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

The viscosity of carbon dioxide was measured by comparing its rate of flow through a long narrow tube (using Poiseuille’s formula) with that of argon. For the same pressure differential, the same volume of carbon dioxide passed through the tube in 55 s as argon in 83 s. The viscosity of argon at
Calculate the thermal conductivity of argon (CV,m = 12.5 J K−1 mol−1, σ = 0.36 nm2) at room temperature (20°C).
Calculate the diffusion constant of argon at 25°C and(a) 1.00 Pa,(b) 100 kPa,(c) 10.0 MPa.If a pressure gradient of 0.10 atm cm−1 is established in a pipe, what is the flow of gas due to diffusion?
The mobility of a chloride ion in aqueous solution at 25°C is 7.91 × 10−8 m2 s−1 V−1. Calculate the molar ionic conductivity.
The mobility of a Rb+ ion in aqueous solution is 7.92 × 10−8 m2 s−1 V−1 at 25°C. The potential difference between two electrodes placed in the solution is 35.0 V. If the electrodes are 8.00 mm apart, what is the drift speed of the Rb+ ion?
Consider molecules that are confined to move in a plane (a twodimensional gas). Calculate the distribution of speeds and determine the mean speed of the molecules at a temperature T.
What fraction of the total current is carried by Li+ when current flows through an aqueous solution of LiBr at 25°C?
The limiting molar conductivities of KCl, KNO3, and AgNO3 are 14.99 mS m2 mol−1, 14.50 mS m2 mol−1, and 13.34 mS m2 mol−1, respectively (all at 25°C). What is the limiting molar conductivity of AgCl at this temperature?
What is the proportion of gas molecules having(a) More than,(b) Less than the root mean square speed?(c) What are the proportions having speeds greater and smaller than the mean speed?
At 25°C the molar ionic conductivities of Li+, Na+, and K+ are 3.87 mS m2 mol−1, 5.01 mS m2 mol−1, and 7.35 mS m2 mol−1, respectively. What are their mobilities?
The mobility of a NO3− ion in aqueous solution at 25°C is 7.40 × 10−8 m2 s−1 V−1. Calculate its diffusion coefficient in water at 25°C.
Calculate the fractions of molecules in a gas that have a speed in a range ∆v at the speed nc* relative to those in the same range at c* itself? This calculation can be used to estimate the fraction of very energetic molecules (which is important for reactions). Evaluate the ratio for n = 3 and n
The diffusion coefficient of CCl4 in heptane at 25°C is 3.17 × 10−9 m2 s−1. Estimate the time required for a CCl4 molecule to have a root mean square displacement of 5.0 mm.
Derive an expression that shows how the pressure of a gas inside an effusion oven (a heated chamber with a small hole in one wall) varies with time if the oven is not replenished as the gas escapes. Then show that t1/2, the time required for the pressure to decrease to half its initial value, is
Estimate the effective radius of a sucrose molecule in water 25°C given that its diffusion coefficient is 5.2 × 10−10 m2 s−1 and that the viscosity of water is 1.00 cP.
The diffusion coefficient for molecular iodine in benzene is 2.13 × 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)?
What are the root mean square distances travelled by an iodine molecule in benzene and by a sucrose molecule in water at 25°C in 1.0 s?
Use mathematical software to calculate P in a one-dimensional random walk, and evaluate the probability of being at x = nλ for n = 6, 10, 14, . . . , 60. Compare the numerical value with the analytical value in the limit of a large number of steps. At what value of n is the discrepancy no more
Interstellar space is a medium quite different from the gaseous environments we commonly encounter on Earth. For instance, a typical density of the medium is about 1 atom cm−3 and that atom is typically H; the effective temperature due to stellar background radiation is about 10 000 K. Estimate
The diffusion coefficient of a particular kind of t-RNA molecule is D = 1.0 × 10−11 m2 s−1 in the medium of a cell interior. How long does it take molecules produced in the cell nucleus to reach the walls of the cell at a distance 1.0 µm, corresponding to the radius of the cell?
The rate of the reaction A + 2 B → 3 C + D was reported as 1.0 mol dm−3 s−1. State the rates of formation and consumption of the participants.
