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Physical Chemistry 8th Edition Peter Atkins, Julio De Paula - Solutions

Consider Stirling’s approximation for ln N! in the derivation of the Boltzmann distribution. What difference would it make if(a) A cruder approximation, N! = NN,(b) The better approximation in Comment 16.2 were used instead?
Explore whether a magnetic field can influence the heat capacity of a paramagnetic molecule by calculating the electronic contribution to the heat capacity of an NO2 molecule in a magnetic field. Estimate the total constantvolume heat capacity using equipartition, and calculate the percentage
Explain the origin of the symmetry number.
Estimate the values of γ = Cp/CV for gaseous ammonia and methane. Do this calculation with and without the vibrational contribution to the energy. Which is closer to the expected experimental value at 25°C?
Estimate the rotational partition function of HCl at(a) 25°C and(b) 250°C.
Calculate the temperature dependence of the heat capacity of p-H2 (in which only rotational states with even values of J are populated) at low temperatures on the basis that its rotational levels J = 0 and J = 2 constitute a system that resembles a two-level system except for the degeneracy of the
Use concepts of statistical thermodynamics to describe the molecular features that determine the magnitudes of the constant-volume molar heat capacity of a molecular substance.
Give the symmetry number for each of the following molecules:(a) CO,(b) O2,(c) H2S, and(d) SiH4,(e) CHCl3.
Calculate the rotational partition function of H2O at 298 K from its rotational constants 27.878 cm−1, 14.509 cm−1, and 9.287 cm−1. Above what temperature is the high-temperature approximation valid to within 10 per cent of the true value?
Calculate the standard molar entropy of N2(g) at 298 K from its rotational constant B = 1.9987 cm−1 and its vibrational wavenumber ∇ = 2358 cm−1. The thermochemical value is 192.1 J K−1 mol−1. What does this suggest about the solid at T = 0?
Describe how liquids are investigated by using concepts of statistical thermodynamics.
From the results of Exercise 17.5a, calculate the rotational contribution to the molar entropy of gaseous water at 25°C.Data in Exercise 17.5a,Calculate the rotational partition function of H2O at 298 K from its rotational constants 27.878 cm−1, 14.509 cm−1, and 9.287 cm−1. Above what
Calculate the rotational partition function of CH4(a) By direct summation of the energy levels at 298 K and 500 K, and(b) By the hightemperature approximation. Take B = 5.2412 cm−1.
The bond length of O2 is 120.75 pm. Use the high-temperature approximation to calculate the rotational partition function of the molecule at 300 K.
Plot the molar heat capacity of a collection of harmonic oscillators as a function of T/θV, and predict the vibrational heat capacity of ethyne at(a) 298 K,(b) 500 K.The normal modes (and their degeneracies in parentheses) occur at wavenumbers 612(2), 729(2), 1974, 3287, and 3374 cm−1.
Calculate and plot as a function of temperature, in the range 300 K to 1000 K, the equilibrium constant for the reaction CD4(g) + HCl(g) ⇌ CHD3(g) + DCl(g) using the following data (numbers in parentheses are degeneracies): ∇(CHD3)/cm−1 = 2993(1), 2142(1), 1003(3), 1291(2), 1036(2);
A CO2 molecule is linear, and its vibrational wavenumbers are 1388.2 cm−1, 667.4 cm−1, and 2349.2 cm−1, the last being doubly degenerate and the others non-degenerate. The rotational constant of the molecule is 0.3902 cm−1. Calculate the rotational and vibrational contributions to the molar
Although expressions like <ε> = −d ln q/dβ are useful for formal manipulations in statistical thermodynamics, and for expressing thermodynamic functions in neat formulas, they are sometimes more trouble than they are worth in practical applications. When presented with a table of energy
The ground level of Cl is 2P3/2 and a 2P1/2 level lies 881 cm−1 above it. Calculate the electronic contribution to the heat capacity of Cl atoms at(a) 500 K and(b) 900 K.
The ground state of the Co2+ ion in CoSO4·7H2O may be regarded as 4T9/2. The entropy of the solid at temperatures below 1 K is derived almost entirely from the electron spin. Estimate the molar entropy of the solid at these temperatures.
Calculate the residual molar entropy of a solid in which the molecules can adopt(a) Three,(b) Five,(c) Six orientations of equal energy at T = 0.
Derive expressions for the internal energy, heat capacity, entropy, Helmholtz energy, and Gibbs energy of a harmonic oscillator. Express the results in terms of the vibrational temperature, θV and plot graphs of each property against T/θV.
