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Physical Chemistry Thermodynamics And Kinetics 10th Edition Peter Atkins, Julio De Paula - Solutions

Distinguish between light-emitting diodes and diode lasers.
Calculate the scattering factor for a hydrogenic atom of atomic number Z in which the single electron occupies (a) the 1s orbital, (b) the 2s orbital. Plot f as a function of (sin θ)/λ.
Show that the volume of a monoclinic unit cell is V=abc sin β.
Calculate the separations of the planes {112}, {110}, and {224} in a crystal in which the cubic unit cell has side 562pm.
Explore how the scattering factor of Problem 18.2 changes when the actual 1s wavefunction of a hydrogenic atom is replaced by a Gaussian function.Data in Problem 18.2Show that if a substance responds nonlinearly to two sources of radiation, one of frequency ω1 and the other of frequency ω2, then
Derive an expression for the volume of a hexagonal unit cell.
The magnitude of the magnetic moment of Mn2+ in its complexes is typically 5.3μB. How many unpaired electrons does the ion possess?
Does titanium expand or contract as it transforms from hcp to bcc? The atomic radius of titanium is 145.8pm in hcp but 142.5pm in bcc.
Discuss what is meant by ‘scattering factor’. How is it related to the number of electrons in the atoms scattering X-rays?
Calculate the lattice enthalpy of CaO from the following data: Sublimation of Ca(s) Ionization of Ca(g) to Ca²+(g) Dissociation of O₂(g) Electron attachment to O(g) Electron attachment to O-(g) Formation of CaO(s) from Ca(s) and O₂(g) AH/(kJ mol-¹) +178 +1735 +249 -141 +844 -635
The unit cells of SbCl3 are orthorhombic with dimensions a=812pm, b=947pm, and c=637pm. Calculate the spacing, d, of the {321} planes.
The volume of a monoclinic unit cell is abc sin β (see Problem 18A.2). 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 122.82° and the mass density of the solid is 1.152 g cm−3. Calculate the dimensions of the cell.Data in
Estimate the spin-only molar susceptibility of CuSO4·5H2O at 25 °C.
A transistor is a semiconducting device that is commonly used either as a switch or an amplifier of electrical signals. Prepare a brief report on the design of a nanometre-sized transistor that uses a carbon nanotube as a component. A useful starting point is the work summarized by Tans, et al.
Describe the consequences of the phase problem in determining structure factors and how the problem is overcome.
Show that the separation of the {hkl} planes in an orthorhombic crystal with sides a, b, and c is given by eqn 18A.1c.Data in eqn 18A.1c. 1 hkl h² k² 1² + a² b² c² +- Orthorhombic lattice Separation of planes (18A.1c)
What are the values of the angle θ of the first three diffraction lines of bcc iron (atomic radius 126pm) when the X-ray wavelength is 72pm?
In an X-ray investigation, the following structure factors were determined (with F−h00=Fh00):Construct the electron density along the corresponding direction. h F1,00 0 10 1 -10 2 8 3 -8 4 6 5 -6 6 7 4 -4 8 2 9 -2
Verify that the radius ratio for (a) sixfold coordination is 0.414, and (b) for eightfold coordination is 0.732.
Construct the Patterson synthesis from the information in Exercise 18A.12(a).Data in Exercise 18A.12(a).In an X-ray investigation, the following structure factors were determined (with F−h00=Fh00):Construct the electron density along the corresponding direction. h Fh00 0 10 1 2 -10
Radiation from an X-ray source consists of two components of wavelengths 154.433pm and 154.051pm. Calculate the difference in glancing angles (2θ) of the diffraction lines arising from the two components in a diffraction pattern from planes of separation 77.8pm.
What is the value of the scattering factor in the forward direction for Br−?
Calculate the structure factors for a face-centred cubic structure (C) in which the scattering factors of the ions on the two faces are twice that of the ions at the corners of the cube.
Use mathematical software to draw a graph of the scattering factor f against (sin θ)/λ for an atom of atomic number Z for which ρ(r)=3Z/4πR3 for 0≤r≤R and ρ(r)=0 for r>R, with R a parameter that represents the radius of the atom. Explore how f varies with Z and R.
What speed should neutrons have if they are to have a wavelength of 65pm?
Describe the origin of the thermodynamic force. To what extent can it be regarded as an actual force?
Calculate the wavelength of electrons that have reached thermal equilibrium by collision with a moderator at 380K.
The diffusion coefficient of glucose in water at 25 °C is 6.73 × 10−10m2 s−1. Estimate the time required for a glucose molecule to undergo a root-mean-square displacement of 5.0mm.
