3-1 Simple mixture
5.1.2 The thermodynamics of mixing
Example 5.1 Calculating the Gibbs energy of mixing
Calculate the effect on the chemical potentials of ice and water of increasing
the pressure from 1.00 bar to 2.00 bar at 0C. The density of ice is 0.917 g
cm3 and that of liquid water is 0.999 g cm3 under these conditions.
Self-test 4.1
Suppose that 2.0 mol H2 at 2.0 atm and 25C and 4.0 mol N2 at 3.0 atm and
25C are mixed at constant volume. Calculate mixG. What would be the value
of mixG had the pressures been identical initially?
5.2.4 Liquid mixtures
Example 5.2 Investigating the validity of Raoults and Henrys law
The vapour pressures of each component in a mixture of propanone (acetone,
A) and trichloromethane (chloroform, C) were measured at 35C with the
following results:
Confirm that the mixture conforms to Raoults law for the component in large
excess and to Henrys law for the minor component. Find the Henrys law
constants.
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Self-test 5.4
The vapour pressure of chloromethane at various mole fractions in a mixture
at 25C was found to be as follows:
Estimate Henrys law constant.
A brief illustration
Estimate the molar solubility of oxygen in water at 25C and a partial pressure of 21
kPa, its partial pressure in the atmosphere at sea level. the mass density of this dilute
solution is essentially that of pure water at 25C, or
Self-test 5.4
The mass percentage composition of dry air at sea level is approximately N2:
75.5; O2: 23.2; Ar: 1.3. Calculate the partial pressures and the molar
solubility of nitrogen in water exposed to air at 25C.
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5.2.5Colligative properties
Example 5.4 Using osmometry to determine the molar mass of a macromolecule
The osmotic pressures of solutions of poly(vinyl chloride), PVC, in
cyclohexanone at 298 K are given below. The pressures are expressed in
terms of the heights of solution (of mass density = 0.980 g cm3) in balance
with the osmotic pressure. Determine the molar mass of the polymer.
Self-test 5.6
Estimate the depression of freezing point of the most concentrated of these
solutions, taking Kf as about 10 K/(mol kg1).
3-1 Simple mixture
Physical Chemistry II
Chapter 5 Simple mixture
Exercise
5.1(a) 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 mol1 and 80.235 cm3 mol1, respectively. What is the
volume of a solution of mass 1.000 kg?
5.1(b) 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 mol1 and 176.14 cm3 mol1,
respectively. The molar masses of the A and B are 241.1 g mol1 and 198.2 g
mol1. What is the volume of a solution of mass 1.000 kg?
5.2(a) At 25C, the density of a 50 per cent by mass ethanolwater solution
is 0.914 g cm3. Given that the partial molar volume of water in the solution
is 17.4 cm3 mol1, calculate the partial molar volume of the ethanol.
5.2(b) At 20C, the density of a 20 per cent by mass ethanol/water solution
is 968.7 kg m3. Given that the partial molar volume of ethanol in the solution
is 52.2 cm3 mol1, calculate the partial molar volume of the water.
5.3(a) At 300 K, the partial vapour pressures of HCl (that is, the partial
pressure of the HCl vapour) in liquid GeCl4 are as follows:
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Show that the solution obeys Henrys law in this range of mole fractions, and
calculate Henrys law constant at 300 K.
5.3(b) At 310 K, the partial vapour pressures of a substance B dissolved in a
liquid A are as follows:
Show that the solution obeys Henrys law in this range of mole fractions, and
calculate Henrys law constant at 310 K.
5.4(a) Predict the partial vapour pressure of HCl above its solution in liquid
germanium tetrachloride of molality 0.10 mol kg1. For data, see Exercise
5.3a. 5.4(b) Predict the partial vapour pressure of the component B above
its solution in A in Exercise 3(b) when the molality of B is 0.25 mol kg1. The
molar mass of A is 74.1 g mol1.
5.5(a) The vapour pressure of benzene is 53.3 kPa at 60.6C, 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. 5.5(b) The
vapour pressure of 2-propanol is 50.00 kPa at 338.8C, but it fell to 49.62 kPa
when 8.69 g of an involatile organic compound was dissolved in 250 g of
2-propanol. Calculate the molar mass of the compound.
5.6(a) 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.