The heat capacity ratio of a gas determines the speed of sound in it through the formula cs = (γRT/M)1/2, where γ = Cp /CV and M is the molar mass of the gas. Deduce an expression for the speed of sound in a perfect gas of(a) Diatomic,(b) Linear triatomic,(c) Nonlinear triatomic molecules at high
Which of the following molecules may be polar: CIF3, O3, H2O2?
The relative permittivities of methanol (m.p. −95°C) corrected for density variation are given below. What molecular information can be deduced from these values? Take ρ = 0.791 g cm−3 at 20°C. Ꮎ/°C -185 Ꮛ 3.2 -170 -150 -140 3.6 4.0 5.1 -110 -80 67 57 -50 -20 4 43 0 38 20 34
Suppose an H2O molecule (µ = 1.85 D) approaches an anion. What is the favourable orientation of the molecule? Calculate the electric field (in volts per metre) experienced by the anion when the water dipole is(a) 1.0 nm,(b) 0.3 nm,(c) 30 nm from the ion.
Explain why the polarizability of a molecule decreases at high frequencies.
The electric dipole moment of toluene (methylbenzene) is 0.4 D. Estimate the dipole moments of the three xylenes (dimethylbenzene). Which answer can you be sure about?
An H2O molecule is aligned by an external electric field of strength 1.0 kV m−1 and an Ar atom (α′ = 1.66 × 10−24 cm3) is brought up slowly from one side. At what separation is it energetically favourable for the H2O molecule to flip over and point towards the approaching Ar atom?
Calculate the magnitude and direction of the dipole moment of the following arrangement of charges in the xy-plane: 3e at (0,0), −e at (0.32 nm, 0), and −2e at an angle of 20° from the x-axis and a distance of 0.23 nm from the origin.
Account for the theoretical conclusion that many attractive interactions between molecules vary with their separation as 1/r6.
The molar polarization of fluorobenzene vapour varies linearly with T−1, and is 70.62 cm3 mol−1 at 351.0 K and 62.47 cm3 mol−1 at 423.2 K. Calculate the polarizability and dipole moment of the molecule.
Values of the molar polarization of gaseous water at 100 kPa as determined from capacitance measurements are given below as a function of temperature. Calculate the dipole moment of H2O and its polarizability volume. T/K Pm/(cm³ mol-¹) 384.3 420.1 57.4 53.5 444.7 50.1 484.1 46.8 522.0 43.1
At 0°C, the molar polarization of liquid chlorine trifluoride is 27.18 cm3 mol−1 and its density is 1.89 g cm−3. Calculate the relative permittivity of the liquid.
Account for the hydrophobic interaction and discuss its manifestations.
The polarizability volume of H2O is 1.48 × 10−24 cm3; calculate the dipole moment of the molecule (in addition to the permanent dipole moment) induced by an applied electric field of strength 1.0 kV cm−1.
The refractive index of CH2I2 is 1.732 for 656 nm light. Its density at 20°C is 3.32 g cm−3. Calculate the polarizability of the molecule at this wavelength.
The polarizability volume of H2O at optical frequencies is 1.5 × 10−24 cm3: estimate the refractive index of water. The experimental value is 1.33; what may be the origin of the discrepancy?
The dipole moment of chlorobenzene is 1.57 D and its polarizability volume is 1.23 × 10−23 cm3. Estimate its relative permittivity at 25°C, when its density is 1.173 g cm−3.
Calculate the vapour pressure of a spherical droplet of water of radius 10 nm at 20°C. The vapour pressure of bulk water at that temperature is 2.3 kPa and its density is 0.9982 g cm−3.
Calculate the potential energy of the interaction between two linear quadrupoles when they are(a) Collinear,(b) Parallel and separated by a distance r.
The contact angle for water on clean glass is close to zero. Calculate the surface tension of water at 20°C given that at that temperature water climbs to a height of 4.96 cm in a clean glass capillary tube of internal radius 0.300 mm. The density of water at 20°C is 998.2 kg m−3.
Calculate the pressure differential of water across the surface of a spherical droplet of radius 200 nm at 20°C.
Suppose the repulsive term in a Lennard-Jones (12,6)-potential is replaced by an exponential function of the form e−r/d. Sketch the form of the potential energy and locate the distance at which it is a minimum.
Consider the collision between a hard-sphere molecule of radius R1 and mass m, and an infinitely massive impenetrable sphere of radius R2. Plot the scattering angle θ as a function of the impact parameter b. Carry out the calculation using simple geometrical considerations.
Calculate the number-average molar mass and the mass-average molar mass of a mixture of equal amounts of two polymers, one having M = 62 kg mol−1 and the other M = 78 kg mol−1.
Suggest reasons why the techniques described in the preceding question produce different mass averages.
