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

Confirm that the bonding and antibonding combinations ψA ± ψB are mutually orthogonal in the sense that their mutual overlap is zero.
Use the electron configurations of NO and N2 to predict which is likely to have the shorter bond length.
The overlap integral between two H1s orbitals on nuclei separated by a distance R is S = {1 + (R/a0) + 1/3 (R/a0)2 }e−R/a0. Plot this function for 0 ≤ R < ∞.
Give the ground-state electron configurations and bond orders of(a) Li2,(b) Be2, and(c) C2.
Discuss the steps involved in the construction of sp3, sp2, and sp hybrid orbitals.
In the ‘free electron molecular orbital’ (FEMO) theory, the electrons in a conjugated molecule are treated as independent particles in a box of length L. Sketch the form of the two occupied orbitals in butadiene predicted by this model and predict the minimum excitation energy of the molecule.
Write down the secular determinants for(a) Linear H3,(b) Cyclic H3 within the Hückel approximation.
Construct the molecular orbital energy level diagrams of ethene on the basis that the molecule is formed from the appropriately hybridized CH2 or CH fragments.
Give the ground-state electron configurations of(a) CO,(b) NO, and(c) CN−.
The partial molar volumes of acetone (propanone) and chloroform (trichloromethane) in a mixture in which the mole fraction of CHCl3 is 0.4693 are 74.166 cm3 mol−1 and 80.235 cm3 mol−1, respectively. What is the volume of a solution of mass 1.000 kg?
At 300 K, the partial vapour pressures of HCl (that is, the partial pressure of the HCl vapour) in liquid GeCl4 are as follows:Show that the solution obeys Henry’s law in this range of mole fractions, and calculate Henry’s law constant at 300 K. ХНСI PHO/kPa 0.005 32.0 0.012 76.9 0.019 121.8
How is Raoult’s law modified so as to describe the vapour pressure of real solutions?
Explain the origin of colligative properties.
The vapour pressure of benzene is 53.3 kPa at 60.6°C, but it fell to 51.5 kPa when 19.0 g of an involatile organic compound was dissolved in 500 g of benzene. Calculate the molar mass of the compound.
Describe the general features of the Debye–Hückel theory of electrolyte solutions.
The addition of 100 g of a compound to 750 g of CCl4 lowered the freezing point of the solvent by 10.5 K. Calculate the molar mass of the compound.
The osmotic pressure of an aqueous solution at 300 K is 120 kPa. Calculate the freezing point of the solution.
The table below lists the vapour pressures of mixtures of iodoethane (I) and ethyl acetate (A) at 50°C. Find the activity coefficients of both components on(a) The Raoult’s law basis,(b) The Henry’s law basis with iodoethane as solute. XI 0 P₁/kPa 0 PA/kPa 37.38 X1 0.5478 P₁/kPa
Consider a container of volume 5.0 dm3 that is divided into two compartments of equal size. In the left compartment there is nitrogen at 1.0 atm and 25°C; in the right compartment there is hydrogen at the same temperature and pressure. Calculate the entropy and Gibbs energy of mixing when the
What proportions of hexane and heptane should be mixed (a) By mole fraction,(b) By mass in order to achieve the greatest entropy of mixing?
A water carbonating plant is available for use in the home and operates by providing carbon dioxide at 5.0 atm. Estimate the molar concentration of the soda water it produces.
The osmotic pressure of solutions of polystyrene in toluene were measured at 25°C and the pressure was expressed in terms of the height of the solvent of density 1.004 g cm−3:Calculate the molar mass of the polymer. c/(g dm-³) h/cm 2.042 6.613 0.592 1.910 9.521 2.750 12.602 3.600
The mean activity coefficients for aqueous solutions of NaCl at 25°C are given below. Confirm that they support the Debye–Hückel limiting law and that an improved fit is obtained with the extended law. b/(mmol kg ¹) 土 1.0 0.9649 2.0 0.9519 5.0 0.9275 10.0 0.9024 20.0 0.8712
The enthalpy of fusion of anthracene is 28.8 kJ mol−1 and its melting point is 217°C. Calculate its ideal solubility in benzene at 25°C.
