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chemistry
physical chemistry

Physical Chemistry 7th Edition Peter Atkins - Solutions

Calculate the change in chemical potential of a perfect gas when its pressure is increased isothermally from 92.0 kPa to 252.0 kPa at 50°C.
Explain how colligative properties are used to determine molar mass.
The fugacity coefficient of a certain gas at 290 K and 2.1 MPa is 0.68. Calculate the difference of its molar Gibbs energy from that of a perfect gas in the same state.
Explain what is meant by a regular solution.
The partial molar volumes of two liquids A and B in a mixture in which the mole fraction of A is 0.3713 are 188.2 cm3 mol-1 and 176.14 cm3 rnol-1 respectively. The molar masses of A and Bare 241.1 g mol-1 and 198.2 g mol-1. What is the volume of a solution of mass 1.000 kg?
Estimate the change in the Gibbs energy of 1.0 dm3 of water when the pressure acting on it is increased from 100 kPa to 300 kPa.
Calculate the change in the molar Gibbs energy of oxygen when its pressure is increased isothermally from 50.0 kPa to 100.0 kPa at 500 K.
Calculate the difference in molar entropy(a) between liquid water and ice at -5°C,(b) between liquid water and its vapour at 95°C and 1.00 atm. The differences in heat capacities on melting and on vaporization are 37.3 T K-1 rnol " and -41.9 T K-I mol:", respectively. Distinguish between the
A block of copper of mass 2.00 kg (Cp, m = 24.44 T K-I mol-1) and temperature O°C is introduced into an insulated container in which there is 1.00 mol H20 (g) at 100°C and 1.00 atm.(a) Assuming all the steam is condensed to water, what will be the final temperature of the system, the heat
At 20°C, the density of a 20 per cent by mass ethanol-water solution is 968.7 kg m-3. Given that the partial molar volume of ethanol in the solution is 52.2 cm3 mol-1, calculate the partial molar volume of the water.
A Carnot cycle uses 1.00 mol of a monatomic perfect gas as the working substance from an initial state of 10.0 atm and 600 K. It expands isothermally to a pressure of 1.00 atm (Step 1), and then adiabatically to a temperature of 300 K (Step 2). This expansion is followed by an isothermal
At 310 K, the partial vapour pressures of a substance B dissolved in a liquid A are as follows: xB 0.010 0.015 0.020 Pg/kPa 82.0 122.0 166.1 Show that the solution obeys Henry's law in this range of mole fractions, and calculate Henry's law constant at 310 K.
Discuss the implications for phase stability of the variation of chemical potential with temperature and pressure.
Predict the partial vapour pressure of the component B above its solution in A in Exercise 5.3b when the molalityofB is 0.25 mol kg-I. The molar mass of A is 74.1 g mol-1.
Discuss what would be observed as a sample of water is taken along a path that encircles and is close to its critical point.
The vapour pressure of2-propanol is 50.00 kPa at 338.8°C, but it fell to 49.62 kPa when 8.69 g of an in volatile organic compound was dissolved in 250 g of 2-propanol. Calculate the molar mass of the compound.
The use of supercritical fluids for the extraction of a component from a complicated mixture is not confined to the decaffeination of coffee. Consult library and internet resources and prepare a discussion of the principles, advantages, disadvantages, and current uses of supercritical fluid
The addition of 5.00 g of a compound to 250 g of naphthalene lowered the freezing point of the solvent by 0.780 K. Calculate the molar mass of the compound.
The osmotic pressure of an aqueous solution at 288 K is 99.0 kPa. Calculate the freezing point of the solution.
Consider a container of volume 250 cm ' that is divided into two compartments of equal size. In the left compartment there is argon at lOOkPa and O°C; in the right compartment there is neon at the same temperature and pressure. Calculate the entropy and Gibbs energy of mixing when the partition is
Distinguish between a first-order phase transition, a second-order phase transition, and a le-transition at both molecular and macroscopic levels.
Calculate the Gibbs energy, entropy, and enthalpy of mixing when 1.00 mol C6HI4 (hexane) is mixed with 1.00 mol C7HI6 (heptane) at 298 K; treat the solution as ideal.
The vapour pressure of a substance at 20.0DCis 58.0 kPa and its enthalpy of vaporization is 32.7 k] mol-1. Estimate the temperature at which its vapour pressure is 66.0 kPa.
What proportions of benzene and ethylbenzene should be mixed (a) By mole fraction, (b) By mass in order to achieve the greatest entropy of mixing?
