PHYSICAL
CHEMISTRY
MET (201)
GENERAL QUESTIONS ON THE
COURSE CONTENTS
PREPRED BY: DR. FAWZI A. A. ELREFAIE
For
The Second Year Metallurgy
Faculty of Engineering
Cairo University
(2014/2015)
1
PHYSICAL
CHEMISTRY
MET (201)
Questions on
INTRODUCTION AND
DEFINATION OF TERMS
2
Group (1)
1. Discuss briefly the concept of microscopic and
macroscopic thermodynamic state. How does the
concept of macroscopic thermodynamic state facilitate
the study of the thermodynamic branch of physical
chemistry?
2. Discuss, in-detail, the concept of simple equilibrium
between a closed thermodynamic system and its
surroundings.
3. Complete: in 1660 ………. found that, for gases, at
constant T: ……….. . Similarly, the V-T relation of
gases at 1787 was established by ……… as: …………
.accordingly, the equation of state for ideal gases can be
written as: ………… .
4.
Define the coefficient of thermal expansion of gases as
given by Joseph Gay Lussac at 1802.
5.
Based on the previous scientific research results, show
how it was established that:
T(K) = T(°C)+273.16
What is the name of the T(K) scale.
3
6.
Complete:
R= 0.082 ………/………
R = 8.3144 ……. /………
R = 1.987 ……..../……....
7.
The volume of one mole of ideal gas at STP is ……. .
8.
What is the difference between thermodynamic
extensive and intensive properties?
9.
Define the homogeneous
thermodynamic systems.
4
and
heterogeneous
Group (2)& Group (3)
10. Complete the following statements:
 The kinetic energy is conserved in ……….. .
 The dynamic energy is conserved in ……... .
 The relationship between the dynamic energy and
heat changes of a system led to the foundation for the
development of the ………. subject.
11. Discuss in-detail the caloric theory of heat proposed by
Cornet Rumford in 1798. Why is it discredited in 1799
by Humphrey Davy?
12. Discuss in-detail Joule’s experiments in 1840 that
established the relationship between heat and work
which placed a fuim quantitative basis for the
thermodynamic subject.
13. Complete:
 The mechanical equivalent of heat = ………… .
14. State the preliminary formulation of the 1st law of
thermodynamics. Using the similarity between this
statement and the case of a body of mass m moving in a
gravitational field from height h1 to height h2, and also
a particle with charge q moving in an electric field from
5
a point at potential Q1 to a point at potential Q2 to show
that W=U2-U1for a thermodynamic system changing its
state form U1 to U2 under the influence of work W done
on it where U is a state property. Based on this relation
show that:
∆𝑼 = 𝒒 + 𝑾
for a closed system if the work done on the system is W
and heat transferred to the system is q.
15. Complete: for infinitesimal changes, you can write
dU=……….+………..
16. Since U is a state property; thus U=U(P,V), U=U(P,T),
and U=U(V,T); thus:
dU= ……… + ………,
dU= …….... + ………,
dU= …….... + …….…,
𝑼
∫𝑼 𝟐 𝒅𝑼 = … … … …,∮ 𝒅𝑼 = … … … …
𝟏
17. Define the heat capacity term of a body. Why it is
necessary to define the heat capacity term at constant
volume or pressure, and why it is impossible to define
the heat capacity term at constant T?
6
18. Derive the following relation:
𝒅𝑽
𝒅𝑼
𝑪𝒑 − 𝑪𝒓 = ( )𝑷 (𝑷 + ( )𝑻 )
𝒅𝑻
𝒅𝑽
Based on this equation, prove that:
𝑪𝒑 − 𝑪𝒗 = 𝑹
𝒅𝑽
19. Based on Joule’s experiment, show that ( )𝑷 = 𝟎for
𝒅𝑻
ideal gases.
20. Show that: 𝑯 = 𝑼 + 𝑷𝑽.
21. Show that:
 for reversible constant volume process of a closed
system:𝒅𝑼 = 𝒒𝒗 = 𝑪𝒗 𝒅𝑻,
 for a reversible constant pressure process of a closed
system:𝒅𝑯 = 𝒒𝒑 = 𝑪𝒑 𝒅𝑻.
22. Show that for reversible adiabatic process of a closed
system:
𝑷𝑽𝜸 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝑻𝑽𝜸−𝟏 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝑻𝑷(𝟏−𝜸)/𝜸 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕
𝒅𝑷
𝒅𝑷
( )for isothermal process is greater than ( )for
𝒅𝑽
𝒅𝑽
adiabatic process.
