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Unformatted text preview: Problems “The Conversion of Energy,” C. M. Summers, Sci. Am September 1971. “Energy Conversion: Better Living Through Themodynamics,” W. D. Metz, Science
188, 820 (1975). “HeatFall and Entropy,” J. P. Lowe, J. Chem. Educ, 59, 353 (1982). “Demons, Engines, and the Second Law,” C. H. Bennett, Sci. Am. November, 1987. “The Conversion of Chemical Energy,” D. J. Wink, J. Chem. Educ. 69, 109 (1992). “Energy Conversion,” P. Berdahl, Encyclopedia of Applied Physics, Trigg, G. L., Ed.,
VCH Publishers, New York (1993), Vol. 6, p. 229. “Steam Engines,” S. Luchter, Encyclopedia of Applied Physics, Trigg, G. L. Ed., VCH
Publishers, New York (1997), Vol. 19, p. 563. The Third Law of Thermodynamics and Residual Entropy “Ice,” L. K. Runnels, Sci. Am. December 1966. “The Third Law of Thermodynamics and the Residual Entropy of Ice,” M. M. Julian,
F. H. Stillinger, and R. R. Festa, J. Chem. Educ. 60, 65 (1983). “How Thermodynamic Data and Equilibrium Constants Changed When the Standard
State Pressure Became 1 Bar,” R. S. Treptow, J. Chem. Educ. 76, 212 (1999). Problems  I ~ . Probability 5.1 Determine the probability that all the molecules of a gas will be found in one half of a container
when the gas consists of (a) 1 molecule, (b) 20 molecules, and (c) 2 million molecules. 5.2 Suppose that your friend told you of the following extraordinary event. A block of metal weighing
500 g was seen rising spontaneously from the table on which it was resting to a height of 1.00
cm above the table. He stated that the metal had absorbed thermal energy from the table that
was then used to raise itself against gravitational pull. (a) Does this process violate the ﬁrst law
of thermodynamics? (b) How about the second law? Assume that the room temperature was
298 K and that the table was large enough that its temperature was unaffected by this transfer
of energy. (Hint: First calculate the decrease in entropy as a result of this process and then
estimate the probability for the occurrence of such a process. The acceleration due to gravity is
9.81 m s‘ .) The Carnot Heat Engine 5.3 Compare the generation of electricity by a hydroelectric plant to the use of a heat engine. Which
method is more efﬁcient? Why? 5.4 Convert the P—Vdiagram for the Carnot cycle to a T—S diagram. What is the area of the enclosed
portion? 5.5 The internal engine of a 1200kg car is designed to run on octane (Cngg), whose enthalpy of
combustion is 5510 k] mol‘l. If the car is moving up a slope, calculate the maximum height (in
meters) to which the car can be driven on 1.0 gallon of the fuel. Assume that the engine cylinder
temperature is 2200 °C and the exit temperature is 760 °C, and neglect all forms of friction. The
mass of 1 gallon of fuel is 3.1 kg. [Hint The work done in moving the car over a vertical
distance is mgh, where m is the mass of the car in kg, 9 the acceleration due to gravity
(9.81 m 5’2), and h the height in meters] . 5.6 A heat engine operates between 210 °C and 35 °C. Calculate the minimum amount of heat that
must be withdrawn from the hot source to obtain 2000 J of work. 159 160 Chapter 5: The Second Law of Thermodynamics The Second Law of Thermodynamics
5.7 Comment on the statement: “Even thinking about entropy increases its value in the universef’ 5.8 One of the many statements of the second law of thermodynamics is: Heat cannot ﬂow from a
colder body to a warmer one without external aid. Assume two systems, 1 and 2, at T1 and T2
(T2 > T1). Show that if a quantity of heat q did ﬂow spontaneously from 1 to 2, the process
would result in a decrease 1n entropy of the universe (You may assume that the heat ﬂows very
slowly so that the process can be regarded as reversible. Assume also that the loss of heat by
system 1 and the gain of heat by system 2 do not affect T1 and T2. ) 5.9 A ship sailing 1n the Indian Ocean takes the warmer surface water at 32 °C to run a heat engine
that powers the ship and discharges the used water back to the surface of the sea. Does this
scheme violate the second law of thermodynamics? If so, what change would you make to
make it work? 5.10 Molecules of a gas at any temperature T above the absolute zero are in constant motion. Does
this “perpetual motion” violate the laws of thermodynamics? 5.11 According to the second law of thermodynamics, the entropy of an irreversible process in an
isolated system must always 1ncrease. On the other hand, it is well known that the entropy of
living systems remains small. (For example, the synthesis of highly complex protein molecules
