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chemistry
physical chemistry

Physical Chemistry 7th Edition Peter Atkins - Solutions

Equivalent lattice points within the unit cell of a Brava is lattice have identical surroundings. What points within a body-centred cubic unit cell are equivalent to the point (1/2, 0, 1/2)?
Find the Miller indices of the planes that intersect the crystallographic axes at the distances (1a, 3b, -c) and (2a, 3b, 4c).
Calculate the separations of the planes {121}, {221l, and {244} in a crystal in which the cubic unit cell has side 523 pm.
The glancing angle of a Bragg reflection from a set of crystal planes separated by 128.2 pm is 19.76°. Calculate the wavelength of the X-rays.
What are the values of 28 of the first three diffraction lines of FCC gold {atomic radius 144 pm) when the X-ray wavelength is 154 pm?
A synchrotron source produces X-radiation at a range of wavelengths. Consider two components of wavelengths 95.401 and 96.035 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
Calculate the volume of the hexagonal unit cell of sodium nitrate, for which the dimensions are a = 1692.9 pm and c= 506.96 pm.
An orthorhombic unit cell of a compound of molar mass 135.01 g mol-l has the dimensions a = 589 pm, b = 822 pm, and c= 798 pm. The density of the solid is estimated as 2.9 g cm-1 Determine the number of formula units per unit cell and calculate a more precise value of the density.
An orthorhombic unit cell has dimensions a = 679 pm, b = 879 pm, and c= 860 pm. Calculate the spacing, d, of the (322) planes.
A substance known to have a cubic unit cell gives reflections with radiation of wavelength 137 pm at the glancing angles 10.7°, 13.6°, 17.7°, and 21.9°. The reflection at 17.7" is known to be due to the (111) planes. Index the other reflections.
Calcium carbonate crystals in the form of aragonite have orthorhombic unit cells of dimensions a= 574.1 pm, b = 796.8 pm, and c= 495.9 pm. Calculate the glancing angles for the (100), (010), and (111) reflections using radiation of wavelength 83.42 pm (from aluminum).
A powder diffraction photograph from tungsten shows lines that index as (110), (200), (211), (220), (310), (222), (321), (400), ... Identify the (Brava is) lattice type of the unit cell.
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), (1,1,1), and (+,+,+). Calculate the structure factors Fhkl when all the atoms are identical.
Calculate the packing fraction for equilateral triangular rods stacked as shown in 2.
Verify that the radius ratios for eightfold coordination are 0.732.
From the data in Table 20.3 determine the radius of the smallest cation that can have(a) Six fold and(b) Eightfold coordination with the K+ ion.
Calculate the atomic packing factor for a side-centred (C) cubic unit cell.
Is there an expansion or a contraction as iron transforms from hep to bcc? The atomic radius of iron is 126 pm in hcp but 122 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 from the C atoms in an isolated benzene molecule.
Calculate the wavelength of neutrons that have reached thermal equilibrium by collision with a moderator at 300 K.
Calculate the lattice enthalpy of Mgbr, from the following data:
Sections of the solid fuel rocket boosters of the space shuttle Challenger were sealed together with O-ring rubber seals of circumference 11 m. These seals failed at O°C, a temperature well above the crystallization temperature of the rubber. Speculate on why the failure occurred.
Young's modulus for iron at room temperature is 215 GPa what strain will be produced when a mass of 10.0 kg is suspended from an iron wire of diameter 0.10 mm?
Poisson's ratio for lead is 0.41. What change in volume takes place when a cube of lead of volume 1.0 dm3 is subjected to an uneasily stress that produces a strain of 2.0 per cent?
The band gap in silicon is 1.12 eV. Calculate the minimum frequency of electromagnetic radiation that results in promotion of electrons from the valence to the conduction band.
The magnetic moment of MN in its complexes is typically 5.3, uB How many unpaired electrons does the ion possess?
Calculate the molar susceptibility of cyclohexane given that its volume susceptibility is -7.9 x 10-7 and its density 811 kg m-3 at 25°C.
