Download Quantum Mechanics and Thermodynamics in Chemistry: A Spring 2008 Course and more Study notes Physical Chemistry in PDF only on Docsity! CHEM 3520 Spring 2008 1 Unit I Introduction A. Review of Quantum Mechanics 1. Molecular energy levels a. Consider a system with ⎩ ⎨ ⎧ electrons nuclei n N b. This system will be (completely) described by a wavefunction Ψ(r, R) depending on the coordinates of all the nuclei and all the electrons. c. According to Born-Oppenheimer approximation, the wavefunction is separated: )();(),( RRrRr nuel ψψ=Ψ □ Solve for electronic wavefunction (ψel) considering a fixed position of the nuclei (i.e., at one nuclear configuration). □ Electrons create a potential in which nuclei move, and this potential is called a potential energy surface for polyatomic molecule or a potential energy curve for a diatomic molecule. □ The nuclear motion (which is also quantized) can be separated in vibrational, rotational and translation motions (or degrees of freedom). 1 nucleus (atom) 2 nuclei (diatomic) N (>2) nuclei (polyatomic) ψel depends on 3n coordinates ψel depends on 3n coordinates ψel depends on 3n coordinates ψnu depends on 3N = 6 coordinates ψnu depends on 3N coordinates Linear Nonlinear 3 translational 3 translational 3 translational 2 rotational 2 rotational 3 rotational 1 vibrational 3N – 5 vibrational 3N – 6 vibrational CHEM 3520 Spring 2008 2 2. Translational energy levels a. The energy levels predicted by the one-dimensional particle-in-a-box model: 2 22 8ma nhEn = where n = 1,2,… ○ a is the length of the box b. The energy levels predicted by the three-dimensional particle-in-a-box model: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ++= 2 2 2 2 2 22 8 c n b n a n m hE zyxnnn zyx ○ a, b, and c are the lengths of the three-dimensional box 3. Rotational energy levels for diatomic molecules a. The energy levels predicted by the rigid-rotator model: )1( 2 2 += JJ I EJ h where 2eRI μ= and J = 0,1,2,… ○ The degeneracy of Jth level is gJ = 2J + 1. b. The energies (in wavenumbers) predicted by the rigid-rotator model: )1(~)( += JJBJF where cI hB 28 ~ π = CHEM 3520 Spring 2008 5 d. Equations of state are mathematical formulations connecting system’s variables (or quantities). e. The quantities (variables or properties) describing macroscopic systems can be divided in: □ Extensive properties ○ are directly proportional to the size of the system ○ Examples: V, m, E (or U) □ Intensive properties ○ are independent of the size of the system ○ Examples: P, T, ρ ,V f. The molar properties are indicated by a bar over the symbol. □ V (L/mol) versus V (L); U versus U; E versus E 3. Pressure a. It is defined as force divided by the area to which the force is applied. b. The pressure exerted by the atmosphere is measured with a barometer; the pressure of a sample of gas inside a container is measured with a manometer. hg A Ahg A mg A FP ρρ ==== or hgPP ρ+= ex c. The units of pressure: 22 ms kg m NPa1 == ; 1 bar = 105 Pa 1 atm = 1.01325×105 Pa = 101.325 kPa = 1.01325 bar = 760 mmHg 1 tor = 1 mmHg = (1/760) atm CHEM 3520 Spring 2008 6 4. Temperature a. It is a measure of how much (kinetic) energy the particles of a system have. b. It is not a form of energy; it indicates the direction of flow of energy (as heat) between two objects. c. The SI units are K (Kelvin) that gives the temperature on the thermodynamic temperature scale (perfect-gas temperature scale): 1 K is defined as 1/273.16 of the triple point of water. d. The Celsius scale of temperature °C (degree Celsius): 15.273K/C/ −= Tt o 5. The zeroth law of thermodynamics a. It is related to the temperature and the energy of a system. b. Enunciation: If an object A is in thermal equilibrium with an object B, and B is in thermal equilibrium with an object C, then C is also in thermal equilibrium with A. c. This zeroth law justifies the concept of temperature and the use of a thermometer. □ Example: If A is in thermal equilibrium with B (i.e., the mercury liquid in a capillary), and C is in thermal equilibrium with B then A and C are in thermal equilibrium (i.e., have same temperature) and no change will occur when they are in contact.
