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All Textbook Solutions for Physical Chemistry
What molality of NaCl is necessary to have the same ionic strength as a 0.100-mCa3(PO4)2 solution?8.62E8.63ECalculate the molar enthalpy of formation of I(aq) if that of H2(g)+I2(s)2H+(aq)+2I(aq) is 110.38kJ.8.65EHydrofluoric acid, HF(aq), is a weak acid that is not completely dissociated in solution. a Using the thermodynamic data in Appendix 2, determine H, S, and G for the dissociation process. b Calculate the acid dissociation constant, Ka, for HF(aq) at 25C. Compare it to a handbook value of 3.5104.8.68E8.69E8.70E8.71E8.72EThe mean activity coefficient for an aqueous 0.0020-molal solution of KCl at 25C is 0.951. How well does the Debye-Huckel limiting law, equation 8.50, predicts this coefficient? As an additional exercise, calculate using equations 8.52 and 8.53 where both a(K+) and a(Cl)=31010m and using equation 8.44.Human blood plasma is approximately 0.9NaCl. What is the ionic strength of blood plasma?Under what conditions does the extended Debye-Huckel law, equation 8.52, become the Debye-Hckel limiting law?8.76EApproximate the expected voltage for the following electrochemical reaction using a the given molal concentrations and b the calculated activities using simple Debye-Hckel theory. The value for both Zn2+ and Cu2+ is 61010m. Zn(s)+Cu2+(aq,0.05molal)Zn2+(aq,0.1molal)+Cu(s) Explain why you get the answers you do.8.78E8.79E8.80Ea The salt NaNO3 can be thought of as NaCl+KNO3KCl. Demonstrate that 0 values show this type of additivity by calculating 0 for NaNO3 from 0 values of NaCl,KNO3andKCl found Table 8.4. Compare your calculated value with the 0 value for NaNO3 in the table. b Predict approximate 0 values for NH4NO3 and CaBr2 using the values given in Table 8.4.8.82EWhat is the estimated velocity for Cu2+ ions moving through water in a Daniell cell in which the electric field is 100.0V/m? Assume that a for Cu2+ is 4A and the viscosity of water is 0.00894poise. Comment on the magnitude of your answer.8.84E8.85E8.86ECalculate a the solubility product constant for Ag2CO3 and b the value of Kw using the following data: Ag2CO3(s)+2e2Ag(s)+CO32(aq)E=0.47VAg+(aq)+eAg(s)E=0.7996VO2(g)+2H2O(l)+4e4OH(aq)E=0.401VO2(g)+4H+(aq)+4e2H2O(l)E=1.229VFor an object having mass m falling in the z direction, the kinetic energy is 12mz and the potential energy is mgz, where g is the gravitational acceleration constant approximately 9.8m/s2 and z is the position. For this one-dimensional motion, determine the Lagrangian function L and write the Lagrangian equation of motion.For the system in exercise 9.1, determine the Hamiltonian equation of motion.9.3E9.4E9.5EList some unexplainable phenomena from the classical science and describe what could not be explained about them at the time.Draw, label, and explain the functions of the parts of a spectroscope.Convert a a wavelength of 218A to cm1, b a frequency of 8.0771013s1 to cm1, c a wavelength of 3.31m to cm1.9.9E9.10EExplain why no lines in the Balmer series of the hydrogen atom spectrum have wavenumbers larger than about 27,434cm1. This is called the series limit.What are the series limits see the previous problem for the Lyman series (n2=1) and the Brackett series (n2=4).The following are the numbers n2 for some of the series of lines in the hydrogen atom spectrum: Lyman :1 Balmer :2 Paschen :3 Brackett :4 Pfund :5 Calculate the energy changes, in cm1, of the lines in each of the stated series for each of the given values for n1: a Lyman, n1=5; b Balmer, n1=8; c Paschen, n1=4; d Brackett, n1=8; e Pfund, n1=6.The Balmer series is isolated from the other series of the hydrogen atom spectrum. This is not the case for all series. Determine the first value of n for which the hydrogen spectral series overlap.Given that the wavelengths of the first three lines of the Balmer series are 656.2,486.1, and 434.0nm, calculate an average value of R.Some scientists study Rydberg atoms, atoms whose electron has a large value of the n quantum number. Some Rydberg hydrogen atoms may have consequences in interstellar chemistry. Predict the radius of a Rydberg hydrogen atom that has n=100.9.17E9.18Ea How much radiant energy is given off, in watt/meter2, by an electric stove heating element that has a temperature of 1000K? b If the area of the heating element is 250cm2, how much power, in watts, is being emitted?Stefans law, equation 9.18, suggests that any body of matter, regardless of the temperature, is emitting energy. At what temperature would a piece of matter have to be in order radiate energy at a flux of 1.00W/m2? At the flux of 10.00W/m2? 