Explain how Fick’s first law arises from the concentration gradient of gas molecules.
The viscosity of benzene varies with temperature as shown in the following table. Use the data to infer the activation energy for viscosity. Ꮎ/l°C n/cP 10 0.758 20 30 40 50 0.652 0.564 0.503 0.442 60 0.392 70 0.358
Calculate the thermal conductivity of argon (CV,m=12.5 JK−1mol−1, σ=0.36nm2) at 298K.
Discuss the difference between the hydrodynamic radius of an ion and its ionic radius and explain why a small ion can have a large hydrodynamic radius.
The viscosity of water at 20 °C is 1.002 cP and 0.7975 cP at 30 °C. What is the energy of activation for the transport process?
Account physically for the form of the diffusion equation.
A layer of 20.0 g of sucrose is spread uniformly over a surface of area 5.0 cm2 and covered in water to a depth of 20 cm. What will be the molar concentration of sucrose molecules at 10 cm above the original layer at (i) 10 s, (ii) 24h? Assume diffusion is the only transport process and take D =
An empirical expression that reproduces the viscosity of water in the range 20–100 °C is where η20 is the viscosity at 20°C.Explore (by using mathematical software) the possibility of fitting an exponential curve to this expression and hence identifying an activation energy for the
Provide molecular interpretations for the dependencies of the diffusion constant and the viscosity on the temperature, pressure, and size of gas molecules.
Calculate the diffusion constant of argon at 20 °C and (i) 1.00Pa, (ii) 100 kPa, (iii) 10.0MPa. If a pressure gradient of 1.0barm−1 is established in a pipe, what is the flow of gas due to diffusion?
Discuss the mechanism of proton conduction in water. How could the model be tested?
The conductivity of aqueous ammonium chloride at a series of concentrations is listed in the following table. Deduce the molar conductivity and determine the parameters that occur in Kohlrausch’s law. c/(moldm-³) K/(mScm-¹) 1.334 131 1.432 139 1.529 147 1.672 156 1.725 164
Suppose the concentration of a solute decays linearly along the length of a container. Calculate the thermodynamic force on the solute at 25 °C and 10 cm and 20 cm given that the concentration falls to half its value in 10 cm.
What might be the effect of molecular interactions on the transport properties of a gas?
Confirm that eqn 19C.11 is a solution of the diffusion equation with the correct initial value.Data in eqn 19C.11 no А(πDt)¹/2 c(x,t)=- -e-x²/4Dt One-dimensional diffusion (19C.11)
Calculate the flux of energy arising from a temperature gradient of 10.5Km−1 in a sample of argon in which the mean temperature is 280K.
Why is a proton less mobile in liquid ammonia than in water?
Suppose the concentration of a solute follows a Gaussian distribution (proportional to e−x2) along the length of a container. Calculate the thermodynamic force on the solute at 20 °C and 5.0 cm given that the concentration falls to half its value in 5.0 cm.
Confirm that is a solution of the diffusion equation with convection (eqn 19C.10) with all the solute concentrated at x=x0 at t=0 and plot the concentration profile at a series of times to show how the distribution spreads and its centroid drifts.Data in eqn 19C.10
A lump of sucrose of mass 10.0 g is suspended in the middle of a spherical flask of water of radius 10 cm at 25 °C. What is the concentration of sucrose at the wall of the flask after (a) 1.0h, (b) 1.0 week? Take D=5.22×10−10m2 s−1.
In a double-glazed window, the panes of glass are separated by 1.0 cm. What is the rate of transfer of heat by conduction from the warm room (28 °C) to the cold exterior (−15 °C) through a window of area 1.0m2? What power of heater is required to make good the loss of heat?
The thermodynamic force has a direction as well as a magnitude, and in a three-dimensional ideal system eqn 19C.7 becomes F=–RTΔ(ln c). What is the thermodynamic force acting to bring about the diffusion summarized by eqn 19C.12 (that of a solute initially suspended at the centre of a flask of
What are the drift speeds of Li+, Na+, and K+ in water when a potential difference of 100V is applied across a 5.00 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
A dilute solution of a weak (1,1)-electrolyte contains both neutral ion pairs and ions in equilibrium (AB ⇌ A+ +B−). Prove that molar conductivities are related to the degree of ionization by the equations:where Λm° is the molar conductivity at infinite dilution and K is the constant in
Calculate the relation between 〈x2〉1/2 and 〈x4〉1/4 for diffusing particles at a time t if they have a diffusion constant D.