The data below apply to the reaction, (CH3)3CBr + H2O → (CH3)3COH + HBr. Determine the order of the reaction, the rate constant, and the molar concentration of (CH3)3CBr after 43.8 h. t/h 0 3.15 6.20 10.00 18.30 30.80 [(CH3)3CBr]/(10-2 mol dm-³) 10.39 8.96 7.76 6.39 3.53 2.07
The rate of formation of C in the reaction 2 A + B → 2 C + 3 D is 1.0 mol dm−3 s−1. State the reaction rate, and the rates of formation or consumption of A, B, and D.
The following data have been obtained for the decomposition of N2O5(g) at 67°C according to the reaction 2 N2O5(g) → 4 NO2(g) + O2(g). Determine the order of the reaction, the rate constant, and the half-life. It is not necessary to obtain the result graphically, you may do a calculation using
The gas-phase decomposition of acetic acid at 1189 K proceeds by way of two parallel reactions: What is the maximum percentage yield of the ketene CH2CO obtainable at this temperature? (1) CH3COOH (2) CH3COOH → CH4 + CO₂ → H₂C=C=O + H₂O k₁ = 3.74 s ¹ k₂=4.65 s-¹
A first-order decomposition reaction is observed to have the following rate constants at the indicated temperatures. Estimate the activation energy. k/(10-s-) Ꮎ/°C 2.46 0 45.1 20.0 576 40.0
The rate law for the reaction in Exercise 22.2a was reported as d[C]/dt = k[A][B][C]. Express the rate law in terms of the reaction rate; what are the units for k in each case?Data in Exercise 22.2aThe rate of formation of C in the reaction 2 A + B → 2 C + 3 D is 1.0 mol dm−3 s−1. State the
The ClO radical decays rapidly by way of the reaction, 2 ClO → Cl2 + O2. The following data have been obtained:Determine the rate constant of the reaction and the half-life of a ClO radical. t/(10-³ s) [CIO]/(10-6 mol dm³) 8.49 8.09 7.10 5.79 5.20 0.12 0.62 0.96 1.60 3.20 4.00 5.75 4.77 3.95
At 518°C, the rate of decomposition of a sample of gaseous acetaldehyde, initially at a pressure of 363 Torr, was 1.07 Torr s−1 when 5.0 per cent had reacted and 0.76 Torr s−1 when 20.0 per cent had reacted. Determine the order of the reaction.
Sucrose is readily hydrolysed to glucose and fructose in acidic solution. The hydrolysis is often monitored by measuring the angle of rotation of plane polarized light passing through the solution. From the angle of rotation the concentration of sucrose can be determined. An experiment on the
The second-order rate constant for the reactionis 0.11 dm3 mol−1 s−1. What is the concentration of ester after(a) 10 s,(b) 10 min when ethyl acetate is added to sodium hydroxide so that the initial concentrations are [NaOH] = 0.050 mol dm−3 and [CH3COOC2H5] = 0.100 mol dm−3?
At 518°C, the half-life for the decomposition of a sample of gaseous acetaldehyde (ethanal) initially at 363 Torr was 410 s. When the pressure was 169 Torr, the half-life was 880 s. Determine the order of the reaction.
The rate constant for the first-order decomposition of N2O5 in the reaction 2 N2O5(g) → 4 NO2(g) + O2(g) is k = 3.38 × 10−5 s−1 at 25°C. What is the half-life of N2O5? What will be the pressure, initially 500 Torr, at(a) 10 s,(b) 10 min after initiation of the reaction?
Cyclopropane isomerizes into propene when heated to 500°C in the gas phase. The extent of conversion for various initial pressures has been followed by gas chromatography by allowing the reaction to proceed for a time with various initial pressures:where p0 is the initial pressure and p is the
A second-order reaction of the type A + B → P was carried out in a solution that was initially 0.050 mol dm−3 in A and 0.080 mol dm−3 in B. After 1.0 h the concentration of A had fallen to 0.020 mol dm−3.(a) Calculate the rate constant.(b) What is the half-life of the reactants?
Consider the dimerization 2 A ⇌ A2, with forward rate constant ka and backward rate constant kb.(a) Derive the following expression for the relaxation time in terms of the total concentration of protein, [A]tot, = [A] + 2[A2]:(b) Describe the computational procedures that lead to the
A reaction 2 A → P has a second-order rate law with k = 3.50 × 10−4 dm3 mol−1 s−1. Calculate the time required for the concentration of A to change from 0.260 mol dm−3 to 0.011 mol dm−3.