Predict the molar susceptibility of nitrogen dioxide at 298 K. Why does the molar susceptibility of a sample of nitrogen dioxide gas decrease as it is compressed?
Data on a single crystal of NiS047H20 give Xm= 6.00 X 10-8 m3 mol-l at 298 K. Determine the effective number of unpaired electrons in this compound and compare your result with the theoretical value.
Estimate the spin-only molar susceptibilityofMnS04·4H20 at 298 K.
Estimate the ratio of populations of the Ms states of a system with S = I in 15.0 T at 298 K.
In the early days of X-ray crystallography there was an urgent need to know the wavelengths of X-rays. One technique was to measure the diffraction angle from a mechanically ruled grating. Another method was to estimate the separation of Justice Planes from the measured density of a crystal. The
The unit cell dimensions of NaCI, KCI, NaBr, and KBr, all of which crystallize in face-centred cubic lattices, are 562.8 pm, 627.7 pm, 596.2 pm, and 658.6 pm, respectively. In each case, anion and cation are in contact along an edge of the unit cell. Do the data support the contention that ionic
Elemental silver reflects X-rays of wavelength 154.18 pm at angles of 19.076°, 22.171° and 32.256°. However, there are no other reflections at angles of less than 33°. Assuming a cubic unit cell, determine its type and dimension. Calculate the density of silver.
In their book X-rays and crystal structures (which begins 'It is now two years since Dr. Laue conceived the idea ... ') the Braggs give a number of simple examples of X-ray analysis. For instance, they report that the reflection from (100) planes in KCI occurs at 5° 23', but for NaCI it occurs at
The carbon-carbon bond length in diamond is 154.45 pm. If diamond were considered to be a close-packed structure of hard spheres with radii equal to half the bond length, what would be its expected density? The diamond lattice is face-centred cubic and its actual density is 3.516 g cm-1, Can you
The structures of crystalline macromolecules may be determined by X-ray diffraction techniques by methods similar to those for smaller molecules. Fully crystalline polyethylene has its chains aligned in an orthorhombic unit cell of dimensions 740 pm X 493 pm X 253 pm. There is 1:\0 repeating CH2CH2
The scattering of electrons or neutrons from a pair of nuclei separated by a distance Rij and orientated at a definite angle to the incident beam can be calculated. When the molecule consists of a number of atoms, we sum over the contribution from all pairs, and find that the total intensity has an
D. Sellmann, M.W. Wemple, W. Dona Bauer, and F.W. Heinemann (Inorg. Chem. 36, 1397 (1997)) describe the synthesis and reactivity of the ruthenium nitride compound [N(C4H9)4] [Ru(N)(S2C6H4)2J. The ruthenium complex anion has the 1:'1'0 1,2-benzenedithiolate legends (3) at the base of a rectangular
In an intrinsic semiconductor, the band gap is so small that the Fermi-Dirac distribution results in some electrons populating the conduction band. It follows from the exponential form of the Fermi-Dirac distribution that the conductance G, the inverse of the resistance (with units of siemens, 1 S
J.J Dannenberg, D. Leotard, P. Hal Vick, and I.C. Rayez (J. Phys. Chem. 100, 9631 (996)) carried out theoretical studies of organic molecules consisting of chains of unsaturated four-membered rings. The calculations suggest that such compounds have large numbers of unpaired spins, and that they
P.G. Radaelli, M. Maurizio, M. Perroux, S. de Brion, J, L. Tholence, Q. Huang, and A. Santoro (Science 265,380 (994)) report the synthesis and structure of a material that becomes superconducting at temperatures below 45 K. The compound is based on a layered compound Hg2 Ba2 YCu2 Oso which has a
Show that the volume of a triclinic unit cell of sides a, b, and c and angles a, B3, and y is V= abc (1- cos2a- cos2B – cos2y+ 2 cos a cos B cos y)1/2 Use this expression to derive expressions for monoclinic and orthorhombic unit cells. For the derivation, it may be helpful to use the result from
The coordinates of the four I atoms in the unit cell ofKI04 are (0,0,0), (O, ½ , ½ ), (½, ½, ½, ), (½,O,¾)· By calculating the phase of the I reflection in the structure factor, show that the I atoms contribute no net intensity to the (114) reflection.