100.00W/m2?9.21EBetelgeuse pronounced beetle juice is a reddish looking star in the constellation Orion, while Rigel rye gel is a bluish star in the same part of the sky. Use Figure 9.12 to argue which star is hotter.An average human body has a surface area of 0.65m2. At a body temperature of 37C, how many watts or J/s of power does a person emit? Understanding such emissions is important to NASA and other space agencies when designing space suits.9.24EThe slope of the plot of energy versus wavelength for the Rayleigh-Jeans law is given by a rearrangement of equation 9.20: dd=8kT4 What are the value and units of this slope for a blackbody having the following temperatures and at the following wavelengths? a 1000K,500nm; b 2000K,500nm; c 2000K,5000nm; d 2000K,10,000nm. Do the answers indicate the presence of an ultraviolet catastrophe?a Use Wien displacement law to determine the max of the Sun if its surface temperature is 5800K. b The human eye sees light most efficiently if the light has a wavelength of 5000o(1o=1010m), which is in the green-blue portion of the spectrum. To what blackbody temperature does that correspond? c Compare your answers from the first two parts and comment.9.27ESunburn is caused by ultraviolet UV radiation. Why does red light not cause sunburn?Calculate the energy of photon having: a a wavelength of 5.42106m; b a frequency of 6.691013s1; c a wavelength of 3.27nm; d a frequency of 106.5MHz 1Hz=1hertz=1s1; this unit is often used for frequency; e a wavenumber of 4321cm1.9.30EIntegrate Plancks law equation 9.23 from the wavelength limits =0 to = to get equation 9.24. You will have to rewrite the expression by redefining the variable and its infinitesimal and use the following integral: 0(x3ex1)dx=4159.43Calculate the power of light in the wavelength range =350nm351nm that is, let d be =1nm in the Plancks law, and let be 350.5nm at temperatures of 1000K, 3000K and 10,000K.9.33EWork functions are typically given in units of electron volts, or eV. 1eV equals 1.6021019J. Determine the minimum wavelength of light necessary to overcome the work function of the following metals minimum implies that the excess kinetic energy, 12mv2, is zero: Li, 2.90eV; Cs, 2.14eV; Ge, 5.00eV.Determine the speed of an electron being emitted by rubidium (=2.16eV) when light of the following wavelengths is shined on the metal in vacuum: a550nm, b450nm,c350nm.Lithium has a work function of 2.90eV. Light having a wavelength of 1850A is shined on Li. Determine the kinetic energy of the electron ejected.9.37EAssume that an electron can absorb more than one photon in the photoelectric effect. a How many photons of wavelength 776.5nm does an electron in Fe need to absorb to escape the iron surface ((Fe)=4.67eV)? b What is the resulting velocity of the emitted electron?The photoelectric effect is used today to make light-sensitive detectors; when light hits a sample of metal in a sealed compartment, a current of electrons may flow if the light has the proper wavelength. Cesium is a desirable component for such detectors. Why?9.40E9.41E9.42E9.43E9.44EUse equation 9.34 to determine the radii, in meters and angstroms, of the fourth, fifth, and sixth energy levels of the Bohr hydrogen atom.9.46ECalculate the energies of an electron in the fourth, fifth, and sixth energy levels of the Bohr hydrogen atom.9.48EShow that the collection of constants given in equation 9.40 gives the correct numerical value of the Rydberg constant.9.50EEquations 9.33 and 9.34 can be combined and rearranged to find the quantized velocity of an electron in the Bohr hydrogen atom. a Determine the expression for the velocity of an electron. b From your expression, calculate the velocity of an electron in the lowest quantized state. How does it compare to the speed of light? (c=2.9979108m/s) c Calculate the angular momentum L=mvr of the electron in the lowest energy state of the Bohr hydrogen atom. How does this compare with the assumed value of the angular momentum from equation 9.33?a Compare equations 9.31, 9.34, and 9.41 and propose a formula for the radius of a hydrogen-like atom that has