The times of flow of dilute solutions of polystyrene in benzene through a viscometer at 25°C are given in the table below. From these data, calculate the molar mass of the polystyrene samples. Since the solutions are dilute, assume that the densities of the solutions are the same as those of pure
The radius of gyration of a long chain molecule is found to be 7.3 nm. The chain consists of C-C links. Assume the chain is randomly coiled and estimate the number of links in the chain.
Calculate the speed of operation (in r.p.m.) of an ultracentrifuge needed to obtain a readily measurable concentration gradient in a sedimentation equilibrium experiment. Take that gradient to be a concentration at the bottom of the cell about five times greater than at the top. Use rtop = 5.0 cm,
A solution consists of solvent, 30 per cent by mass, of a dimer with M = 30 kg mol−1 and its monomer. What average molar mass would be obtained from measurement of(a) Osmotic pressure,(b) Light scattering?
Identify the terms in and limit the generality of the following expressions:(a) ∆S = − 1/2 kN ln{(1 + ν)1+ν(1 − ν)1−ν},(b) Rrms = (2N)1/2l, and(c) Rg = (N/6)1/2l.
What is the relative rate of sedimentation for two spherical particles of the same density, but which differ in radius by a factor of 10?
The concentration dependence of the osmotic pressure of solutions of a macromolecule at 20°C was found to be as follows: Determine the molar mass of the macromolecule and the osmotic virial coefficient. c/(g dm-3) П/Pa 1.21 134 2.72 321 5.08 655 6.60 898
It is observed that the critical micelle concentration of sodium dodecyl sulfate in aqueous solution decreases as the concentration of added sodium chloride increases. Explain this effect.
Human haemoglobin has a specific volume of 0.749 × 10−3 m3 kg−1, a sedimentation constant of 4.48 Sv, and a diffusion coefficient of 6.9 × 10−11 m2 s−1. Determine its molar mass from this information.
Find the drift speed of a particle of radius 20 µm and density 1750 kg m−3 which is settling from suspension in water (density 1000 kg m−3) under the influence of gravity alone. The viscosity of water is 8.9 × 10−4 kg m−1 s−1.
The rate of sedimentation of a recently isolated protein was monitored at 20°C and with a rotor speed of 50 000 r.p.m. The boundary receded as follows: Calculate the sedimentation constant and the molar mass of the protein on the basis that its partial specific volume is 0.728 cm3 g−1 and
Discuss the physical origins of the surface Gibbs energy.
For some proteins, the isoelectric point must be obtained by extrapolation because the macromolecule might not be stable over a very wide pH range. Estimate the pH of the isoelectric point from the following data for a protein: pH Drift speed/(um s-¹) 4.5 -0.10 5.0 -0.20 5.5 -0.30 6.0 -0.35
At 20°C the diffusion coefficient of a macromolecule is found to be 8.3 × 10−11 m2 s−1. Its sedimentation constant is 3.2 Sv in a solution of density 1.06 g cm−3. The specific volume of the macromolecule is 0.656 cm3 g−1. Determine the molar mass of the macromolecule.
The data from a sedimentation equilibrium experiment performed at 300 K on a macromolecular solute in aqueous solution show that a graph of ln c against r2 is a straight line with a slope of 729 cm−2. The rotational rate of the centrifuge was 50 000 r.p.m. The specific volume of the solute is
Calculate the radial acceleration (as so many g) in a cell placed at 6.0 cm from the centre of rotation in an ultracentrifuge operating at 80 000 r.p.m.
A polymer chain consists of 700 segments, each 0.90 nm long. If the chain were ideally flexible, what would be the r.m.s. separation of the ends of the chain?
Calculate the contour length (the length of the extended chain) and the root mean square separation (the end-to-end distance) for polyethylene with a molar mass of 280 kg mol−1.
Construct a two-dimensional random walk by using a random number generating routine with mathematical software or electronic spreadsheet. Construct a walk of 50 and 100 steps. If there are many people working on the problem, investigate the mean and most probable separations in the plots by direct
Equivalent lattice points within the unit cell of a Bravais lattice have identical surroundings. What points within a face-centred cubic unit cell are equivalent to the point (1/2,0,0)?
Describe the procedure for identifying the type and size of a cubic unit cell.
Find the Miller indices of the planes that intersect the crystallographic axes at the distances (2a, 3b, 2c) and (2a, 2b, ∞c).
Calculate the separations of the planes {111}, {211}, and {100} in a crystal in which the cubic unit cell has side 432 pm.
Describe the phase problem and explain how it may be overcome.
The glancing angle of a Bragg reflection from a set of crystal planes separated by 99.3 pm is 20.85°. Calculate the wavelength of the X-rays.
Describe the caesium-chloride and rock-salt structures in terms of the occupation of holes in expanded close-packed lattices.