The treatment in Problem 20.28 applies only to one-dimensional solids. In three dimensions, the variation of density of states is more like that shown in Fig. 20.70. Account for the fact that in a three-dimensional solid the greatest density of states lies near the centre of the band and the lowest
An NO molecule has thermally accessible electronically excited states. It also has an unpaired electron, and so may be expected to be paramagnetic. However, its ground state is not paramagnetic because the magnetic moment of the orbital motion of the unpaired electron almost exactly cancels the
What features in an X-ray diffraction pattern suggest a helical conformation for a biological macromolecule? Use Fig. 20.26 to deduce as much quantitative information as you can about the shape and size of a DNA molecule.
The tip of a scanning tunneling microscope can be used to move atoms on a surface. The movement of atoms and ions depends on their ability to leave one position and stick to another, and therefore on the energy changes that occur. As an illustration, consider a two-dimensional square lattice of
The viscosities of solutions of polyisobutylene in benzene were measured at 24Â°C (the θ temperature for the system) with the following results: Use the information in Table 19.4 to deduce the molar mass of the polymer.
Standard polystyrene solutions of known average molar masses continue to be used as for the calibration of many methods of characterizing polymer solutions. M. Kolinsky and J. Janca (J. Polym. Sci, Polym Chem 12, 1181 (1974)) studied polystyrene in tetrahydrofuran (THF) for use in calibrating a gel
On the assumption that the tension required to keep a sample at a constant length is proportional to the temperature (t = aT, the analogue of p ∞ T), show that the tension can be ascribed to the dependence of the entropy on the length of the sample. Account for this result in terms of the
Radius of gyration is defined in eqn 19.32. Show that an equivalent definition is that Rg is the average root mean square distance of the atoms or groups (all assumed to be of the same mass), that is, that R g = (1/N)Σj RJ, where Rj is the distance of atom j from the centre of mass.
The pure rotational microwave spectrum of HCI has absorption lines at the following wave numbers (in cm-1): 21.19, 42.37, 63.56, 84.75, 105.93, 127.12148.31169.49,190.68,211.87,233.06,254.24, 275.43, 296.62, 317.80, 338.99,360.18,381.36,402.55,423.74,444.92,466.11, 487.30, 508.48. Calculate the
For H2 at very low temperatures, only translational motion contributes to the heat capacity. At temperatures above eR = luBlk, the rotational contribution to the heat capacity becomes significant. At still higher temperatures, above ev = hulk, the vibrations contribute. But at this latter
Show that the mean interaction energy of N atoms of diameter d interacting with a potential energy of the form C61R6 is given by U=-2NzC6/3Vd3, where V is the volume in which the molecules are confined and all effects of clustering are ignored. Hence, find a connection between the van der Waals
Find the drift speed of a particle of radius 15.5m and density 1250 kg m-1 which is settling from suspension in water (density 1000 kg m-1) under the influence of gravity alone. The viscosity of water is 8.9 x 10-4 kg m-I S-1.
Calculate (a) The thermal wavelength, (b) The translational partition function of an Ar atom in a cubic box of side 1.00 cm at (i) 300 K and (ii) 3000 K.
Use the kinetic theory to justify the following observations: (a) The rate of a reaction in the gas phase depends on the energy with which two molecules collide, which in turn depends on their speeds; (b) In the Earth's atmosphere, light gases, such as H, and He, are rare but heavier gases, such as
Provide a molecular interpretation for the observation that the viscosity of a gas increases with temperature, whereas the viscosity of a liquid decreases with increasing temperature.
Limit the generality of the following expressions:(a) J =-D (dc/dx),(b) D = kT/I, and(c) D = kT/6πηa.
Discuss how nuclear magnetic resonance spectroscopy, inelastic neutron scattering, and dynamic light scattering may be used to measure the mobility of molecules in liquids.