7
23. Two closed systems that each has the same P-V state,
draw schematically the P-V figures for each on the
same diagram.
8
Group (4)
24. Based on Lewis and Randall reasoning, show that:
∆𝑺 = 𝒒/𝑻
25. Discuss in-detail the main aspects of spontaneous
processes.
26. Show how the concept of entropy in induce to measure
(quantify) the degree of irreversibility of a process.
27. Give three examples of natural processes.
28. Define the reversible processes.
29. Use the water evaporation (or condensation) in closed
system of water and water vapor at uniform temperature
T contained in a cylinder fitted with a frictionless
piston placed in thermal contact with a heat reservoir at
constant temperature T as an example to illustrate the
irreversible and reversible process.
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Group (5) & Group (6)
30. By using water vapor-water system at temperature T
contained in a heat bath at temperature T, prove that:
through the evaporation process:
∆𝑺𝒔𝒚𝒔 = 𝒒𝒓𝒆𝒗 /𝑻
𝒒𝒊𝒓𝒓
∆𝑺𝒔𝒖𝒓𝒓 = −
𝑻
∆𝑺𝒕𝒐𝒕𝒂𝒍 = ∆𝑺𝒊𝒓𝒓
where;∆𝑺𝒕𝒐𝒕𝒂𝒍 = 𝟎 for reversible processes and is
greater than zero for irreversible processes.
31. Calculate ∆𝑺𝒔𝒚𝒔 , ∆𝑺𝒔𝒖𝒓𝒓 and ∆𝑺𝒊𝒓𝒓 for the reversible
isothermal compression process of an ideal gas in a
closed system.
32. Calculate ∆𝑺𝒔𝒚𝒔 , ∆𝑺𝒔𝒖𝒓𝒓 and ∆𝑺𝒊𝒓𝒓 for reversible and
irreversible adiabatic expansion of an ideal gas.
33. Complete the following statements:
 When a system undergoes a spontaneous process, the
entropy of the system …………
 The entropy is a ………… function.
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 When a system undergoes a reversible process, no
entropy is ……, i.e. ∆𝑺𝒊𝒓𝒓 =……., entropy is simply
……… from one part of the system to …… part.
34. Define the heat engine. Explain the following
statement: “it is impossible to create perpetual motion
machine of the second type”.
35. Define the efficiency of a heat engines, i.e. efficiency =
………
, for a heat engine that works according to Carnot
………
………
cycle: 𝜼 =
.
………
36. Examine the following question: is it possible to have
tow heat engines working between T2 and T1following
Carnot cycle one of them is more efficient than the
other. Based on the previous discussion, state the
principle of Thomson and principle of Clauses
“Preliminary formulations of the second law of
Thermodynamics”
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Group (7)
37. Starting with Carnot cycles, establish the
thermodynamics temperature scale (Kelvin scale);
prove that this scale is identical to the perfect gas
temperature scale.
38. Prove that:(−𝑾) ≤ 𝑻(𝑺𝑩 − 𝑺𝑨 ) − (𝑼𝑩 − 𝑼𝑨 ). Write
down the relations that gives (−𝑾𝒎𝒂𝒙 )and𝑾𝒅𝒆𝒈𝒓𝒂𝒅𝒆𝒅 .
39. Combine the statements of the 1st and 2nd law of
thermodynamics and show that:
 for closed systems at constant U, V, and 𝒏𝒊 𝒔,
 equilibrium occurs at maximum S; and
 for closed systems at constant S, V, and 𝒏𝒊 𝒔, and
 equilibrium occurs at minimum U.
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Group (8)
40. What is the physical significances of internal energy
and entropy? Discuss in-detail how the entropy change
of the (system + surroundings) is related to the disorder
on the atomic scale at the freezing temperature of a
liquid and for supercooled liquid.
41. Complete the following statements:
 The development of a ……… relationship between
……… and the “degree of mixed-up-ness” can be
obtained from the condition of “elementary ………”
and the basis of “…………”.
 Statistical mechanics is developed based on the
assumption that “the equilibrium state of a system is
simply the most …………………………”.
 A postulate of the quantum theory is that “if particle
is confined to move within ………, then its energy is
……. . As the volume increase, the energy gaps …. .
Based on this, the energy levels available in a solid
are more ……… spaced than are in a ………”.
