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chemistry
physical chemistry

Physical Chemistry 7th Edition Peter Atkins - Solutions

Planets lose their atmospheres over time unless they are replenished. A complete analysis of the overall process is very complicated and depends upon the radius of the planet, temperature, atmospheric composition, and other factors. Prove that the atmosphere of planets cannot be in an equilibrium
Consider a protein P with four distinct sites, with each site capable of binding one legend 1. Show that the possible varieties (configurations) of the species PL; (with P Lo denoting P) are given by the binomial coefficients C(4,i).
Discuss the limitations of the expressions qR = kT/heB, qY = kT/hev and qE =gE
Explain the origin of residual entropy.
One of the excited states of the C2 molecule has the valence electron configuration 1σ2g 1σ 2u 1π3u 1π1g. Give the multiplicity and parity of the term.
The electronic spectrum of the 1Br molecule shows two low-lying, well defined convergence limits at 14660 and 18345 cm-1. Energy levels for the iodine and bromine atoms occur at 0,7598 and 0,3685 cm-1, respectively. Other atomic levels are at much higher energies. What possibilities exist for the
A transition of particular importance in 02 gives rise to the 'Schumann-Rung c band' in the ultraviolet region. The wave numbers (in cm-1) of transitions from the ground state to the vibrational levels of the first excited state (3Σ-) are 50 062.6, 50 725.4, 51369.0, 51988.6, 52579.0, 53143.4,
Assume that the electronic states of the re electrons of a conjugated molecule can be approximated by the wave functions of a particle in a one-dimensional box, and that the dipole moment can be related to the displacement along this length by μ= -ex. Show that the transition probability for the
Calculate σd for a hydrogenic atom with atomic number Z.
NMR spectroscopy may be used to determine the equilibrium constant for dissociation of a complex between a small molecule, such as an enzyme inhibitor I, and a protein, such as an enzyme E: El→ E + I K[ = [E] [IJ/[EI]
Use concepts of statistical thermodynamics to describe the molecular features that lead to the equations of state of perfect and real gases.
Use concepts of statistical thermodynamics to describe the molecular features that determine the magnitudes of equilibrium constants and their variation with temperature.
Use the equipartition theorem to estimate the constant-volume molar heat capacity of (a) 03 (b) C2H6, (c) CO2 in the gas phase at 25°e.
Estimate the value of y= Cp/Cv for carbon dioxide. 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 0, at(a) 25°C(b) 250°e.
Give the symmetry number for each of the following molecules:(a) CO2, (b) 03' (c) 503' (d) SF6, and (e) Al2C16.
Calculate the rotational partition function of S0, at 298 K from its rotational constants 2.027 36 cm3, 0.34417 cm3, and 0.293 535 cm3. Above what temperature is the high-temperature approximation valid to within 10 per cent of the true value?
From the results of Exercise 17.5b, calculate the rotational contribution to the molar entropy of sulfur dioxide at 25°e.
Calculate the rotational partition function of CH3CN (a) By direct summation of the energy levels at 298 K and 500 K, and (b) By the high temperature approximation. Take A = 5.28 cm-1 and B = 0.307 cm-1.
The NOF molecule is an asymmetric rotor with rotational constants 3.1752 cm-1, 0.3951 cm-1, and 0.3505 cm-1. Calculate the rotational partition function of the molecule at (a) 25°C, (b) 100°e
Plot the molar entropy of a collection of harmonic oscillators as a function of T/ by and predict the standard molar entropy of ethyne at(a) 298 K,(b) 500 K. For data, see the preceding exercise.
An 03 molecule is angular, and its vibrational wave numbers are 1110 cm-1, 705 cm-1, and 1042 cm-1, The rotational constants of the molecule are 3.553 cm-1, 0.4452 cm-1, and 0.3948 cm-I. Calculate the rotational and vibrational contributions to the molar Gibbs energy at 298 K.