Determine the ratios of (a) The mean speeds, (b) The mean kinetic energies of His atoms and Hg atoms at 25°e.
The best laboratory vacuum pump can generate a vacuum of about 1 n Torr. At 25°C and assuming that air consists of N, molecules with a collision diameter of395 pm, calculate (a) The mean speed of the molecules,(b) The mean free path, (c) The collision frequency in the gas.
At what pressure does the mean free path of argon at 25°C become comparable to the diameters of the atoms themselves?
At an altitude of 15 km the temperature is 217 K and the pressure 12.1 kPa. What is the mean free path of N, molecules? (a= 0.43 NM2)
How many collisions per second does an N2 molecule make at an altitude of 15 km? (See Exercise 21.4b for data.)
Calculate the mean free path of carbon dioxide molecules using a= 0.52 nm at 25°C and (a) 15 atm, (b) 1.0 bar, (c) 1.0 Torr.
Use the Maxwell distribution of speeds to estimate the fraction of CO, molecules at 300 K that have speeds in the range 200 to 250 m S-l
A solid surface with dimensions 3.5 mm x 4.0 cm is exposed to helium gas at III Pa and 1500 K. How many collisions do the He atoms make with this surface in 10 s?
An effusion cell has a circular hole of diameter 3.00 mm. If the molar mass of the solid in the cell is 300 g mol-1 and its vapour pressure is 0.224 Pa at 450 K, by how much will the mass of the solid decrease in a period of24.00 h?
A manometer was connected to a bulb containing nitrogen under slight pressure. The gas was allowed to escape through a small pinhole, and the time for the manometer reading to drop from 65.1 cm to 42.1 cm was 18.5 s. When the experiment was repeated using a fluorocarbon gas, the same fall took
A container of internal volume 22.0 m3 was punctured, and a hole of radius 0.050 mm was formed. If the nitrogen pressure within the vehicle is initially 122 kPa and its temperature 293 K, how long will the pressure take to fall to 105 kPa?
Calculate the flux of energy arising from a temperature gradient of 3.5 K m-I in a sample of hydrogen in which the mean temperature is 260 K.
Use the experimental value of the thermal conductivity of nitrogen (Table 21.2) to estimate the collision cross-section of N, molecules at 298 K.
Two sheets of copper of area 1.50 m are separated by 10.0 cm. What is the rate of transfer of heat by conduction from the warm sheet (50°e) to the cold sheet (-10°C). What is the rate of loss of heat?
Use the experimental value of the coefficient of viscosity for nitrogen (Table 21.2) to estimate the collision cross-section of the molecules at 273 K.
Calculate the inlet pressure required to maintain a flow rate of 8.70 cm3 S-I of nitrogen at 300 K flowing through a pipe of length 10.5 m and diameter 15 mm. The pressure of gas as it leaves the tube is 1.00 bars the volume of the gas is measured at that pressure.
Calculate the viscosity of benzene vapour at(a) 273 K,(b) 298 K,(c) 1000 K. Take a~ 0.88 nM-1.
Calculate the thermal conductivities of(a) Neon,(b) Nitrogen at 300 K and 15 mbar. Each gas is confined in a cubic vessel of side 15 cm, one wall being at 305 K and the one opposite at 295 K. What is the rate of flow of energy as heat from one wall to the other in each case?
The viscosity of a chlorofluorocarbon (CFC) was measured by comparing its rate of flow through a long narrow tube (using Poiseuille formula) with that of argon. For the same pressure differential, the same volume of the CFC passed through the tube in 72.0 s as argon in 18.0 s. The viscosity of
Calculate the thermal conductivity of nitrogen (C; m = 20.8 J K-1 mol-1, a= 0.43 nm3) at room temperature (20°C).
Calculate the diffusion constant of nitrogen at 25°C and(a) 10,0 Pa,(b) 100 kPa,(c) 15.0 MPa. If a pressure gradient 01'0.20 bar m-1 is established in a pipe, what is the flow of gas due to diffusion?
The mobility of an acetate ion in aqueous solution at 25°C is 4.24 x 10-8 m3 S-1 V-1. Calculate the molar ionic conductivity.