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5.6(b) 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.
5.7(a) The osmotic pressure of an aqueous solution at 300 K is 120 kPa.
Calculate the freezing point of the solution. 5.7(b) The osmotic pressure of
an aqueous solution at 288 K is 99.0 kPa. Calculate the freezing point of the
solution.
5.8(a) 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 25C; in the right compartment there is hydrogen at the same
temperature and pressure. Calculate the entropy and Gibbs energy of mixing
when the partition is removed. Assume that the gases are perfect.
5.8(b) Consider a container of volume 250 cm3 that is divided into two
compartments of equal size. In the left compartment there is argon at 100
kPa and 0C; 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 removed. Assume that the gases are perfect.
5.9(a) Air is a mixture with a composition given in Example 1.3. Calculate the
entropy of mixing when it is prepared from the pure (and perfect) gases.
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5.9(b) Calculate the Gibbs energy, entropy, and enthalpy of mixing when
1.00 mol C6H14 (hexane) is mixed with 1.00 mol C7H16 (heptane) at 298 K;
treat the solution as ideal.
5.10(a) 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?
5.10(b) 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?
5.11(a) Use Henrys law and the data in Table 5.1 to calculate the solubility
(as a molality) of CO2 in water at 25C when its partial pressure is (a) 0.10
atm, (b) 1.00 atm.
5.11(b) The mole fractions of N2 and O2 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 25C.
5.12(a) 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.5.12(b) 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.
5.13(a) The enthalpy of fusion of anthracene is 28.8 kJ mol1 and its melting
point is 217C. Calculate its ideal solubility in benzene at 25C.
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5.13(b) Predict the ideal solubility of lead in bismuth at 280C given that its
melting point is 327C and its enthalpy of fusion is 5.2 kJ mol1.
5.14(a) The osmotic pressure of solutions of polystyrene in toluene were
measured at 25C and the pressure was expressed in terms of the height of
the solvent of density 1.004 g cm3:
Calculate the molar mass of the polymer. 5.14(b) The molar mass of an
enzyme was determined by dissolving it in water, measuring the osmotic
pressure at 20C, and extrapolating the data to zero concentration. The
following data were obtained:
Calculate the molar mass of the enzyme.
5.15(a) Substances A and B are both volatile liquids with = 300 Torr,
= 250 Torr, and KB = 200 Torr (concentration expressed in mole fraction).
When xA = 0.9, bB = 2.22 mol kg1, pA = 250 Torr, and pB = 25 Torr. Calculate
the activities and activity coefficients of A and B. Use the mole fraction,
Raoults law basis system for A and the Henrys law basis system (both mole
fractions and molalities) for B.
5.15(b) Given that p*(H2O) = 0.02308 atm and p(H2O) = 0.02239 atm in a
solution in which 0.122 kg of a non-volatile solute (M = 241 g mol1) is
dissolved in 0.920 kg water at 293 K, calculate the activity and activity
coefficient of water in the solution.
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5.16(a) A dilute solution of bromine in carbon tetrachloride behaves as an
ideal-dilute solution. The vapour pressure of pure CCl4 is 33.85 Torr at 298 K.
The Henrys law constant when the concentration of Br2 is expressed as a
mole fraction is 122.36 Torr. Calculate the vapour pressure of each
component, the total pressure, and the composition of the vapour phase
when the mole fraction of Br2 is 0.050, on the assumption that the conditions
of the ideal-dilute solution are satisfied at this concentration.
5.16(b) Benzene and toluene form nearly ideal solutions. The boiling point of
pure benzene is 80.1C. Calculate the chemical potential of benzene relative
to that of pure benzene when xbenzene = 0.30 at its boiling point. If the activity
coefficient of benzene in this solution were actually 0.93 rather than 1.00,
what would be its vapour pressure?
5.17(a) By measuring the equilibrium between liquid and vapour phases of
an acetone(A)/methanol(M) solution at 57.2C at 1.00 atm, it was found that
xA = 0.400 when yA = 0.516. Calculate the activities and activity coefficients
of both components in this solution on the Raoults law basis. The vapour
pressures of the pure components at this temperature are: = 105 kPa and
= 73.5 kPa. (xA is the mole fraction in the liquid and yA the mole fraction in
the vapour.)
