BCH 330 Exam 3
Give the Schrodinger equation and explain each parameter (except H^)
-H(^)Y=EY, where H(^) is the Hamiltonian operator, used to calculate the energy (E) of the system. these values are typically calculated using H(^)=h^2/2m(d2/dx2+d2/dy2+d2/dz2)+U(x,y,z)=(h2/2m)v2+U(x,y,z), where h(h-bar) is planck's constant (h) divided by 2pi. -v2 is the gradient squared, which accounts for acceleration in the 3D space -Yis called an eigenfunction of the operator; the prefix eigen means single -E is called an eigenvalue
Explain how blackbody radiation led to the idea of "quanta"
-all bodies release radiation -a "blackbody" absorbs all radiation -heating a cavity with a blackbody surface containing one tiny pinhole to a given temperature makes it emit radiation -Stephan-Boltzmann law describes the R=sigma [T+273K]^4, where sigma=5.67x10^-8 Wm^-20K^-4. -if the body is grey, it can emit or absorb radiation, both can occur with equal capability -the emissivity (E) ranges from 0 to 1 and characterizes the degree of greyness (1=black, 0=white) R=Esigma[T+273K]^4
When light interacts with an infinite simple crystal lattice, one can determine the distance between the atoms. Provide a figure that shows how the light interacts with the crystal and the resulting altered property of the light.
-for an infinite crystal lattice consisting of identical planes spaced at interval of distance d, the allowed states of motion along the z axis, in the absence of input light is: S=$pzdz=nzt
Define orthogonality
-in a 3D system that has 3 axes, the relation of one axis relative to the other two is called orthogonality -orthogonality tells there is no overlap or connection between the occupied dimension and the remaining two dimensions because they are placed in teh furthest distance therefore, the occupied axis provides info about the energy
Give a specific example of a reciprocal lattice coordinate.
-x-ray crystallography illustrates the concept of reciprocal lattice by diffracting the molecule in different patterns and intensities. -one will be able to determine the molecular structure and shape of an unknown molecule. (1/a,1/b,1/c)=(1/3,1/3,1/3) nxnynz=333
What is the effect on the number of nodes as the energy is increased?
E=hv -if the energy is increased, the frequency (v) will also increase from the equation, where h is the Planck's constant -the particle leaves the reciprocal space as the energy increases, the reciprocal gets bigger, which results in a bigger number of nodes to obtain a better structure
Different conjugated pi electron containing molecules absorb at different frequencies. Explain the relation between molecular length and the frequency absorbed radiation.
-a larger number of conjugated double bonds (delocalized extended structure composed of alternating double and single bonds) lead to progressively lower absorption energies -this is used to advantage in chloroplasts, which contain a series of different UV and infrared regions. as a result, the entire range of light is received and focused into photosynthesis
Explain using the concept of molecular transition or plane polarized light, how light is (a) absorbed and (b) emitted by the molecule. Explain by using relevant terms and explanations.
-a molecule is perturbed by light because its distribution of electrical charge is altered by the oscillating electrical field -the dipole of a molecule is characterized using the molecular dipole operator (Mcap): Mcap=SUMi ei rcapi -sum is taken over all charges ei at all of their respective positon operators rcapi -goal is to calculate the rate at which light induces transitions between the wavefunction (Ya) and all other accessible states -consider the transitions between two state Ya(ground state) and Yb(excited state) -because light is time-dependent, the time-dependent Schrodinger equation must be solved -the Hamiltonian is: H^=H+Vcap(t) -all effects of light are handled by Vcap(t) -transitions can be either from a to b (absorption) or b to a (emission) -two contributions go into the net function; this is called linear combination: Y(t)=Ca(t)Yaexp(-iEat/h)+Cb(t)Yb exp(-i Eo t/h) -The two time-dependent Schrodinger equations, for absorption and emission are: Ground State Absortption ihdCa/dt=<Ya/Vcap/Ya>Ca + <Ya/Vcap/Yb> Cb exp [-(Eb-Ea)t/h] Emission Excited State ihdCb/dt=<Yb/Vcap/Ya>Ca exp[-(Ea-Eb)t/h] + <Yb/Vcap/Yb> Cb -the integrals symbolized by the brackets are taken over spatial coordinates. the operator that produces the interactions between the molecule and light is Vcap Vcap=Mcap Eo exp(iwt), where Mcap is transition and E is energy -vibrational energy levels are decomposed into a series of rotational levels
Explain how the action integral leads to the law one uses to determine the interatomic distance. Give the law, provide its name define all the relevant parameters.