from individual amino acids 1s a process that leads to a decrease 1n entropy.) Is the second law
invalid for living systems? Explain. 5.12 On a hot summer day, a person tries to cool himself by opening the door of a refrigerator. Is this
a wise action, thermodynamically speaking? 1 (' Entropy Changes 5.13 The molar heat of vaporization of ethanol is 39.3 H mol’l, and the boiling point of ethanol is
78.3 °C. Calculate the value of AvapS for the vaporization of 0.50 mole of ethanol. 5.14 Calculate the values of AU, AH, and AS for the following process: 1 mole of liquid water 1 mole of steam
at 25°C and 1 atm —> at 100°C and 1 atm The molar heat of vaporization of water at 373 K is 40.79 kJ mol‘l, and the molar heat
capacity of water is 75.3 J K‘1 mol“. Assume the molar heat capacity to be temperature
independent and idealgas behavior. 5.15 Calculate the value of AS in heating 3.5 moles of a monatomic ideal gas from 50 °C to 77 °C at
constant pressure. 5.16 A quantity of 6.0 moles of an ideal gas is reversibly heated at constant volume from 17 °C to 35 °C. Calculate the entropy change. What would be the value of AS if the heating were carried
out irreversibly? 5.17 One mole of an ideal gas is ﬁrst, heated at constant pressure from T to 3T and second, cooled
back to T at constant volume. (a) Derive an expression for AS for the overall process. (b) Show
that the overall process is equivalent to an isothermal expansion of the gas at T from V to 3V, where Vis the original volume. (c) Show that the value of AS for the process in (a) 1s the same
as that for (b). 5.18 A quantity of 35.0 g of water at 25.0 °C (called A) is mixed with 160.0 g of water at 86.0 °C
(called B). (a) Calculate the ﬁnal temperature of the system, assuming that the mixing is carried
out adiabatically. (b) Calculate the entropy change of A, B, and the entire system. 5.19 The heat capacity of chlorine gas is given by 5,. = (31.0 + 0.008T) J K1 mol‘l Problems Calculate the entropy change when 2 moles of gas are heated from 300 K to 400 K at constant
pressure. 5.20 A sample of neon (Ne) gas initially at 20 °C and 1.0 atm is expanded from 1.2 L to 2.6 L and
simultaneously heated to 40 °C. Calculate the entropy change for the process. 5.21 One of the early experiments in the development of the atomic bomb was to demonstrate that
235U and not 238U is the ﬁssionable isotope. A mass spectrometer was employed to separate
235UF5 from 238UF6. Calculate the value of AS for the separation of 100 mg of the mixture of gas, given that the natural abundances of 235U and 238U are 0.72% and 99.28%, respectively,
and that of 19F is 100%. 5.22 One mole of an ideal gas at 298 K expands isothermally from 1.0 L to 2.0 L (a) reversibly and
(b) against a constant external pressure of 12.2 atm. Calculate the values of ASSyS, ASsm, and
AS“;v in both cases. Are your results consistent with the nature of the processes? 5.23 The absolute molar entropies of 02 and N2 and 205 J K‘1 mol‘1 and 192 J K‘1 mol“,
respectively, at 25 °C. What is the entropy of a mixture made up of 2.4 moles of 02 and 9.2
moles of N2 at the same temperature and pressure? 5.24 A quantity of 0.54 mole of steam initially at 350°C and 2.4 atm undergoes a cyclic process for
which q = ~74 J. Calculate the value of AS for the process. 5.25 Predict whether the entropy change is positive or negative for each of the following reactions at 298 K: (a) 4FC(S) + 302(9) > 2F6203(S)
(b) 0(g)+0(9) + 02(9) (c) NH4C1(s) —> NH3(g) + HCl(g) .
((1) H201) +C12(g) > 2HC1(9) 5.26 Use the data in Appendix B to calculate the values of ArS° of the reactions listed in the previous
problem. 5.27 A quantity of 0.35 mole of an ideal gas initially at 156°C is expanded from 1.2 L to 7.4 L.
Calculate the values of w, q, AU, and AS if the process is carried out (a) isothermally and
reversibly, and (b) isothermally and irreversibly against an external pressure of 1.0 atm. 5.28 One mole of an ideal gas is isothermally expanded from 5.0 L to 10 L at 300 K. Compare the
entropy changes for the system, surroundings, and the universe if the process is carried out
(a) reversibly, and (b) irreversibly against an external pressure of 2.0 atm. 5.29 The heat capacity of hydrogen may be represented by
6,. = (1.554 + 0.0022T) J K'1 moll Calculate the entropy changes for the system, surroundings, and the universe for the (a) reversible heating, and (b) irreversible heating of 1.0 mole of hydrogen from 300 K to 600 K.
[Hints In (b), assume the surroundings to be at 600 K.] 5.30 Consider the reaction N2(g)+02(g) > 2N0(g) Calculate the values of AIS° for the reaction mixture, surroundings, and the universe at 298 K.
