Download CHEMISTRY 313 PHYSICAL CHEMISTRY I Additional Problems for

CHEMISTRY 313
PHYSICAL CHEMISTRY I
Additional Problems for Exam 3
Exam 3 date: Wednesday, December 12, 2007.
Section I. Chapter 4
I.1. (a) What characterizes a first-order phase transition? Give two examples of a firstorder phase transition. (b) What characterizes a second-order phase transition? Give
two examples. (c) What characterizes a λ-transition? Give two examples.
I.2. Using the phase diagram in Fig. 1, state what would be observed during each step
when a sample of carbon dioxide, initially at 1.0 atm and 298 K, is subjected to the
following cycle: (a) isobaric heating to 320 K, (b) isothermal compression to 100 atm, (c)
isobaric cooling to 210 K, (d) isothermal decompression to 1.0 atm, (e) isobaric heating
to 298 K.
Figure 1: Phase diagram for carbon dioxide as given in Physical Chemistry, 8th ed., by
P. Atkins and J. dePaula (Freeman, New York, 2006).
I.3. The molar volume of a certain solid is 142.0 mL/mol at 1.00 bar and 427.15 K, its
melting temperature. The molar volume of the liquid at this temperature and pressure
is 152.6 mL/mol. At 1.2 MPa, the melting temperature changes to 429.26 K. Calculate
the enthalpy and entropy of fusion of the solid.
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I.4. The vapor pressure of a liquid in the temperature range 200 - 260 K was found to
fit the expression ln(P/P0 ) = 16.255 − 2501.8K/T , where P is the pressure in Torr, T is
the temperature in Kelvin, and P0 is one Torr. Calculate the enthalpy of vaporization
for the liquid.
I.5. The vapor pressure of benzene between 10◦ C and 30◦ C fits the expression log(P/P0 ) =
7.960 − 1780K/T , where P is the pressure in Torr, T is the temperature in Kelvin, and
P0 is one Torr. Calculate (a) the enthalpy of vaporization and (b) the normal boiling
point of benzene.
I.6. Suppose the incident sunlight at ground level has a power density of 0.87 kW/m2
at noon. What is the maximum rate of loss of water from a lake of area 1.0 × 104 m2 .
Assume that all the radiant energy is absorbed.
I.7. Naphthalene (C10 H8 ) melts at 80.2◦ C. If the vapor pressure of the liquid is 0.013 bar
at 85.8◦ C and 0.053 bar at 119.3◦ C, use the Clausisu-Clapeyron equation to calculate
(a) the enthalpy of vaporization, (b) the normal boiling point and (c) the entropy of
vaporization at the normal boiling point.
I.8. The normal boiling point of hexane is 69.0◦ C. Esitmate (a) its enthalpy of vaporization and (b) its vapor pressure at 25◦ C and 60◦ C.
I.9. Construct the phase diagram for benzene near its triple point at 36 Torr and 5.50◦ C
using the following data: ∆f us H = 10.6 kJ/mol, ∆vap H = 30.8 kJ/mol, ρ(s) = 0.891
g/mL, ρ(l) = 0.879 g/mL.
I.10. In an investigation of thermophysical properties of toluene [R. D. Goodwin, J. Phys.
Chem. Ref. Data 18, 1565 (1989)] presented expression for two coexistence curves. The
solid-liquid coexistence curve is given by
P = Pt + 1000 × (5.60 + 11.727x)x ,
where x = T /T3 − 1, P is the pressure in bar, T is the temperature in K, Tt is the triple
point temperature of 178.15 K, and Pt is the triple point pressure of 0.4362 µbar. The
liquid-vapor coexistence curve is obtained from
ln(P/P0 ) = −10.418/Tr + 21.157 − 15.996Tr + 14.015Tr2 − 5.0120Tr3 + 4.7224(1 − Tr )1.70 ,
where Tr = T /Tc with Tc = 593.95 K, P0 = 1.0000 bar, and P being the pressure in bar.