Use the Gibbs–Duhem equation to derive the Gibbs–Duhem–Margules equationwhere f is the fugacity. Use the relation to show that, when the fugacities are replaced by pressures, if Raoult’s law applies to one component in a mixture it must also apply to the other. a Infa Ə In xA P,T a lnfB a
Calculate the ionic strength of a solution that is 0.10 mol kg−1 in KCl(aq) and 0.20 mol kg−1 in CuSO4(aq).
Estimate the mean ionic activity coefficient and activity of a solution that is 0.010 mol kg−1 CaCl2(aq) and 0.030 mol kg−1 NaF(aq).
Use the Gibbs–Helmholtz equation to find an expression for d ln xA in terms of dT. Integrate d ln xA from xA = 0 to the value of interest, and integrate the right–hand side from the transition temperature for the pure liquid A to the value in the solution. Show that, if the enthalpy of
The mean activity coefficients of HBr in three dilute aqueous solutions at 25°C are 0.930 (at 5.0 mmol kg−1), 0.907 (at 10.0 mmol kg−1), and 0.879 (at 20.0 mmol kg−1). Estimate the value of B in the extended Debye–Hückel law.
K. Sato, F.R. Eirich, and J.E. Mark (J. Polymer Sci., Polym. Phys. 14, 619 (1976)) have reported the data in the table below for the osmotic pressures of polychloroprene (ρ = 1.25 g cm−3) in toluene (ρ = 0.858 g cm−3) at 30°C. Determine the molar mass of polychloroprene and its second
Haemoglobin, the red blood protein responsible for oxygen transport, binds about 1.34 cm3 of oxygen per gram. Normal blood has a haemoglobin concentration of 150 g dm−3. Haemoglobin in the lungs is about 97 per cent saturated with oxygen, but in the capillary is only about 75 per cent
Label the regions of the phase diagram in Fig. 6.38. State what substances (if compounds give their formulas) exist in each region. Label each substance in each region as solid, liquid, or gas. Fig. 6.38. Temperature, T 0 T 0.2 0.33 0.4 X₂ 0.6 0.67 0.8 1
What factors determine the number of theoretical plates required to achieve a desired degree of separation in fractional distillation?
It is found that the boiling point of a binary solution of A and B with xA = 0.6589 is 88°C. At this temperature the vapour pressures of pure A and B are 127.6 kPa and 50.60 kPa, respectively.(a) Is this solution ideal?(b) What is the initial composition of the vapour above the solution?
Phosphorus and sulfur form a series of binary compounds. The best characterized are P4S3, P4S7, and P4S10, all of which melt congruently. Assuming that only these three binary compounds of the two elements exist,(a) draw schematically only the P/S phase diagram. Label each region of the diagram
Consider the phase diagram in Fig. 6.44, which represents a solid–liquid equilibrium. Label all regions of the diagram according to the chemical species that exist in that region and their phases. Indicate the number of species and phases present at the points labelled b, d, e, f, g, and k.
State the number of components in the following systems. (a) NaH2PO4 in water at equilibrium with water vapour but disregarding the fact that the salt is ionized.(b) The same, but taking into account the ionization of the salt.
Blue CuSO4·5H2O crystals release their water of hydration when heated. How many phases and components are present in an otherwise empty heated container?
Methylethyl ether (A) and diborane, B2H6 (B), form a compound that melts congruently at 133 K. The system exhibits two eutectics, one at 25 mol per cent B and 123 K and a second at 90 mol per cent B and 104 K. The melting points of pure A and B are 131 K and 110 K, respectively. Sketch the phase
Iron(II) chloride (melting point 677°C) and potassium chloride (melting point 776°C) form the compounds KFeCl3 and K2FeCl4 at elevated temperatures. KFeCl3 melts congruently at 380°C and K2FeCl4 melts incongruently at 399°C. Eutectics are formed with compositions x = 0.38 (melting point 351°C)
Indicate on the phase diagram in Fig. 6.41 the feature that denotes incongruent melting. What is the composition of the eutectic mixture and at what temperature does it melt?Fig. 6.41 Temperature, 9/°C T, 2 0 0.2 500° 400° 300℃ 0.4 XB a: e 0.6 b 0.8 T, 1
Use the phase diagram in Fig. 6.40 to state(a) The solubility of Ag in Sn at 800°C and(b) The solubility of Ag3Sn in Ag at 460°C,(c) The solubility of Ag3Sn in Ag at 300°C.Fig. 6.40 Temperature, 0/°C 1000 800 600 400 200 0 Liquid O. b Ag,Sn 60 a 20 40 80 Mass percentage Ag/% 100
Show that two phases are in mechanical equilibrium only if their pressures are equal.
