Organic Chemistry structure and function 6th edition K. Peter C. Vollhardt, Neil E. Schore - Solutions

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Consider a simple model of the helium atom in which two electrons, each with mass m, move around the nucleus (charge +2e) in the same circular orbit. Each electron has orbital angular momentum Ii (that is, the orbit is the smallest-radius Bohr orbit), and the two electrons are always on opposite
Repeat the calculation of Problem 41.39 for a one-electron on with nuclear charge Z. (See Problem 41.54.) How does the probability of the electron being found in the classically forbidden region depend on Z?
We know from Chapter 18 that the average kinetic energy of an ideal gas atom or molecule at Kelvin temperature T is 3/2kT. For what value of T does this energy correspond to? (a) The bond energy of the van der Waals bond in He2, (7.9 x 10-4eV) and (b) The bond energy of the covalent bond in H2
An Ionic Bond (a) Calculate the electric potential energy for a K+ ion and a Br– ion separated by a distance of 0.29nm, the equilibrium separation in the KBr molecule. Treat the ions as point charges. (b) The ionization energy of the potassium atom is 4.3eV. Atomic bromine has an electron
A hypothetical NH molecule makes a rotational-level transition from l = 3 to l = 1 and gives off a photon of wavelength 1.780nm in doing so. What is the separation between the two atoms in this molecule if we model them as point masses? The mass of hydrogen is 1.67 x 10-27 kg, and the mass of
The water molecule has an l = 1 rotational level 1.01 x 10-5eV above the 1 = 0 ground level. Calculate the wavelength and frequency of the photon absorbed by water when it undergoes a rotational-level transition from l = 0 to l = 1. The magnetron oscillator in a microwave oven generates microwaves
In Example 42.2 the moment of inertia for CO was calculated using Eq. (42.6). (a) In CO, how far is each atom from the center of mass of the molecule? (b) Use I = m1r12? + m2r22 to calculate the moment of inertia of CO about an axis through the center of mass and perpendicular to the line joining
Two atoms of cesium (Cs) can form a Cs2 molecule. The equilibrium distance between the nuclei in a Cs2 molecule is 0.447nm. Calculate the moment of inertia about an axis through the center of mass of the two nuclei and perpendicular to the line joining them. The mass of a cesium atom is 2.21 x
The rotational energy levels of CO are calculated in Example 42.2. If the energy of the rotating molecule is described by the classical expression K = ½ Iw2, for the l = 1 level what are (a) The angular speed of the rotating molecule; (b) The linear speed of each atom (use the result of Exercise
A lithium atom has mass 1.17 x 10-26kg, and a hydrogen atom has mass 1.67 x 10-27 kg. The equilibrium separation between the two nuclei in the LiH molecule is 0.159 nm. (a) What is the difference in energy between the l = 3 and l = 4 rotational levels? (b) What is the wavelength of the photon
When a hypothetical diatomic molecule having atoms 0.8860 nm apart undergoes a rotational transition from the l = 2 state to the next lower state, it gives up a photon having energy 8.841 x 10-4eV. When the molecule undergoes a vibrational transition from one energy state to the next lower energy
(a) Show that the energy difference between rotational levels with angular-momentum quantum numbers l and l – 1 is H2/I. (b) In terms of l, h, and I, what is the frequency of the photon emitted in the pure rotation transition l → 1 – 1?
The rotational energy levels of the CO molecule are calculated in Example 42.2 and the vibrational-level energy differences are given in Example 42.3. The vibrational and rotational energies are given by Eq. (42.9). Calculate the wave• length of the photon absorbed by CO in each of the following
The vibration frequency for the molecule HF is 1.24 x 1014 Hz. The mass of a hydrogen atom is 1.67 x 10-27 kg, and the mass of a fluorine atom is 3.15 x 10-26 kg. (a) What is the force constant k' for the inter atomic force? (b) What is the spacing between adjacent vibrational energy levels in
Potassium bromide (KBr) has a density of2.75 x 103 kg/m3 and the same crystal structure as NaC1. The mass of a potassium atom is 6.49 x 10-26 kg, and the mass of a bromine atom is 1.33 x 10-25 kg. (a) Calculate the average spacing between adjacent atoms in a KBr crystal. (b) How does the value
Density of NaC1 the spacing of adjacent atoms in a crystal of sodium chloride is 0.282 nm. The mass of a sodium atom is 3.82 x 10-26 kg, and the mass of a chlorine atom is 5.89 x 10-26 kg. Calculate the density of sodium chloride.