The molar volume of a certain solid is 142.0 cm-1 mol-1 at 1.00 atm and 427.15 K, its melting temperature. The molar volume of the liquid at this temperature and pressure is 152.6 cm-1 mol-1. At 1.2 MPa the melting temperature changes to 429.26 K. Calculate the enthalpy and entropy of fusion of the
The mole fractions of Nz and 0z in air at sea level are approximately 0.78 and 0.21. Calculate the molalities of the solution formed in an open flask of water at 25°C.
The vapour pressure of a liquid in the temperature range 200 K to 260 K was found to fit the expression in (p/Torr) = 18.361 - 3036.8/ (TIK). Calculate the enthalpy of vaporization of the liquid.
After some weeks of use, the pressure in the water carbonating plant mentioned in the previous exercise has fallen to 2.0 atm. Estimate the molar concentration of the soda water it produces at this stage.
The vapour pressure of a liquid between 15°C and 35°C fits the expression log(p/Torr) = 8.750 -1625/(T/K). Calculate. (a) The enthalpy of vaporization and (b) The normal boiling point of the liquid.
Predict the ideal solubility of lead in bismuth at 280°C given that its melting point is 327°C and its enthalpy of fusion is 5.2 k] mol-1.
When a certain liquid freezes at -3.65°C its density changes from 0.789 g cm-3 to 0.801 g cm-3 Its enthalpy of fusion is 8.68 k] mol-1. Estimate the freezing point of the liquid at 100 MPa.
The molar mass of an enzyme was determined by dissolving it in water, measuring the osmotic pressure at 20°C, and extrapolating the data to zero concentration. The following data were obtained:c/(mg cm-3)3.2214.6185.1126.722h/cm5.7468.2389.11911.990Calculate the molar mass of the enzyme.
Given that p*(HzO) = 0.02308 atm and p (HzO) = 0.02239 atm in a solution in which 0.122 kg of a non-volatile solute (M = 241 g mol-1) is dissolved in 0.920 kg water at 293 K, calculate the activity and activity coefficient of water in the solution.
Suppose the incident sunlight at ground level has a power density of 0.87 kW m-2 at noon. What is the maximum rate of loss of water from a lake of area 1.0 ha? (1 ha = 104 m2.) Assume that all the radiant energy is absorbed.
Benzene and toluene form nearly ideal solutions. The boiling point of pure benzene is 80.1*C, Calculate the chemical potential of benzene relative to that of pure benzene when xbmzenc = 0.30 at its boiling point. If the activity coefficient of benzene in this solution were actually 0.93 rather than
By measuring the equilibrium between liquid and vapour phases of a solution at 30°C at 1.00 atm, it was found that xA = 0.220 when lA = 0.314. Calculate the activities and activity coefficients of both components in this solution on the Raoult's law basis. The vapour pressures of the pure
Calculate the ionic strength of a solution that is 0.040 mol kg-I in K3 [Fe (CN) 6J (aq), 0.030 mol kg-1 in KCI (aq), and 0.050 mol kg3 in NaBr (aq).
Calculate the masses of (a) KNOJ and, separately, (b) Ba (N03lz to add to a 0.110 mol kg-l solution of KNOJ (aq) containing 500 g of solvent to raise its ionic strength to 1.00.
Estimate the mean ionic activity coefficient and activity of a solution that is 0.020 mol kg-I NaCI (aq) and 0.035 mol kg-I Ca (N03Maq).