Why is your result reassuring to Earth’s inhabitants? 161 162 Chapter 5: The Second Law of Thermodynamics The Third Law of Thermodynamics and Residual Entropy 5.31 The AfFI" values can be negative, zero, or positive, but the S° values can be only zero or
positive. Explain. 5.32 Choose the substance with the greater molar entropy in each of the following pairs: (a) H20(l),
H20(g), (b) NaCl(s), CaClz(s), (c) N2 (0.1 atm), N2 (1 atm), (d) C (diamond), C (graphite),
(e) 02(9), 03 (9), (1') 63131101 (CzHSOH), dimethly ether (C2H60), (g) N204(g), 2N02(9),
(h) Fe(s) at 298 K, Fe(s) at 398 K. (Unless otherwise stated, assume the temperature is 298 K.) 5.33 A chemist ft’mnd a discrepancy between the third law entropy and the calculated entropy from
statistical thermodynamics for a compound. (a) Which value is larger? (b) Suggest two reasons
that may give rise to this discrepancy. . 5.34 Calculate the molar residual entropy of a solid in which the molecules can adopt (a) three,
(b) four, and (c) ﬁve orientations of equal energy at absolute zero. 5.35 Account for the measured residual entropy of 10.1 J K‘1 mol‘l for the CH3D molecule. 5.36 Explain why the value of S" (graphite) is greater than that of S° (diamond) at 298 K (see
Appendix B). Would this inequality hold at 0 K? Additional Problems 5.37 Entropy has sometimes been described at “time’s arrow” because it is the property that
determines the forward direction of time. Explain. 5.38 State the condition(s) under which the following equations can be applied: (a) AS = AH / T ,
(b) S0 = 0, (c) dS = deT/ T, ((1) d8 = dq/T. 5.39 Without referring to any table, predict whether the entropy change is positive, nearly zero, or
negative for each of the following reactions: ' (a) N2(g)+02(g) ’ 2N0(9) (b) . 2Mg(S) +02(g) —> 2Mg0(S) (c) 2H202(1) 9 2H200) +0201) (‘1) H2(g)+C02(9) —’ H20(g)+C0(9) 5.40 Calculate the entropy change when neon at 25 °C and 1.0 atm in a container of volume 0.780 L is
allowed to expand to 1.25 L and is simultaneously heated to 85 °C. Assume ideal behavior.
(Hint: Because S is a state function, you can ﬁrst calculate the value of AS for expansion and
then calculate the value of AS for heating at constant ﬁnal volume.) 5.41 Photosynthesis makes use of photons of visible light to bring about chemical changes. Explain
why heat energy in the form of infrared photons is ineffective for photosynthesis. 5.42 One mole of an ideal monatomic gas is compressed from 2.0 atm to 6.0 atm while being cooled
from 400 K to 300 K. Calculate the values of AU, AH, and AS for the process. 5.43 The three laws of thermodynamics are sometimes stated colloquially as follows: First law: You
cannot get something for nothing; Second law: The best you can do is get even; Third law: You
cannot get even. Provide a scientiﬁc basis for each of these statements. (Hint: One consequence
of the third law is that it is impossible to attain the absolute zero of temperature.) 5.44 Use the following data to determine the normal boiling point, in kelvins, of mercury. What
assumptions must you make to do the calculation?
Hg(l): Afﬁ°= 0 (by definition)
S°= 77.4 J K"1 moll
Hg(g): AfH°= 60.78 kJ mol" '
§°= 174.7 J K“ moll Problems 5.45 Referring to Trouton’s rule, explain why the ratio is considerably smaller than 90 J K‘1 mol‘1
for liquid HF. 5.46 Give a detailed example of each of the following, with an explanation: (a) a thermodynamically
spontaneous process; (b) a process that would violate the ﬁrst law of thermodynamics; (c) a process that would violate the second law of thermodynamics; (d) an irreversible process; (e) an equilibrium process. i’ 5.47 In the reversible adiabatic expansion of an ideal gas, there are two contributions to entropy
changes: the expansion of the gas and the cooling of the gas. Show that these two contributions
are equal in magnitude but opposite in sign. Show also that for an irreversible adiabatic gas
expansion, these two contributions are no longer equal in magnitude. Predict the sign of AS. 5.48 A refrigerator set at 0 °C discharges heat into the kitchen at 20 °C. (a) How much work would be
required to freeze 500 mL of water (about an ice tray’s volume)? (b) How much heat would be
discharged during this process? (The molar enthalpy of fusion of water is 6.01 k] mol‘l, and
the refrigerator operates at 35% eﬂiciency.) 5.49 Superheated water is water heated above 100 °C without boiling. As for supercooled water (see
Example 5.7), superheated water is thermodynamically unstable». Calculate the values of ASsys,
A551,", and AS“,v when 1.5 moles of superheated water at 110 °C and 1.0 atm are converted to
steam at the same temperature and pressure. (The molar enthalpy of vaporization of water is 40‘. 79 kJ mol‘l, and the molar heat capacities of water and steam in the temperature range
100—110°C are 75.5 J K‘1 mol‘1 and 34.4 J K‘1 mol‘l, respectively. 5.50 Toluene (C7H3) has a dipole moment, whereas benzene (C5H5) is nonpolar: CH3?
m.pt. 5.5°C —95°C
b.pt. 80.1°C 110.6°C Explain why, contrary to our expectation, benzene melts at a much higher temperature than
toluene. Why is the boiling point of toluene higher than that of benzene? 163 ...
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 Fall '07
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