The first electronically excited state of O is Ill. and lies 7918.1 cm-1 above the ground state, which is Calculate the electronic contribution to the molar Gibbs energy of 0, at 400 K.
The first electronically excited state of O is Ill. and lies 7918.1 cm-1 above the ground state, which is Calculate the electronic contribution to the molar Gibbs energy of 0, at 400 K.
Estimate the contribution of the spin to the molar entropy of a solid sample of a d-metal complex with S =3/2
Calculate the value of K at 298 K for the gas-phase isotopic exchange reaction 2 79Br81Br79Br79 ( 79Br79Br+ 81Br81Br the Br2 molecule has a non-degenerate ground state, with no other electronic states nearby. Base the calculation on the wave number of the vibration 079Br81Br, which is 323.33 cm-1
The NO molecule has a doubly degenerate electronic ground state and a doubly degenerate excited state at 121.1 Cm-I Calculate the electronic contribution to the molar heat capacity of the molecule at(a) 50 K,(b) 298 K, and(c) 500 K.
The energy levels of a CH3 group attached to a larger fragment are given by the expression for a particle on a ring, provided the group is rotating freely. What is the high-temperature contribution to the heat capacity and entropy of such a freely rotating group at 25°C? The moment of inertia of
J.G. Dojahn, E.C.M. Chen, and W.E. Wentworth (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homo nuclear diatomic halogen anions. The ground state ofF2 is 2L: with a fundamental vibrational wave number of 450.0 cm-1 and equilibrium
Treat carbon monoxide as a perfect gas and apply equilibrium statistical thermodynamics to the study of its properties, as specified below, in the temperature range 100-1000 K at 1 bar. V = 2169.8 cm-1, B =1.931 cm-1, and Do = 11.09 eV; neglect anharmonicity and centrifugal distortion.(a) Examine
The exchange of deuterium between acid and water is an important type of equilibrium, and we can examine it using spectroscopic data on the molecules. Calculate the equilibrium constant at(a) 298 K and(b) 800 K for the gas-phase exchange reaction HzO + DCl ( HDO + HCl from the following data:
Suppose that an intermolecular potential has a hard-sphere core of radius 'I and a shallow attractive well of uniform depth E out to a distance '2' Show, by using eqn 17.42 and the condition E« kT, that such a model is approximately consistent with a van der Waals equation of state when b« Vm,
A more formal way of arriving at the value of the symmetry number is to note that is the order (the number of elements) of the rotational subgroup of the molecule, the point group of the molecule with all but the identity and the rotations removed. The rotational subgroup of H20 is {E, C2}, so 0=
Show how the heat capacity of a linear rotor is related to the following sum:Where the E (J) are the rotational energy levels and g (J) their degeneracy's then go on to show graphically that the total contribution to the heat capacity of a linear rotor can be regarded as a sum of contributions due
Determine whether a magnetic field can influence the value of an equilibrium constant. Consider the equilibrium 12(g) ;:=' 2 I(g) at 1000 K, and calculate the ratio of equilibrium constants K( 'B)/ K, where K( 'B) is the equilibrium constant when a magnetic field 'B is present and removes the
An average human DNA molecule has 5 x 108 binucleotides (rungs on the DNA ladder) of four different kinds. If each rung were a random choice of one of these four possibilities, what would be the residual entropy associated with this typical DNA molecule?
R. Viswanathan, R.W. Schmude, Jr., and K.A. Gingerich (J. Phys. Chem. 100,10784 (1996)) studied thermodynamic properties of several boron-silicon gas-phase species experimentally and theoretically. These species can occur in the high-temperature chemical vapour deposition (CVD) of silicon-based
J. Hutter, H.P. Luthi, and F. Diederich (J. Amer. Chem. Soc. 116,750 (1994)) examined the geometric and vibrational structure of several carbon molecules of formula Cn' Given that the ground state of C3, a molecule found in interstellar space and in flames, is an angular singlet with moments of
Explain how the permanent dipole moment and the polarizability of a molecule arise.
