Introductory Chemistry version 1.0 1st edition David W. Ball - Solutions

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Evaluate how many lines there are in a true rotational spectrum of CO molecules whose natural vibration frequency is w = 4.09 ∙ 1014 s–1 and moment of inertia I = 1.44 ∙ 10–39 g ∙ cm2.
Find the number of rotational levels per unit energy interval, dN/dE, for a diatomic molecule as a function of rotational energy E. Calculate that magnitude for an iodine molecule in the state with rotational quantum number J = 10. The distance between the nuclei of that molecule is equal to 267 pm.
Find the ratio of energies required to excite a diatomic molecule to the first vibrational and to the first rotational level. Calculate that ratio for the following molecules:Here co is the natural vibration frequency of a molecule, d is the distance between nuclei.
The natural vibration frequency of a hydrogen molecule is equal to 8.25. 1014 s-1, the distance between the nuclei is 74 pm. Find the ratio of the number of these molecules at the first excited vibrational level (v = t) to the number of molecules at the first excited rotational level (J = t) at a
Derive Eq. (6.4c), making use of the Boltzmann’s distribution. From Eq. (6.4c) obtain the expression for molar vibration heat capacity Cv vib of diatomic gas. Calculate Cv vib for C12 gas at the temperature 300 K. The natural vibration frequency of these molecules is equal to 1.064 • 1014 s-1.
In the middle of the rotation-vibration band of emission spectrum of HC1 molecule, where the "zeroth" line is forbidden by the selection rules, the interval between neighbouring lines is ∆w = 0.79.1013 s-1. Calculate the distance between the nuclei of an HC1 molecule.
Calculate the wavelengths of the red anal violet satellites, closest to the fixed line, in the vibration spectrum of Raman scattering by F2 molecules if the incident light wavelength is equal to λ0 = 404.7 nm and the natural vibration frequency of the molecule is w = 2.15 ∙ 1014 s -1.
Find the natural vibration frequency and the quasielastic force coefficient of an S2 molecule if the wavelengths of the red and violet satellites, closest to the fixed line, in the vibration spectrum 'of Raman scattering are equal to 346.6 and 330.0 nm.
Find the ratio of intensities of the violet and red satellites, closest to the fixed line, in the vibration spectrum of Raman scattering by CI2 molecules at a temperature T = 300 K if the natural vibration frequency of these molecules is w = 1.06 ∙ 1014 s -1. By what factor will this ratio
Consider the possible vibration modes in the following linear molecules: (a) CO2 (O – C – O); (b) C2H2 (H – C – C – H).
Find the number of natural transverse vibrations of a string of length l in the frequency interval from w to w + dw if the propagation velocity of vibrations is equal to v. All vibrations are supposed to occur in one plane.
There is a square membrane of area S. Find the number of natural vibrations perpendicular to its plane in the frequency interval from w to w + dw if the propagation velocity of vibrations is equal to v.
Find the number of natural transverse vibrations of a right angled parallelepiped of volume V in the frequency interval from w to w + dw if the propagation velocity of vibrations is equal to v.
Assuming the propagation velocities of longitudinal and transverse vibrations to be the same and equal to v, find the Debye temperature (a) For a unidimensional crystal, i.e. a chain of identical atoms, incorporating n o atoms per unit length; (b) For a two-dimensional crystal, i.e. a plane square
Calculate the Debye temperature for iron in which the propagation velocities of longitudinal and transverse vibrations are equal to 5.85 and 3.23 km/s respectively.
Evaluate the propagation velocity of acoustic vibrations in aluminium whose Debye temperature is O = 396 K.
Derive the formula expressing molar heat capacity of a unidimensional crystal, a chain of identical atoms, as a function of temperature T if the Debye temperature of the chain is equal to Simplify the obtained expression for the case T >> Θ.
In a chain of identical atoms the vibration frequency w depends on wave number k as w = w max sin (ka/2), where w max is the maximum vibration frequency, k = 2π/λ, is the wave number corresponding to frequency w a is the distance between neighbouring atoms. Making use of this dispersion
Calculate the zero-point energy per one gram of copper whose Debye temperature is Θ = 330 K.
Fig. 6.10 shows heat capacity of a crystal vs temperature in terms of the Debye theory. Here C cl is classical heat capacity, Θ is the Debye temperature. Using this plot, find:(a) The Debye temperature for silver if at a temperature T = 65 K its molar heat capacity is equal to 15J/ (mol.
Demonstrate that molar heat capacity of a crystal at a temperature T
Can one consider the temperatures 20 and 30 K as low for a crystal whose heat capacities at these temperatures are equal to 0.226 and 0.760 J / (mol. K)?
Calculate the mean zero-point energy per one oscillator of a crystal in terms of the Debye theory if the Debye temperature of the crystal is equal to Θ.