-action integral is used in quantization in terms of frequency to get energy -if one shoots light on a crystal lattice at a right angle, the vibration frequency will be detected, and the energy going out will be quantized -this energy produces intense spots in the detector -therefore, one measures the spots intensity to determine the shape of a molecule and the interatomic distance by braggs Law= n(wavelength)=2dsintheta, where n is the quantum number, wavelength, d is the interatomic distance, and theta is the diffraction angle
The light-induced dipole (transition dipole) oscillates in a preferred direction relative to the chromophore, usually in the plane of aromatic rings. In some cases two predominant transition dipole vectors exist, in which case the oscillations can either occur in phase or out of phase with respect to each other. a. using vectors that are 180 degrees out of phase with each other (up and down) show how this leads to a possible difference in interactions between the chromophore and polarized light.
-consider the lowest excited electronic state of one chromophore with all higher electronic excited states of the neighboring chromophores -suppose one illuminates the system with light of frequency Voa, which can only excite the lowest electronic transition -in the chromophore that absorbs, there will be a light-induced dipole Moa. the magnitude of this dipole determines the absorption intensity -figure shows the consequences for two schematic structures. In a parallel stack, all of the dipoles are aligned in a mutually repelling fashion. this result will be a shorter dipole, which means less absorption (hypochromism) -the opposite result occurs in an end-to-end arrangement -the induced dipoles attract the transition dipole, which lengthens and leads to hyperchromism -this phenomenon has been used to determine the stabilities of nucleic acid double helices (relative to unpaired strands)
Why do the Pz, Px, Py orbitals point in three different directions? Why does each have a node at the origin?
-each has a node at the origin, because the particle is constrained to a one-dimensional well -the x dimension is orthogonal to the y and z dimensions, the y dimension is orthogonal to x and z -z is orthogonal to x and y -each only occupies one dimension yet each contributes to the net energy a=axEx+ayEy+azEz b=bxEx+byEy+bzEz -the vectors Ex,Ey, and Ez are an othonormal set, they are all orthogonal to each other -show three orthogonal P orbitals
Why is the integral a loop integral and what's the relation to the concept of oscillation. Draw a diagram to illustrate.
-emission or absorption of light by a harmonic oscillator could, in principle, occur at frequencies corresponding to any multiple deltan Vo of the fundamental frequency Vo, where deltanVo=(n2-n1)Vo. -n1 and n2 are any two orbitals -quantum mechanical selection rules only accept frequencies in which the quantum number n changes by deltan=+-1 -the harmonic oscillator is restricted to an equilibrium position x=0 by a restoring force that moves along the x axis is described by the harmonic oscillator -kx=-4pi^2 mVo^2 x, where k is the restoring force constant -the motion is described by: x=xo sin 2pi Vot
Explain how the energy, frequency and wavelength of radiation are related. Give appropriate formulas
-energy emitted or absorbed from a system occurs in a small singular unit called a "quantum" -emission and absorption are symmetric processes -wavelength of emitted light is inversely proportional to its energy; a larger wavelength corresponds to a smaller energy (E). -relation between wavelength and frequency of radiation expressed as: v=c/wavelength, where c is the speed of light (3x10^8 ms-1) -unit of frequency, the Hertz, is equal to 1 cycle per second -key correlary equation is Planck's law: E=hv, where h is Planck's constant
Explain what a complex function is and what its complex conjugate function is.
