbital of hydrogen a What are the quantum numbers b Using the above quantum numbers write down an equation for the 3pZ orbital O v Solve for the nodes of the 3pZ orbital d Sketch the 3pZ orbital and its nodes labeling each node with the corresponding nodal equation above Genearal equations for the radial and angular wavefunctions 12 3 I 2 Rd n l 13 i r1 L113 1 errnun 2nn l nao nao 21 1 l w 1 quot39 W inn Y1 9 4 0 m P cos0 e The rstfew Legendre mctions Po x 1 P1 x x P1106 11 x2 P2 x 6262 1 P21x 3x1 1 x2 P22x 31 262 P30 i6 3quot P3106 35 1x1 x2 P32x15x1 262 The rstfew Laguerre polynomials L11x 1 L12x 22 x 130 6 1500 63 3x x2 Lime 244 x L55x 120 Hydrogen like atom Consider the ion Liz which consists of a nucleus of charge 3 and 1 electron It is thus just like a H atom but with higher nuclear charge Write down the Hamiltonian operator for Liz including all terms Measurements on the Hydrogen 3pX orbital a What linear combination of orbitals gives rise to the 3pX orbital Write down in spherical coordinates the wavefunction for this orbital b What are the possible outcomes for measuring the energy The magnitude of the angular momentum L The z component of the angular momentum LZ c What is the most probable value of the measured distance between the electron and the nucleus CHGN 351 FALL 1999 PRACTICE MIDTERM 2 Systems amp Methods of Quantum Mechanics True or False WP 9 9 o N D ID I bu The HO eigenstates form an orthonormal set Tunneling is only possible for a HO For a 1D harmonic oscillator HO every eigenstate with an odd quantum number is actually an even function The zeropoint vibrational energy of HCl is higher than that of DCl In the Morse potential the energy level spacing gets larger and larger as the vibrational quantum number increases In the harmonic oscillator model for molecular vibrations a photon with exactly 27m of energy can be absorbed to change the molecular vibrational state The reduced mass of a diatomic is always less than the mass of either atom The set of eigenstates for a particle in a central potential can always be chosen to also be eigenstates of the total angular momentum 2 The eigenfunctions of L are real functions The rotational spectrum of a molecule can be used to determine its moment of inertia The energies of the Hatom depend only on the n quantum number and are the same values given by the Bohr model The radial part of the wave lnction for the hydrogen atom is the same for the 5s and the 5p states The angular part of the 3pZ orbital is a spherical harmonic with 13 and m0 The 3s orbital of an H atom is a polynomial times a Gaussian in r The variational method gives an energy which is always higher than the true ground state energy More general knowledge and principles Give the eigenstates and associated energies for the following Hamiltonian you can leave normalization constants as unspeci ed constants 2 x2 with oonSoo and OSySa Name a quantum mechanical system where the energy spacing between adjacent eigenstates a increases with increasing quantum number b stays constant with increasing quantum number c decreases with increasing quantum number Particle in a spherical box The particle in a spherical box has V0 for rltb and Voo for rgtb For this system a Explain why w Rrf9 1 where Rr satis es 1 i r2 M 2m r2 6r 6r 2mr2 What is the function f9 1 b Solve the above equation for Rr for the 10 states Hints The substitution Rrgrr reduces the above to an easily solved equation Use the boundary condition that w is nite at F0 and use a second boundary condition Show that VrRr ERr qFNsin krr for the 10 states where n h2 k2mEh2m and E 2 8mb with n 123 CHGN 351 FALL 1999 PRACTICE MIDTERM 1 Principles of Quantum Mechanics True 01 False 9 994 gt1 9 wao Ho N D ID I bu In a photoelectric experiment the work function is the sum of the incident photon energy and the electron kinetic energy The Bohr model predicts an ionization energy for hydrogen atom that is nite Any wave on a classical string can be expressed as a linear combination of the normal modes The state function w can never be negative The probability density for a particleina box can never be negative The state function wx for a particlein a box size a must be of the form A sinn11xa where n is a positive integer and A a constant The state function w for a particlein a box must be a real lnction The system s state function wmust be an eigen lnction of the system s Hamiltonian The smaller the box a particle is in the higher the kinetic energy in the ground state The ground state of a particle in atwodimensional box is degenerate A measurement of the momentum for a particleinabox ground state can only give two possible values The timedependent state function is always equal to a function of time multiplied by a function of the coordinates The momentum and Hamiltonian operators commute for a free particle If I is an eigen lnction of the linear operator 121 then c I is an eigen lnction of 121 where c is an arbitrary constant If we measure the property A when the system s state function is not an eigenfunction of 121 then we can get a result that is not an eigenvalue of 121 Phenomena and significance of the quantum theory 1 Write a couple of sentences explaining what deBroglie s hypothesis is some experimental veri cations of the hypothesis and some signi cant consequences 2 List a few ways in which an electron behaves like a macroscopic particle and in which it behaves like a macroscopic wave Similarly list a few ways in which an electron doesn39t behave like a particle and doesn t behave like a wave Calculating expectation values the harmonic oscillator ground state We will discuss the harmonic oscillator in class but here we give without derivation a stationary state solution to the harmonic oscillator Schrodinger equation For this problem calculate expectation values based on the wavefunction of the ground state w0x cc11 4 eXpOLX22 where at k u hbar2 2 k is the spring constant and u is the reduced mass The parameter X describes the coordinate of the particle in a harmonic potential VX kXZZ and can take any value from oo to 00 a Show this satis es the harmonic oscillator Schrodinger s Equation where the potential energy term in the Hamiltonian is given by VX kXZZ b Determine ltXgt and ltpgt for the ground state c Determine the average potential energy szZ and the average kinetic energy pZZu Some integrals you may nd help ll n It Ie dx xze dx a m M a ne 2 Ixe 2 dx is a very simple integral that requires no algebra remember symmetry foo