Radiation of atomic hydrogen falls normally on a diffraction grating of width l = 6.6 mm. The 50th line of the Balmer series in the observed spectrum is close to resolution at a diffraction angle θ (in accordance with Rayleigh's criterion). Find that angle.
Evaluate the maximum values of energy and momentum of a phonon (acoustic quantum) in copper whose Debye temperature is equal to 330 K.
Employing Eq. (6.4g), find at T = 0: (a) The maximum kinetic energy of free electrons in a metal if their concentration is equal to n; (b) The mean kinetic energy of free electrons if their maximum kinetic energy Tmax is known.
What fraction (in per cent) of free electrons in a metal at T = 0 has a kinetic energy exceeding half the maximum energy?
Find the number of free electrons per one sodium atom at T = 0 if the Fermi level is equal to EF = 3.07 eV and the density of sodium is 0.97 g/cm3.
Up to what temperature has one to heat classical electronic gas to make the mean energy of its electrons equal to that of free electrons in copper at T = 0? Only one free electron is supposed to correspond to each copper atom.
Calculate the interval (in eV units) between neighbouring levels of free electrons in a metal at T = 0 near the Fermi level, if the concentration of free electrons is n = 2.0.1022 cm -3 and the volume of the metal is V = 1.0 cm2.
Making use of Eq. (6.4g), find at T = 0: (a) The velocity distribution of free electrons; (b) The ratio of the mean velocity of free electrons to their maximum velocity.
On the basis of Eq. (6.4g) find the number of free electrons in a metal at T = 0 as a function of de Broglie wavelengths.
Calculate the electronic gas pressure in metallic sodium, at T = 0, in which the concentration of free electrons is n = 2.5.1023 cm -3. Use the equation for the pressure of ideal gas.
Find the velocity of photoelectrons liberated by electromagnetic radiation of wavelength λ = 18.0 nm from stationary He + ions in the ground state.
Find the refractive index of metallic sodium for electrons with kinetic energy T = 135 eV. Only one free electron is assumed to correspond to each sodium atom.
At what minimum kinetic energy must a hydrogen atom move for its inelastic head-on collision with another, stationary, hydrogen atom to make one of them capable of emitting a photon? Both atoms are supposed to be in the ground state prior to the collision.
From the conditions of the foregoing problem find how much (in per cent) the energy of the emitted photon differs from the energy of the corresponding transition in a hydrogen atom.
Fig. 6.11 illustrates logarithmic electric conductance as a function of reciprocal temperature (T in kK units) for some n-type semiconductor. Using this plot, find the width of the forbidden band of the semiconductor and the activation energy of donor levels.
The resistivity of an impurity-free semiconductor at room temperature is p = 50Ω.cm. It becomes equal to Pl = 40Ω ∙ cm when the semiconductor is illuminated with light, and t = 8 ms after switching off the light source the resistivity becomes equal to P2 = 45Ω ∙ cm.
In Hall Effect measurements a plate of width h = 10 mm and length l = 50 mm made of p-type semiconductor was placed in a magnetic field with induction B = 5.0kG. A potential difference V =10V was applied across the edges of the plate. In this case the Hall field is VH = 50 mV and resistivity p =
In Hall Effect measurements in a magnetic field with induction B = 5.0kG the transverse electric field strength in an impurity-free germanium turned out to be η = 10 times less than the longitudinal electric field strength. Find the difference in the mobilities of conduction electrons and
The Hall Effect turned out to be not observable in a semiconductor whose conduction electron mobility was η = 2.0 times that of the hole mobility. Find the ratio of hole and conduction electron concentrations in that semiconductor.
Employing Thomson's model, calculate the radius of a hydrogen atom and the wavelength of emitted light if the ionization energy of the atom is known to be equal to E = 13.6 eV.
An alpha particle with kinetic energy 0.27 MeV is deflected through an angle of 60° by a golden foil. Find the corresponding value of the aiming parameter.
To what minimum distance will an alpha particle with kinetic energy T = 0.40 MeV approach in the case of a head-on collision to (a) A stationary Pb nucleus; (b) A stationary free Li7 nucleus?
An alpha particle with kinetic energy T = 0.50 MeV is deflected through an angle of 0 = 90° by the Coulomb field of a stationary Hg nucleus. Find:(a) The least curvature radius of its trajectory; (b) The minimum approach distance between the particle and the nucleus.
A proton with kinetic energy T and aiming parameter b was deflected by the Coulomb field of a stationary Au nucleus. Find the momentum imparted to the given nucleus as a result of scattering.
A proton with kinetic energy T = 10 MeV flies past a stationary free electron at a distance b i0 pm. Find the energy acquired by the electron, assuming the proton's trajectory to be rectilinear and the electron to be practically motionless as the proton flies by.