Calculate the wavelengths of (a) A 6.20-keV x ray; (b) A 37.6-eVelectron; (c) A 0.020,V neutron.
The maximum wavelength of light that a certain silicon photocell can detect is 1.11µm. (a) What is the energy gap (in electron volts) between the valence and conduction bands for this photocell? (b) Explain why pure silicon is opaque.
The gap between valence and conduction bands in diamond is 5.47eV. (a) What is the maximum wavelength of a photon that can excite an electron from the top of the valence band into the conduction band? In what region of the electromagnetic spectrum does this photon lie? (b) Explain why pure diamond
The gap between valence and conduction bands in silicon is 1.12eV. A nickel nucleus in an excited state emits a gamma-ray photon with wavelength 9.31 x 10-4nm. How many electrons can be excited from the top of the valence band to the bottom of the conduction band by the absorption of this gamma ray?
What is the value of the constant A in Eq. (42.10) that makes ψ(x, y, z) normalized?
Calculate the density of states g (E) for the free-electron model of a metal if E = 7.0eV and V = 1.0 cm3. Express your answer in units of states per electron volt.
Calculate Vrms for free electrons with average kinetic energy 3/2kT at a temperature of 300 K. How does your result compare to the speed of an electron with a kinetic energy equal to the Fermi energy of copper, calculated in Example 42.9? Why is there such a difference between these speeds?
(a) Show that the wave function ψ given in Eq. (42.10) is a solution of the three-dimensional Schrodinger equation, Eq. (40.29), with the energy as given by Eq. (42.11). (b) What are the energies of the ground level and the lowest two excited levels? What is the degeneracy of each of these
Silver has a Fermi energy of 5.48eV. Calculate the electron contribution to the molar heat capacity at constant volume of silver, Cv, at 300 K. Express your result (a) As a multiple of R and(b) As a fraction of the actual value for silver, Cv = 25.3 J/mol · K.(c) Is the value of Cv due principally
The Fermi energy of sodium is 3.23eV. (a) Find the average energy Egv of the electrons at absolute zero. (b) What is the speed of an electron that has energy Eav? (c) At what Kelvin temperature T is kT equal to Ep? (This is called the Fermi temperature for the metal. It is approximately the
For a solid metal having a Fermi energy of 8.500eV, what is the probability, at room temperature, that a state having an energy of 8.520eV is occupied by an electron?
Pure germanium has a band gap of 0.67eV. The Fermi energy is in the middle of the gap. (a) For temperatures of 250 K, 300 K, and 350 K, calculate the probability f(E) that a state at the bottom of the conduction band is occupied. (b) For each temperature in part (a), calculate the probability that
Germanium has a band gap of 0.67eV. Doping with arsenic adds donor levels in the gap 0.01eV below the bottom of the conduction band. At a temperature of 300 K, the probability is 4.4 x 10-4 that an electron state is occupied at the bottom of the conduction band. Where is the Fermi level relative to
For a certain p-n junction diode, the saturation current at room temperature (20oC) is 0.750mA. What is the resistance of this diode when the voltage across it is (a) 85.0mV(b) -50.0mV?
(a) A forward-bias voltage of 15.0 mV produces a positive current of 9.25mA through a p-n junction at 300 K. What does the positive current become if the forward-bias voltage is reduced to 10.0mV? (b) For reverse-bias voltages of -15.0mV and -10.0mV, what is the reverse-bias negative current?
A p-n junction has a saturation current of 3.60mA. (a) At a temperature of 300 K, what voltage is needed to produce a positive current of 40.0mA? (b) For a voltage equal to the negative of the value calculated in part (a), what is the negative current?