Copper Kα radiation consists of two components of wavelengths 154.433 pm and 154.051 pm. Calculate the separation of the diffraction lines arising from the two components in a powder diffraction pattern recorded in a circular camera of radius 5.74 cm (with the sample at the centre) from planes of
Genuine pearls consist of concentric layers of calcite crystals (CaCO3) in which the trigonal axes are oriented along the radii. The nucleus of a cultured pearl is a piece of mother-of-pearl that has been worked into a sphere on a lathe. The oyster then deposits concentric layers of calcite on the
Explain how X-ray diffraction can be used to determine the absolute configuration of molecules.
The compound Rb3TlF6 has a tetragonal unit cell with dimensions a = 651 pm and c = 934 pm. Calculate the volume of the unit cell.
Explain how metallic conductors and semiconductors are identified and explain their electrical and optical properties in terms of band theory.
The orthorhombic unit cell of NiSO4 has the dimensions a = 634 pm, b = 784 pm, and c = 516 pm, and the density of the solid is estimated as 3.9 g cm−3. Determine the number of formula units per unit cell and calculate a more precise value of the density.
Calculate the coefficient of thermal expansion of diamond given that the (111) reflection shifts from 22° 2′ 25″ to 21° 57′ 59″ on heating a crystal from 100 K to 300 K and 154.0562 pm X-rays are used.
Describe the characteristics of the Fermi–Dirac distribution. Why is it appropriate to call the parameter µ a chemical potential?
The unit cells of SbCl3 are orthorhombic with dimensions a = 812 pm, b = 947 pm, and c = 637 pm. Calculate the spacing, d, of the (411) planes.
To what extent are the electric and magnetic properties of molecules analogous? How do they differ?
The volume of a monoclinic unit cell is abc sin β. Naphthalene has a monoclinic unit cell with two molecules per cell and sides in the ratio 1.377:1:1.436. The angle β is 122° 49′ and the density of the solid is 1.152 g cm−3. Calculate the dimensions of the cell.
Potassium nitrate crystals have orthorhombic unit cells of dimensions a = 542 pm, b = 917 pm, and c = 645 pm. Calculate the glancing angles for the (100), (010), and (111) reflections using CuKα radiation (154 pm).
Construct the electron density along the x-axis of a crystal given the following structure factors: h 0 F₁ +30.0 h 10 11 12 13 14 15 F₁ +6.5 +5.2 -4.3 -1.2 +0.1 +2.1 1 2 +8.2 +6.5 3 4 5 6 7 +4.1 8 9 +5.5 -2.4 +5.4 +3.2 -1.0 +1.1
The coordinates, in units of a, of the atoms in a body-centred cubic lattice are (0,0,0), (0,1,0), (0,0,1), (0,1,1), (1,0,0), (1,1,0), (1,0,1), and (1,1,1). Calculate the structure factors Fhkl when all the atoms are identical.
Calculate the packing fraction for close-packed cylinders.
Verify that the radius ratios for sixfold coordination is 0.414.
Aided by the Born–Mayer equation for the lattice enthalpy and a Born–Haber cycle, show that formation of CaCl is an exothermic process (the sublimation enthalpy of Ca(s) is 176 kJ mol−1). Show that an explanation for the nonexistence of CaCl can be found in the reaction enthalpy for the
Calculate the atomic packing factor for diamond.
Is there an expansion or a contraction as titanium transforms from hcp to body-centred cubic? The atomic radius of titanium is 145.8 pm in hcp but 142.5 pm in bcc.
In a Patterson synthesis, the spots correspond to the lengths and directions of the vectors joining the atoms in a unit cell. Sketch the pattern that would be obtained for a planar, triangular isolated BF3 molecule.
What velocity should neutrons have if they are to have wavelength 50 pm?
Lead has Tc = 7.19 K and Hc = 63 901 A m−1. At what temperature does lead become superconducting in a magnetic field of 20 kA m−1?
Cotton consists of the polymer cellulose, which is a linear chain of glucose molecules. The chains are held together by hydrogen bonding. When a cotton shirt is ironed, it is first moistened, then heated under pressure. Explain this process.
Young’s modulus for polyethylene at room temperature is 1.2 GPa. What strain will be produced when a mass of 1.0 kg is suspended from a polyethylene thread of diameter 1.0 mm?
Poisson’s ratio for polyethylene is 0.45. What change in volume takes place when a cube of polyethylene of volume 1.0 cm3 is subjected to a uniaxial stress that produces a strain of 1.0 per cent?
Calculate the packing fractions of(a) A primitive cubic lattice,(b) A bcc unit cell,(c) An fcc unit cell.
Is arsenic-doped germanium a p-type or n-type semiconductor?

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