Describe the experimental procedures available for determining the electric dipole moment of a molecule.
Describe the formation of a hydrogen bond in terms of molecular orbitals.
Describe how molecular beams are used to investigate intermolecular potentials.
Which of the following molecules may be polar 503' XeF4, 5F/
Calculate the resultant of two dipole moments of magnitude 1.5 D and 0.80 D that make an angle of 109.5 to each other.
Calculate the magnitude and direction of the dipole moment of the following arrangement of charges in the xy-plane: 4e at (0, 0), -2e at (162 pm, 0), and -2e at an angle of30° from the x-axis and a distance of 143 pm from the origin.
The molar polarization of the vapour of a compound was found to vary linearly with t-1 and is 75.74 cm-1 mol-1 at 320.0 K and 71.43 cm3 mol-1 at 421.7 K. Calculate the polarizability and dipole moment of the molecule.
At O° C, the molar polarization of a liquid is 32.16 cm3 mol-1 and its density is 1.92 g cm-3. Calculate the relative permittivity of the liquid. Take M = 55.0 g mol-1.
The polarizability volume ofNH3 is 2.22 X 10-30m3 calculate the dipole moment of the molecule (in addition to the permanent dipole moment) induced by an applied electric field of strength 15.0 kV m-1.
The refractive index of a compound is 1.622 for 643 nm light. Its density at 20°C is 2.99 g cm>'. Calculate the polarizability of the molecule at this wavelength. Take M = 65.5 g mol-1.
The polarizability volume of a liquid of molar mass 72.3 g mol-1 and density 865 kg mol-1 at optical frequencies is 2.2 x 10-30 m': estimate the refractive index of the liquid.
The dipole moment of bromobenzene is 5. 17 X 10-30 C m and its polarizability volume is approximately 1.5 x 10-29 m3. Estimate its relative permittivity at 25°C, when its density is 1491 kg m-3.
Calculate the vapour pressure of a spherical droplet of water of radius 20.0 nm at 35.0°e. The vapour pressure of bulk water at that temperature is 5.623 kPa and its density is 994.0 kg m-1.
The contact angle for water on clean glass is close to zero. Calculate the surface tension of water at 30°C given that at that temperature water climbs to a height of 9.11 cm in a clean glass capillary tube of internal radius 0.320 mm. The density of water at 30°C is 0.9956 g cm-3.
Calculate the pressure differential of ethanol across the surface of a spherical droplet of radius 220 nm at 20°e. The surface tension of ethanol at that temperature is 22.39 mN m-1
Suppose an Hp molecule (11= 1.85 D) approaches an anion. What is the favorable orientation of the molecule? Calculate the electric field (in volts per meter) experienced by the anion when the water dipole is(a) 1.0 nm,(b) 0.3 nm,(c) 30 nm from the ion.
The relative permittivity of chloroform was measured over a range of temperatures with the following results:The freezing point of chloroform is -64Â°e. Account for these results and calculate the dipole moment and polarizability volume of the molecule.
In his classic book Polar molecules, Debye reports some early measurements of the polarizability of ammonia. From the selection below, determine the dipole moment and the polarizability volume of the molecule.
F. Luo, G.C. MeBane, 0. Kim, C.F. Giese, and W.R. Gentry (J. Chem Phys. 98,3564 (1993) reported experimental observation of the He2 complex, a species that had escaped detection for a long time. The fact that the observation required temperatures in the neighbourhood of 1mK is consistent with
From data in Table 18.1 calculate the molar polarization, relative permittivity, and refractive index of methanol at 20°e. Its density at that temperature is 0.7914 g cm-1,
Show that, in a gas (for which the refractive index is close to I), the refractive index depends on the pressure as nr = 1+ const X p, and find the constant of proportionality. Go on to show how to deduce the polarizability volume of a molecule from measurements of the refractive index of a gaseous
The cohesive energy density, V ,is defined as U/V, where U is the mean potential energy of attraction within the sample and V its volume. Show that v= ½ N fV(R)dr, where 91<': is the number density of the molecules and VCR) is their attractive potential energy and where the integration ranges
The dependence of the scattering characteristics of atoms on the energy of the collision can be modeled as follows. We suppose that the two colliding atoms behave as impenetrable spheres, as in Problem 18.16, but that the effective radius of the heavy atoms depends on the speed v of the light atom.