Methane (melting point 91 K) and tetrafluoromethane (melting point 89 K) do not form solid solutions with each other, and as liquids they are only partially miscible. The upper critical temperature of the liquid mixture is 94 K at x(CF4) = 0.43 and the eutectic temperature is 84 K at x(CF4) = 0.88.
Refer to the information in Exercise 6.15(b) and sketch the cooling curves for liquid mixtures in which x(B2H6) is(a) 0.10,(b) 0.30,(c) 0.50,(d) 0.80, and(e) 0.95.Data in Exercise 6.15(b) Describe the phase changes that take place when a liquid mixture of 4.0 mol B2H6 (melting point 131 K) and
Some polymers can form liquid crystal mesophases with unusual physical properties. For example, liquid crystalline Kevlar (3) is strong enough to be the material of choice for bulletproof vests and is stable at temperatures up to 600 K. What molecular interactions contribute to the formation,
Hexane and perfluorohexane show partial miscibility below 22.70°C. The critical concentration at the upper critical temperature is x = 0.355, where x is the mole fraction of C6F14. At 22.0°C the two solutions in equilibrium have x = 0.24 and x = 0.48, respectively, and at 21.5°C the mole
A sample consisting of 1.00 mol Ar is expanded isothermally at 0°C from 22.4 dm3 to 44.8 dm3(a) Reversibly,(b) Against a constant external pressure equal to the final pressure of the gas, and(c) Freely (against zero external pressure). For the three processes calculate q, w, ∆U, and ∆H.
A strip of magnesium of mass 15 g is dropped into a beaker of dilute hydrochloric acid. Calculate the work done by the system as a result of the reaction. The atmospheric pressure is 1.0 atm and the temperature 25°C.
A vessel of volume 22.4 dm3 contains 2.0 mol H2 and 1.0 mol N2 at 273.15 K. Calculate(a) The mole fractions of each component,(b) Their partial pressures, and(c) Their total pressure.
Deduce the relation between the pressure and mass density, ρ, of a perfect gas of molar mass M. Confirm graphically, using the following data on dimethyl ether at 25°C, that perfect behaviour is reached at low pressures and find the molar mass of the gas. p/kPa p/(kg m-³) 12.223 0.225 25.20
Explain the term ‘partial pressure’ and explain why Dalton’s law is a limiting law.
(a) A car tyre (i.e. an automobile tire) was inflated to a pressure of 24 lb in−2 (1.00 atm = 14.7 lb in−2) on a winter’s day when the temperature was –5°C. What pressure will be found, assuming no leaks have occurred and that the volume is constant, on a subsequent summer’s day when the
What is the significance of the critical constants?
A sample of 255 mg of neon occupies 3.00 dm3 at 122 K. Use the perfect gas law to calculate the pressure of the gas.
The molar mass of a newly synthesized fluorocarbon was measured in a gas microbalance. This device consists of a glass bulb forming one end of a beam, the whole surrounded by a closed container. The beam is pivoted, and the balance point is attained by raising the pressure of gas in the container,
A diving bell has an air space of 3.0 m3 when on the deck of a boat. What is the volume of the air space when the bell has been lowered to a depth of 50 m? Take the mean density of sea water to be 1.025 g cm−3 and assume that the temperature is the same as on the surface.
A manometer consists of a U-shaped tube containing a liquid. One side is connected to the apparatus and the other is open to the atmosphere. The pressure inside the apparatus is then determined from the difference in heights of the liquid. Suppose the liquid is water, the external pressure is 770
At 500°C and 93.2 kPa, the mass density of sulfur vapour is 3.710 kg m−3. What is the molecular formula of sulfur under these conditions?
At 273 K measurements on argon gave B = −21.7 cm3 mol−1 and C = 1200 cm6 mol−2 , where B and C are the second and third virial coefficients in the expansion of Z in powers of 1/Vm. Assuming that the perfect gas law holds sufficiently well for the estimation of the second and third terms of
Calculate the mass of water vapour present in a room of volume 400 m3 that contains air at 27°C on a day when the relative humidity is 60 per cent.