42. Examine the effect of quantization of energy on the
hypothetical systems in the distribution of the three
identical particles A,B and C among the equally spaced
energy levels 0,1,2, assuming that the ground level (є ̥ )
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being taken as zero, the levels are spaced by energy
value of U, the total energy of the system is 3U, and the
particles be distributed in three different sites in each
energy level. Based on this example, define the
microstates (complexion) and the macrostate.
43. If n particles are distributed among the energy levels є̥ ,
є1, є2, ……, єi, ……, єr such that there are n ̥ in levelє̥ ,
n1 in level є1,ni in level єi,………; prove that:
𝒏
є
𝒏𝒊 = ( ) 𝒆𝒙𝒑⁡(− 𝒊 )
𝑷
𝑲𝑻
whereP (partition function) = ∑𝒓𝒊 𝒆𝒙𝒑⁡(−є𝒊 /𝑲𝑻).
14
Group (9)
44. For systems with very large number of particles, prove
that:
𝑼 = −𝑲𝑻𝒍𝒏𝜴 + 𝒏𝑲𝑻𝒍𝒏𝑷
45. Schematically show diagrammatically the influence of
temperature on the most probable distribution of
particles among energy levels in a closed system of
constant volume. Relate T to β on the diagram.
46. Show that for a particles in a closed-system at constant
V in thermal equilibrium with a heat reservoir at
temperature T:
𝑺 = 𝑲𝒍𝒏𝜴
what is the physical meaning of this equation?
47. Prove that when an amount of heat δq is transferred
from A to B at constant total energy, thermal
equilibrium will prevail between the two bodies if TA=
TB.
48. Consider two pure crystals A and B which are at the
same temperature and each has 4 atoms, and let a
mixing process to start; calculate Ωemf for the mixing
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process, and also calculate the most probable
complexions. Comment on your results; compare your
results by thermal equilibrium.
49. Calculate ∆Stotalfor ideal mixing process.
50. Explain why if the A-B bond in much different than the
A-A bond or B-B bond in the binary mixing of A and
B, ideal mixing does not occur.
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Group (10)
51. Show how the combination of the 1st and 2nd laws of
thermodynamics leads to the following function:
𝑼 = 𝒇(𝑺, 𝑽)
if the work performed by the system is (WP-V) only.
52. What are the criteria for equilibrium that are provided
due to the combination of the 1st and 2nd laws of
thermodynamics?
53. What are the most convenient pair of independent
variables that can be used for:
 experimental point of view;
 theroritical point of view in thermodynamic
calculations?
54. Define the enthalpy function, H. show that:
𝑯 = 𝒇(𝑺, 𝑷)
∆𝑯 = 𝒒𝒑
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55. Define the Helmholtz free energy function (work
function), A. Show that:
𝑨 = 𝒇(𝑽, 𝑻)
−𝑾 ≤ −∆𝑨
(𝑾𝒎𝒂𝒙 ) = −∆𝑨
For spontaneous processes that occur at constant T and
V; dA ˂ 0, equilibrium occur at the minimum value of
A, i.e. at dA = 0.
56. Consider n atoms of an element occurring in a
crystalline phase and a vapor phase both contained in a
constant volume vessel which is immersed in constanttemperature heat reservoir, both constitute very large
adiabatic system. Starting the inner system having all n
atoms in the crystalline phase; i.e. partially occupying
the total volume, V, of the system which is constant.
Discuss the dependence of U, S, -TS, A on 𝒏𝒗 during
the evaporation process, then identify the conditions at
which P(eq,T) occur. Illustrate your answer by using
diagrams. Also by relating the total entropy change of
the whole adiabatic system, show that equilibrium will
occur the maximum value of∆𝑺𝒕𝒐𝒕𝒂𝒍 .
18
Group (11)
57. Define the Gibbs free energy in terms of H and TS; and
in terms of U, PV, and TS. Then prove that:
𝐺 = 𝑓 (𝑃, 𝑇),
(𝑊 ′ ) ≤ −∆𝐺,
(𝑊 ′ ) = −∆𝐺, and
𝑑𝐺 = 0
What are the conditions required for the validity of each
of the previous equality and inequality.
58. State the four forms of the fundamental equations.
59. Knowing that: 𝑼 = 𝒇(𝑺, 𝑽), 𝑯 = 𝒇(𝑺, 𝑷), 𝑨 =
𝒇(𝑽, 𝑻),and 𝑮 = 𝒇(𝑷, 𝑻), complete the following:
 T = ………., T = ………..
 P = ………., P = ………..
 V = ………, V = ………..
 S = ………., S = ………..
