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

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Explore the consequences of replacing the equation of state of a perfect gas by the van der Waals equation of state for the pressure-dependence of the molar Gibbs energy. Proceed in three steps. First, consider the case when a=0 and only repulsions are significant. Then consider the case when b=0
A block of copper of mass 500 g and initially at 293K is in thermal contact with an electric heater of resistance 1.00 kΩ and negligible mass. A current of 1.00 A is passed for 15.0 s. Calculate the change in entropy of the copper, taking Cp,m = 24.4 JK−1 mol−1. The experiment is then repeated
Calculate the change in entropy of the system when 10.0 g of ice at −10.0 °C is converted into water vapour at 115.0 °C and at a constant pressure of 1bar. The constant-pressure molar heat capacity of H2O(s) and H2O(l) is 75.291 JK−1mol−1 and that of H2O(g) is 33.58 JK−1mol−1.
How many phases are present at each of the points marked in Fig. 4.1a?Data in Fig. 4.1a? Pressure (a) d b Temperature O a
The protein lysozyme unfolds at a transition temperature of 75.5 °C and the standard enthalpy of transition is 509 kJmol−1. Calculate the entropy of unfolding of lysozyme at 25.0 °C, given that the difference in the constant pressure heat capacities upon unfolding is 6.28 kJK−1mol−1 and can
Describe how the concept of chemical potential unifies the discussion of phase equilibria.
What is the physical reason for the fact that the chemical potential of a pure substance decreases as the temperatures is raised?
In a study of the vapour pressure of chloromethane, A. Bah and N. Dupont-Pavlovsky (J. Chem. Eng. Data 40, 869 (1995)) presented data for the vapour pressure over solid chloromethane at low temperatures. Some of that data is as follows: Estimate the standard enthalpy of sublimation of
Estimate the difference between the normal and standard melting points of ice.
Why does the chemical potential change with pressure even if the system is incompressible (that is, remains at the same volume when pressure is applied)?
The difference in chemical potential between two regions of a system is +7.1 kJmol−1. By how much does the Gibbs energy change when 0.10mmol of a substance is transferred from one region to the other?
What is the physical reason for the fact that the chemical potential of a pure substance increases as the pressure is raised?
Water is heated from 25 °C to 100 °C. By how much does its chemical potential change?
Explain why four phases cannot be in equilibrium in a one-component system.
What is the maximum number of phases that can be in mutual equilibrium in a two-component system?
How may differential scanning calorimetry (DSC) be used to identify phase transitions?
Pressure was exerted with a piston on water at 20 °C. The vapour pressure of water under 1.0 bar is 2.34 kPa. What is its vapour pressure when the pressure on the liquid is 20MPa?
By how much does the chemical potential of copper change when the pressure exerted on a sample is increased from 100 kPa to 10MPa?
Figure 4B.1 gives a schematic representation of how the chemical potentials of the solid, liquid, and gaseous phases of a substance vary with temperature. All have a negative slope, but it is unlikely that they are truly straight lines as indicated in the illustration. Derive an expression for the
Combine the ‘barometric formula’, p= p0e−a/H, where H= 8 km, for the dependence of the pressure on altitude, a, with the Clausius–Clapeyron equation, and predict how the boiling temperature of a liquid depends on the altitude and the ambient temperature. Take the mean ambient
The Clapeyron equation does not apply to second-order phase transitions, but there are two analogous equations, the Ehrenfest equations, that do. They are: Where α is the expansion coefficient, κT the isothermal compressibility, and the subscripts 1 and 2 refer to two different phases.
Specify and analyse critically the assumptions that underlie the kinetic model of gases.
Determine the ratios of (i) The mean speeds, (ii) The mean kinetic energies of He atoms and Hg atoms at 25 °C.
Could 131 g of xenon gas in a vessel of volume 1.0dm3 exert a pressure of 20 atm at 25 °C if it behaved as a perfect gas? If not, what pressure would it exert?
Provide molecular interpretations for the dependencies of the mean free path on the temperature, pressure, and size of gas molecules.