The reaction mechanism involves an intermediate A. Deduce the rate law for the reaction. A₂A+A (fast) A+B → P (slow)
Show that t1/2 ∝ 1/[A]n−1 for a reaction that is nth-order in A.
The rate constant for the decomposition of a certain substance is 2.80 × 10−3 dm3 mol−1 s−1 at 30°C and 1.38 × 10−2 dm3 mol−1 s−1 at 50°C. Evaluate the Arrhenius parameters of the reaction.
Set up the rate equations for the reaction mechanism: Show that the mechanism is equivalent to under specified circumstances. A А ka, К В Кы, К, С ka, B C
The base-catalysed bromination of nitromethane-d3 in water at room temperature (298 K) proceeds 4.3 times more slowly than the bromination of the undeuterated material. Account for this difference. Use kf(C-H) = 450 N m−1.
The effective rate constant for a gaseous reaction that has a Lindemann–Hinshelwood mechanism is 2.50 × 10−4 s−1 at 1.30 kPa and 2.10 × 10−5 s−1 at 12 Pa. Calculate the rate constant for the activation step in the mechanism.
Consider the dimerization A ⇔ A2 with forward rate constant ka and backward rate constant kb. Show that the relaxation time is: T= 1 kb + 4k₁ [A]eq
Derive an integrated expression for a second-order rate law v = k[A][B] for a reaction of stoichiometry 2 A + 3 B → P.
Derive an equation for the steady-state rate of the sequence of reactions A ⇌ B ⇌ C ⇌ D, with [A] maintained at a fixed value and the product D removed as soon as it is formed.
One of the hazards of nuclear explosions is the generation of 90Sr and its subsequent incorporation in place of calcium in bones. This nuclide emits β rays of energy 0.55 MeV, and has a half-life of 28.1 y. Suppose 1.00 µg was absorbed by a newly born child. How much will remain after(a) 18 y,(b)
Bearing in mind distinctions between the mechanisms of stepwise and chain polymerization, describe ways in which it is possible to control the molar mass of a polymer by manipulating the kinetic parameters of polymerization.
Distinguish between competitive, non-competitive, and uncompetitive inhibition of enzymes. Discuss how these modes of inhibition may be detected experimentally.
Discuss experimental procedures that make it possible to differentiate between quenching by energy transfer, collisions, or electron transfer.
When benzophenone is illuminated with ultraviolet light it is excited into a singlet state. This singlet changes rapidly into a triplet, which phosphoresces. Triethylamine acts as a quencher for the triplet. In an experiment in methanol as solvent, the phosphorescence intensity varied with amine
The enzyme-catalysed conversion of a substrate at 25°C has a Michaelis constant of 0.035 mol dm−3. The rate of the reaction is 1.15 × 10−3 mol dm−3 s−1 when the substrate concentration is 0.110 mol dm−3. What is the maximum velocity of this enzymolysis?
The number of photons falling on a sample can be determined by a variety of methods, of which the classical one is chemical actinometry. The decomposition of oxalic acid (COOH)2, in the presence of uranyl sulfate, (UO2)SO4, proceeds according to the sequence (1) UO2+ + hν → (UO2+)*(2)
In a photochemical reaction A → 2 B + C, the quantum efficiency with 500 nm light is 2.1 × 102 mol einstein−1 (1 Einstein = 1 mol photons). After exposure of 300 mmol of A to the light, 2.28 mmol of B is formed. How many photons were absorbed by A?
In an experiment to measure the quantum efficiency of a photochemical reaction, the absorbing substance was exposed to 490 nm light from a 100 W source for 45 min. The intensity of the transmitted light was 40 per cent of the intensity of the incident light. As a result of irradiation, 0.344 mol of
The following mechanism has been proposed for the thermal decomposition of acetaldehyde (ethanal): (1) CH3CHO → ·CH3 + CHO(2) ·CH3 + CH3CHO → CH4 + ·CH2CHO(3) ·CH2CHO → CO + ·CH3(4) ·CH3 + ·CH3 → CH3CH3Find an expression for the rate of formation of methane and the rate of
Calculate the ratio of the mean cube molar mass to the mean square molar mass in terms of(a) The fraction p,(b) The chain length.
