Review Sheet for Introduction Physical Chemistry | CHEM 3510, Study notes of Physical Chemistry

Download Review Sheet for Introduction Physical Chemistry | CHEM 3510 and more Study notes Physical Chemistry in PDF only on Docsity! CHEM 3510 Fall 2007 1 Unit I Introduction A. Introduction to Physical Chemistry 1. Physical Chemistry is the part of chemistry dealing with application of physical methods to investigate chemistry. 2. Physical Chemistry main subdivisions are: a. Quantum Mechanics □ deals with structure and properties of molecules b. Spectroscopy □ deals with the interaction between light and matter c. Computational Chemistry □ deals with modeling chemical properties of reactions using computers d. Statistical Mechanics □ deals with how knowledge about molecular energy levels (or microscopic world) transforms into properties of the bulk (or macroscopic world) e. Thermodynamics □ deals with properties of systems and their temperature dependence and with energetics of chemical reactions f. Electrochemistry □ deals with processes in with electrons are either a reactant or a product of a reaction g. Chemical Kinetics □ deals with the rates of chemical reactions or physical processes CHEM 3510 Fall 2007 2 B. Classical Physics Review 1. Classical Physics was introduced in the 17th century by Isaac Newton. 2. At the end of 19th century, classical physics (mechanics, thermodynamics, kinetic theory, electromagnetic theory) was fully developed and was divided into: a. the corpuscular side or particle domain (the matter) b. the undulatory side or wave domain (the light) 3. Some useful classical physics equations: a. Total energy E: VKE += ○ K is the kinetic energy (or energy arising from motion) ○ V is the potential energy (or energy arising from position) b. Kinetic energy K: m pmK 2 v 2 1 22 == ○ m is the mass ○ v is the velocity (or speed) ○ p is the momentum c. Frequency ν (Greek letter nu): π ων λ ν 2 ~ === cc ○ λ is the wavelength (Greek letter lambda) ○ c is the speed of light ○ ν~ is the wavenumber (read “nu tilde”) ○ ω is the angular frequency (Greek letter omega) 4. Classical mechanics was successful in explaining the motion of everyday objects but fails when applied to very small particles. These failures led to the development of Quantum Mechanics. CHEM 3510 Fall 2007 5 g. Look more closely to the solutions: Number of wavelength that fits in 2l: n = 1 n = 2 n = 3 n = 4 □ By generalization: n l n 2 =λ ○ This is called the eigenvalue condition. □ The solutions are a set a functions called eigenfunctions or characteristic functions. xBx l nBxX n nnn λ ππ 2sinsin)( == □ Also, angular frequencies 0 vv22 ωπ λ ππνω n l n n nn ==== (where l v 0 πω = ) are called eigenvalues or characteristic values. 1 2 3 4 CHEM 3510 Fall 2007 6 h. Solving for T(t) but keeping in mind that l nπβ = 0)(v)( 222 2 =+ tT dt tTd β – Similar to above, the solution is: tEtDtT nn ωω sincos)( += where: l n n vv πβω == i. Coming back to ),( txu : )()(),( tTxXtxu = ⇒ l xntGtFtxu nn πωω sin)sincos(),( += ; n = 1, 2,… – There is a ),( txu function for each n so a better notation would be: l xntGtFtxu nnnnn πωω sin)sincos(),( += ; n = 1, 2,… – The sum of all ),( txun solutions is also a solution of the equation (This is called the principle of superposition.) The general solution is: ∑ ∞ = += 1 sin)sincos(),( n nnnn l xntGtFtxu πωω – Make the transformation: )cos(sincos φωωω +=+ tAtGtF where φ (Greek letter phi) is the phase angle and A is the amplitude of the wave. – Rewrite the general equation as: ⇒ ∑∑ ∞ = ∞ = =+= 11 ),(sin)cos(),( n n n nnn txul xntAtxu πφω □ Each ),( txun is called: ○ a normal mode ○ a standing wave ○ a stationary wave ○ an eigenfunction of this problem j. The time dependence of each mode represents a harmonic motion of frequency: n n n l n λπ ων v 2 v 2 === where the angular frequency is: n nn l nv λ πππβω v2v2v ==== . CHEM 3510 Fall 2007 7 k. Solutions: □ First term is l xt l A πφπ sin)vcos( 11 + ○ First term is called fundamental mode or first harmonic. ○ The frequency is: l2/v1 =ν □ Second term is l xt l A πφπ 2sin)v2cos( 22 + ○ Second term is called first overtone or second harmonic. ○ The frequency is: l/v2 =ν ○ The midpoint has a zero displacement at all times, and it is called a node. □ Third term is l xt l A πφπ 3sin)v3cos( 33 + ○ Third term is called second overtone or third harmonic. ○ The frequency is: l2/v33 =ν ○ This term has two nodes. □ Fourth term is l xt l A πφπ 4sin)v4cos( 44 + CHEM 3510 Fall 2007 10 c. Again, the general function is a superposition of normal modes ),,( tyxunm but in this case one obtains nodal lines (lines where the amplitude is 0) instead of nodes. d. Examples: m = 1 n = 1 m = 2 n = 1 m = 1 n = 2 m = 2 n = 2 e. The case of a square membrane ( ba = ), the frequencies of the normal modes are given by: 22v mn anm += πω □ For the cases of n = 1, m = 2 and n = 2, m = 1 one can see that: a πωω v52112 == although ),,(),,( 2112 tyxutyxu ≠ f. This is an example of a degeneracy. □ The frequency 2112 ωω = is double degenerate or two-fold degenerate. □ This phenomenon appears because of the symmetry ( ba = ). CHEM 3510 Fall 2007 11 D. Unit Review 1. Important Terminology frequency wavelength wavenumber angular frequency independent variables boundary conditions separation of variables eigenfunctions eigenvalues stationary wave traveling wave node degeneracy CHEM 3510 Fall 2007 12 2. Important Formulas VKE += m pmK 2 v 2 1 22 == π ων λ ν 2 ~ === cc