(a) Calculate the root mean square speeds of H2 and O2 molecules at 20 °C. (b) Calculate the root mean square speeds of CO2 molecules and He atoms at 20 °C.
What is the significance of the critical constants?
(a) Use the Maxwell–Boltzmann distribution of speeds to estimate the fraction of N2 molecules at 400K that have speeds in the range 200 to 210ms−1.(b) Use the Maxwell–Boltzmann distribution of speeds to estimate the fraction of CO2 molecules at 400K that have speeds in the range 400 to
(a) Calculate the most probable speed, the mean speed, and the mean relative speed of CO2 molecules in air at 20 °C.(b) Calculate the most probable speed, the mean speed, and the mean relative speed of H2 molecules in air at 20 °C.
Explain how the van der Waals equation accounts for critical behaviour.
Assume that air consists of N2 molecules with a collision diameter of 395 pm. Calculate(i) The mean speed of the molecules,(ii) The mean free path,(iii) The collision frequency in air at 1.0 atm and 25 °C.
A vessel of volume 22.4dm3 contains 2.0mol H2 and 1.0mol N2 at 273.15K initially. All the H2 reacted with sufficient N2 to form NH3. Calculate the partial pressures and the total pressure of the final mixture.
In an attempt to determine an accurate value of the gas constant, R, a student heated a container of volume 20.000dm3 filled with 0.251 32g of helium gas to 500°C and measured the pressure as 206.402 cm of water in a manometer at 25°C. Calculate the value of R from these data. (The density of
Ozone is a trace atmospheric gas which plays an important role in screening the Earth from harmful ultraviolet radiation, and the abundance of ozone is commonly reported in Dobson units. One Dobson unit is the thickness, in thousandths of a centimetre, of a column of gas if it were collected as a
Derive an expression for 〈vn〉1/n from the Maxwell–Boltzmann distribution of speeds. You will need standard integrals given in the Resource section.
Balloons are still used to deploy sensors that monitor meteorological phenomena and the chemistry of the atmosphere. It is possible to investigate some of the technicalities of ballooning by using the perfect gas law. Suppose your balloon has a radius of 3.0m and that it is spherical.(a) What
Given that the density of air at 0.987bar and 27°C is 1.146 kg m−3,calculate the mole fraction and partial pressure of nitrogen and oxygen assuming that(i) Air consists only of these two gases,(ii) Air also contains 1.0 mole percent Ar.
Plot different Maxwell–Boltzmann speed distributions by keeping the molar mass constant at 100 gmol−1 and varying the temperature of the sample between 200K and 2000K.
Evaluate numerically the fraction of molecules with speeds in the range 100ms−1 to 200ms−1 at 300K and 1000K.
The composition of the atmosphere is approximately 80 per cent nitrogen and 20 per cent oxygen by mass. At what height above the surface of the Earth would the atmosphere become 90 per cent nitrogen and 10 per cent oxygen by mass? Assume that the temperature of the atmosphere is constant at 25 °C.
Calculate the frequency of molecular collisions per square centimetre of surface in a vessel containing(a) Hydrogen,(b) Propane at 25°C when the pressure is (i) 100 Pa, (ii) 0.10 µTorr.
(a) Describe the advantages and limitations of each of the spectroscopic techniques designated by the acronyms AES, EELS, HREELS, RAIRS, SERS, SEXAFS, SHG, UPS, and XPS.(b) Describe the advantages and limitations of each of the microscopy, diffraction, and scattering techniques designated by the
What pressure of argon gas is required to produce a collision rate of 4.5 × 1020 s−1 at 425 K on a circular surface of diameter 1.5 mm?