A hypothetical diatomic molecule of oxygen (mass = 2656 x 10-26kg) and hydrogen (mass = 1.67 x 10-27 kg) emits a photon of wavelength 2.39µm when it makes a transition from one vibrational state to the next lower state. If we model this molecule as two point masses at opposite ends of a mass
A hypothetical diatomic molecule of oxygen (mass = 2656 x 10-26kg) and hydrogen (mass = 1.67 x 10-27 kg) emits a photon of wavelength 2.39µm when it makes a transition from one vibrational state to the next lower state. If we model this molecule as two point masses at opposite ends of a mass less
(a) For the sodium chloride molecule (NaC1) discussed at the beginning of Section 42.1, what is the maximum separation of the ions for stability if they may be regarded as point charges? That is, what is the largest separation for which the energy of an Na+ ion and a C1– ion, calculated in this
The rotational spectrum of HC1 contains the following wavelengths (among others): 60.4µm, 69.0µm, 80.4µm, 96.4µm, and 120.4µm. Use this spectrum to find the moment of inertia of the HCI molecule about an axis through the center of mass and perpendicular to the line joining the two nuclei.
(a) Use the result of Problem 42.38 to calculate the equilibrium separation of the atoms in HC1 molecule. The mass of a chlorine atom is 5.81 x 10-26 kg, and the mass of a hydrogen atom is 1.67 x 10-27 kg. (b) The value of I changes by ±1 in rotational transitions. What is the value of l for the
When a NaF molecule makes a transition from the 1 = 3 to the I = 2 rotational level with no change in vibrational quantum number or electronic state, a photon with wavelength 3.83mm is emitted. A sodium atom has mass 3.82 x 10-26 kg, and a fluorine atom has mass 3.15 x 10-26 kg. Calculate the
Consider a gas of diatomic molecules (moment of inertia I) at an absolute temperature T. If Eg is a ground-state energy and E ex is the energy of an excited state, then the Maxwell-Boltzmann distribution (see Section 38.6) predicts that the ratio of the numbers of molecules in the two states is(a)
Our galaxy contains numerous molecular clouds, regions many light-years in extent in which the density is high enough and the temperature low enough for atoms to form into molecules. Most of the molecules are H2, but a small fraction of the molecules are carbon monoxide (CO). Such a molecular cloud
Our galaxy contains numerous molecular clouds, regions many light-years in extent in which the density is high enough and the temperature low enough for atoms to form into molecules. Most of the molecules are H2, but a small fraction of the molecules are carbon monoxide (CO). Such a molecular cloud
When an OH molecule undergoes a transition from the n = 0 to the n = 1 vibrational level, its internal vibrational energy increases by 0.463eV. Calculate the frequency of Vibration and the force constant for the inter atomic force. (The mass of an oxygen atom is 2.66 x 10-26 kg, and the mass of a
The force constant for the inter nuclear force in a hydrogen molecule (H2) is k' = 576 N/m. A hydrogen atom has mass 1.67 x 10-27 kg. Calculate the zero-point vibrational energy for H2 (that is, the vibrational energy the molecule has in the n = 0 ground vibrational level). How does this energy
Suppose the hydrogen atom in HF (see Exercise 42.13) is replaced by an atom of deuterium, an isotope of hydrogen with a mass of 3.34 x 10-27 kg. The force constant is determined by the electron configuration, so it is the same as for the normal HF molecule.(a) What is the vibrational frequency of
Suppose the hydrogen atom in HF (see Exercise 42.13) is replaced by an atom of deuterium, an isotope of hydrogen with a mass of 3.34 x 10-27 kg. The force constant is determined by the electron configuration, so it is the same as for the normal HF molecule.(a) What is the vibrational frequency of
Compute the Fermi energy of potassium by making the simple approximation that each atom contributes one free electron. The density of potassium is 851 kg/m', and the mass of a single potassium atom is 6.49 x 10-26 kg.