A particle with kinetic energy T is deflected by a spherical potential well of radius R and depth U0, i.e. by the field in which the potential energy of the particle takes the form where r is the distance from the centre of thewill. Find the relationship between the aiming parameter b of the
A stationary ball of radius R is irradiated by a parallel stream of particles whose radius is r, assuming the collision of a particle and the ball to be elastic, find: (a) The deflection angle 0 of a particle as a function of its aiming parameter b; (b) The fraction of particles which after a
A narrow beam of alpha particles with kinetic energy 1.0 MeV falls normally on a platinum foil 1.0μm thick. The scattered particles are observed at an angle of (10° to the incident beam direction by means of a counter with a circular inlet area 1.0 cm2 located at the distance 10 cm from the
A narrow beam of alpha particles with kinetic energy T = 0.50 MeV and intensity I = 5.0 ∙ 105 particles per second falls normally on a golden foil. Find the thickness of the foil if at a distance r = 15 cm from a scattering section of that foil the flux density of scattered particles at the
A narrow beam of alpha particles falls normally on a silver foil behind which a counter is set to register the scattered particles. On substitution of platinum foil of the same mass thickness for the silver foil, the number of alpha particles registered per unit time increased η = 1.52 times.
A narrow beam of alpha particles falls normally on a silver foil behind which a counter is set to register the scattered particles. On substitution of platinum foil of the same mass thickness for the silver foil, the number of alpha particles registered per unit time increased η = 1.52 times.
A narrow beam of alpha particles with kinetic energy T = 0.50 MeV falls normally on a golden foil whose mass thickness is pd = 1.5 mg/cm2. The beam intensity is I0 = 5.0 ∙ 105 particles per second. Find the number of alpha particles scattered by the foil during a time interval x = 30 min into
A narrow beam of alpha particles with kinetic energy T = 600 keV falls normally on a golden foil incorporating n = 1.1 ∙ l019 nuclei/cm2. Find the fraction of alpha particles scattered through the angles 0 < θo = 20°.
A narrow beam of protons with kinetic energy T = 1.4 MeV falls normally on a brass foil whose mass thickness Pd = 1.5 mg/cm2. The weight ratio of copper and zinc in the foil is equal to 7: 3 respectively. Find the fraction of the protons scattered through the angles exceeding 00 = 30°.
Find the effective cross section of a uranium nucleus corresponding to the scattering of alpha particles with kinetic energy T = 1.5 MeV through the angles exceeding 0o = 60°.
The effective cross section of a gold nucleus corresponding to the scattering of mono energetic alpha particles within the angular interval from 90° to 180° is equal to ∆o = 0.50 kb. Find: (a) The energy of alpha particles; (b) The differential cross section of scattering
In accordance with classical electrodynamics an electron moving with acceleration w loses its energy due to radiation as dE/dt = - 2e2/3c3 W2, where e is the electron charge, c is the velocity of light. Estimate the time during which the energy of an electron performing almost harmonic oscillations
Making use of the formula of the foregoing problem, estimate the time during which an electron moving in a hydrogen atom along a circular orbit of radius r = 50 pm would have fallen onto the nucleus. For the sake of simplicity assume the vector w to be permanently directed toward the centre of the
Demonstrate that the frequency w of a photon emerging when an electron jumps between neighbouring circular orbits of a hydrogen-like ion satisfies the inequality wn, > w > wn+1, where wn and wn+1 are the frequencies of revolution of that electron around the nucleus along the circular orbits. Make
A particle of mass rn moves along a circular orbit in a Centro-symmetrical potential field U(r) = kr2/2. Using the Bohr quantization condition, find the permissible orbital radii and energy levels of that particle.
Calculate for a hydrogen atom and a He + ion: (a) The radius of the first Bohr orbit and the velocity of an electron moving along it; (b) The kinetic energy and the binding energy of an electron in the ground state; (c) The ionization potential, the first excitation potential and the
Calculate the angular frequency of an electron occupying the second Bohr orbit of He+ ion.
For hydrogen-like systems find the magnetic moment μn corresponding to the motion of an electron along the n-th orbit and the ratio of the magnetic and mechanical moment’s μn/Mn. Calculate the magnetic moment of an electron occupying the first Bohr orbit.
Calculate the magnetic field induction at the centre of a hydrogen atom caused by an electron moving along the first Bohr orbit.
Calculate and draw on the wavelength scale the spectral intervals in which the Lyman, Balmer, and Paschen series for atomic hydrogen are confined. Show the visible portion of the spectrum.
To what series does the spectral line of atomic hydrogen belong if its wave number is equal to the difference between the wave numbers of the following two lines of the Balmer series: 486.1 and 410.2 nm? What is the wavelength of that line?
For the case of atomic hydrogen find: (a) The wavelengths of the first three lines of the Balmer series; (b) The minimum resolving power λ/δλ of a spectral instrument capable of resolving the first 20 lines of the Balmer series.