In a study of the catalytic properties of a titanium surface it was necessary to maintain the surface free from contamination. Calculate the collision frequency per square centimetre of surface made by O2 molecules at(a) 100 kPa,(b) 1.00 Pa and 300 K.Estimate the number of collisions made with a
Nickel is face-centred cubic with a unit cell of side 352 pm. What is the number of atoms per square centimetre exposed on a surface formed by(a) (100),(b) (110),(c) (111) planes?Calculate the frequency of molecular collisions per surface atom in a vessel containing(a) Hydrogen,(b) Propane at 25°C
Consider the analysis of surface plasmon resonance data (as in biosensor analysis) and discuss how a plot of a0/Req against a0 may be used to evaluate Rmax and K.
Describe the essential features of the Langmuir–Hinshelwood, Eley–Rideal, and Mars van Krevelen mechanisms for surface-catalysed reactions.
The volume of oxygen gas at 0°C and 101 kPa adsorbed on the surface of 1.00 g of a sample of silica at 0°C was 0.284 cm3 at 142.4 Torr and 1.430 cm3 at 760 Torr. What is the value of Vmon?
The following data have been obtained for the adsorption of H2 on the surface of 1.00 g of copper at 0°C. The volume of H2 below is the volume that the gas would occupy at STP (0°C and 1 atm).Determine the volume of H2 necessary to form a monolayer and estimate the surface area of the copper
The enthalpy of adsorption of CO on a surface is found to be −120 kJ mol−1. Estimate the mean lifetime of a CO molecule on the surface at 400 K.
The average time for which an oxygen atom remains adsorbed to a tungsten surface is 0.36 s at 2548 K and 3.49 s at 2362 K. Find the activation energy for desorption. What is the pre-exponential factor for these tightly chemisorbed atoms?
(a) Discuss the main structural features of the electrical double layer. (b) Distinguish between the electrical double layer and the Nernst diffusion layer.
The adsorption of a gas is described by the Langmuir isotherm with K = 0.85 kPa−1 at 25°C. Calculate the pressure at which the fractional surface coverage is(a) 0.15,(b) 0.95.
For how long on average would an H atom remain on a surface at 298 K if its desorption activation energy were(a) 15 kJ mol−1,(b) 150 kJ mol−1?Take τ0 = 0.10 ps. For how long on average would the same atoms remain at 1000 K?
Hydrogen iodide is very strongly adsorbed on gold but only slightly adsorbed on platinum. Assume the adsorption follows the Langmuir isotherm and predict the order of the HI decomposition reaction on each of the two metal surfaces.
Nitrogen gas adsorbed on charcoal to the extent of 0.921 cm3 g−1 at 490 kPa and 190 K, but at 250 K the same amount of adsorption was achieved only when the pressure was increased to 3.2 MPa. What is the enthalpy of adsorption of nitrogen on charcoal?
In an experiment on the Pt|H2|H+ electrode in dilute H2SO4 the following current densities were observed at 25°C. Evaluate α and j0 for the electrode.How would the current density at this electrode depend on the overpotential of the same set of magnitudes but of opposite sign? η/mV j/(m^
In an experiment on the adsorption of oxygen on tungsten it was found that the same volume of oxygen was desorbed in 27 min at 1856 K and 2.0 min at 1978 K. What is the activation energy of desorption? How long would it take for the same amount to desorb at(a) 298 K,(b) 3000 K?
The standard potentials of lead and tin are −126 mV and −136 mV respectively at 25°C, and the overvoltage for their deposition are close to zero. What should their relative activities be in order to ensure simultaneous deposition from a mixture?
The transfer coefficient of a certain electrode in contact with M3+ and M4+ in aqueous solution at 25°C is 0.39. The current density is found to be 55.0 mA cm−2 when the overvoltage is 125 mV. What is the overvoltage required for a current density of 75 mA cm−2?
Determine the exchange current density from the information given in Exercise 25.16a.Data in Exercise 25.16a.The transfer coefficient of a certain electrode in contact with M3+ and M4+ in aqueous solution at 25°C is 0.39. The current density is found to be 55.0 mA cm−2 when the overvoltage is
To a first approximation, significant evolution or deposition occurs in electrolysis only if the overpotential exceeds about 0.6 V. To illustrate this criterion determine the effect that increasing the overpotential from 0.40 V to 0.60 V has on the current density in the electrolysis of 1.0 M
Estimate the limiting current density at an electrode in which the concentration of Mg2+ ions is 1.5 mmol dm−3 at 25°C. The thickness of the Nernst diffusion layer is 0.32 mm. The ionic conductivity of Mg2+ at infinite dilution and 25°C is 10.60 mS m2 mol−1.