Hydrogen is found in two naturally occurring isotopes; normal hydrogen (containing a single proton in its nucleus) and deuterium (having a proton and a neutron). Assuming that both molecules are the same size and that the proton and neutron have the same mass (which is almost the case), find the
Metallic lithium has a bee crystal structure. Each unit cell is a cube of side length a = 0.35nm. (a) For a bee lattice, what is the number of atoms per unit volume? Give your answer in terms of a.(b) Use the result of part (a) to calculate the zero-temperature Fermi energy EF0 for metallic
The one-dimensional calculation of Example 42.5 (Section 42.3) can be extended to three dimensions. For the three dimensional fcc NaC1 lattice, the result for the potential energy of a pair of Na+ and CI?? ions due to the electrostatic interaction with all of the ions in the crystal is U = ??
Consider a system of N free electrons within a volume V. Even at absolute zero such a system exerts a pressure p on its surroundings due to the motion of the electrons. To calcu1ate this pressure, imagine that the volume increases by a small amount dV. The electrons will do an amount of work pdV on
When the pressure p on a material increases by an amount ?p, the volume of the material will change from V to V + ?V, where ?V is negative. The bulk modulus B of the material is defined to be the ratio of the pressure change ?p to the absolute value |?V/V| of the fractional volume change. The
In the discussion of free electrons in Section 42.5, we assumed that we could ignore the effects of relativity. This is not a safe assumption if the Fermi energy is greater than about 1/100 mc2 (that is, more than about 1% of the rest energy of an electron).(a) Assume that the Fermi energy at
A variable DC battery is connected in series with a 125-Ω resistor and a p-n junction diode that has a saturation current of 0.625mA at room temperature (20oC). When a voltmeter across the 125-0 resistor reads 35.0 V, what are? (a) The voltage across the diode and (b) The resistance of
Van der Waals bonds arise from the interaction between two permanent or induced electric dipole moments in a pair of atoms or molecules.(a) Consider two identical dipoles, each consisting of charges +q and ?? q separated by a distance d and oriented as shown in Fig. 42.40a. Calculate the electric
(a) Consider the hydrogen molecule (H2) to be a simple harmonic oscillator with an equilibrium spacing of 0.074 nm, and estimate the vibrational energy-level spacing for H2. The mass of a hydrogen atom is 1.67 x 10-27 kg. (b) Use the results of part (a) to calculate the vibrational energy level
Molecules Condensed MatterHow many protons and how many neutrons are there in a nucleus of the most common isotope of (a) Silicon 2814Si; (b) Rubidium 8537Rb; (c) Thallium 20581?
Consider the three nuclei of Exercise 43.1. Estimate (a) The radius,(b) The surface area, and (c) The volume of each nucleus. Determine (d) The mass density (in kg/m3) and (e) The nucleon density (in nucleons per cubic meter) for each nucleus. Assume that the mass of each nucleus is A atomic mass
Hydrogen atoms are placed in an external magnetic field. The protons can make transitions between states in which the nuclear spin component is parallel and anti parallel to the field by absorbing or emitting a photon. What magnetic-field magnitude is required for this transition to be induced by
Neutrons are placed in a magnetic field with magnitude 2.30 T. (a) What is the energy difference between the states with the nuclear spin angular momentum components parallel and anti parallel to the field? Which state is lower in energy, the one with its spin component parallel to the field or the
Hydrogen atoms are placed in an external 1.65-T magnetic field. (a) The protons can make transitions between states where the nuclear spin component is parallel and anti parallel to the field by absorbing or emitting a photon. Which state has lower energy: the state with the nuclear spin component
The most common isotope of uranium, 23892U, has atomic mass 238.050783 u. Calculate (a) The mass defect; (b) The binding energy (m MeV); (c) The binding energy per nucleon.
What is the maximum wavelength of a γ ray that could break a deuteron into a proton and a neutron? (This process is called photodisintegration.)
Calculate (a) The total binding energy and (b) The binding energy per nucleon of 12c (c) What percent of the rest mass of this nucleus is its total binding energy?
A photon with a wavelength of 3.50 x 10-13 m strikes a deuteron, splitting it into a proton and a neutron. (a) Calculate the kinetic energy released in this interaction. (b) Assuming the two particles share the energy equally, and taking their masses to be 1.00 u, calculate their speeds after the
Calculate the mass defect, the binding energy (in MeV), and the binding energy per nucleon of (a) The nitrogen nucleus, 147N and (b) The helium nucleus, 42He. (c) How do the results of parts (a) and (b) compare?