Radiation of atomic hydrogen falls normally on a diffraction grating of width l = 6.6 mm. The 50th line of the Balmer series in the observed spectrum is close to resolution at a diffraction angle θ (in accordance with Rayleigh's criterion). Find that angle. Discuss.
What element has a hydrogen-like spectrum whose lines have wavelengths four times shorter than those of atomic hydrogen?
How many spectral lines are emitted by atomic hydrogen excited to the n-th energy level?
What lines of atomic hydrogen absorption spectrum fall within the wavelength range from 94.5 to 130.0 nm?
Find the quantum number n corresponding to the excited state of He+ ion if on transition to the ground state that ion emits two photons in succession with wavelengths 108.5 and 30.4 nm.
Calculate the Rydberg constant R if He + ions are known to have the wavelength difference between the first (of the longest wavelength) lines of the Balmer and Lyman series equal to ∆λ = = 133.7 nm.
What hydrogen-like ion has the wavelength difference between the first lines of the Balmerand Lyman series equal to 59.3 nm?
Find the wavelength of the first line of the He + ion spectral series whose interval between the extreme lines is ∆w = 5.18 ∙ 1015 s-l.
Find the binding energy of an electron in the ground state of hydrogen-like ions in whose spectrum the third line of the Balmer series is equal to 108.5 nm.
The binding energy of an electron in the ground state of He atom is equal to E o - 24.6 eV. Find the energy required to remove both electrons from the atom.
Find the velocity of photoelectrons liberated by electromagnetic radiation of wavelength λ = 18.0 nm from stationary He + ions in the ground state. Discuss.
At what minimum kinetic energy must a hydrogen atom move for its inelastic head-on collision with another, stationary, hydrogen atom to make one of them capable of emitting a photon? Both atoms are supposed to be in the ground state prior to the collision.
A stationary hydrogen atom emits a photon corresponding to the first line of the Lyman series. What velocity does the atom acquire?
From the conditions of the foregoing problem find how much (in per cent) the energy of the emitted photon differs from the energy of the corresponding transition in a hydrogen atom.
A stationary He+ ion emitted a photon corresponding to the first line of the Lyman series. That photon liberated a photoelectron from a stationary hydrogen atom in the ground state. Find the velocity of the photoelectron.
Find the velocity of the excited hydrogen atoms if the first line of the Lyman series is displaced by ∆λ, = 0.20 nm when their radiation is observed at an angle θ = 45 ° to their motion direction.
According to the Bohr-Sommerfeld postulate the periodic motion of a particle in a potential field must satisfy the following quantization rule: where q and p are generalized coordinate and momentum of the particle,n are integers. Making use of this rule, find the permitted values of energy for a
Taking into account the motion of the nucleus of a hydrogen atom, find the expressions for the electron's binding energy in the ground state and for the Rydberg constant. How much (in per cent) do the binding energy and the Rydberg constant, obtained without taking into account the motion of the
For atoms of light and heavy hydrogen (H and D) find the difference (a) Between the binding energies of their electrons in the ground state; (b) Between the wavelengths of first lines of the Lyman series.
Calculate the separation between the particles of a system in the ground state, the corresponding binding energy, and the wavelength of the first line of the Lyman series, if such a system is (a) A Mesonic hydrogen atom whose nucleus is a proton (in a mesonic atom an electron is replaced by a
Write the nuclear equation for the beta emission of iodine-131, the isotope used to diagnose and treat thyroid problems.
A common uranium compound is uranyl nitrate hexahydrate [UO2(NO3)2ּ6H2O]. What is the formula mass of this compound?
Plutonium forms three oxides: PuO, PuO2, and Pu2O3. What are the formula masses of these three compounds?
A banana contains 600 mg of potassium, 0.0117% of which is radioactive potassium-40. If 1 g of potassium-40 has an activity of 2.626 ( 105 Bq, what is the activity of a banana?
Write the nuclear equation that represents radioactive decay of polonium-208 by alpha particle emission and identify the daughter isotope.
Write the nuclear equation that represents the radioactive decay of technetium-133 by beta particle emission and identify the daughter isotope. A gamma ray is emitted simultaneously with the beta particle.
The half-life of titanium-44 is 60.0 y. A sample of titanium contains 0.600 g of titanium-44. How much remains after 240.0 y?
The half-life of titanium-44 is 60.0 y. A sample of titanium contains 0.600 g of titanium-44. How much remains after 100.0 y?
A sample of radon has an activity of 60,000 Bq. If the half-life of radon is 15 h, how long before the sample’s activity is 3,750 Bq?
A sample of radon has an activity of 60,000 Bq. If the half-life of radon is 15 h, how long before the sample’s activity is 10,000 Bq?

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Introductory Chemistry version 1.0 1st edition David W. Ball - Solutions

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