Calculate the viscosity of air at (i) 273K, (ii) 298K, (iii) 1000K. Take σ≈0.40nm2. (The experimental values are 173µP at 273K, 182µP at 20 °C, and 394µP at 600 °C.)
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?
The isomerization of cyclopropane over a limited pressure range was examined in Problem 20B.14. If the Lindemann mechanism of unimolecular reactions is to be tested we also need data at low pressures. These have been obtained (H.O. Pritchard et al., Proc. R. Soc. A 217, 563 (1953)): Test the
Michaelis and Menten derived their rate law by assuming a rapid pre-equilibrium of E, S, and ES. Derive the rate law in this manner, and identify the conditions under which it becomes the same as that based on the steady-state approximation (eqn 20H.1). Data in eqn 20H.1 V= k₂ [E]o 1+
A solid compound of molar mass 100 gmol−1 was introduced into a container and heated to 400 °C. When a hole of diameter 0.50mm was opened in the container for 400 s, a mass loss of 285mg was measured. Calculate the vapour pressure of the compound at 400 °C.
Consider the base-catalysed reaction Deduce the rate law. (1) AH+B B (2) A+ AH->product BH+A (both fast) (slow) K
Summarize the characteristic of zeroth-order, first-order, secondorder, and pseudofirst-order reactions.
Show that the definition of Ea given in eqn 20D.3 reduces to eqn 20D.1 for a temperature-independent activation energy. Data in eqn 20D.3 E₂=RT² dink, dT Definition Activation energy (20D.3)
Predict how the total pressure varies during the gas-phase reaction 2 ICl(g)+H2(g)→I2(g)+2 HCl(g) in a constant-volume container.
Describe the strategy of a temperature-jump experiment. What parameters of a reaction are accessible from this technique?
The equilibrium NH3(aq)+H2O(l)⇌NH4+(aq)+OH−(aq) at 25 °C is subjected to a temperature jump which slightly increased the concentration of NH4+(aq) and OH−(aq). The measured relaxation time is 7.61ns. The equilibrium constant for the system is 1.78×10−5 at 25 °C, and the equilibrium
Define the terms in and discuss the validity of the expression ln kr=ln A−Ea/RT.
Use mathematical software or a spreadsheet to examine the time dependence of [I] in the reaction mechanism . In all of the following calculations, use [A]0=1mol dm−3 and a time range of 0 to 5 s.(a) Plot [In] against t for ka=10 s−1 and kb=1 s−1.(b) Increase the ratio kb/ka steadily by
The rate constant for the decomposition of a certain substance is 3.80×10−3 dm3 mol−1 s−1 at 35 °C and 2.67×10−2 dm3 mol−1 s−1 at 50 °C. Evaluate the Arrhenius parameters of the reaction.
Distinguish between reaction order and molecularity.
Discuss the range of validity of the expression kr=kakb[A]/(kb+ka′[A]) for the effective rate constant of a unimolecular reaction according to the Lindemann–Hinshelwood mechanism.
The following initial-rate data were obtained on the rate of binding of glucose with the enzyme hexokinase present at a concentration of 1.34mmol dm−3. What is(a) The order of reaction with respect to glucose, (b) The rate constant? [C6H12O6]/(mmol dm-³) 1.00 vo/(moldm³
The effective rate constant for a gaseous reaction which has a Lindemann–Hinshelwood mechanism is 2.50 × 10−4 s−1 at 1.30 kPa and 2.10 × 10−5 s−1 at 12Pa. Calculate the rate constant for the activation step in the mechanism.
Consider the quenching of an organic fluorescent species with τ0 = 6.0ns by a d-metal ion with kQ = 3.0×108 dm3 mol−1 s−1. Predict the concentration of quencher required to decrease the fluorescence intensity of the organic species to 50 per cent of the unquenched value.
Ultraviolet radiation photolyses O3 to O2 and O. Determine the rate at which ozone is consumed by 305nm radiation in a layer of the stratosphere of thickness 1.0 km. The quantum yield is 0.94 at 220K, the concentration about 8 nmol dm−3, the molar absorption coefficient 260 dm3 mol−1 cm−1,
When can a reaction order not be ascribed?
The rate of the reaction A + 2 B → 3 C + D was reported as 2.7mol dm−3 s−1. State the rates of formation and consumption of the participants.
Write the rate law that corresponds to each of the following expressions:(a) [A]=[A]0− krt,(b) ln([A]/[A]0)=−krt,(c) [A]=[A]0/(1+krt[A]0).