The mobility of a u- ion in aqueous solution is 4.01 x 10-8 m2 S-1 V-1 at 25°C. The potential difference between two electrodes placed in the solution is 12.0 V. If the electrodes are 1.00 cm apart, what is the drift speed of the ion?
What fraction of the total current is carried by er when current flows through an aqueous solution of NaCI at 25°C?
The limiting molar conductivities of NaI, NaCH3CO" and Mg(CH3C02)2 are 12.69 mS m2 mol-1, 9.10 mS m2 mol-1, and 18.78 mS m2 mol-1, respectively (all at 25°C). What is the limiting molar conductivity of MgI2 at this temperature?
At 25°C the molar ionic conductivities of F3, er, and Bc are 5.54 mS m2 mol-1, 7.635 mS m2 mol-1, and 7.81 mS m2 mol-1, respectively. What are their mobilities?
The mobility of a CH1COi ion in aqueous solution at 25°C is 4.24 x 10-8 m2 S-1 V-1 Calculate its diffusion coefficient in water at 25°C.
The diffusion coefficient of I in hexane at 25°C is 4.05 x 10-9 m2 S-1. Estimate the time required for an iodine molecule to have a root mean square displacement of 1.0 cm.
Estimate the effective radius of a glycine molecule in water at 25°C given that its diffusion coefficient is 1.055 X 10-9 m2 S-1 and that the viscosity of water is 1.00 cP.
The diffusion coefficient for CCl4 in heptane is 3.17 X 10-9 m2 S-1' How long does a molecule take to jump through about one molecular diameter (approximately the fundamental jump length for translational motion)?
About how long, on average, does it take for the molecules in Exercise 21.31a to drift to a point?(a) 1.0 mm,(b) 1.0 cm from their starting points
Instead of the arrangement in Fig. 21.8, the speed of molecules can also be measured with a rotating slotted-disc apparatus, which consists of five coaxial 5.0 cm diameter discs separated by 1.0 cm, the slots in their rims being displaced by 2.0° between neighbours. The relative intensities, I, of
A population consists of people of the following heights (in meters, numbers of individuals in brackets): 1.80 (1), 1.82 (2), 1.84 (4), 1.86 (7), 1.88 (10), 1.90 (15), 1.92 (9), 1.94 (4), 1.96 (0), 1.98 (1). What are? (a) The mean height, (b) The root mean square height of the population?
A Knudsen cell was used to determine the vapour pressure of germanium at 1000°C. During an interval of7200 s the mass loss through a hole of radius 0.50 mm amounted to 43 ug, what is the vapour pressure of germanium at 1000°C? Assume the gas to be monatomic.
An atomic beam is designed to function with (a) Cadmium, (b) Mercury. The source is an oven maintained at 380 K, there being a small slit of dimensions 1.0 cm x 1.0 x 10-3 cm. The vapour pressure of cadmium is 0.13 Pa and that of mercury is 12 Pa at this temperature. What is the atomic current (the
The resistances of a series of aqueous NaCI solutions, formed by successive dilution of a sample, were measured in a cell with cell constant (the constant C in the relation K= C/R) equal to 0.2063 cm-I The following values were found: Verify that the molar conductivity follows the Kohlrausch law
What are the drift speeds of Li+ Na+ and K+ in water when a potential difference of 10V is applied across a 100-cm conductivity cell? How long would it take an ion to move from one electrode to the other? In conductivity measurements it is normal to use alternating current: what are the
In a moving boundary experiment on KC] the apparatus consisted of a tube of internal diameter 4.146 mm, and it contained aqueous K Cl at a concentration of 0.021 mol dm>'. A steady current of 18.2 mA was passed, and the boundary advanced as follows: Find the transport number of K its mobility,
A dilute solution of potassium permanganate in water at 25°C was prepared. The solution was in a horizontal tube of length 10 cm, and at first there was a linear gradation of intensity of the purple solution from the left (where the concentration was 0.100 mol dm-1) to the right (where the

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