5.17(b) By measuring the equilibrium between liquid and vapour phases of a
solution at 30C at 1.00 atm, it was found that xA = 0.220 when yA = 0.314.
Calculate the activities and activity coefficients of both components in this
solution on the Raoults law basis. The vapour pressures of the pure
components at this temperature are: = 73.0 kPa and = 92.1 kPa. (xA is
the mole fraction in the liquid and yA the mole fraction in the vapour.)
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5.18(a) Calculate the ionic strength of a solution that is 0.10 mol kg1 in
KCl(aq) and 0.20 mol kg1 in CuSO4(aq).
5.18(b) Calculate the ionic strength of a solution that is 0.040 mol kg1 in
K3[Fe(CN)6](aq), 0.030 mol kg1 in KCl(aq), and 0.050 mol kg1 in NaBr(aq).
5.19(a) Calculate the masses of (a) Ca(NO3)2 and, separately, (b) NaCl to
add to a 0.150 mol kg1 solution of KNO3(aq) containing 500 g of solvent to
raise its ionic strength to 0.250. 5.19(b) Calculate the masses of (a) KNO3
and, separately, (b) Ba(NO3)2 to add to a 0.110 mol kg1 solution of KNO3(aq)
containing 500 g of solvent to raise its ionic strength to 1.00.
5.20(a) Estimate the mean ionic activity coefficient and activity of CaCl2 in a
solution that is 0.010 mol kg1 CaCl2(aq) and 0.030 mol kg1 NaF(aq).
5.20(b) Estimate the mean ionic activity coefficient and activity of NaCl in a
solution that is 0.020 mol kg1 NaCl(aq) and 0.035 mol kg1 Ca(NO3)2(aq).
5.22(a) At 90C, the vapour pressure of methylbenzene is 53.3 kPa and that
of 1,2-dimethylbenzene is 20.0 kPa. What is the composition of a liquid
mixture that boils at 90C when the pressure is 0.50 atm? What is the
composition of the vapour produced? 5.22(b) At 90C, the vapour pressure
of 1,2-dimethylbenzene is 20 kPa and that of 1,3-dimethylbenzene is 18 kPa.
What is the composition of a liquid mixture that boils at 90C when the
pressure is 19 kPa? What is the composition of the vapour produced?
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5.23(a) The vapour pressure of pure liquid A at 300 K is 76.7 kPa and that of
pure liquid B is 52.0 kPa. These two compounds form ideal liquid and gaseous
mixtures. Consider the equilibrium composition of a mixture in which the
mole fraction of A in the vapour is 0.350. Calculate the total pressure of the
vapour and the composition of the liquid mixture. 5.23(b) The vapour
pressure of pure liquid A at 293 K is 68.8 kPa and that of pure liquid B is 82.1
kPa. These two compounds form ideal liquid and gaseous mixtures. Consider
the equilibrium composition of a mixture in which the mole fraction of A in the
vapour is 0.612. Calculate the total pressure of the vapour and the
composition of the liquid mixture.
5.24(a) It is found that the boiling point of a binary solution of A and B with
xA = 0.6589 is 88C. 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? 5.24(b) It is
found that the boiling point of a binary solution of A and B with xA = 0.4217 is
96C. At this temperature the vapour pressures of pure A and B are 110.1 kPa
and 76.5 kPa, respectively. (a) Is this solution ideal? (b) What is the initial
composition of the vapour above the solution?
5.25(a) Dibromoethene (DE, = 22.9 kPa at 358 K) and dibromopropene
(DP, = 17.1 kPa at 358 K) form a nearly ideal solution. If zDE = 0.60, what
is (a) ptotal when the system is all liquid, (b) the composition of the vapour
when the system is still almost all liquid? 5.25(b) Benzene and toluene form
nearly ideal solutions. Consider an equimolar solution of benzene and toluene.
At 20C the vapour pressures of pure benzene and toluene are 9.9 kPa and
2.9 kPa, respectively. The solution is boiled by reducing the external pressure
below the vapour pressure. Calculate (a) the pressure when boiling begins, (b)
the composition of each component in the vapour, and (c) the vapour
pressure when only a few drops of liquid remain. Assume that the rate of
vaporization is low enough for the temperature to remain constant at 20C.