Calculate the average polymer length in a polymer produced by a chain mechanism in which termination occurs by a disproportionation reaction of the form M· + ·M → M + :M.
Derive an expression for the time dependence of the degree of polymerization for a stepwise polymerization in which the reaction is acid catalysed by the -COOH acid functional group. The rate law is d[A]/dt = −k[A]2[OH].
The photochemical chlorination of chloroform in the gas has been found to follow the rate law d[CCl4]/dt = k[Cl2] 1/2Ia1/2. Devise a mechanism that leads to this rate law when the chlorine pressure is high.
Nitrogen dioxide reacts bimolecularly in the gas phase to give 2 NO + O2. The temperature dependence of the second-order rate constant for the rate law d[P]/dt = k[NO2]2 is given below. What are the P factor and the reactive cross-section for the reaction? T/K k/(cm³ mol-¹ s-¹) Take o=0.60
The light-induced electron transfer reactions in photosynthesis occur because chlorophyll molecules (whether in monomeric or dimeric forms) are better reducing agents in their electronic excited states. Justify this observation with the help of molecular orbital theory.
Calculate the collision frequency, z, and the collision density, Z, in ammonia, R = 190 pm, at 25°C and 100 kPa. What is the percentage increase when the temperature is raised by 10 K at constant volume?
Distinguish between a diffusion-controlled reaction and an activation controlled reaction.
Collision theory demands knowing the fraction of molecular collisions having at least the kinetic energy Ea along the line of flight. What is this fraction when(a) Ea = 10 kJ mol−1,(b) Ea = 100 kJ mol−1 at (i) 300 K and (ii) 1000 K?
Calculate the percentage increase in the fractions in Exercise 24.2a when the temperature is raised by 10 K. Data in Exercise 24.2aCollision theory demands knowing the fraction of molecular collisions having at least the kinetic energy Ea along the line of flight. What is this fraction when(a)
Discuss the physical origin of the kinetic salt effect.
Describe how the following techniques are used in the study of chemical dynamics: infrared chemiluminescence, laser-induced fluorescence, multiphoton ionization, resonant multiphoton ionization, reaction product imaging, and femtosecond spectroscopy.
A typical diffusion coefficient for small molecules in aqueous solution at 25°C is 5 × 10−9 m2 s−1. If the critical reaction distance is 0.4 nm, what value is expected for the second-order rate constant for a diffusion-controlled reaction?
The rate constant of the reaction I−(aq) + H2O2(aq) → H2O(l) + IO−(aq) varies slowly with ionic strength, even though the Debye–Hückel limiting law predicts no effect. Use the following data from 25°C to find the dependence of log kr on the ionic strength:Evaluate the limiting value of kr
Justify the following statements:(a) Reactions with attractive potential energy surfaces proceed more efficiently if the energy is in relative translational motion.(b) Reactions with repulsive potential surfaces proceed more efficiently if the excess energy is present as vibrations.
Calculate the magnitude of the diffusion-controlled rate constant at 298 K for a species in(a) Water,(b) Pentane.The viscosities are 1.00 × 10−3 kg m−1 s−1 , and 2.2 × 10−4 kg m−1 s−1 , respectively.
Calculate the magnitude of the diffusion-controlled rate constant at 298 K for the recombination of two atoms in water, for which η = 0.89 cP. Assuming the concentration of the reacting species is 1.0 mmol dm−3 initially, how long does it take for the concentration of the atoms to fall to half
Discuss how the following factors determine the rate of electron transfer in homogeneous systems: the distance between electron donor and acceptor, and the reorganization energy of redox active species and the surrounding medium.
For the gaseous reaction A + B → P, the reactive cross-section obtained from the experimental value of the pre-exponential factor is 9.2 × 10−22 m2. The collision cross-sections of A and B estimated from the transport properties are 0.95 and 0.65 nm2, respectively. Calculate the P-factor for
Two neutral species, A and B, with diameters 588 pm and 1650 pm, respectively, undergo the diffusion-controlled reaction A + B → P in a solvent of viscosity 2.37 × 10−3 kg m−1 s−1 at 40°C. Calculate the initial rate d[P]/dt if the initial concentrations of A and B are 0.150 mol dm−3 and
The reaction of propylxanthate ion in acetic acid buffer solutions has the mechanism A− + H+→ P. Near 30°C the rate constant is given by the empirical expression k2 = (2.05 × 1013)e−(8681 K)/T dm3 mol−1 s−1. Evaluate the energy and entropy of activation at 30°C.