atomic charge Z. b What is the radius of a U91+ ion if the electron has a quantum number of 100? Ignore any possible relativistic effects.Label each of the properties of an electron as a particle property, a wave property, both, or neither. a mass, b de Broglie wavelength, c diffraction, d velocity, e momentum.The de Broglie equation for a particle can be applied to an electron orbiting a nucleus if one assumes that the electron must have an exact integral number of wavelengths as it covers the circumference of the orbit having radius r:n=2r. From this, derive Bohrs quantized angular momentum postulate.What is the wavelength of a baseball having mass 100.0g traveling at a speed of 160km/hr? What is the wavelength of an electron traveling at the same speed?Electron microscopes operate on the fact that electrons act as waves. A typical electron kinetic energy is 100keV 1eV=1.6021019J. What is the wavelength of such an electron? Ignore relativistic effects.9.57E9.58EDetermine under what conditions of temperature and wavelength the Rayleigh-Jeans law approximates Plancks law.9.60E9.61EState the postulates of quantum mechanics introduced throughout the chapter in your own words.10.2EState whether the following functions are acceptable wavefunctions over the range given. If they are not, explain why not. aF(x)=x2+1,0x10 bF(x)=x+1,x+ cf(x)=tan(x),x d=ex2,x+State whether the following functions are acceptable wavefunctions over the range given. If they are not, explain why not. a=ex2,x+ bF(x)=sin4x,x cx=y2,x0 dThe function that looks like this: eThe function that looks like this:10.5E10.6EEvaluate the operations in parts a, b, and f in the previous problem.The following operators and functions are defined: A=x()B=sin()C=1()D=10()p=4x32x2q=0.5r=45xy2s=2x3 Evaluate: a Ap b Cq c Bs d Dq e A(Cr) f A(Dq)10.9EIndicate which of these expressions yield eigenvalue equations, and if so indicate the eigenvalue. a ddxsinx2b d2dx2sinx2 c iddxsinx2d iddxeimx, where m is a constant e ddx(ex)f (22md2dx2+0.5)sin2x3 g ddy(ey2)Indicate which of these expressions yield an eigenvalue equation, and if so indicate the eigenvalue. a ddxcos4xb d2dx2cos4x c px(sin2x3)d x(2asin2xa) e 3(4lnx2), where 3=3f ddsincos g d2d2sincosh ddtanWhy is multiplying a function by a constant considered an eigenvalue equation?10.13EUsing the original definition of the momentum operator and the classical form of kinetic energy, derive the one-dimensional kinetic energy operator K=22m2x2Under what conditions would the operator described as multiplication by i the square root of 1 be considered a Hermitian operator?A particle on a ring has a wavefunction =12eim where equals 0 to 2 and m is a constant. Evaluate the angular momentum p of the particle if p=i How does the angular momentum depend on the constant m?Calculate the uncertainty in position, x, of a baseball having mass 250g moving at 160km/hr with an uncertainty in velocity of 4km/hr. Calculate the uncertainty in position for an electron moving at the same speed.For an atom of mercury, an electron in the 1s shell has a velocity of about 58(0.58) of the speed of light. At such speeds, relativistic corrections to the behavior of the electron are necessary. If the mass of the electron at such speeds is 1.23me where me is the rest mass of the electron and the uncertainty in velocity is 10,000m/s, what is the uncertainty in position of this electron?Classically, a hydrogen atom behaves as if it were 74pm across. What is the uncertainty in momentum, p, and the resulting uncertainty in velocity, v, if x=74pm for an electron in a classical H atom?The largest known atom, francium, has an atomic diameter of 540pm. What is the uncertainty in momentum, p, and the resulting uncertainty in velocity, , if x=540pm for an electron in Fr?How is the Bohr theory of the hydrogen atom inconsistent with the uncertainty principle? In fact, it was this inconsistency, along with the theorys limited application to non-hydrogen-like systems, that limited Bohrs theory.Though not strictly equivalent, there is a similar uncertainty relationship between the observables time and energy: Et2 In emission spectroscopy, the width of the lines which gives a measure of E in a spectrum can be related to the lifetime that is, t of the excited state. If the width of a spectral line of a certain electronic transition is 1.00cm1, what is the minimum