(a) Plot the solid-liquid and liquid-vapor phase boundaries. (b) Estimate the standard
melting point of toluene. (c) Estimate the standard boiling point of toluene. (d) Compute
the standard enthalpy of vaporization of toluene, given that the molar volumes of the
liquid and vapor at the normal boing point are 0.12 L/mol and 30.3 L/mol, respectively.
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I.11. Fig. 2 gives a schematic representation of how the chemical potentials of the
solid, liquid and gaseous phases of a substance vary with temperature. All have negative
slope, but its is unlikely that they are truly straight lines as indicated in the illustration.
Derive an expression for the curvatures (specifically, the second derivatives with respect to
temperature) of these lines. Is there a restriction on the curvature of these lines? Which
state of matter shows the greatest curvature? Can you explain this at the microscopic
level?
Figure 2: Schematic representation of the variation of the chemical potential of a solid,
liquid, and gas phase of a neat substance as a function of temperature. This figure was
taken from Physical Chemistry, 8th ed., by P. Atkins and J. dePaula (Freeman, New
York, 2006).
Section II. Chapter 5
II.1. Explain the difference between an ideal and an ideal-dilute solution.
II.2. The partial molar volumes of acetone and chloroform in a mixture in which the
mole fraction of chloroform is 0.4693 are 74.166 mL/mol and 80.235 mL/mol, respectively. What is the volume of a solution of mass 1.000 kg?
II.3. 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 mL/mol and 176.14 mL/mol, respectively. The molar
masses of A and B are 241.1 g/mol and 198.2 g/mol. What is the volume of a solution
of mass 1.000 kg?
II.4. At 25◦ C, the density of a 50% by mass ethanol/water solution is 0.914 g/mL. Given
that the partial molar volume of water in the solution is 17.4 mL/mol, calculate the
partial molar volume of ethanol.
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II.5. At 20◦ C, the density of a 20% by mass ethanol/water solution is 968.7 kg/m3 . Given
that the partial molar volume of ethanol in the solution is 52.2 mL/mol, calculate the
partial molar volume of the water.
II.6. 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.
II.7. The vapor pressure of 2-propanol is 0.05000 bar at 338.8◦ C, but it fell to 0.04962
bar when 8.69 g of an involatile organic compound was dissolved in 250 g of 2-propanol.
Calculate the molar mass of the compound.
II.8. Consider a container of volume 250 mL that is divided into two compartments of
equal size. In the left compartment there is argon at 10.0 mbar and 0◦ C. In the right
compartment, there is neon. The system is under both mechanical and thermal equilibrium. Calculate the entropy and Gibbs energy of mixing when the partition is removed.
Assume that the gases are perfect.
II.9. What proportions of hexane and heptane should be mixed (a) by mole fraction and
(b) by mass in order to achieve the greatest entropy of mixing?
II.10. Substances A and B are both volatile liquids with PA∗ = 300 Torr and PB∗ = 250
Torr, and KB = 200 Torr (concentration expressed in mole fraction). When xA = 0.9,
bB = 2.22 mol/kg, PA = 250 Torr and PB = 25 Torr. Calculate the activities and activity coefficients of A and B. Assume and ideal-dilute solution, with A being the solvent.
II.11. By measuring the equilibrium between the liquid (x) and vapor (y) phases of an
acetone (A)/methanol (M) solution at 57.2◦ C and 1.00 bar, it was found that xA = 0.400
when yA = 0.516. Calculate the activities and the activity coefficients for both components in this solution assuming an ideal solution. The vapor pressures of the pure
components at this temperature are PA∗ = 0.105 bar and PB∗ = 0.0735 bar.
II.12. By measuring the equilibrium between the liquid (x) and vapor (y) phases of a solution at 30.0◦ C and 1.00 bar, it was found that xA = 0.220 when yA = 0.314. Calculate
the activities and the activity coefficients for both components in this solution assuming
an ideal solution. The vapor pressures of the pure components at this temperature are
PA∗ = 0.073 bar and PB∗ = 0.0921 bar.
II.13. Calculate the ionic strength of a solution that is 0.040 mol/kg of K3 [Fe(CN)6 ] (aq),
0.030 mol/kg in KCl (aq) and 0.050 mol/kg in Na2 SO4 (aq).
