Physical Chemistry - ACS Quantum Mechanics Final

who assumed the energy of all oscillators in a blackbody was quantized? what was it said to be quantized by?

Planck; e=nhv where n=quantum number, h=Planks constant, and v=frequency of the oscillator

what did Einstein propose through use of Plank's quantization of energy theory?

that radiation itself existed as packets of energy (called photons) with e=hv

what is the empirical equation explaining the observed spectrum of hydrogen?

v=Rh((1/n1^2)-(1/n2^2)) where Rh is the Rydberg constant, and n1 and n2 are quantum numbers

the angular momentum of the hydrogen atom is quantized by units of what?

h/2Pi or hbar

what is the relation of momentum to wavelength? (de Broglie relation)

wavelength=h/p or h/m*v

where v is velocity, m is mass, and h is planck's constant

what is the Schrodinger equation?

a partial differential equation describing the wave properties of matter. solutions are called wave functions.

equation for the theory that two electrons cannot occupy the same spatial orbital unless they are of opposite spin? (Pauli exclusion principle)

Ψ(1,2)= -Ψ(2,1)

what is the equation for the Heisengburg uncertainty principle?

ΔxΔp is greater than or equal to 0.5hbar

what does the correspondence principle state?

classical and quantum mechanical results merge in the limit of high quantum numbers

What is the time independent schrodinger equation?

HΨ=EΨ where H is the hamiltonian operator and E is the energy

when is a function an eigen function?

example: A is an eigen function if applying A to the function f is the same as the multiplication of f by a constant, a. Or Af=af

the wavefunctions and energies of systems are _____ of the Hamiltonian operator

eigen functions. In other words, applying the Hamiltonian to the wave function is the same as multiplying the wavefunction by the constant, E or energy.

what is an operator?

A rule for changing one function into another function

what makes an operator linear?

it (represented by A) satisfies the equation: A(cf+dg)=Acf+Adg where f and g are functions and c and d are constants

what makes an operator Hermitian?

it (represented by A) satisfies the equation: INTEGRAL(fAg dT) = INTEGRAL(gAf dT)

what is the Born-Oppenheimer approximation?

because the electrons in molecules move much more quickly than the nucleus, we assume the nucleus is fixed

what is the Fanck-Condon principle?

because nuclei are much more massive than electrons, en electronic transition takes place in the presence of a fixed nucleus

why isn't every solution to the Schrondinger equation acceptable?

because of boundary conditions for each given problem. Also, the wave function must be continuous, continually differentiablex, single-valued (i.e. can't have 2 possible Y values for one X value), finite-valued (i.e. can't go to infinity), and able to be normalized over the appropriate range

when a set is orthogonal what happens?

INTEGRAL(Ψ*Ψ dT)=0

wave functions that are solutions to a given Hamiltonian are always?

orthonormal sets (i.e. they're orthogonal and normalized)

what is an expectation value?

for an observatble corresponding to a quantum mechanical operator (A) in the state described by Ψ,

=INTEGRAL(Ψ*AΨ dT)

de Broglie postulated that the wavelength of a particle is inversely proportional to it's momentum. The constant of proportionality is?

h

because nuclear motions are much slower than those ot the electrion, the molecular Schrodinger equation for electron motion can be solved by assuming that the nuclei are at fixed locations. This is?

the Born-Oppenheimer approximation

According to the Heisenburg uncertainty priciple, if the operators for two physical properties do not commute then?

the product of the two uncertainties must be greater than of equal to h/4Pi or hbar/2

the requirement that wavefunctions for electrons in atoms and molecules be antisymmetric with respect to interchange of any pair of electrons is?

the Pauli exclusion principle

for electrons emitted due to the photoelectric effect, what are the kinetic energy and current functions of?

kinetic energy is a function of frequency and current is a function of intensity

in he Schrodinger equation the quantity H (hamiltonian) represents the?

total energy operator

the total energy of a particle with mass =m moving in the x-direction with momentum=p is p^2/2m. The Hamiltonian operator for this system is?

since the operator for momentum (p) is -ih (d/dx), the Hamiltonian is -hbar^2(d^2/dx^2)/2m which reduces (after changing hbar to h) to (-h^2/8mPi^2)(d^2/dx^2)

according to the postulates of quantum mechanics, would e^x be an acceptable wave function in the region -∞

No. Because as X approaches ∞, so does e^x. So it isn't finite.

when the hydrogen atomic 2p is acted upon by the operator for the z-component of the angular momentum (Lz) we find that?