The mean activity coefficients of KCI in three dilute aqueous solutions at 25°C are 0.927 (at 5.0 mmol kg-I), 0.902 (at 10.0 mmol kg-I), and 0.816 (at 50.0 mmol kg"). Estimate the value of B in the extended Debye-Huckel law,
In a study of the properties of an aqueous solution of Th(N03)4 (by A. Apelblat, D. Azoulay, and A. Sahar,]. Chem. Sac. Faraday Trans., I, 1618, (1973», a freezing point depression of 0.0703 K was observed for an aqueous solution of molality 9.6 mmol kg-I. What is the apparent number of ions per
At 18°C the total volume V of a solution formed from MgS04 and 1.000 kg of water fits the expression v = 1001.21 + 34.69(x - 0.070)2, where v = V/cm3 and x = blb-1. Calculate the partial molar volumes of the salt and the solvent when in a solution of molality 0.050 mol kg-1•
The following table gives the mole fraction of methylbenzene (A) in liquid and gaseous mixtures with butanone at equilibrium at 303.15 K and the total pressure p. Take the vapour to be perfect and calculate the partial pressures of the two components. Plot them against their respective mole
Use the Gibbs-Duhem equation to show that the partial molar volume (or any partial molar property) of a component B can be obtained if the partial molar volume (or other property) of A is known for all compositions up to the one of interest. Do this by proving that Use the following data (which are
The osmotic coefficient' ф defined as ф = - (xA/xB) In aA By writing r =xSlxA, and using the Gibbs-Duhem equation, show that we can calculate the activity of B from the activities of A over a composition range by using the formula
Show that the freezing-point depression of a real solution in which the solvent of molar mass M has activity aA obeys dInaA/d (T) = - M/Kf and use the Gibbs-Duhem equation to show that dInaB/d (T) = - 1/bBKf where aB is the solute activity and bB is its molality. Use the Debye-HUckel limiting law
For the calculation of the solubility c of a gas in a solvent, it is often convenient to use the expression c=Kp, where K is the Henry's law constant. Breathing air at high pressures, such as in scuba diving, results in an increased concentration of dissolved nitrogen. The Henry's law constant for
The form of the Scat chard equation given in Impact 15.2 applies only when the macromolecule has identical and independent binding sites. For non-identical independent binding sites, the Scat chard equation is Plot v/rAJ for the following cases.(a) There are four independent sites on an enzyme
Polymer scientists often report their data in rather strange units. For example, in the determination of molar masses of polymers in solution by osmometry, osmotic pressures are often reported in grams per square centimeter (g cm") and concentrations in grams per cubic centimeter (g cm3).(a) With
What proportions of ethanol and water should be mixed in order to produce 100 cm:' of a mixture containing 50 per cent by mass of ethanol? What change in volume is brought about by adding 1.00 cm3 of ethanol to the mixture? (Use data from Fig. 5.1.)
For a first-order phase transition, to which the Clapeyron equation does apply, prove the relation C3 = CP - aVurH/urs V Where Cs = (∂q/∂T) s is the heat capacity along the coexistence curve of two phases.
Plot the vapour pressure data for a mixture of benzene (B) and acetic acid (A) given below and plot the vapour pressure/composition curve for the mixture at 50°e. Then confirm that Raoult's and Henry's laws are obeyed in the appropriate regions. Deduce the activities and activity coefficients of
Comelli and Francesconi examined mixtures of propionic acid with various other organic liquids at 313.15 K (F. Comelli and R. Francesconi, Chem. Eng. Data 41,101 (1996)). They report the excess volume of mixing propionic acid with oxane as VE = x1X: 2{aO + a, (x, - X:2)}, where x, is the mole
Chen and Lee studied the liquid-vapour equilibria of cyclohexanol with several gases at elevated pressures 0.-T. Chen and M.-]. Lee,]. Chem, Eng Data 41, 339 (1996)). Among their data are the following measurements of the mole fractions of cyclohexanol in the vapour phase (y) and the liquid phase
The excess Gibbs energy of solutions of methylcyclohexane (MCH) and tetrahydrofuran (THF) at 303.15 K was found to fit the expression GE=RTx (1- x) {0.4857 - 0.1077(2x -1) + 0.0191(2x- 1)2} where x is the mole fraction of the methylcyclohexane. Calculate the Gibbs energy of mixing when a mixture of
The excess Gibbs energy of a certain binary mixture is equal to gRTx (1- x) where g is a constant and x is the mole fraction of a solute A.
Explain how the perfect gas equation of state arises by combination of Boyle's law, Charles's law, and Avogadro's principle.
Explain how the compression factor varies with pressure and temperature and describe how it reveals information about intermolecular interactions in real gases.
Describe the formulation of the van der Waals equation and suggest a rationale for one other equation of state in Table 1.7.
Estimate the coefficients a and b in the Dieterici equation of state from the critical constants of xenon. Determine pressure exerted by 1.0 mol Xe when it is confined to 1.0 dm3 at 25°C.
(a) Could 25 g of argon gas in a vessel of volume 1.5 dm3 exert a pressure of 2.0 bar at 30°C if it behaved as a perfect gas? If not, what pressure would it exert? (b) What pressure would it exert if it behaved as a van der Waals gas?
A perfect gas undergoes isothermal compression, which reduces its volume by 1.80 dm3. The final pressure and volume of the gas are 1.97 bar and 2.14 dm3, respectively. Calculate the original pressure of the gas in(a) bar,(b) Torr.
A sample of hydrogen gas was found to have a pressure of 125 kPa when the temperature was 23°C. What can its pressure be expected to be when the temperature is 11°C?