uncertainty in the lifetime of the transition? Watch your units.The uncertainty principle is related to the order of the two operators operating on a wavefunction. Evaluate the expressions x (pxsinx) and px( x sinx) and demonstrate that you get different results.10.24E10.25EFor a particle in a state having the wavefunction =2asinxa in the range x=0toa, what is the probability that the particle exists in the following intervals? a x=0to0.02ab x=0.24ato0.26a c x=0.49ato0.51ad x=0.74ato0.76a e x=0.98ato1.00a Plot the probabilities versus x. What does your plot illustrate about the probability?10.27EA particle on a ring has a wavefunction =eim, where =0to2 and m is a constant. a Normalize the wavefunction, where d is d. How does the normalization constant depend on the constant m? b What is the probability that the particle is in the ring indicated by the angular range =0to2/3? Does this answer make sense? How does the probability depend on constant m?10.29E10.30E10.31ENormalize the following wavefunctions over the range indicated. You may have to use the integral table in Appendix 1. a =x,x=0to1b =x,x=0to2 c =sin2x,x=0tod =sin2x,x=0to2 e =sin3/2x,x=0to210.33E10.34EFor an unbound or free particle having mass m in the complete absence of any potential energy that is, V=0, the acceptable one-dimensional wavefunctions are =Aei(2mE)1/2x/h+Bei(2mE)1/2x/h, where A and B are constants and E is the energy of the particle. Is this wavefunction normalizable over the interval x+? Explain the significance of your answer.10.36E10.37E10.38EEvaluate the expression for the total energies for a particle having mass m and a wavefunction =2sinx, if the potential energy V is 0 and if the potential energy V is 0.5 assume arbitrary units. What is the difference between the two eigenvalues for the energy, and does this difference make sense?10.40EVerify that the following wavefunctions are indeed eigenfunctions of the Schrdinger equation, and determine their energy eigenvalues. a =eiKx where V=0 and K is a constant b =eiKx where V=k, k is some constant potential energy, and K is a constant c =2asinxa where V=0.In exercise 10.41a, the wavefunction is not normalized. Normalize the wavefunction and verify that it still satisfies the Schrdinger equation. The limits on x are 0 and 2. How does the expression for the energy eigenvalue differ?10.43E10.44EExplain why n=0 is not allowed for a particle-in-a-box.10.46E10.47E10.48ECarotenes are molecules with alternating CC and C=C bonds in which the electrons are delocalized across the entire alternating bond system. As such, the electrons can be approximated as being a particle-in-a-box. Lycopene is a carotene found in tomatoes and watermelon. Assuming that the alternating carbon-carbon bond system is 2.64nm wide: a What is the energy of the n=11 level? b What is the energy of the n=12 level? c What is the E between n=11 and n=12? d If, by the Bohr frequency condition, E=h, what frequency and wavelength of light corresponds to the transition from n=11 to n=12?The electronic spectrum of the molecule butadiene, CH2=CHCH=CH2, can be approximated using the one dimensional particle-in-a-box if one assumes that the conjugated double bonds span the entire four-carbon chain. If the electron absorbing a photon having wavelength 2170 is going from the level n=2 to the level n=3, what is the approximate length of the C4H6 molecule? The experimental value is about 4.8.10.51E10.52EShow that the normalization constants for the general form of the wavefunction =sin(nx/a) are the same and do not depend on the quantum number n.10.54E10.55EAn official baseball has a mass of 145g. a Assuming that a baseball in New Orleans Superdome width =310m is acting as a particle-in-a-box, what is its energy in the n=1 state? b Assuming that the energy in part a is all kinetic energy (=12mv2), what is the velocity of the baseball in the n=1 state? c A hit baseball can travel as fast as 44.7m/s. Calculate the classical kinetic energy of the hit baseball and, assuming that this energy is quantized, determine the quantum number of the hit baseball.Is the uncertainty principle consistent with our description of the wavefunctions of the 1D particle-in-a-box? Hint: Remember that position is not an eigenvalue operator for the particle-in-a-box wavefunctions.10.58E10.59EInstead of x=0 to a, assume that the limits on the 1-D box were