60. Write down Maxwell’s relations.
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61. Drive the equation of state that relates the internal
energy of a closed one-component system to the
experimentally measurable quantities T, and P. By
using this relation prove that: dU = CvdT for ideal gas.
Also prove U is independent of its pressure.
62. Prove that:
𝝏𝑯
𝝏𝑽
( )𝑻 = −𝑻( )𝑷 + 𝑽
𝝏𝑷
𝝏𝑻
By using this relation, prove that 𝒅𝑯 = 𝑪𝒑 𝒅𝑻 for ideal
gases. Also prove that the enthalpy of an ideal gas is
independent of its pressure.
63. Prove the Gibbs-Helmholtz equation.
64. Prove that: 𝑪𝒑 − 𝑪𝒗 = 𝒗𝑻𝜶𝟐 /𝜷; why 𝑪𝒑 − 𝑪𝒗 > 1.
65. Define the chemical potential in four different forms for
systems which undergo composition change. For
systems which undergo composition change, write
down the four fundamental equations.
66. Prove that: 𝜹𝑾 = −𝑷𝒅𝑽 + ∑ 𝝁𝒊 𝒅𝒏𝒊 , what is the
meaning of each term of the previous equation?
20
Group (12)
67. Explain the difference between Einstein crystal and
Debye crystal. Show how to relateev to T for both
crystals; and finally express the relation between Cvand
T for both crystals. For Debye crystal, express the
relation between Cv and T at very low temperature.
68. State the Kopp’s rule.
69. Prove that for the reaction 𝑨(𝒔) → 𝑨(𝒍) :
𝑻𝟐
∆𝑯(𝑻𝟐 ) − ∆𝑯(𝑻𝟏 ) = ∫ ∆𝑪𝒑 𝒅𝑻
𝑻𝟏
Where over the temperature range T1-T2 no phase
transformations occur. Schematically represent the
variation of the enthalpies of the solid and the liquid
phases of a substance with temperature.
21
Group (13)
70. Define the following terms: a standard state, a reference
state, a standard enthalpy of transition, the standard
enthalpy of a reaction, the standard enthalpy of
formation, and conceptual reaction.
71. Discuss in-detail: Hess’s law, and Kirchhoff’s law.
72. Prove that: for isothermal processes:
𝑷𝟐
∆𝑯 = ∫ (𝟏 − 𝜶𝑻)𝑽𝒅𝑷
𝑷𝟏
𝑷𝟐
∆𝑺 = − ∫ 𝜶𝑽𝒅𝑷
𝑷𝟏
then prove that, in general:
𝑻𝟐
𝑷𝟐
∆𝑯 = ∫ 𝑪𝒑 𝒅𝑻 + ∫ 𝑽(𝟏 − 𝜶)𝒅𝑷
𝑻𝟏
𝑷𝟏
𝑻𝟐
∆𝑺 = ∫
𝑷𝟐
𝑪𝒑 𝒅𝒍𝒏𝑻 − ∫ 𝜶𝑽𝒅𝑷
𝑻𝟏
𝑷𝟏
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73. State the Nernst postulate for the third law of
thermodynamics; Plank’s pointed out that this
statement, why? To overcome this incompetence,
Plank’s formulated another postulate, write this
postulate.
74. Give five examples that show a homogenous phase will
not reach to complete internal equilibrium at 0K.
75. Give an example to experimentally verify the third law
of thermodynamics. Discuss in-detail this example.
76. Complete the following statements:


∆𝑯𝒇𝒖𝒔
𝑻𝒇𝒖𝒔
∆𝑯𝒗𝒂𝒑
𝑻𝒗𝒂𝒑
= ……… e.u.
(𝑇𝑟𝑜𝑢𝑡𝑒𝑛′ 𝑠⁡𝑙𝑎𝑤) = ………. e.u.
77. The change in the standard entropy of a reaction is
nearly equal to the entropy change resulting from
……………
23
Group (14)
𝑷𝑽
78. Define the real gases in terms of =
. Let
𝑹𝑻
𝒁 = 𝒎𝒃 + 𝟏, as for hydrogen gas, relate P to V at
constant T.
79. Define the reduced functions. Plot Z to P for different
real gases at constant T; repeat the same plot for
different ideal gases at constant T. Schematically plot Z
versus Pr for different real gases at the same Tr.
80. Write down Van Der Waal’s equation for a real gas at
constant T which is less than Tcr; what is the meaning
of the constants a andb of Van Der Waal’s equation.
What is the type of Van Der Waal’s equation?
Calculate the values of Pcr, Vcr, and Tcr in terms of a, b,
and R.