-imaginary numbers often used in derivations and manipulations of equations containing vectors -complex numbers are written in the form: z=x+iy. --imaginary root i=(-1)^1/2 -numbers are divided into scalars and vectors: scalars with magnitude, vectors with both magnitude and direction -relation to vectors involves the complex plane idea: the cartesian axis system version of this idea involves a real x axis and an imaginary y axis --polar coordinate version involves an axis system in which the unit vector has length r and phase angle (theta) ranging from 0 to 2 pi (the standard unit circle setup in trigonometry) -- length of polar coordinate vector r is calculated in terms of cartesian coordinates: r=(x^2+y^2)^1/2 --phase angle (theta)=tan-1(y/x) --complex exponential is defined in terms of trigonometry functions (exp)(z)=e^x (cosy+siny) e^z=e^2e^iy -cartesian to polar transformation function allows one to express the complex cartesian number z in the corresponding polar coordinate form" z=x+iy=re^(itheta) -complex variable (w) can always be converted to a real number by multiplying it by its complex conjugate (w*): w+u+iv w*=u-iv ww*=u^2+v^2
Compare and contrast isotropic and anisotropic system. Describe the spatial relationships (x and y)
-in a crystal, the physical and mechanical properties usually differ with the orientation when the properties of a material vary with different crystallographic orientations, the material is said to be anisotropic -a fluorophore attached to a small molecule undergoes rapid rotational diffusion but when attached to larger molecule the fluorophore reorients more slowly -this phenomenon is quantified by considering a beam of (excitation) light that travels along the x direction, with its E vector polarized along z -when light emerges from the fluorophore as emission, the intensity along the z axis is called Ill and the intensity along the x-axis is I (perpendicular). -the anisotropy is defined as: A=(Ill-Il_)/(Ill+2Il_) -the relation between fluorescence lifetime and observable motion is called the Perrin equation: Ao=AF/[1+(Tr/Tc)], whre Ao is the steady state anisotropy, and Af is the anisotropy in the absence of internal motion -the value of Ao is only sensitive to the role of rotational diffusion (1/Tc) if the lifetime of the fluorophore (TF) is comparable to the rotational correlation time -Alternately, when the properties of a material are teh same in all directions, the material is said to be isotropic -for many polycrystalline materials the grain orientations are random before working of material is done -therefore, even if the individual grains are anisotropic, the property differences tend to average out and, overall, the material is isotropic. Isotropic means spherical -when you have a piece of DNA with negative charge and put Na+ around it, when its away from DNA have nucleus and electrons that are both spherical, with an averaged NMR peak. as you get the Na+ closer to the DNA, get electrons being pushed away and nucleus pulled in a non-spherical form. the non-spherical forms give several NMR peaks (maybe one broad peak) -in the context of fluorescence, a large molecule will not be able to rotate much before light comes back off. this is called an anisotropic response, because it has not been averaged fully. a smaller molecule will be able to rotate more, producing a spherical average (isotropic response). Rotational diffusion rate= 1/correlation time
Explain how quanta are related to Planck's constant
-max planck attempted to quantify the way blackbody radiation depended on wavelength --graph of curves showing this trend for set of temperatures (spectral emittance vs wavelength) -Planck involved a new principle in which matter is interpreted as consisting of a collection of linear oscillators -energy emitted or absorbed from a system occurs in a small singular unit called a "quantum" -emission and absorption are symmetric processes -wavelength is inversely proportional to its energy; a larger wavelength corresponds to a smaller energy -Hertz and Maxwell developed an equation that expresses the relation between the wavelength and the frequency of the radiation -v=c/wavelength, where c is the speed of light (3x10^8 ms-1) -unit of frequency, Hertz, is equal to 1 cycle per second, i.e. the number of oscillations per unit time -key correlary to this equation is Planck's law: E=hv, where h is plancks constant -electromagnetic wave consists of coupled electric and magnetic waves, oscillating at the same frequency -wave travels as the electrical (E) and magnetic (M) vectors oscillate at right angles to and in phase with each other, perpendicular to the direction the wave properties
Show what a benzene ring does to the peak position (frequency) of H atoms at the periphery of atoms above the ring. Explain.
-nearby electrons affect the resonance frequency of the nuclear spin in an atom -for ex. , ring currents are produced by induction of pi electrons circulating within double bonds and aromatic rings by the bulk (Bo) field -electronic perturbations can lead to large offsets in S values, e.g. about 7 ppm downfield, "deshielded" in the spectrum of benzene -the precise positioning of a proton with respect to two benzene rings leads to three different chemical shifts -three different ring current environments produce three different chemical shifts -In picture: the widespread S values in the pi alkane bridge-containing benzene system --the precise position of the methylene protons leads to three distinct chemical shifts
What is the essence of the Heisenberg uncertainty principle? What are the specific parameters? Use the analogy of Schrodinger's cat to explain the duality expressed by this idea.