The most common isotope of boron is 115B. (a) Determine the total binding energy of 115B from Table 43.2 in Section 43.1.(b) Calculate this binding energy from Eq. (43.11). (Why is the fifth term zero?) Compare to the result you obtained in part (a). What is the percent difference? Compare the
The most common isotope of copper is 6329Cu. The measured mass of the neutral atom is 62.929601 u. (a) From the measured mass, determine the mass defect, and use it to find the total binding energy and the binding energy per nucleon. (b) Calculate the binding energy from Eq. (43.11). (Why is the
What nuclide is produced in the following radioactive decays? (a) A decay of 2394Pu; (b) β– decay of 2411Na; (c) β+ decay of 158O.
(a) Is the decay n → p + β - + vc energetically possible? If not, explain why not. If so, calculate the total energy released. (b) Is the decay p → n + β+ + vc energetically possible? If not, explain why not. If so, calculate the total energy released.
The a decay of 238U is accompanied by a γ ray of measured wavelength 0.0248 nm. This decay is due to a transition of the nucleus between two energy levels. What is the difference in energy (in MeV) between these two levels?
238U decays spontaneously by a emission to 234Th. Calculate (a) The total energy released by this process and (b) The recoil velocity of the 234Th nucleus. The atomic masses are 238.050788 u for 238U and 234.043601 u for 234Th.
The atomic mass of 14c is 14.003242 u. Show that the β– decay of 14C is energetically possible, and calculate the energy released in the decay.
What particle (a particle, electron, or positron) is emitted in the following radioactive decays? (a) 2714Si → 2713Al; (b) 23892U → 23490Th; (c) 7433As → 7434Se.
(a) Calculate the energy released by the electron-capture decay of 5727Co (see Example 43. 7). (b) A negligible amount of this energy goes to the resulting ~e atom as kinetic energy. About 90% of the time, the ~e nucleus emits two successive gamma-ray photons after the electron-capture process, of
The isotope 90Sr undergoes β – decay with a half-life of 28 years. (a) What nucleus is produced by this decay? (b) If a nuclear power plant is contaminated with 90Sr, how long will it take for the radiation level to decrease to 1.0% of its initial value?
If a 6.13-g sample of an isotope having a mass number of 124 decays at a rate of 0.350Ci, what is its half-life?
Radioactive isotopes used in cancer therapy have a "shelf-life," like pharmaceuticals used in chemotherapy. Just after it has been manufactured in a nuclear reactor, the activity of a sample of 60co is 5000Ci. When its activity falls below 3500Ci, it is considered too weak a source to use in
A 12.0-g sample of carbon from living matter decays at the rate of 180.0decays/min due to the radioactive 14C in it what will be the decay rate of this sample in(a) 1000 years and (b) 50,000 years?
(a) Assuming that the sample all went to the thyroid gland, what will be the decay rate in that gland 24 d (about 2 ½ weeks) later? (b) If the decay rate in the thyroid 24 d later is actually measured to be 17.0Bq, what percentage of the tracer went to that gland? (c) What isotope remains after
Tritium (3H) undergoes β– decay with a half-life of 12.3 years. It is also highly toxic to living things. (a) What nucleus is produced in the β– decay of tritium? (b) Suppose some tritium gas is released into the atmosphere in a nuclear power plant accident. How long will it take
As a health physicist, you are being consulted about a spill in a radiochemistry lab. The isotope spilled was 500µCi of 131Ba, which has a half-life of 12 days. (a) What mass of 131Ba was spilled? (b) Your recommendation is to clear the lab until the radiation level has fallen 1.00µCi. How long
Measurements on a certain isotope tell you that the decay rate decreases from 8318 decays/min to 3091 decays/min in 4.00 days. What is the half-life of this isotope?
The isotope 226Ra undergoes a decay with a half-life of 1620 years. What is the activity of 1.00 g of 226Ra? Express your answer in Bq and in Ci.
The ratio of 14c to 12C in living matter is measured to be 14C/12C = 1.3 x 10-12 at the present time. A 12.0-g sample of carbon produces 180decays/min due to the small amount of 14c in it from this information calculates the half-life of 14C.
