CHE 349 Physical Chemistry for Life Sciences Lecture 18 Particle In a Box The particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers For example in classical systems a particle trapped in a large box can move at any speed within the box and it is no more likely to be found at one position than another The particle may only occupy certain positive energy levels The particle can never have zero energy meaning that the particle can never sit still Depending on the particle s energy level it is more likely to be found at certain positions The energy difference between two levels in the particle in a box equation 8 than at others 28 2 2 1 28 2 2 2 2 2 2 Harmonic Oscillator For a particle in a 2D 3D box of sides a b and c the wavefunctions are Translational Partition Function from a 2 D 3 D particle in a box equation A harmonic oscillator is a good approximation for the vibration of diatomic molecules and the vibration of bonds in polyatomic molecules Vibrational frequencies are characteristic of a particular type of bond such as C H C C C C and C N but they can also be used to determine force constants k for the particular vibration When the force constant k for a double bond vibration decreases the frequency decreases it means that the bond has become weaker the double bond character has decreased These differences in bond strength for the same type of bond in different molecules are a useful clue to the distribution of electrons in the molecules As you may or may not recall bond strength is influenced by factors such as electronegativity molecular geometry and the presence of lone pairs or multiple bonds If atoms forming a bond have different electronegativities the more electronegative atom tends to attract electrons more strongly leading to an uneven distribution of electron density and a polar bond The molecular geometry a k a arrangement of atoms in a molecule affects how electron pairs are distributed For example in a double bond the electrons are more localized between the two bonded atoms compared to a single bond impacting bond strength Lone pairs can create regions of higher electron density affecting the overall electron distribution in the molecule Multiple bonds involve stronger interactions than single bonds due to the increased electron density between the bonded atoms Take a mass m connected to an infinitely heavy wall by a spring Assume there s no gravity The equilibrium length of the spring is referred to as When we pull on the mass and extend the spring to length L there s going to be a restoring force to bring the spring back to its original position same would happen if you were to compress the spring The displacement is positive if the spring is extended and the displacement is negative if the spring is compressed NOTE The force constant is the measure of the stiffness or strength of the electronic bond holding the nuclei together It s not affected by any changes in the reduced mass thus making the force constant independent of the reduced mass 0 depends on the strength of the bond and the masses Simple Harmonic Oscillator Harmonic Oscillator Schr dinger equation Each oscillator has a fundamental vibration frequency k force constant for the bond x the displacement from an equilibrium position 0 0 22 2 2 12 2 Harmonic Potential Energy equation 12 2 Quantum Oscillator Energy equation 12 0 Vibration Frequency of the oscillator 0 12 1 2 3 00 108 0 1 2 1 2 1 2 1 2 0 1 2 Ratio of New Mass 1 to Original Mass 1 Equation 0 0 reduced mass of the oscillator k force constant for the bond Wavenumber of a light wave c velocity of light wavefunction 1 1 original first mass new first mass Ratio of New Mass 2 to Original Mass 2 Equation 2 2 original second mass new second mass 0 1 2 1 2 1 2 1 2 0 1 2 Simple Harmonic Oscillator Eigenfunctions and Eigenvalues 2 2 1 4 4 2 1 2 1 2 1 42 1 2 2 0 2 1 4 2 2 52 0 1 32 0 0 12 0 NOTE According to quantum mechanics the lowest energy level is zero point energy called the Reduced Mass 12 0 The concept of reduced mass helps us to describe vibrations in a diatomic molecule connected by a spring and and are found at The equilibrium distance between the two masses will be referred to as When we pull on either mass and extend it to length L there s going to be a restoring The displacement from equilibrium will be referred to as x The equation for this 1 2 0 and 1 2 force to bring the spring back to its original position Imagine there are two masses the positions 1 2 0 displacement is 2 1 1 2 1 2 2 2 2 2 1 2 2 2 1 1 2 1 2 2 2 1 2 2 1 2 1 2 2 2 1 2 1 2 1 1 1 2 Expression for the reduced mass 1 2 1 2 1 2 1 10 2 10 1 10 2 4 1 10 2 1 molecular weight mass of atom 1 molecular weight mass of atom 2 and will decrease even further So is half of each of the masses So will significantly decrease So Examples for the reduced mass that the reduced mass the reduced mass reduced mass 3 Now take 2 Now take 1 Take and and left The plus sign means that after extension the spring pulls the mass to the right The negative sign means that after extension the spring pulls the mass back to the 1 2 If the two masses are equal we ll find that the If one of the masses gets smaller we ll find that If one of the masses gets even smaller we ll find 5 2 8 0 9 Quantum Mechanical Tunneling Tunneling is a quantum mechanical phenomenon when a particle is able to penetrate through a potential energy barrier that is higher in energy than the particle s kinetic energy The probability of tunneling depends on the energy and mass of the particle and on the height and width of the barrier For a quantum particle to appreciably tunnel through a barrier three conditions must be met The height of the barrier must be finite and the thickness of the barrier should be The potential energy of the barrier exceeds the kinetic energy of the particle thin E V The particle has wave properties because the wavefunction is able to penetrate through the barrier This suggests that quantum tunneling only apply to microscopic objects such protons or electrons and does not apply to macroscopic objects For a constant energy the wavelength of the wavefunction of the particle increases as the mass of the particle decreases As the mass of a particle increases the probability of quantum tunneling …