-quanta can take all possible paths and we don't know what they did until we interrogate them -however, detectors affect the system and change it, so measurements become imperfect as expressed by Heisenberg's uncertainty principle -one must use a detector to acquire measurements but the detector "perturbs" the measurement, so there will always be an error -consider the situation in which a vial of poison is in a box with a cat, along with a vial of radioisotope that emits a nuclear particle randomly. if the particle is emitted, the vial breaks and the cat dies. the grand question is: at any given moment, is the cat dead or alive? one doesn't until one opens the box --both possibilities, dead and alive, are true until one perturbs the system by doing the experiment of visualizing the cat by opening the box. --this paradox is a metaphor for how we interpret the very essence of physical reality. one is forced into thinking very carefully about measurements, their uncertainty and ambiguity --information and knowledge can be dissapated and lost. as time progresses and ordered system will move toward disorder. -quantum info can be focused to specified state then applied to a sample of molecules -as time goes forward the acquired focused energy in the molecule becomes less coherent. this is relaxation
Explain the concept of reciprocal lattice and why it is used.
-reciprocal lattice is used to define the structure of a molecule -particle will be in a 3D lattice and the lattice is placed perpendicular to each plane of a crystal lattice -the point of connection will be the origin where the distance from each point to the origin is reciprocal to the specific lattice planes -further distance indicates a finer structure or better decision. -reciprocal lattice is used because one gets all info of direct space from reciprocal space info, which is the position, intensity, and phase
Placing a nucleus in a magnetic field quantizes the energy levels. Explain how this is accomplished and show pictorially what happens. Give any relevant equations.
-specific quantized values of angular momentum occur as integral multiples of h/2pi, which is shown symbolically by placing a bar through the stem of the h and is called h-bar -the angular momentum vector (w) projects from the origin to the orbital surface, perpendicular to it -the angular frequency w has length kh/2pi -if the angle between w and z axis is theta, the z-axis length of p is w2=kcostheta(h/2pi). this leads to costheta=m/k -since magnetic quantum number (m) cannot be 0, orbitals occur in the x-y plane at m=+-1,+-2, ...+-k -an electron with charge e and mass mo, in an orbital of angular momentum has the following angular frequency: we=khe/(4pimoc) -this component is aligned in the z-axis direction Me,z=khe/(4piMoc) -in magnetic field Ho, a magnetic field due to interaction energy splits the peaks into +- m different values. this is called the Zeeman effect. -magnetic field rotation direction rule leads to the conservation of energy in time-space as the magnetic vector relaxes -3D relaxation processes have the form of attenuated periodic complex functions. spin or spin network evolves through space over time -chemical shift has units of ppm units, calculated relative to the frequency of a suitable internal standard
Explain what a stationary state is and the relation to the "action integral"
-stationary state is a state in which oscillations occur but the mass is contained to a single geometric center -consider a Hamiltonian that makes use of the coordinates (q1...q3N) and conjugate (connected) momenta (p1...p3N) Sk=$pk dqk=nkh, for k=1,2,3..3N, where k is the index for the number of degrees of freedom. -Sk is called an action integral -the symbol $ refers to integration over one cycle of the motion, i.e. the coordinates and momenta are followed through one cycle of motion
Explain why these concepts are central to the interpretation of quantum mechanics.
-target of many quantum mechanics calculations, a wavefunction, only has physical meaning when multiplied by its complex conjugate (Y*) to produce Y^2 -according to Heisenburg, a key interpretation is that Y^2 is the probability of whatever one has calculated -EX: quantities such as the electron distribution, coherance of a spectroscopic signal, superimposed positions of x-rays diffracting from a crystal -reality (Y^2) can be decomposed into two contributing underlying sub-functions Y and Y*, which cannot be observed directly --to decipher the details mathematically one often varies the sampling frequency and phase angle and determines how the results are affected
Using potential energy diagrams, show the relation between energies associated with the transitions between electronic, vibrational and rotational energy levels. Provide approximate values.