x=+(a/2) to (a/2). Derive acceptable wavefunction for this particle-in-a-box. You may have to consult an integral table to determine the normalization constant. What are the quantized energies for the particle?In a plot of ||2, the maximum maxima in the plot is/are called the most probable positions. What is/are the most probable positions for a particle-in-a-box when: a n=1 b n=2 c n=3 d Do you see a trend?10.62E10.63EThe average value of radius in a circular system, r, is determined by the expression r=0r4r2dr. a Evaluate r for =(1a3)1/2er/a, where a is a constant. See the integral table in Appendix 1. b What is the numerical value of r if a=5.291011m?10.65E10.66E10.67E10.68E10.69EAssume that for a particle on a ring the operator for the angular momentum, p, is i(/). What is the eigenvalue for momentum for a particle having unnormalized equal to e3i? The integration limits are 0 to 2. What is the average value of the momentum, p for a particle having this wavefunction? How are these results justified?Mathematically, the uncertainty A in some observable A is given by A=A2A2. Use this formula to determine x and px for =(2/a)sin(x/a) and show that the uncertainty principle holds.10.72E10.73EVerify that the wavefunctions in equation 10.20 satisfy the three-dimensional Schrdinger equation.An electron is confined to a box of dimensions 2A3A5A. Determine the wavefunctions for the five lowest energy states.a What is the ratio of energy levels having the same quantum numbers in a box that is 1nm1nm1nm and a box that is 2nm2nm2nm? b Are the degeneracies of each set of quantum numbers the same for both cubical boxes? Explain your answer.Consider a one-dimensional particle-in-a-box and a three-dimensional particle-in-a-box that have the same dimensions. a What is the ratio of the energies of a particle having the lowest possible quantum numbers in both boxes? b Does this ratio stay the same if the quantum numbers are not the lowest possible values?10.78E10.79E10.80E10.81EWhat are x,y, and z for 111 of a 3-D particle-in-a-box? The operators for y and z are similar to the operator for x, except that y is substituted for x wherever it appears, and the same for z. What point in the box is described by these average values?10.83E10.84E10.85E10.86E10.87E10.88ESubstitute (x,t)=eiEt/(x) into the time-dependent Schrdinger equation and show that it does solve that differential equation.Write (x,t)=eiEt/(x) in terms of sine and cosine, using Eulers theorem: ei=cos+isin. What would a plot of (x,t) versus time look like?10.91E10.92E10.93E10.95EConvert 3.558mdyn/A into units of N/m.11.2E11.3E11.4E11.5E11.6E11.7E11.8E11.9E11.10E11.11E11.12Ea For a pendulum having classical frequency of 1.00s1, what is the energy difference in J between quantized energy levels? b Calculate the wavelength of light that must be absorbed in order for the pendulum to go from one level to another. c Can you determine in what region of the electromagnetic spectrum such a wavelength belongs? d Comment on your results for parts a and b based on your knowledge of the state of science in early twentieth century. Why wasnt the quantum mechanical behavior of nature noticed?11.14EThe OH bond in water vibrates at a frequency of 3650cm1. What wavelength and frequency in s1 of light would be required to change the vibrational quantum number from n=0 to n=4, assuming OH acts as a harmonic oscillator?Show that 2 and 3 for the harmonic oscillator are orthogonal.11.17E11.18E11.19EUse the expression for 1 in equations 11.17 and normalize the wavefunction. Use the integral defined for the Hermite polynomials in Table 11.2. Compare your answer with the wavefunction defined by equation 11.19.11.21E11.22EConsider Figure 11.4 and choose the correct phrase: As the vibrational quantum number increases, the extension of the vibration increases/decreases/stays the same while the average length of the oscillator itself increases/decreases/stays the same. Explain your choices.Based on the trend shown in Figure 11.5, draw the probability distribution of a harmonic oscillator wavefunction that has a very high value of n. Explain how this is consistent with the correspondence principle.11.25E11.26E11.27E11.28E11.29E11.30ECompare the mass of the electron, me, with a the reduced mass of a hydrogen atom; b the reduced mass of a deuterium atom (deuterium=2H); c the reduced mass of a carbon 12 