II.14. Calculate the necessary amount of Ca(NO3 )2 added to a 0.150 mol/kg solution
of KNO3 (aq) containing 500 g of solvent to create a solution with an ionic strength of
0.250. How much NaCl (instead of Ca(NO3 )2 ) would be required to create the same ionic
strength solution?
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II.15. Estimate the mean ionic activity coefficient and activity of a solution that is 0.01
mol/kg CaCl2 (aq) and 0.030 mol/kg NaF (aq).
Section III. Chapter 6
III.1. Draw phase diagrams for the following type systems. Label the regions and intersections of the diagrams, stating what materials (possibly compounds or azeotropes)
are present and whether they are solid, liquid or gas. (a) One component, pressuretemperature diagram, liquid density greater than that of solid. (b) Two component,
temperature-composition solid-liquid diagram, one compound AB formed that melts congruently, negligible solid-solid solubility.
III.2. Draw phase diagrams for the following types of systems. Label the regions and
intersections of the diagrams, stating what materials (possibly compounds or azeotropes)
are present and whether they are solid, liquid or gas. (a) Two component, temperaturecomposition, solid-liquid diagram, one compound of formula AB2 that melts incongruently, negligible solid-solid solubility. (b) Two component, constant temperature composition, liquid-vapor diagram, formation of an azeotrope at xB = 0.333, complete miscibility.
III.3. Label the regions of the phase diagram in Fig. 3a. State what substances (if
compounds give their formulas) exist in each region. Label each substance in each region
as solid, liquid or gas.
III.4. Label the regions of the phase diagram in Fig. 3b. State what substances (if
compounds give their formulas) exist in each region. Label each substance in each region
as solid, liquid or gas.
III.5. State the number of components in the following systems. (a) NaH2 PO4 in water
at equilibrium with water vapor but disregarding the fact that the salt is ionized. (b)
The same, but taking into account the ionization of the salt.
III.6. State the number of components for a system in which AlCl3 is dissolved in water,
accounting for the hydrolysis of water and the precipitation of Al(OH)3 . (Disregard the
vapor and assume that all of the AlCl3 dissolves.)
III.7. Blue CuSO4 · 5H2 O crystals release their water of hydration when heated. How
many phases and components are present in an otherwise empty heated container?
III.8. Ammonium chloride decomposes when it is heated. (a) How many components
and phases are present when the salt is heated in an otherwise empty container? (b) Now
suppose that additional ammonia is also present. How many components and phases are
present?
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Figure 3: . Two component, temperature-composition, solid-liquid diagrams for (a)
Problem III.3 and (b) Problem III.4. Both of these diagrams indicate that compounds
are formed. These diagrams taken from Physical Chemistry, 8th ed., by P. Atkins and
J. dePaula (Freeman, New York, 2006).
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III.9. A saturated solution of Na2 SO4 with excess of the solid is present at equilibrium
with its vapor in a closed vessel. (a) How many phases and components are present. (b)
What is the variance of the system? Identify the independent variables.
III.10. Suppose that the solution referred to in Problem III.9 is not saturated. (a) How
many phases and components are present. (b) What is the variance of the system? Identify the independent variables.
III.11. The vapor pressure of pure liquid A at 293 K is 0.0688 bar and that of pure liquid
B is 0.0821 bar. 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 vapor is
0.612. Calculate the total pressure of the vapor and the composition of the liquid mixture.
III.12. Dibromoethene (DE, P∗ = 0.0229 at 358 K) and dibromopropene (DP, P∗ =
0.0171 at 358 K) form a nearly ideal solution. If zDE = 0.60, what is the total pressure
when the system is all liquid? What is the composition of the vapor when the system is
still almost all liquid?
III.13. It is found that the boling point of a binary solution of A and B with xA = 0.6589
is 88◦ C. At this temperature the vapor pressures of pure A and B are 0.1276 bar and
0.0506 bar, respectively. (a) Is this solution ideal? (b) What is the initial composition of
the vapor above the solution?
