CHM320 EXAM #2 USEFUL INFORMATION

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1 CHM30 EXAM # USEFUL INFORMATION Constants mass of electron: m e = kg. Rydberg constant: R H = cm 1 = J. speed of light: c = m/s Planck constant: Js κ = 1 4πǫ 0 = kgm 3 /c s charge of electron: e = C Bohr radius: a 0 = 5.9 pm Avogadro s number: N A = mol 1 Unit Conversion 1 J = 1 kgm /s 1 ev = J. 1 J = cm 1 1 amu = kg. 1 Å= m. 1 N (newton) = 1 kg m/s Formula (1) Energy of a particle in classical mechanics: E = kinetic (T) + potential (V) E = T +V = p m +V where p is the linear momentum (p = mv). Force is related to the potential energy as F = V. For one-dimensional system, F = dv dx. 1

2 () Electromagnetic wave (classical view): E(x,t) = E 0 sin(kx ωt) where k = π (wavenumber) and ω = πν (angular frequency). E λ 0 is called the amplitude of wave. The period of oscillation is Γ = 1. λ is the wavelength and ν is the frequency of ν oscillation. The frequency and wavelength of light are related through the speed of light (c) as ν = c λ (3) Photoelectric Effect: T e = E photon Φ where T e is the kinetic energy of ejected electron and Φ is the work function. The photon energy is given by E photon = hν = h c λ = hc ν where c is the speed of light and ν = 1 λ. (4) Rydberg formula for hydrogen atom emission spectra: ( 1 ) ν(cm 1 ) = 1 λ = R H n 1 1 n where n > n 1 and R H is the Rydberg constant in cm 1. Lymann, Balmer and Paschen series of emission lines involve n 1 = 1,, and 3, respectively. (5) de Broglie hypothesis: λ = h p (6) Classical physics of angular motion: 1. angular momentum: l = pr.. kinetic energy: T = l mr 3. centripetal force: F cent = l mr 3

3 (7) Bohr model of hydrogen atom: Postulate: Angular momentum is quantized, l = n h ( κme e 4 ) 1 E n = h n = R H n, n = 1,,3,.. where κ = 1 4πǫ 0 and R H is the Rydberg constant. (8) Uncertainty Principle: x p = h (9) Schrodinger Equation (1-dimensional): h d ψ(x) +V(x)ψ(x) = Eψ(x) m dx (10) Probability: Probability to find the particle with wavefunction ψ(x) between x = a and x = b is given by P = b a ψ(x) dx Normalization condition: Since the particle should be found somewhere, sum of all probability must be equal to 1. ψ(x) dx = 1 (11) Particle in a box (1-dimensional, 0 < x < a) ( ) nπx ψ(x) = a sin a where a is the box length. E n = ( h 8ma ) n 3

4 (1) Harmonic Oscillator (1-dimensional, classical) Harmonic oscillator experiences Hooke s law force: F = kx where x is the displacement from the rest position and k is force constant. The potential energy corresponding to Hooke s law force is Newton s equation for harmonic oscillator: General solution: V(x) = 1 kx d x(t) dt + k m x(t) = 0 x(t) = Acos(ωt+φ) where A and φ are constants that depend on the initial condition. ω = frequency of oscillation. Note that ω = πν. k m is (angular) (13) Vibration of diatomic molecule: Small amplitude vibration can be modeled by harmonic oscillator. The mass of the oscillator should be the reduced mass of diatomic molecule. µ = m Am B m A +m B where m A and m B are the masses of two atoms of diatomic molecule. (i) Schrodinger equation of molecular vibration: h d ψ(x) + 1 µ dx µω x ψ(x) = Eψ(x) where ω = k is the frequency of vibration and k is the force constant. Thus, µ µω = k in the Schrodinger equation. (ii) Wavefunctions: ψ 0 (x) = ( ) 1/4 β e βx /, ψ 1 (x) = π ( ) 4β 3 1/4 xe βx /, ψ (x) = π ( ) 1/4 β (βx 1)e βx / 4π where β = µω h. (iii) Energy levels: ( E n = hω n+ 1 ), where n = 0,1,,3,... 4

5 (14) Infra-red spectrum of diatomic molecule Under the harmonic oscillator model, IR absorption frequency should be ν obs = 1 k π µ = ω π Note that the energy of IR photon that is absorbed by the molecule is given by E photon = hν obs = h c λ obs = hc ν obs (15) Quantum mechanical operators: (i) kinetic (T) and potential (V) energies (ii) position (x) and momentum (p) d ˆT = h mdx, ˆV = V(x) ˆx = x, ˆp = i h d dx (16) Free particle: For a free particle (V=0) in space with the mass m and the energy E, the unnormalized wavefunctions are ψ(x) = e ikx, and ψ(x) = e ikx where k = me, or E = h k. h m (17) Quantum mechanical average (Expectation value): If a system is described by a normalized wavefunction, Ψ, then the average values of an observable corresponding to the operator  is given by a = Ψ ÂΨdτ allspace Note that dτ is the volume element (dx for 1D, dxdydz for 3D, for example) (18) Orthogonality of wavefunctions: Eigenfunctions of an operator are orthogonal to each other. φ n(x)φ m (x)dx = 0, (n m) where φ n (x) and φ m (x) are eigenfunctions of given operator. (19) Commutator of two operators,  and ˆB: [Â, ˆB] = ˆB ˆB 5

6 Math formula (1) Integral formula: sin axdx = x sin(ax), 4a cos axdx = x + sin(ax), 4a () Integral formula: x sin axdx = x3 6 (a x 1)sin(ax) xcos(ax) 8a 3 4a xsin axdx = x 4 cos(ax) xsin(ax) 8a 4a (3) Trigonemetric relation: sinαsinβ = 1 [cos(α β) cos(α+β)] sin(a±b) = sinacosb ±cosasinb (4) Gaussian integral: In general, π e αx dx = α x n e ax dx = (n 1) π n a n a (5) Integral formula: 0 x e ax dx = a 3 6