2p is not an eigen function of Lz, therefore that is no definite value

the quantum mechanical operator for velocity is given by?

-(i*hbar/m)(d/dx)

the quantum mechanical operator for momentum is given by?

-(i*hbar)(d/dx)

the commutator of two operators (a and b) equals zero. the physical properties (A and B) associated with the two operators ?

may be simultaneously determined with unlimited precision

what equation did de Broglie NOT use in making his determination?

the schrodinger equation

a nodal surface is best described as what?

separating regions of different sign for a wave function (think of the graph of the sin function)

an eigenfunction of the operator d/dx is?

cos(ax)

if Ψ(x) is the normalized wave function for a particle in one dimension, the average value for the momentum can be calculated using the momentum operator from what integral?

INTEGRAL(Ψ(x)*pΨ dx)

What is the formula for energy levels for a particle in a 1-D box

En=n^2(h^2/8mL^2)

where L is the length of the box

How do conjugated systems relate to particles in a box?

the particles are electrons and the box is the conjugated pi system of the molecule

what is tunneling?

when the probability distribution of a particle in a system extends into the region where the potential barrier height is higher than the energy of the particle

what is the velocity equation for the harmonic oscillator?

v=0.5kx^2 where k is the force constant and x is the distance

what is the equation for the quantized energy levels of the harmonic oscillator?

E=(hbar)(omega)(n+0.5)

where omega is 2Pi*frequency and n is the quantum number

energy levels of the harmonic oscillator are?

equally spaced, there is a zero point energy for a quantum number of 0 and the energy goes to infinity as the quantum number goes to infinity

the Morse potential is an example of what?

an anharmonic function because it involves an exponential in the potential function (this is closer to what the potential curve for real molecules looks like)

force constants tell us?

how stiff a bond is

what must happen for a line to appear in IR spectra?

the transition dipole moment must not equal zero (there must be a change in dipole moment), Δn=1 or -1

what can you observe in the microwave region of electromagnetic spectra?

lines for molecules that are useful for determining the bond length and moments of inertia arising from molecular rotation

what model is used for the rotation of molecules?

the rigid rotor model which assumes that the internuclear distance remains constant as the molecule rotates

what is the equation for the quantized energy levels of the rigid rotor?

E=J(J+1)(hbar^2/2I)

where J is the angular momentum and is greater than or equal to the mass, and I is the moment of inertia

how many degenerate energy levels do rotational states have?

2J+1

what is the operator for the M2 component of angular momentum (J)?

-ihbar(d/dΨ)

here, Ψ is an angle of rotation I think

for a particle in a one dimensional box, the average value of the position of the particle is what for any allowed state?

L/2

the most important defect of the simple harmonic oscillator approximation for the upper vibrational levels of a molecule is?

all energy levels are equally spaced

when you increase temperature, what happens to the intensities and peak positions of an electromagnetic spectrum of a molecule?

the peak positions stay the same but the most intense peak will be at a higher wavelength compared to the spectrum at a lower temperature

in the rotational spectrum of DCl, the rotational line spacings compared to those of HCl are approximately?

halved

if the box length for a particle in a one dimensional box is doubled, what happens to the energy?

it will be 1/4 of the original energy (because in the equation for energy, L is in the denominator and squared)

difference between the one-dimensional quantum mechanical descriptions of the harmonic oscillator and the particle in a box with infinite walls?

the particle in a box wave functions do not penetrate the box walls; the harmonic oscillator wavefunctions penetrate into the classically forbidden regions

the force constants for HCl and DCl are?

roughly the same

the lowest dip in the Morse potential is where what is happening?

this is the attractive region of the interaction where the molecule is most stable

what system does not have a zero point energy?

a two particle rigid rotor

Slater orbitals are _____ with respect to the exchange of any two electrons

antisymmetric

for the hydrogen atom, the energy of a state does not depend on what?

the azimuthal quantum number

the splitting between the two lines in the yellow region of the UV Spectra of Na are due to?

spin orbit coupling

what factor in the quantum mechanical equations of multi-electron atoms makes it so they cannot be solved analytically?

the electron-electron repulsion term

How many nodes are present in a 4d orbital of a hydrogen atom?