-the Jablonski diagram shows the relation between the ground and excited energy states -the triplet state (T) differs from the singlet state (s) in terms of spin state -importantly, the T1 state does not easity convert to the So state. this slows the release of energy, so phosphorescence can occur several hours after absorption, while fluorescence is relatively rapid (milliseconds) -the diagram shows the ground and excited energy states of absorption and emission -v represents different vibrational levels -each vibrational substrate is associated with a different energy level -rotational transition occurs within the same vibrational state -in each jump, vibration produces 5-10 kcal/mole rotation with 1 kcal per mole and electronic with 80 kcal per mole
Why would one use the blue portion of the Bunsen burner flame, as opposed to the yellow portion? Relate your answers to the wavelength and energy of the situation.
-the color of the flame corresponds to the wavelength emitted -wavelength of light is inversely proportional to its energy; larger wavelength corresponds to smaller energy (E) -yellow flame transfers less heat than a blue flame, i.e. less energy and greater wavelength
Draw two different coordinate systems and show right- and left- circularly polarized light propagating from the origin (0,0) to the right. Show how the E vectors propagate in each case.
-the two types of circularly polarized light are shown in the figure. the spectrum is produced by subtracting the absorption of the left- (CW) and right- (CCW) CP light. Circular dichroism: CD(wavelength)=DeltaE(wavelength) Cl Delta E(wavelength)=EL-ER
Define spherical harmonics
-the whole set of harmonic functions defined on the surface of a sphere; can be defined by the Laplace's equation Vbar^2phi=0, where Vbar^2 is teh Laplace's operator, and phi is the longitudinal angle -the equation can be used to solve the correct form of the electric potential in an electromagnetic field
Describe how an NMR signal is generated within relaxed nuclear spin, then excite and show what happens with time. Show how we get a peak from the raw data.
-to initiate, one flips the nuclear magnetic vector by 90 degrees using a radio frequency pulse -result is the rotation about the z axis in the x-y plane -NMR signal is the result of relaxation back to being aligned with bulk magnetic field while rotating around the z axis -"right-hand rule" from electromagnetism requires that the relaxation pathway follow a chiral corkscrew-like route -magnetic field rotation direction rule leads to conservation of energy in time-space as magnetic vector relaxes -3D relaxation processes have the form of attenuated period complex functions: spin evolves through space over time. -chemical shift has units of parts per million (ppm) units, calculated relative to the frequency of a suitable internal standard -individual spin sub-components correspond to frequencies that are larger and smaller than the central peak component in the Guassian population
Show how Beer's Law is derived and explain how it is used in practical application.
-two time-dependent Schrodinger equations, for absorption and emission are: Ground State Absorption ihdCa/dt=<Ya/Vcap/Yb>Ca+<Ya/Vcap/Yb>Cb exp [-(Eb-Ea)t/h] Emission Excited State ihdCb/dt=<Yb/Vcap/Ya>Ca exp [-(Ea-Eb) t/h]+<Yb/Vcap/Ya>Cb -the integrals symbolized by the brackets are taken over spatial coordinates. the operator that produces the interaction between the molecule and light is (Vcap) Vcap=Mcap Eo exp (iwt) -Fourier transforms provide solutions to the absorption and emission equations ikdCa/dt=Cb<Ya/Mcap/Yb> Eo exp [-i(Eb/h-Ea/h-w)t] ihdCb/dt=Ca<Yb/Mcap/Ya> Eo exp [-i(Ea/h-Eb/h-w)t], where the terms Eoexp[-i(Eavrb/h-Ea/h-w)t] determine whether the vectors are large enough to interact -transitions from a to b (absorption) only occurs when: hv=Eb-Ea -for polarized light and an orientation molecule, the probability that illumination with light frequency V will transform the molecule from dPb/dt=d/dt Sdv/Cb(t)/^2=(1/2h^2)l<Yb/Mcap/Ya>Eb/^2 -the parameter Bab is defined as the transition rate per unit energy density of the radiation Caka the Einstein coefficient for simulated absorption -I(v) is the energy density incident on the sample frequency v dPb/dt=Bab I(v) -one must average over all orientations between the Eb vetor of the incident light and the transition dipole -dI(v)=hv(NaBab-NbBba)I(v) -Na and Nb are the numbers of moles in each state. this is the Beer-Lambert equation
Give an explicit mathematical expression for H^ and explain its purpose.
In a simple harmonic oscillator, the Hamiltonian is: H^=h2/2m(d2/dx2)+(1/2)kx2, where En=(n+(1/2))hw), n=1,2,...,En is the quantized energy and w=(k/m)^(1/2) is the angular momentum of the oscillator