81. Explain in-detail one of the P-V curve at T ˂ Tcr of a
real gas starting from a state at which the system is at
its gaseous state to a state at which the system is in the
liquid phase. In your explanation define, the following:
 The saturated vapor pressure,
24
 The isothermal compressibility factor of the gaseous
phase as compared to the same parameter of the
liquid phase in equilibrium with the gaseous phase at
the same temperature,
 The variation of molar volume of the gas with
temperature, and
 The variation of molar volume of the liquid phase
with temperature.
25
Group (15)
82. Drive the free energy-pressure relation of a system of
an ideal gas which undergo isothermal process.
83. Define the mole fraction of a species (𝒊) in a mixture of
perfect gases.
84. Write the statement of Dalton’s law; based on that
prove that𝑷(𝒊) = 𝒙(𝒊)𝑷.
85. For a mixture of ideal gases which undergoes
isothermal process; prove that:
𝑮(𝒊) − 𝑮° (𝒊) = 𝑹𝑻𝒍𝒏(𝒙(𝒊)𝑷)
∆𝑯(𝒊) = 𝟎
∆𝑯𝒎 = 𝟎
∆𝑮𝒎 = 𝑹𝑻 ∑ 𝒙(𝒊)𝒍𝒏𝒙(𝒊)
∆𝑺𝒎 = −𝑹 ∑ 𝒙(𝒊)𝒍𝒏𝒙(𝒊)
26
86. Define the fugacity term. Prove that⁡𝒇 = 𝑷𝟐 /𝑷𝒊𝒅𝒆𝒂𝒍 , for
a gaseous system whose equation of state is given by:
𝒇
𝒛−𝟏
𝑷(𝒗 − 𝒃) = 𝑹𝑻. Also prove that 𝒅𝒍𝒏 ( ) = ( ) 𝒅𝑷.
𝑷
𝑷
87. For Van der Waal’s isothermal shows in the following
figure, show how to apply the integration:
𝑷𝒊
𝑮(𝒊) = 𝑮(𝑨) + ∫ 𝑽𝒅𝑷
𝑷𝑨
At the points B, C, D, E, F, G, H, I, J, K, L, M, N, and
O.
Show that G-P curve you obtain on G-P diagram.
Explain the G-P results you obtain.
88. Prove that the enthalpy of vaporization of Van der
Waal’s liquid at𝑻𝒄𝒓 = 𝟎.
27
Group (16)
89. By considering the reaction𝑨(𝒈) + 𝑩(𝒈) = 𝟐𝑪(𝒈) and by
starting with 1 mole of A and 1 mole of B and assuming
that at time t, the number of moles of A, B and C are
n(A), n(B), and 2(1-n), prove that:
𝒏
∆𝑮 = (𝟏 − 𝒏)∆𝑮° (𝑹) + 𝟐𝑹𝑻(𝒏𝒍𝒏 (𝟐) + (𝟏 − 𝒏)𝒍𝒏⁡(𝟏 − 𝒏),
if⁡∆𝐺 ° (𝑅) = −1000⁡𝑐𝑎𝑙, show the dependence of
𝒏
(𝟏 − 𝒏)∆𝑮° (𝑹), 𝟐𝑹𝑻 (𝒏𝒍𝒏 ( ) + (𝟏 − 𝒏) 𝒍𝒏(𝟏 − 𝒏)),
𝟐
and ∆G on n. Prove that the equilibrium of this reaction
°
occurs at: ∆𝑮 =
𝑷(𝑪)𝟐
−𝑹𝑻𝒍𝒏 ( )
.
𝑷 𝑨 .𝑷(𝑩)
90. By considering a gaseous mixture of A, B, and C in
which the chemical reaction 𝑨 + 𝑩 = 𝑪; and if the
starting point is 1 mole A and 1 mole B, and by
considering that:
𝑮\ = 𝒏(𝑨)𝑮(𝑨) + 𝒏(𝑩)𝑮(𝑩) + ⋯
Prove that at equilibrium:
28
𝑷(𝑪)
∆𝑮 = −𝑹𝑻𝒍𝒏
= −𝑹𝑻𝒍𝒏𝑲𝒑
𝑷(𝑨)⁡𝑷(𝑩)
°
91. Starting by Gibbs-Helmholtz, drive the van’t Hoff
equation for the reaction R. show that if ∆𝐻 ° (𝑅) is
independent of temperature, Kp varies linearly with
(1/T). What is the name of Kp-(1/T) relation in this
case?