atom having a +5 charge, that is, C5+. Suggest a conclusion to the trend presented by parts a-c.Reduced mass is not reserved only for atomic systems. A solar system or a planet/satellite system, for example, can have its behavior described by first determining its reduced mass. If the mass of Earth is 2.4351024kg and that of the moon is 2.9951022kg, what is the reduced mass of the Earth-moon system? This is not to imply any support of a planetary model for atoms11.33EAn OH bond has a frequency of 3650cm1. Using equation 11.27 twice, set up a ratio and determine the expected frequency of an OD bond, without calculating the force constant. D=deuterium(2H). Assume that the force constant remains the same.11.35E11.36E11.37E11.38E11.39EWhat are the energies and angular momenta of the first five energy levels of benzene in the 2-D rotational motion approximation? Use the mass of the electron and a radius of 1.51A to determine I.11.41EA 25-kg child is on a merry-go-round/calliope, going around and around in a large circle that has a radius of 8meters. The child has an angular momentum of 600kgm2/s. a From these facts, estimate the approximate quantum number for the angular momentum the child has. b Estimate the quantized amount of energy the child has in this situation. How does this compare to the childs classical energy? What principle does this illustrate?11.43Ea Using the expression for the energy of a 2-D rigid rotor, construct the expression for the energy difference between two adjacent levels, E(m+1)E(m). b For HCl, E(1)E(0)=20.7cm1. Calculate E(2)=E(1), assuming HCl acts as a 2-D rigid rotor. c This energy difference is determined experimentally as 41.4cm1. How good would you say a 2-D model is for this system?11.45E11.46E11.47EThe quantized angular momentum is choose one: dependent on/independent of mass. Explain your choice.11.49E11.50E11.51ECan you evaluate r for the spherical harmonic Y22? Why or why not?Show that 1,0 and 1,1 for 3-D rotational motion are orthogonal. The form of d is found in equation 11.42.11.54E11.55Ea Using the he expression for the energy of a 3-D rigid rotor, construct the expression for the energy difference between two adjacent levels, E(l+1)E(l). b For HCl, E(1)E(0)=20.7cm1. Calculate E(2)E(1), assuming HCl acts as a 3-D rigid rotor. c This energy difference is determined experimentally as 41.4cm1. How good would you say a 3-D model is for this system?11.57EIn exercise 11.57 regarding C60, what are the numerical values of the total angular momenta of the electron for each state having quantum number l? What are the z components of the angular momentum for each state?Draw the graphical representations see Figure 11.15 of the possible values for l and ml for the first four energy levels of the 3-D rigid rotor. What are the degeneracies of each state?11.60EWhat is the physical explanation of the difference between a particle having the 3-D rotational wavefunction 3,2 and an identical particle having the wavefunction 3,2?List the charges on hydrogen-like atoms whose nuclei are of the following elements. a lithium, b carbon, c iron, d samarium, e xenon, f francium, g uranium, h seaborgium11.63E11.64E11.65ECalculate the difference between the Bohr radius defined as a and the Bohr radius defined as a0.To four significant figures, the first four lines in the Balmer series in the hydrogen atom (n2=2) spectrum appear at 656.5, 486.3, 434.2, and 410.3nm. a From these numbers, calculate an average value of RH, Rydberg constant. b At what wavelengths would similar transitions appear for He+?What would the wavelengths of the Balmer series for deuterium be?Construct an energy level diagram showing all orbitals for the hydrogen atom up to n=5, labeling each orbital with its appropriate quantum numbers. How many different orbitals are in each shell?11.70EWhat is the degeneracy of an h subshell? An n subshell?What is the numerical value of the total angular momentum of an electron in an f subshell of an H atom? What is it for an f electron in Li2+?What are the values of E, L, and Lz for an F8+ atom whose electron has the following wavefunctions, listed as n,l,ml? a 1,0,0 b 3,2,2c 2,1,1d 9,6,3.11.74EWhy does the wavefunction 4,4,0 not exist? Similarly, why does a 3f subshell not exist? See exercise 11.73 for notation definition.11.76EWhat is the probability of finding an electron in the 1s orbital