The second virial coefficient of methane can be approximated by the empirical equation B′(T) = a + be−c/T2, where a = −0.1993 bar−1, b = 0.2002 bar−1, and c = 1131 K2 with 300 K < T < 600 K. What is the Boyle temperature of methane?
The density of a gaseous compound was found to be 1.23 kg m−3 at 330 K and 20 kPa. What is the molar mass of the compound?
The densities of air at −85°C, 0°C, and 100°C are 1.877 g dm−3, 1.294 g dm−3, and 0.946 g dm−3, respectively. From these data, and assuming that air obeys Charles’s law, determine a value for the absolute zero of temperature in degrees Celsius.
The critical volume and critical pressure of a certain gas are 160 cm3 mol−1 and 40 atm, respectively. Estimate the critical temperature by assuming that the gas obeys the Berthelot equation of state. Estimate the radii of the gas molecules on the assumption that they are spheres.
Express the van der Waals parameters a = 0.751 atm dm6 mol−2 and b = 0.0226 dm3 mol−1 in SI base units.
Show that the van der Waals equation leads to values of Z < 1 and Z > 1, and identify the conditions for which these values are obtained.
In an industrial process, nitrogen is heated to 500 K at a constant volume of 1.000 m3. The gas enters the container at 300 K and 100 atm. The mass of the gas is 92.4 kg. Use the van der Waals equation to determine the approximate pressure of the gas at its working temperature of 500 K. For
Derive the relation between the critical constants and the Dieterici equation parameters. Show that Zc = 2e−2 and derive the reduced form of the Dieterici equation of state. Compare the van der Waals and Dieterici predictions of the critical compression factor. Which is closer to typical
Suppose that 10.0 mol C2H6(g) is confined to 4.860 dm3 at 27°C. Predict the pressure exerted by the ethane from(a) The perfect gas (b) The van der Waals equations of state. Calculate the compression factor based on these calculations. For ethane, a = 5.507 dm6 atm mol−2, b = 0.0651 dm3
The critical constants of methane are pc = 45.6 atm, Vc = 98.7 cm3 mol−1, and Tc = 190.6 K. Calculate the van der Waals parameters of the gas and estimate the radius of the molecules.
Use the van der Waals parameters for chlorine to calculate approximate values of(a) The Boyle temperature of chlorine and(b) The radius of a Cl2 molecule regarded as a sphere.
The equation of state of a certain gas is given by p = RT/Vm + (a + bT)/V2m, where a and b are constants. Find (∂V/∂T)p.
Suggest the pressure and temperature at which 1.0 mol of (a) NH3,(b) Xe,(c) He will be in states that correspond to 1.0 mol H2 at 1.0 atm and 25°C.
A certain gas obeys the van der Waals equation with a = 0.50 m6 Pa mol−2. Its volume is found to be 5.00 × 10−4 m3 mol−1 at 273 K and 3.0 MPa. From this information calculate the van der Waals constant b. What is the compression factor for this gas at the prevailing temperature and pressure?
Derive an expression for the compression factor of a gas that obeys the equation of state p(V – nb) = nRT, where b and R are constants. If the pressure and temperature are such that Vm = 10b, what is the numerical value of the compression factor?
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 3.0 m and that it is spherical.(a) What
Chlorofluorocarbons such as CCl3F and CCl2F2 have been linked to ozone depletion in Antarctica. As of 1994, these gases were found in quantities of 261 and 509 parts per trillion (1012) by volume (World Resources Institute, World resources 1996–97). Compute the molar concentration of these gases
Calculate the work needed for a 65 kg person to climb through 4.0 m on the surface of(a) The Earth and(b) The Moon (g = 1.60 m s−2).
A chemical reaction takes place in a container of cross-sectional area 100 cm2. As a result of the reaction, a piston is pushed out through 10 cm against an external pressure of 1.0 atm. Calculate the work done by the system.
A sample consisting of 1.0mol CaCO3(s) was heated to 800°C, when it decomposed. The heating was carried out in a container fitted with a piston that was initially resting on the solid. Calculate the work done during complete decomposition at 1.0 atm. What work would be done if instead of having a
The heat capacity of chloroform (trichloromethane, CHCl3) in the range 240 K to 330 K is given by Cp,m /(J K−1 mol−1) = 91.47 + 7.5 × 10−2 (T/K). In a particular experiment, 1.00 mol CHCl3 is heated from 273 K to 300 K. Calculate the change in molar entropy of the sample.