1
Based on van’t Hoff relation, show that Kp- relation
𝑇
obeys the le-Chatelier’s principle.
29
Group (17)
92. Prove that 𝑲𝒑 = 𝒌𝒙 𝑷(𝑪+𝑫)−(𝒂+𝒃) for the reaction
𝒂𝑨 + 𝒃𝑩 = 𝒄𝑪 + 𝒅𝑫. Give the value of Kxin terms of
x(A), x(B), x(C), and x(D). Show when Kx is
independent on the system pressure and when it is not.
Also prove that as the pressure of the system increase,
the reaction equilibrium will be shifted to the direction
of the side that have lower number of moles of gases.
93. By considering an ideal gaseous system that consists of
½ mole of A(g) and ½ mole of B(g) each at 1 atmosphere
which when mixed at constant T and P react according
𝟏
𝟏
to the reaction 𝑨(𝒈) + 𝑩(𝒈) = 𝑪(𝒈) ; prove that for an
𝟐
𝟐
extent x of the reaction; the equilibrium occurs at:𝒙 =
∆𝑮°
∆𝑮°
(− )/(𝟒 + 𝟐𝐞𝐱𝐩⁡(− )where ∆𝑮° is the standard
𝑹𝑻
𝑹𝑻
free energy change of the mentioned reaction. If
𝒇(𝑨) = 𝒇(𝑩) = 𝟏/(𝟐 + 𝐞𝐱𝐩 (−
30
∆𝑮°
𝑹𝑻
))at
the
equilibrium condition; then prove that if ∆𝑮° = 𝟎,at
equilibrium:
⁡⁡𝒙 =
𝟏
𝟏
𝟏
𝟏
, 𝑷𝑨 = 𝑷𝑩 = 𝑷𝑪 = , 𝒇𝑨 = 𝒇𝑩 = , ∆𝑮𝒎 = 𝑹𝑻𝒍𝒏( )
𝟔
𝟑
𝟑
𝟑
Group (18)
94. By considering the reaction:
𝟏
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑴(𝒔) + 𝑶𝟐 (𝒈) = 𝑴𝑶(𝒔)And
through
𝟐
the equilibrium that occurs between the equilibrium that
occurs between the vapors of the pure solid phases each
at (P,T) and oxygen, also at (P,T), prove that the
equilibrium of the above mentioned reaction occurs if:
𝟏
𝟐
°(
∆𝑮 𝑹) = −𝑹𝑻𝒍𝒏(𝟏/𝑷(𝑶𝟐 ) )
Calculate for this reaction system, the number of
degrees of freedom at the equilibrium states; show
schematically how this equilibrium is represented on
𝟏
𝒍𝒐𝒈𝑷(𝑶𝟐 ) − ( ) diagram.
𝑻
95. Given that 𝑪𝒑 (𝒊) = 𝒂 + 𝒃𝑻 + 𝑪𝑻𝟐 , prove that:
∆𝒃 𝟐
∆𝑪
°
∆𝑮 = ∆𝑯° + 𝑰(𝑻) − (∆𝒂)𝑻𝒍𝒏𝑻 ( ) 𝑻 − ( )/𝑻
𝟐
𝟐
For any chemical reaction; where I is an integration constant.
31
96. Write the three different relations that may be obtained
by fitting the experimentally measured values of
∆𝑮° (𝑹) at different values of T.
Group (19)
97. Complete:
 Experimentally Ellingham found that the (∆𝑮° − 𝑻)
relation for the oxidation of a series of pure metals by
oxygen to form pure metal oxides is given by the
relation: ∆𝑮° = ⋯ + ⋯
 The intercepts of an Ellingham line with ∆𝑮° axis at
T = 0K represents …… of the oxidation reaction
𝑴 + 𝑶𝟐 = 𝑴𝑶𝟐 and∆𝑺° (𝑹)is the ……..
 ∆𝑆 ° (𝑅) is given by: ∆𝑺° (𝑹) = ⋯
 ∆𝐺 ° (𝑅) is given by: ∆𝑮° (𝑹) = 𝑹𝑻 … …
 At the horizontal line ∆𝐺 ° = 0, 𝑷(𝑶𝟐 ) = ⋯ atm.
 The temperature at which Ellingham line intercepts
with the horizontal line ∆𝐺 ° = 0 represents the …,
…, … equilibrium under 𝑷(𝑶𝟐 ) = ⋯; under this
temperature, … will be spontaneous; if the
temperature is higher, the … reaction of MO will
spontaneously occur.