A typical exchange current density, that for H+ discharge at platinum, is 0.79 mA cm−2 at 25°C. What is the current density at an electrode when its overpotential is(a) 10 mV,(b) 100 mV,(c) −5.0 V?Take α = 0.5.
Suppose that the electrode potential is set at 1.00 V. The exchange current density is 6.0 × 10−4 A cm−2 and α = 0.50. Calculate the current density for the ratio of activities a(Fe3+)/a(Fe2+) in the range 0.1 to 10.0 and at 25°C.
What overpotential is needed to sustain a current of 20 mA at a Pt|Fe3+,Fe2+ electrode in which both ions are at a mean activity a = 0.10? The surface area of the electrode is 1.0 cm2.
How many electrons or protons are transported through the double layer in each second when the Pt,H2 |H+, Pt|Fe3+,Fe2+, and Pb,H2 |H+ electrodes are at equilibrium at 25°C? Take the area as 1.0 cm2 in each case. Estimate the number of times each second a single atom on the surface takes part in a
Finally we come full circle and return to the Langmuir isotherm. Find the form of dµ that, when inserted in the Gibbs adsorption isotherm, leads to the Langmuir isotherm.
State what happens when a platinum electrode in an aqueous solution containing both Cu2+ and Zn2+ ions at unit activity is made the cathode of an electrolysis cell.
What are the conditions that allow a metal to be deposited from aqueous acidic solution before hydrogen evolution occurs significantly at 293 K? Why may silver be deposited from aqueous silver nitrate?
The exchange current density for H+ discharge at zinc is about 50 pA cm−2. Can zinc be deposited from a unit activity aqueous solution of a zinc salt?
Now suppose that the overpotential is in the high overpotential region at all times even though it is oscillating. What waveform will the current across the interface show if it varies linearly and periodically (as a sawtooth waveform) between η− and η+ around η0? Take α = 1/2.
Can magnesium be deposited on a zinc electrode from a unit activity acid solution at 25°C?
Derive an expression for the current density at an electrode where the rate process is diffusion-controlled and ηc is known. Sketch the form of j/jL as a function of ηc. What changes occur if anion currents are involved?
Calculate the maximum (zero-current) potential difference of a nickel–cadmium cell, and the maximum possible power output when 100 mA is drawn at 25°C.
The corrosion current density jcorr at an iron anode is 1.0 A m−2. What is the corrosion rate in millimetres per year? Assume uniform corrosion.
Calculate the thermodynamic limit to the zero-current potential of fuel cells operating on(a) Hydrogen and oxygen,(b) Methane and air, and(c) Propane and air.Use the Gibbs energy information in the Data section, and take the species to be in their standard states at 25°C.
Estimate the magnitude of the corrosion current for a patch of zinc of area 0.25 cm2 in contact with a similar area of iron in an aqueous environment at 25°C. Take the exchange current densities as 1 µA cm−2 and the local ion concentrations as 1 µmol dm−3.
Calculate the entropy of activation for a collision between two structureless particles at 500 K, taking M = 78 g mol−1 and σ = 0.62 nm2.
The rate constant of the reaction H2O2(aq) + I−(aq) + H+(aq) → H2O(l) + HIO(aq) is sensitive to the ionic strength of the aqueous solution in which the reaction occurs. At 25°C, k = 12.2 dm6 mol−2 min−1 at an ionic strength of 0.0525. Use the Debye–Hückel limiting law to estimate the
At what pressure does the mean free path of argon at 25°C become comparable to the size of a 1 dm3 vessel that contains it? Take σ = 0.36 nm2.