Explain the significance of a physical observable being a state function and compile a list of as many state functions as you can identify.
A sample of 70 mmol Kr(g) expands reversibly and isothermally at 373 K from 5.25 cm3 to 6.29 cm3 , and the internal energy of the sample is known to increase by 83.5 J. Use the virial equation of state up to the second coefficient B = −28.7 cm3 mol−1 to calculate w, q, and ∆H for this change
A sample of 4.50 g of methane occupies 12.7 dm3 at 310 K. (a) Calculate the work done when the gas expands isothermally against a constant external pressure of 200 Torr until its volume has increased by 3.3 dm3 .(b) Calculate the work that would be done if the same expansion occurred
Suggest (with explanation) how the internal energy of a van der Waals gas should vary with volume at constant temperature.
A sample of 1.00 mol H2O(g) is condensed isothermally and reversibly to liquid water at 100°C. The standard enthalpy of vaporization of water at 100°C is 40.656 kJ mol−1. Find w,q, ∆U, and ∆H for this process.
Calculate the work done during the isothermal reversible expansion of a van der Waals gas. Account physically for the way in which the coefficients a and b appear in the final expression. Plot on the same graph the indicator diagrams for the isothermal reversible expansion of(a) A perfect gas,(b) A
Calculate the final temperature of a sample of argon of mass 12.0 g that is expanded reversibly and adiabatically from 1.0 dm3 at 273.15 K to 3.0 dm3.
A sample of carbon dioxide of mass 2.45 g at 27.0°C is allowed to expand reversibly and adiabatically from 500 cm3 to 3.00 dm3. What is the work done by the gas?
Calculate the final pressure of a sample of carbon dioxide that expands reversibly and adiabatically from 57.4 kPa and 1.0 dm3 to a final volume of 2.0 dm3 . Take γ = 1.4.
When 1.3584 g of sodium acetate trihydrate was mixed into 100.0 cm3 of 0.2000 m HCl(aq) at 25°C in a solution calorimeter, its temperature fell by 0.397°C on account of the reaction:The heat capacity of the calorimeter is 91.0 J K−1 and the heat capacity density of the acid solution is 4.144 J
When 229 J of energy is supplied as heat to 3.0 mol Ar(g), the temperature of the sample increases by 2.55 K. Calculate the molar heat capacities at constant volume and constant pressure of the gas.
A sample of 4.0 mol O2 is originally confined in 20 dm3 at 270 K and then undergoes adiabatic expansion against a constant pressure of 600 Torr until the volume has increased by a factor of 3.0. Calculate q, w, ∆T, ∆U, and ∆H. (The final pressure of the gas is not necessarily 600 Torr.)
A sample consisting of 1.0 mol of perfect gas molecules with CV = 20.8 J K−1 is initially at 3.25 atm and 310 K. It undergoes reversible adiabatic expansion until its pressure reaches 2.50 atm. Calculate the final volume and temperature and the work done.
The standard enthalpy of formation of ethylbenzene is −12.5 kJ mol−1. Calculate its standard enthalpy of combustion.
The standard enthalpy of combustion of cyclopropane is −2091 kJ mol−1 at 25°C. From this information and enthalpy of formation data for CO2(g) and H2O(g), calculate the enthalpy of formation of cyclopropane. The enthalpy of formation of propene is +20.42 kJ mol−1. Calculate the enthalpy of
A sample consisting of 1.00 mol of a van der Waals gas is compressed from 20.0 dm3 to 10.0 dm3 at 300 K. In the process, 20.2 kJ of work is done on the gas. Given that µ = {(2a/RT) − b}/Cp,m, with Cp,m = 38.4 J K−1 mol−1 , a = 3.60 dm6 atm mol−2, and b = 0.44 dm3 mol−1, calculate ∆H
When 120 mg of naphthalene, C10H8(s), was burned in a bomb calorimeter the temperature rose by 3.05 K. Calculate the calorimeter constant. By how much will the temperature rise when 10 mg of phenol, C6H5OH(s), is burned in the calorimeter under the same conditions?
Calculate the standard enthalpy of solution of AgCl(s) in water from the enthalpies of formation of the solid and the aqueous ions.

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