32
 If ∆𝐶𝑝 = 0, ∆𝐻 ° value will be linearly … of
temperature, and 𝑇∆𝑆 ° will be … dependent.
 If ∆𝑆 ° for two oxidation reaction are identical, the
relative stabilities of the two oxides depend on the
∆𝐻 ° values of the two reactions.
 The more … the value of …, the more stable the
oxide.
 If the two ∆𝐺 ° − 𝑇 lines representing the oxidation of
two metals intersects at temperature TE, the
intersection point represents the equilibrium:
𝐵 + ⋯ = 2𝐴 + ⋯
Below TE …, … are stable while, above TE …, … are
the stable phases.
 At T ˂ TE, … reduces … to form … + BO2.
98. Calculate the (∆G – T) relation for the states (2,T),
(1,T), (10-2, T), (10-6, T); plot these relations.
99. Draw to scale the variation with temperature of the
difference between the Gibbs free energy of 1 mole of
an ideal gas in the state (P = Patm, T) and the Gibbs free
energy of 1 mole of this gas in the state (P = 1atm, T)
where P = 102, 1, 10-2, 10-4, 10-6, 10-8, 10-10, and 1012
atm over the temperature range 0 – 1200K.
33
100. Superimpose the Ellingham line for the oxidation
reaction:
𝟐𝑨 + 𝑶𝟐 = 𝟐𝑨𝑶
whose
∆𝑮°
at
200k = -200,000 joule, and at 1200k = 100,000 joule on
the Richardson lines whose P(O2) = 10, 1, 10-4, 10-8,
and 10-20 atm. From this diagram determine the
temperatures at which the system A, AO2, O2
equilibrate at the mentioned oxygen pressures.
Group (20)
101. Discuss in-detail the effect of the phase transformation
of solid metal to liquid metal, and from solid metal
oxide to liquid metal oxide on the Ellingham line that
represents the oxidation of this metal. Show this effect
diagrammatically on the (∆𝐺 ° − 𝑇) Ellingham line for
the following two cases:
 𝑇𝑚 (𝑀) < 𝑇𝑚 (𝑀𝑂)
 𝑇𝑚 (𝑀) > 𝑇𝑚 (𝑀𝑂)
102. Given that:
For
the
reaction:⁡𝐶 (𝑔𝑟) + 𝑂2 (𝑔) =
𝐶𝑂2 (𝑔)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(𝑖),∆𝐺 ° = −94,200 − 0.2𝑇⁡𝑐𝑎𝑙,
And for the reaction: 2𝐶 (𝑔𝑟) + 𝑂2 (𝑔) = 2𝐶𝑂(𝑔)(𝑖𝑖 ),
∆𝐺 ° = −53,400 − 41.90𝑇⁡𝑐𝑎𝑙.
Show the effect of varying the pressures of the product
gases of reactions (i) and (ii) on the variations of ∆G
34
with T; determine the equilibrium temperature for the
system C, CO (1atm, T), and CO2 (1atm, T).
103. Plot the variation of %CO in a CO/CO2 mixture, which
is in equilibrium with graphite, versus temperature
given that: 𝑃(𝐶𝑂) + 𝑃(𝐶𝑂2 ) = 1. The ∆𝐺 ° − 𝑇
relation for the reaction: 𝑪(𝒈𝒓) + 𝑪𝑶𝟐 (𝒈) = 𝟐𝑪𝑶(𝒈)
is given by: ∆𝐺 ° = 48,800 − 41.7𝑇⁡𝑐𝑎𝑙, explain the
curve.
Group (21)
104. Illustrate the effect of the ratio P(CO)/P(CO2) in a COCO2 gas mixture on the temperature at which the
reaction
equilibrium
𝑀 + 𝐶𝑂2 = 𝑀𝑂 + 𝐶𝑂
is
established.
The ∆𝐺 ° − 𝑇 relation for the reactions:
 𝐶 (𝑔𝑟) + 𝑂2 (𝑔) = 𝐶𝑂2 (𝑔)(𝑖 ),
 𝐶 (𝑔𝑟) + 𝑂2 (𝑔) = 2𝐶𝑂(𝑔)(𝑖𝑖 ),
Is given by:
 ∆𝐺 ° (𝑖 ) = −94,200 − 0.2𝑇⁡𝑐𝑎𝑙
 ∆𝐺 ° (𝑖𝑖 ) = −53,400 − 41.90𝑇⁡𝑐𝑎𝑙
Make your calculations at: P(CO)/P(CO2) = 10-2, 10-1, 1,
10, 102.