Calculate the ratio of the thermal conductivities of gaseous hydrogen at 300 K to gaseous hydrogen at 10 K. Be circumspect, and think about the modes of motion that are thermally active at the two temperatures.
Discuss the mechanism of proton conduction in liquid water.
At an altitude of 20 km the temperature is 217 K and the pressure 0.050 atm. What is the mean free path of N2 molecules? (σ = 0.43 nm2.)
How many collisions does a single Ar atom make in 1.0 s when the temperature is 25°C and the pressure is(a) 10 atm,(b) 1.0 atm,(c) 1.0 µatm?
The nuclide 244Bk (berkelium) decays by producing α particles, which capture electrons and form He atoms. Its half-life is 4.4 h. A sample of mass 1.0 mg was placed in a container of volume 1.0 cm3 that was impermeable to α radiation, but there was also a hole of radius 2.0 µm in the wall. What
Provide a molecular interpretation for the observation that mediated transport across a biological membrane leads to a maximum flux Jmax when the concentration of the transported species becomes very large.
Calculate the mean free path of molecules in air using σ = 0.43 nm2 at 25°C and(a) 10 atm,(b) 1.0 atm,(c) 1.0 µatm.
Use the Maxwell distribution of speeds to estimate the fraction of N2 molecules at 500 K that have speeds in the range 290 to 300 m s−1.
Conductivities are often measured by comparing the resistance of a cell filled with the sample to its resistance when filled with some standard solution, such as aqueous potassium chloride. The conductivity of water is 76 mS m−1 at 25°C and the conductivity of 0.100 mol dm−3 KCl(aq) is 1.1639
A solid surface with dimensions 2.5 mm × 3.0 mm is exposed to argon gas at 90 Pa and 500 K. How many collisions do the Ar atoms make with this surface in 15 s?
An effusion cell has a circular hole of diameter 2.50 mm. If the molar mass of the solid in the cell is 260 g mol−1 and its vapour pressure is 0.835 Pa at 400 K, by how much will the mass of the solid decrease in a period of 2.00 h?
After correction for the water conductivity, the conductivity of a saturated aqueous solution of AgCl at 25°C was found to be 0.1887 mS m−1. What is the solubility of silver chloride at this temperature?
A manometer was connected to a bulb containing carbon dioxide under slight pressure. The gas was allowed to escape through a small pinhole, and the time for the manometer reading to drop from 75 cm to 50 cm was 52 s. When the experiment was repeated using nitrogen (for which M = 28.02 g mol−1)
A space vehicle of internal volume 3.0 m3 is struck by a meteor and a hole of radius 0.10 mm is formed. If the oxygen pressure within the vehicle is initially 80 kPa and its temperature 298 K, how long will the pressure take to fall to 70 kPa?
The mobilities of H+ and Cl− at 25°C in water are 3.623 × 10−7 m2 s−1 V−1 and 7.91 × 10−8 m2 s−1 V−1, respectively. What proportion of the current is carried by the protons in 10−3 m HCl(aq)? What fraction do they carry when the NaCl is added to the acid so that the solution is
Calculate the flux of energy arising from a temperature gradient of 2.5 K m−1 in a sample of argon in which the mean temperature is 273 K.
In a double-glazed window, the panes of glass are separated by 5.0 cm. What is the rate of transfer of heat by conduction from the warm room (25°C) to the cold exterior (−10°C) through a window of area 1.0 m2? What power of heater is required to make good the loss of heat?
Calculate the inlet pressure required to maintain a flow rate of 9.5 × 105 dm3 h−1 of nitrogen at 293 K flowing through a pipe of length 8.50 m and diameter 1.00 cm. The pressure of gas as it leaves the tube is 1.00 bar. The volume of the gas is measured at that pressure.
Calculate the viscosity of air at(a) 273 K,(b) 298 K,(c) 1000 K.Take σ ≈ 0.40 nm2. (The experimental values are 173 µP at 273 K, 182 µP at 20°C, and 394 µP at 600°C.)

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

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