A homeowner uses 4.00 x 10J mJ of natural gas in a year to heat a home. Assume that natural gas is all methane, CH4, and that methane is a perfect gas for the conditions of this problem, which are 1.00 atm and 20°C. What is the mass of gas used?
What pressure difference must be generated across the length of a 15 cm vertical drinking straw in order to drink a water-like liquid of density 1.0 g cm-3?
A manometer like that described in Exercise 1.6a contained mercury in place of water. Suppose the external pressure is 760 Torr, and the open side is 10.0 cm higher than the side connected to the apparatus. What is the pressure in the apparatus? (The density of mercury at 25°C is 13.55 g cm-3)
The following data have been obtained for oxygen gas at 273.15 K. Calculate the best value of the gas constant R from them and the best value of the molar mass of02•
At 100°C and 1.60 kPa, the mass density of phosphorus vapour is 0.6388 kg m-3. What is the molecular formula of phosphorus under these conditions?
Calculate the mass of water vapour present in a room of volume 250 m3 that contains air at 23°C on a day when the relative humidity is 53 per cent.
A gas mixture consists of 320 mg of methane, 175 mg of argon, and 225 mg of neon. The partial pressure of neon at 300 K is 8.87 kPa. Calculate (a) The volume and (b) The total pressure of the mixture.
In an experiment to measure the molar mass of a gas, 250 cm3 of the gas was confined in a glass vessel. The pressure was 152 Torr at 298 K and, after correcting for buoyancy effects, the mass of the gas was 33.5 mg. What is the molar mass of the gas?
A certain sample of a gas has a volume of20.00 dm ' at O°Cand 1.000 atm. A plot of the experimental data of its volume against the Celsius temperature, θ, at constant p, gives a straight line of slope 0.0741 dm3 (oC)-1 from these data alone (without making use of the perfect gas law), determine
A certain gas obeys the van der Waals equation with a =0.76 m6 Pa mol-2, its volume is found to be 4.00 X 10-4 m3 mol-1 at 288 K and 4.0 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?
Calculate the pressure exerted by 1.0 mol H2S behaving as (a) A perfect gas, (b) A van der Waals gas when it is confined under the following conditions: (i) At 273.15 K in 22.414 dm3, (ii) At 500 Kin 150 em3:'. Use the data in Table 1.6.
Express the van der Waals parameters a = 1.32 atm dm6 mol? And b = 0.0436 d3 mol-1 in SI base units.
A gas at 350 K and 12 atm has a molar volume 12 per cent larger than that calculated from the perfect gas law. Calculate (a) The compression factor under these conditions and (b) The molar volume of the gas. Which are dominating in the sample, the attractive or the repulsive forces?
Cylinders of compressed gas are typically filled to a pressure of 200 bar. For oxygen, what would be the molar volume at this pressure and 25°C based on (a) The perfect gas equation, (b) The van der Waals equation. For oxygen, a = 1.364 dm" atm mol-2, b = 3.19 X 10-2 dm3 mol-1
At 300 K and 20 atm, the compression factor of a gas is 0.86. Calculate (a) The volume occupied by 8.2 mmol of the gas under these conditions and (b) An approximate value of the second virial coefficient B at 300 K.
A vessel of volume 22.4 dm ' contains 1.5 mol H2 and 2.5 mol N2 at 273.15 K. Calculate (a) The mole fractions of each component, (b) Their partial pressures, and (c) Their total pressure.
The critical constants of ethane are Pc =48.20 atm, Vc = 148 cm? Mol 2, and T; = 305.4 K. Calculate the van der Waals parameters of the gas and estimate the radius of the molecules.
Use the van der Waals parameters for hydrogen sulfide to calculate approximate values of (a) The Boyle temperature of the gas and (b) The radius of a H2S molecule regarded as a sphere (a = 4.484 dm6 atm mol-2, b = 0.0434 d3 mol-1).
Suggest the pressure and temperature at which 1.0 mol of (a) H2S, (b) CO2, (c) Ar will be in states that correspond to 1.0 mol Nz at 1.0 atm and 25°e.
A scientist proposed the following equation of state: p = RT/Vm - B/v2m + Cv3m Show that the equation leads to critical behaviour. Find the critical constants of the gas in terms of Band C and an expression for the critical compression factor.