The gas-phase association reaction between F2 and IF5 is first-order in each of the reactants. The energy of activation for the reaction is 58.6 kJ mol−1. At 65°C the rate constant is 7.84 × 10−3 kPa−1 s−1. Calculate the entropy of activation at 65°C.
Determine the ratios of(a) The mean speeds,(b) The mean kinetic energies of H2 molecules and Hg atoms at 20°C.
Provide a molecular interpretation for each of the following processes: diffusion, thermal conduction, electric conduction, and viscosity.
A 1.0 dm3 glass bulb contains 1.0 × 1023 H2 molecules. If the pressure exerted by the gas is 100 kPa, what are(a) The temperature of the gas,(b) The root mean square speeds of the molecules?(c) Would the temperature be different if they were O2 molecules?
The element polonium crystallizes in a cubic system. Bragg reflections, with X-rays of wavelength 154 pm, occur at sin θ = 0.225, 0.316, and 0.388 from the (100), (110), and (111) sets of planes. The separation between the sixth and seventh lines observed in the powder diffraction pattern is
What are the relative populations of the states of a two-level system when the temperature is infinite?
Explain how the internal energy and entropy of a system composed of two levels vary with temperature.
Calculate the translational partition function at(a) 300 K and(b) 600 K of a molecule of molar mass 120 g mol−1 in a container of volume 2.00 cm3.
Consider 1.00 × 1022 4He atoms in a box of dimensions 1.0 cm × 1.0 cm × 1.0 cm. Calculate the occupancy of the first excited level at 1.0 mK, 2.0 K, and 4.0 K. Do the same for 3He. What conclusions might you draw from the results of your calculations?
Enumerate the ways by which the parameter β may be identified with 1/kT.
Calculate the ratio of the translational partition functions of D2 and H2 at the same temperature and volume.
Distinguish between the zipper and Zimm–Bragg models of the helix–coil transition.
A certain atom has a threefold degenerate ground level, a nondegenerate electronically excited level at 3500 cm−1, and a threefold degenerate level at 4700 cm−1. Calculate the partition function of these electronic states at 1900 K.
By what factor does the number of available configurations increase when 20 m3 of air at 1.00 atm and 300 K is allowed to expand by 0.0010 per cent at constant temperature?
Calculate the electronic contribution to the molar internal energy at 1900 K for a sample composed of the atoms specified in Exercise 16.4a.Data in Exercise 16.4a.A certain atom has a threefold degenerate ground level, a non degenerate electronically excited level at 3500 cm−1, and a
A certain molecule has a non-degenerate excited state lying at 540 cm−1 above the non-degenerate ground state. At what temperature will 10 per cent of the molecules be in the upper state?
Consider a system of distinguishable particles having only two nondegenerate energy levels separated by an energy that is equal to the value of kT at 10 K. Calculate(a) The ratio of populations in the two states at (1) 1.0 K, (2) 10 K, and (3) 100 K,(b) The molecular partition function at 10 K,(c)
The four lowest electronic levels of a Ti atom are: 3F2, 3F3, 3F4, and 5F1, at 0, 170, 387, and 6557 cm−1, respectively. There are many other electronic states at higher energies. The boiling point of titanium is 3287°C. What are the relative populations of these levels at the boiling point?
At what temperature would the population of the first excited vibrational state of HCl be 1/e times its population of the ground state?
Calculate the standard molar entropy of neon gas at(a) 200 K,(b) 298.15 K.
Calculate the vibrational contribution to the entropy of Cl2 at 500 K given that the wavenumber of the vibration is 560 cm−1.
Identify the systems for which it is essential to include a factor of 1/N! on going from Q to q:(a) A sample of helium gas,(b) A sample of carbon monoxide gas,(c) A solid sample of carbon monoxide,(d) water vapour.
A certain molecule can exist in either a non-degenerate singlet state or a triplet state (with degeneracy 3). The energy of the triplet exceeds that of the singlet by ε. Assuming that the molecules are distinguishable (localized) and independent,(a) Obtain the expression for the molecular

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