The dependence of the scattering characteristics of atoms on the energy of the collision can be modeled as follows. We suppose that the two colliding atoms behave as impenetrable spheres, as in Problem 18.16, but that the effective radius of the heavy atoms depends on the speed v of the light atom.
Molecular orbital calculations may be used to predict the dipole moments of molecules.(a) Using molecular modeling software and the computational method of your choice, calculate the dipole moment of the peptide link, modeled as a trans-N-methylacetamide (18). Plot the energy of interaction between
Derivatives of the compound TIBO (20) inhibit the enzyme reverse transcriptase, which catalyses the conversion of retroviral RNA to DNA. A QSAR analysis of the activity A of a number of TIBO derivatives suggests the following equation: log A = bv + bvS + b2W where 5 is a parameter related to the
Distinguish between number-average, weight-average, and Z-average molar masses. Discuss experimental techniques that can measure each of these properties.
Distinguish between contour length, root mean square separation, and radius of gyration of a random coil.
Distinguish between molecular mechanics and molecular dynamics calculations. Why are these methods generally more popular in the field of polymer chemistry than the quantum mechanical procedures discussed in Chapter 11?
Self-assembled monolayers (SAMs) are receiving more attention than Longmuir-Blodgett (LB) films as starting points for nanofabrication. How do SAMs differ from LB films and why are SAMs more useful than LB films in nanofabrication work?
Explain the physical origins of surface activity by surfactant molecules.
Calculate the number-average molar mass and the mass-average molar mass of a mixture of two polymers, one having M = 62 kg mol-1 and the other M = 78 kg mol-1, with their amounts (numbers of moles) in the ratio 3:2.
The radius of gyration of a long chain molecule is found to be 18.9 nm. The chain consists of links of length 450 pm. Assume the chain is randomly coiled and estimates the number of links in the chain.
A solution consists of 25 per cent by mass of a trimmer with M = 22 kg mol-1 and its monomer. What average molar mass would be obtained from measurement of: (a) Osmotic pressure, (b) Light scattering?
Evaluate the rotational correlation time, 'R = 4na31]/3kT, for a synthetic polymer in water at 20°C on the basis that it is a sphere of radius 4.5 nm.
What is the relative rate of sedimentation for two spherical particles with densities 1.10 g cm and 1.18 g cm and which differ in radius by a factor of8.4, the former being the larger? Use p = 0.794 g cm-1 for the density of the solution.
A synthetic polymer has a specific volume of 8.01 x 10-4 m kg-1, a sedimentation constant of7.46 Sv, and a diffusion coefficient of 7.72 x 10-11 m2 S-1 Determine its molar mass from this information.
At 20°C the diffusion coefficient of a macromolecule is found to be 7.9 X 10-11 m2 S-l its sedimentation constant is 5.1 Sv in a solution of density 997 kg m-1, the specific volume of the macromolecule is 0.721 cm-1 g-1. Determine the molar mass of the macromolecule.
The data from a sedimentation equilibrium experiment performed at 293 K on a macromolecular solute in aqueous solution show that a graph of in c against (r/cm) 2 is a straight line with a slope of821. The rotation rate of the centrifuge was 1080 Hz. The specific volume of the solute is 7.2 x 10-4
Calculate the radial acceleration (as so many g) in a cell placed at 5.50 cm from the centre of rotation in an ultracentrifuge operating at 1.32 kHz.