If you are of average mass, about 360 million nuclei in your body undergo radioactive decay each day. Express your activity in curies.
The radioactive nuclide 199Pt has a half-life of 30.8 minutes. A sample is prepared that has an initial activity of 7.56 x 1011 Bq.(a) How many 199Pt nuclei are initially present in the sample?(b) How many are present after 30.8 minutes? What is the activity at this time? (c) Repeat part (b) for a
Radiocarbon Dating a sample from timbers at an archeological site containing 500g of carbon provides 3070 decays/min. What is the age of the sample?
The unstable isotope 40K is used for dating rock samples. Its half-life is 1.28 x 109y. (a) How many decays occur per second in a sample containing 1.63 x l0-6g of 40K? (b) What is the activity of the sample in curies?
A person exposed to fast neutrons receives a radiation dose of 200rem on part of his hand, affecting 25 g of tissue. The RBE of these neutrons is 10. (a) How many rad did he receive? (b) How many joules of energy did this person receive? (c) Suppose the person received the same rad dosage, but from
A nuclear chemist receives an accidental radiation dose of 5.0Gy from slow neutrons (RBE = 4.0). What does she receive in rad, rem, and J/kg?
To Scan or Not to Scan it has become popular for some people to have yearly whole-body scans (CT scans, formerly called CAT scans) using x rays, just to see if they detect anything suspicious. A number of medical people have recently questioned the advisability of such scans, due in part to the
Food Irradiation Food is often irradiated with either x rays or electron beams to help prevent spoilage. A low dose of 5-75kilorads (k rad) helps to reduce and kill inactive parasites, a medium dose of 100-400krad kills microorganisms and pathogens such as salmonella, and a high dose of
In an industrial accident a 65-kg person receives a lethal whole-body equivalent dose of 5.4 Sv from x rays.(a) What is the equivalent dose in rem? (b) What is the absorbed dose in rad?(c) What is the total energy absorbed by the person's body? How does this amount of energy compare to the
In an industrial accident a 65-kg person receives a lethal whole-body equivalent dose of 5.4 Sv from x rays.(a) What is the equivalent dose in rem? (b) What is the absorbed dose in rad?(c) What is the total energy absorbed by the person's body? How does this amount of energy compare to the amount
A person ingests an amount of a radioactive source with a very long lifetime and activity O.72µCi. The radioactive material lodges in the lungs, where all of the 4.0-MeV a particles emitted are absorbed within a 0.50-kg mass of tissue. Calculate the absorbed dose and the equivalent dose for one
Consider the nuclear reaction where X is a nuclide 21H + 147N→ X +105B (a) What is Z and A for the nuclide X? (b) Calculate the reaction energy Q (in MeV). (c) If the 21H nucleus is incident on a stationary 147N nucleus, what minimum kinetic energy must it have for the reaction to occur?
Consider the nuclear reaction where X is a nuclide 21He + 94Be → X + 42He (a) What are the values of Z and A for the nuclide X? (b) How much energy is liberated? (c) Estimate the threshold energy for this reaction.
The United States uses 1.0 x 1019J of electrical energy per year. H all this energy came from the fission of 235U, which releases 200 MeV per fission event, (a) How many kilograms of 235U would be used per year and (b) How many kilograms of uranium would have to be mined per year to provide that
At the beginning of Section 43.7 the equation of a fission process is given in which 235U is struck by a neutron and undergoes fission to produce 144Ba, 89Kr, and three neutrons. The measured masses of these isotopes are 235.043930 u (235u), 143.922953 u (I44Ba), 88.9l7630u (89Kr), and 1.0086649 u
Consider the nuclear reaction 2814 Si + γ → 2814 Mg + X where X is a nuclide. (a) What is Z and A for the nuclide X? (b) Ignoring the effects of recoil, what minimum energy must the photon have for this reaction to occur? The mass of a 2814Si atom is 27.976927 u, and the mass of a

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Organic Chemistry structure and function 6th edition K. Peter C. Vollhardt, Neil E. Schore - Solutions

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