For a second-order reaction of the form A → n B (with n possibly fractional), the concentration of the product varies with time as [B] A = nkrt[A]02/(1 +krt[A]0). Plot the time dependence of [A] and [B] for the cases n = 1/2 ,1, and 2.
What feature of a reaction would ensure that its rate would respond to a pressure jump?
Set up the rate equations and plot the corresponding graphs for the approach to an equilibrium of the form A ⇌ 2 B.
What might account for the failure of the Arrhenius equation at low temperatures?
The rate of a chemical reaction is found to triple when the temperature is raised from 24 °C to 49 °C. Determine the activation energy.
For many enzymes, the mechanism of action involves the formation of two intermediates: Show that the rate of formation of product has the same form as that shown in eqn 20H.1, but with vmax and KM given by Data in eqn 20H.1 E+S →ES ES →E+S ES ES' ES' → E+P v=k₂[E][S] v=k₁
The mechanism of a composite reaction consists of a fast preequilibrium step with forward and reverse activation energies of 27 kJmol−1 and 35 kJmol−1, respectively, followed by an elementary step of activation energy 15 kJmol−1. What is the activation energy of the composite reaction?
Calculate the fraction condensed and the degree of polymerization at t=5.00h of a polymer formed by a stepwise process with kr=1.39 dm3 mol−1 s−1 and an initial monomer concentration of 10.0m mol dm−3.
The following data were obtained on the initial rates of a reaction of a d-metal complex with a reactant Y in aqueous solution. What is (a) The order of reaction with respect to the complex and Y, (b) The rate constant? For the experiments (i), [Y]=2.7mmol dm−3 and for experiments (ii)
Consider an enzyme-catalysed reaction that follows Michaelis– Menten kinetics with KM =3.0m mol dm−3. What concentration of a competitive inhibitor characterized by KI=20µmol dm−3 will reduce the rate of formation of product by 50 per cent when the substrate concentration is held at 0.10mmol
What are the advantages of ascribing an order to a reaction?
The reaction A ⇌ 2 B is first-order in both directions. Derive an expression for the concentration of A as a function of time when the initial molar concentrations of A and B are [A]0 and [B]0. What is the final composition of the system?
Distinguish between a pre-equilibrium approximation and a steady-state approximation. Why might they lead to different conclusions?
The following kinetic data (v0 is the initial rate) were obtained for the reaction 2 ICl(g)+H2(g)→I2(g)+2 HCl(g): Experiment [IC]/(mmol dm-³) [H₂]/(mmoldm-³) v/(moldm³s-¹) 1 2 3 4 1.5 3.0 3.0 4.7 1.5 1.5 4.5 2.7 3.7x10-7 7.4×10-7 22x10-7 ?
Consider a polymer formed by a chain process. By how much does the kinetic chain length change if the concentration of initiator increases by a factor of 3.6 and the concentration of monomer decreases by a factor of 4.2?
Explain how it is possible for the activation energy of a reaction to be negative.
Some enzymes are inhibited by high concentrations of their own products. Sketch a plot of reaction rate against concentration of substrate for an enzyme that is prone to product inhibition.
The enzyme-catalysed conversion of a substrate at 25 °C has a Michaelis constant of 90µmoldm−3 and a maximum velocity of 22.4µmol dm−3 s−1 when the enzyme concentration is 1.60 nmol dm−3.(a) Calculate kcat and η.(b) Is the enzyme ‘catalytically perfect’?
Summarize the experimental procedures that can be used to monitor the composition of a reaction system.
The rate law for the reaction in Exercise 20A.2(a) was found to be v=kr[A][B]. What are the units of kr when the concentrations are in moles per cubic decimetre? Express the rate law in terms of the rates of formation and consumption of (i) A, (ii) C. Data in Exercise 20A.2(a)The rate of the
Some enzymes are inhibited by high concentrations of their own substrates.(a) Show that when substrate inhibition is important the reaction rate v is given bywhere KI is the equilibrium constant for dissociation of the inhibited enzyme– substrate complex.(b) What effect does substrate inhibition
A reaction 2 A→P has a second-order rate law with kr=4.30×10−4 dm3 mol−1 s−1. Calculate the time required for the concentration of A to change from 0.210mol dm−3 to 0.010mol dm−3.
The rate law for the reaction in Exercise 20A.3(a) was reported as d[C]/dt=kr[A][B][C]. Express the rate law in terms of the reaction rate v; what are the units for kr in each case when the concentrations are in moles per cubic decimetre?Data in Exercise 20A.3(a)The rate of formation of C in the
Explain and illustrate how reaction orders may change under different circumstances.

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