2 angular nodes and 1 radial node

IR activity in character table?

x, y, z

Raman activity in character table?

xy, xz, yz, x^2, y^2, z^2

normal of normal modes for 1. linear and 2. nonlinear molecules

1. 3N-5 2. 3N-6

what is the general form of a term symbol?

(2s+1) X (J)

where 2s+1 is the spin multiplicity, s is the sum of spin angular momentum of each electron, and J is the angular momentum

what is L?

the total orbital angular momentum. can be anything in the range: l1+l2≥L≥l1-l2

what is Hund's maximum multiplicity rule?

an atom in it's ground state adopts an electron configuration with the most unpaired electrons

the state with the ____ S value is most stable; the when there are identical S values for 2 or more states, the ___ L value determines the most stable state; for states with the same S and L, the _____ J when the shell is half-filled (or the ____ J when the shell is more than half filled) determines the most stable state

smallest, smallest, smallest, largest

selection rules for electronic transitions:

ΔJ=0,1,-1 (J=0 to J=0 is forbidden); ΔL=0,1,-1; Δs=0, Δl=1,-1

what are the possible term symbols for the 1s^12s^1 configuration of Helium?

S1, S3

What is the symmetry of BF3?

D3h

What is the symmetry of cyclopropane?

D3h

only molecules with ____ are labeled as ungerade (odd) or gerade (even)

a center of inversion

this symmetry element is not possible with chirality

σh

the full Hamiltonian for a molecule has the form?

H=Telectrons + Tnuclei + Vnuclei/electrons + Velectrons/electrons + Vnuclei/nuclei

the simplified Hamiltonian for a multielectron molecule has the form?

Helectronic=Telectrons + Vnuclei/electrons + Velectrons/electrons + Vnuclei/nuclei

equation for bond order?

bond order = (# of bonding electrons - # of antibonding electrons)/(2)

as number of modes increases, energy _____

increases as well

stable molecules have what type of bond order?

greater than zero

B2 has/doesn't have unpaired electrons

has

what makes an orbital a Pi orbital? What makes is gerade?

it it extends above and below the plane of symmetry of the molecule; if it has a center of symmetry

what are microwave spectra due to?

transitions between rotational quantum states

the distance between energy levels of rotor quantum states _____ with increasing J quantum number

increases

what factors contribute to the intensity of the lines in microwave spectra?

the Boltzmann distribution and the degeneracy of the angular momentum states.

what are IR spectra due to?

vibrational transitions, which only occur when the change in quantum number is 1 or -1 and ΔJ=1 or -1

what can be seen as fine structure under high resolution in IR spectra?

rotational transitions

What are the branches in IR?

the R branch is where ΔJ=1, the P branch is where ΔJ=-1, and the Q branch is where ΔJ=0

what are UV Vis spectra due to?

transitions between electronic quantum states

each electronic transition is accompanied by what?

both vibrational and rotational transitions

what are selection rules for electronic transitions?

ΔS=0, ΔL=1 or -1. ΔJ=0,1,-1 where L is the orbital angular momentum, S is the total spin quantum number, and J is the total angular momentum number

what is the intensity of the lines in UVVis spectroscopy determined by?

the Franck-Condon factors which are proportional to the overlap of the wavefunctions in the ground and excited states

what transitions are shown in UV-Vis?

v" (lower curve) to v' (higher curve); De (bottom of lower curve to bottom of upper curve); D0 (middle/bottom of lower curve to middle/bottom of higher curve)

what happened to electronic spectra as atomic numbers increase?

forbidden transitions start occurring leading to the appearance of weak line and making the spectra more complex

rotational and vibrational raman lines lie where in the spectra?

rotational: closer to the exciting frequency, vibrational: further from the exciting frequency due to larger energy gap

molecules that have a center of symmetry are special in regards to spectroscopy because?

they will be Raman or IR active but not both.

the intensity of an absorption line is determined by?

the value of the transition moment integral