within 0.1A of a hydrogen nucleus?What is the probability of finding an electron in the 1s orbital within 0.1A of an Ne9+ nucleus? Compare your answer to the answer to exercise 11.77 and justify the difference.11.79E11.80EState how many radial, angular, and total nodes are in each of the following hydrogen-like wavefunctions. a 2s b 3s c 3p d 4f e 6g f 7s11.82E11.83EVerify the specific value of a, the Bohr radius, by using the values of the various constants and evaluating equation 11.68.11.85E11.86EEvaluate Lz for 3px, Compare it to the answer in Example 11.25, and explain the difference in the answers.Calculate V for 1s of the H atom and compare it to the total energy.11.89E11.90E11.91E11.92EGraph the first five wavefunctions for the harmonic oscillators and their probabilities. Superimpose these graphs on the potential energy function for a harmonic oscillator and numerically determine the x values of the classical turning points. What is the probability that an oscillator will exist beyond the classical turning points? Do plots of the probability begin to show a distribution as expected by the correspondence principle?11.94ESet up and evaluate numerically the integral that shows that Y11 and Y11 are orthogonal.11.96EIn the Stern-Gerlach experiment, silver atoms were used. This was a good choice, as it turned out. Using the electron configuration of silver atoms, explain why silver was a good candidate for being able to observe the intrinsic angular momentum of the electron. Hint: Dont use the aufbau principle to determine the electron configuration of Ag, because its one of the exceptions. Look up the exact electron configuration in a table.12.2E12.3ESuppose s=12 for an electron. Into how many parts will a magnetic field split a beam of Ag atoms?Using and labels, write two possible wavefunctions for an electron in the 3d2 orbital of He+.List all possible combinations of all four quantum numbers for an electron in a 3d orbital of a hydrogen atom.What are the degeneracies of the H atom wavefunctions when spin is accounted for? Give a general formula.12.8Ea Differentiate between the quantum numbers s and ms. b What will the possible values of ms be for a particle having an s quantum number of 0, 2, and 32?Is the spin orbital 1s for the H atom still spherically symmetric? Explain your answer.Draw a diagram analogous to Figure 11.15, but now representing an electrons spin and its z-component. How many cones are there?Are mathematical expressions for the following potential energies positive or negative? Explain why in each case. a The attraction between an electron and a helium nucleus b The repulsion between two protons in a nucleus c The attraction between a north and a south magnetic pole d The force of gravity between the Sun and Earth e A rock perched on the edge of a cliff with respect to the base of the cliff12.13E12.14Ea Assume that the electronic energy of Li was a product of three hydrogen-like wavefunctions with principal quantum number equal to 1. What would be the total energy of Li? b Assume that two of the principal quantum numbers are 1 and the third principal quantum number is 2. Calculate the estimated electronic energy. c Compare both energies with an experimental value of 3.261017J. Which estimate is better? Is there any reason you might assume that this estimate would be better from the start?Spin orbitals are products of spatial and spin wavefunctions, but correct antisymmetric forms of wavefunctions for multielectron atoms are sums and differences of spatial wavefunctions. Explain why acceptable antisymmetric wavefunctions are sums and differences that is, combinations instead of products of spatial wavefunctions.If 1 and 2 are the individual wavefunctions for electron 1 and electron 2, identify each overall wavefunction as symmetric or antisymmetric with respect to exchange of the two electrons. i.e., 1 becomes 2 and vice versa. a=1+2 b=12 c=12+22 d=sin1+sin2 d=cos1cos2Show that the correct behavior of a wavefunction for He is antisymmetric by exchanging the electrons to show that (1,2)=(2,1).12.19EWhy isnt the electron configuration of beryllium, given as 1s22s2, considered a violation of the Pauli principle? Hint: See example 12.6.12.21EWrite a Slater determinant for the lithide ion, Li.Why does the concept of antisymmetric wavefunctions not need to be considered for the hydrogen atom?