III.14. For the two component, temperature-composition, solid-liquid diagram in Fig. 4,
indicate the feature that denotes incongruent melting. What is the composition of the
eutectic mixture and at what temperature does it melt? Sketch the cooling curves for
the isopleths a and b.
III.15. 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. At 86 K, the phase in equilibrium with
the thtrafluoromethane-rich solution changes from solid methane to a methane-rich liquid. At that temperature, the two liquid solutions that are in mutual equilibrium have
the compositions x(CF4 ) = 0.10 and x(CF4 ) = 0.80. Sketch the phase diagram. Using
this phase diagram, sketch the cooling curves for liquids mixtures in which x(CF4 ) is (a)
0.10, (b) 0.30, (c) 0.50, (d) 0.80 and (e) 0.95.
III.16. Describe the phase changes that take place when a liquid mixture of 4.0 mol B2 H6
(melting point 131 K) and 1.0 mol CH3 OCH3 (melting point 135 K) is cooled from 140
K to 90 K. These substances form a compound (CH3 )2 OB2 H6 that melts congruently at
133 K. The system exhibits one eutectic at x(B2 H6 ) = 0.25 and 123 K and another at
x(B2 H6 ) = 0.90 and 104 K. Sketch the phase diagram for this system and the cooling
curves for liquid mixtures in which x(B2 H6 is (a) 0.10, (b) 0.30, (c) 0.50, (d) 0.80 and (e)
0.95.
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Figure 4: . Two component, temperature-composition, solid-liquid diagram for Problem
III.14. Note that this diagram indicates that compounds are formed. These diagrams
taken from Physical Chemistry, 8th ed., by P. Atkins and J. dePaula (Freeman, New
York, 2006).
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Section IV. Chapter 7
IV.1. At 2257 K and 1.00 bar total pressure, water is 1.77 % dissociated at equilibrium
by way of the reaction 2 H2 O (g) ­ 2 H2 (g) + 2 O2 (g). Calculate (a) K, (b) ∆r Gª ,
and (c) ∆r G.
IV.2. In the gas-phase reaction 2 A + B ­ 3 C + 2 D, it was found that, when 1.00 mole
A, 2.00 mole B and 1.00 mol D were mixed and allowed to come to equilibrium at 25◦ C,
the resulting mixture contained 0.90 mole C at a total pressure of 1.00 bar. Calculate
(a) the mole fractions of each species at equilibrium, (b) Kx , (c) K and (d) ∆r Gª .
IV.3. In the gas phase reaction A + B ­ C + 2 D, it was found that, when 2.00 mole A,
1.00 mole B and 3.00 mole D were mixed and allowed to come to equilibrium at 25◦ C,
the resulting mixture contained 0.79 mole C at a total pressure of 1.00 bar. Calculate
(a) the mole fractions of each species at equilibrium, (b) Kx , (c) K and (d) ∆r Gª .
IV.4. The standard reaction enthalpy of Zn (s) + H2 O (g) → ZnO (s) + H2 (g) is approximately constant at + 224 kJ/mol from 920 K up to 1280 K. The standard reaction
Gibbs energy is 33 kJ/mol at 1280 K. Estimate the temperature at which the equilibrium
constant becomes greater than 1.
IV.5. Calculate the percent change in Kx for the reaction H2 CO (g) ­ CO (g) + H2 (g)
when the total pressure is increased from 1.0 bar ot 2.0 bar at constant temperature.
IV.6. The equilibrium constant for the reaction N2 (g) + O2 (g) ­ 2 NO (g) is 1.69×10−3
at 2300 K. A mixture consisting of 5.0 g of nitrogen and 2.0 g of oxygen in a container
of volume 1.0 L is heated to 2300 K and allowed to come to equilibrium. Calculate the
mole fraction of NO at equilibrium.
IV.7. Estimate the temperature at which CaCO3 decomposes.
IV.8.The standard Gibbs energy of formation of NH3 (g) is -16.5 kJ/mol at 298 K. What
is the reaction Gibbs energy when the partial pressures of N2 , H2 and NH3 (treated as
perfect gases) are 3.0 bar, 1.0 bar and 4.0 bar, respectively? What is the spontaneous
direction of the reaction in this case?
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