A polymer chain consists of 1200 segments, each 1.125 nm long. If the chain were ideally flexible, what would be the Lm 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 polypropylene of molar mass 174 kg mol-1.
In a sedimentation experiment the position of the boundary as a function of time was found to be as follows:T/min 15.529.136.458.2R/cm 5.055.095.125.19The rotation rate of the centrifuge was 45000 r. p. m. Calculate the sedimentation constant of the solute.
The concentration dependence of the viscosity of a polymer solution is found to be as follows:
The osmotic pressure of a fraction of poly (vinyl chloride) in a ketone solvent was measured at 25Â°C. The density of the solvent (which is virtually equal to the density of the solution) was 0.798 g cm3. Calculate the molar mass and the osmotic virial coefficient, B, of the fraction from the
In formamide as solvent, poly (y-benzyl-L-glutamate) is found by light scattering experiments to have a radius of gyration proportional to M; in contrast, polystyrene in butanone has Rg proportional to M1/2. Present arguments to show that the first polymer is a rigid rod whereas the second is a
A polymerization process produced a Gaussian distribution of polymers in the sense that the proportion of molecules having a molar mass in the range M to M + dM was proportional to e-(M-M)2/2YdM. What is the number average molar mass when the distribution is narrow?
Use eqn 19.27 to deduce expressions for(a) The root mean square separation of the ends of the chain,(b) The mean separation of the ends, and(c) Their most probable separation. Evaluate these three quantities for a fully flexible chain with N = 4000 and 1= 154 pm
Evaluate the radius of gyration, Rg, of (a) A solid sphere of radius a,(b) A long straight rod of radius a and length I. Show that in the case of a solid sphere of specific volume v" Rg/nm = 0.056902 x {(vJcm3 g-I) (M/g mol-1)} 1/3. Evaluate Rg for a species with M = 100 kg mol-1, v, = 0.750 cm3
Calculate the excluded volume in terms of the molecular volume on the basis that the molecules are spheres of radius a. Evaluate the osmotic virial coefficient in the case of bushy stunt virus, a = 14.0 mm, and hemoglobin, a = 3.2nm (see Problem 19.18). Evaluate the percentage deviation of the
Suppose that a rod-like DNA molecule of length 250 nm undergoes a conformational change to a closed-circular (cc) form.(a) Use the information in Problem 19.24 and an incident wavelength x = 488 nm to calculate the ratio of scattering intensities by each of these conformations, Imd/1ce' when 8=
Sedimentation studies on haemoglobin in water gave a sedimentation constant S = 4.5 Sv at 20°C. The diffusion coefficient is 6.3 x 10-11 m2 S-1 at the same temperature. Calculate the molar mass of haemoglobin using Vs = 0.75 cm g-I for its partial specific volume and p = 0.998 g cm-1 for the
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:
The melting temperature of a DNA molecule can be determined by differential scanning calorimetry (Impact 12.1). The following data were obtained in aqueous solutions containing the specified concentration [salt of an soluble ionic solid for a series of DNA molecules with varying base pair
Polystyrene is a synthetic polymer with the structure - (CH2- CH (C6Hs)), A batch of polydisperse polystyrene was prepared by initiating the polymerization with t-butyl radicals. As a result, the t-butyl group is expected to be covalently attached to the end of the final products. A sample from
The determination of the average molar masses of conducting polymers is an important part of their characterization. S. Hold croft (J. Poiym. Sci, Polym Phys. 29, 1585 (1991)) has determined the molar masses and Mark-Houwink constants for the electronically conducting polymer,
Explain how planes of lattice points are labelled.
What is meant by a systematic absence? How do they arise?
Describe the structures of elemental metallic solids in terms of the packing of hard spheres. To what extent is the hard-sphere model inaccurate?
Explain how X-ray diffraction can be used to determine the helical configuration of biological molecules.
Describe the characteristics of the Fermi-Dirac distribution. Why is it appropriate to call the parameter u. a chemical potential?

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