Recent communication with the inhabitants of Neptune have revealed that they have a Celsius-type temperature scale, but based on the melting point (OoN)and boiling point (I OOoN)of their most common substance, hydrogen. Further communications have revealed that the Neptunian’s know about perfect
Charles's law is sometimes expressed in the form V = Vo (l + aθ), where θ is the Celsius temperature, a is a constant, and Vo is the volume of the sample at O°e. The following values for a have been reported for nitrogen at O°C: P/Torr 749.7 599.6 333.1 98.6 103a/
A constant-volume perfect gas thermometer indicates a pressure of 6.69 kPa at the triple point temperature of water (273.16 K). (a) What change of pressure indicates a change of 1.00 K at this temperature? (b) What pressure indicates a temperature of 100.00°C? (c) What change of pressure
Calculate the molar volume of chlorine gas at 350 K and 2.30 atm using (a) The perfect gas law and (b) The van der Waals equation. Use the answer to (a) to calculate a first approximation to the correction term for attraction and then use successive approximations to obtain a numerical answer
Calculate the volume occupied by 1.00 mol N2 using the van der Waals equation in the form of a virial expansion at (a) Its critical temperature, (b) Its Boyle temperature, and (c) Its inversion temperature. Assume that the pressure is 10 atm throughout. At what temperature is the gas most
The mass density of water vapour at 327.6 atm and 776.4 K is 133.2 kg m-3. Given that for water Tc = 647.4 K, Pc = 218.3 atm, a = 5.464 dm6 atm mol-2, b= 0.03049 dm3 mol-1, and M= 18.02 g mol-1, calculate (a) The molar volume. Then calculate the compression factor (b) From the data, (c) From
Estimate the coefficients a and b in the Dieterici equation of state from the critical constants of xenon. Calculate the pressure exerted by 1.0 mol Xe when it is confined to 1.0 dm3 at 25°C.
Express the van der Waals equation of state as a virial expansion in powers of 1/Vm and obtain expressions for Band C in terms of the parameters a and b. The expansion you will need is (1- xtI = 1 + x + xl + .... Measurements on argon gave B=-21.7 cm3 mol-1 and C= 1200 cm6 mol-2 for the virial
The second virial coefficient B' can be obtained from measurements of the density p of a gas at a series of pressures. Show that the graph of p/ p against p should be a straight line with slope proportional to B'. Use the data on dimethyl ether in Problem 1.2 to find the values of B' and B at 25°e.
The following equations of state are occasionally used for approximate calculations on gases: (gas A) p Vm = RT(1 + b/V m)' (gas B) p(V m - b) = RT. Assuming that there were gases that actually obeyed these equations of state, would it be possible to liquefy either gas A or B? Would they have a
The discovery of the element argon by Lord Rayleigh and Sir William Ramsay had its origins in Rayleigh's measurements of the density of nitrogen with an eye toward accurate determination of its molar mass. Rayleigh prepared some samples of nitrogen by chemical reaction of nitrogen containing
Atmospheric pollution is a problem that has received much attention. Not all pollution, however, is from industrial sources. Volcanic eruptions can be a significant source of air pollution. The Kilauea volcano in Hawaii emits 200-300 t of 502 per day. If this gas is emitted at 800°C and 1.0 atm,
The barometric formula relates the pressure of a gas of molar mass Mat an altitude h to its pressure Po at sea level. Derive this relation by showing that the change in pressure dp for an infinitesimal change in altitude dh where the density is p is dp =-pgdh. Remember that p depends on the
The preceding problem is most readily solved (see the Solutions manual) with the use of the Archimedes principle, which states that the lifting force is equal to the difference between the weight of the displaced air and the weight of the balloon. Prove the Archimedes principle for the atmosphere
The standard molar entropy of NHJ (g) is 192.45 T K-I mol-1 at 298 K, and its heat capacity is given by eqn 2.25 with the coefficients given in Table 2.2. Calculate the standard molar entropy at (a) lOO°C and (b) 500°C.
Find an expression for the change in entropy when two blocks of the same substance and of equal mass, one at the temperature Th and the other at T" are brought into thermal contact and allowed to reach equilibrium. Evaluate the change for two blocks of copper, each of mass 500 g, with Cp, m = 24.4
The molar heat capacity of lead varies with temperature as follows: TIK 10 15 20 25 30 50 Cp,m/O K-1 rnol-1) 2.8 7.0 10.8 14.1 16.5 21.4 TIK 70 100 150 200 250 298 Cp,m/OK-1 mol-1) 23.3 24.5 25.3 25.8 26.2 26.6 Calculate the standard Third-Law entropy of lead at (a) O°C and (b)

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