35
Calculate the equilibrium temperatures of the system M,
MO, CO, CO2 at the mentioned pressures if ∆𝐺 ° values
for the reaction 𝑀 + 𝑂2 = 𝑀𝑂2 are -600,000
joules/mole O2 at 200K and -300,000 joules/mole O2 at
800K. Also, construct the CO/CO2 scale for this system.
105. By considering the attached figures calculate the
minimum values of P(CO)/P(CO2) at 1000°C; also
calculate the following values:
 P(CO)/P(CO2) whose P(O2) = 10-10atm, at which Fe,
FeO, CO, CO2 equilibrate at 1000°C
 ∆𝐻 ° of the reaction 𝐶𝑎𝑂 + 𝐶𝑂 = 𝐶𝑂 + 𝐶𝑂2 .
Group (22)
106. Define the following: the kinetics of a reaction, the rate
of a reaction, the first-order reaction, the second-order
reaction, the half-life period of a reaction, and
Arrhenius equation.
107. Prove that for a first-order reaction:
𝑪𝒐
𝑲
𝒍𝒐𝒈
=
𝒕
𝑪𝒐 − 𝒙 𝟐. 𝟑𝟎𝟑
𝒕𝟎.𝟓
𝟐. 𝟑𝟎𝟑𝒍𝒐𝒈𝟐
=
𝑲
36
108. Prove that for a second-order reaction:
If 𝐶𝑜 (𝐴) > 𝐶𝑜 (𝐵):
𝑪𝒐 (𝑩)(𝑪𝒐 (𝑨) − 𝒙)
𝑲
[𝑪𝒐 (𝑨) − 𝑪𝒐 (𝑩)]𝒍𝒐𝒈
=
𝒕,
(
)
𝑪𝒐 (𝑨)(𝑪𝒐 𝑩 − 𝒙) 𝟐. 𝟑𝟎𝟑
If 𝐶 (𝐴) = 𝐶 (𝐵) = 𝐶:
𝒙
= 𝑲𝑻
𝑪𝒐 (𝑪𝒐 − 𝒙)
𝟏
𝒕𝒐.𝟓 =
𝑲𝑪𝒐
109. Complete:
 The effect of temperature on reaction rate constant is
given by ……… equation, which is mathematically
expressed by:
k = ………
Logk = ……… - 0.4342 ………/………
Define the constants of this equation.
1
 The slope of the straight line 𝑙𝑜𝑔𝑘 − equal to
𝑇
………
 The half-life period of a reaction of an overall order n
is given by:
𝑡0.5 = ………
or, log(𝑡0.5 ) = ………
 The slope of the straight line log(𝑡0.5 ) = 𝑙𝑜𝑔𝐶𝑜 =⁡….
37
PHYSICAL
CHEMISTRY
MET (201)
GENERAL QUESTIONS (2)
38
110. Write short notes on:
 Simple equilibrium.
 Extensive and intensive properties, give examples.
 Homogenous and heterogeneous systems.
 The historic development of the relationship between
heat and work. Ending by the mechanical equivalent
of heat.
 The first law of thermodynamics and the
impossibility of constructing a perpetual motion
machine of the first kind.
111. Based on experimentation, prove that 𝒖 = 𝒇(𝑻) for
perfect gases.
39
112. For real gases that obey the equation of state:
𝒂
(𝑷 + ( 𝟐 )) (𝒗 − 𝒃) = 𝑹𝑻
𝒗
Prove that: 𝒖 = 𝒇(𝒗, 𝑻).
113. Is it possible to have more efficient heat engine
between t2 and t1 following Carnot cycle? Give
arguments leading to Thomson impossibility statement,
and Clausis impossibility statement.
114. Show how Kelvin established the thermodynamic
temperature scale; prove that it is identical to the ideal
gas temperature scale.
115. Prove that:
𝑞
 𝑑𝑆𝑠𝑦𝑠𝑡𝑒𝑚 = + 𝑑𝑆𝑖𝑟𝑟
𝑇
 (−𝛿𝑤) = 𝑇∆𝑆 − 𝑑𝑢 − 𝑇𝑑𝑆𝑖𝑟𝑟
 (−𝑊)𝑚𝑎𝑥 = 𝑇∆𝑆 − ∆𝑢
116. The combined mathematical expression of the first and
second laws of thermodynamics for reversible
processes.
117. What are the criteria of equilibrium for:
 An isolated system of constant u and v.
40
 An isolated system of constant S and v.
 An isolated system of constant T and v.
 An isolated system of constant T and P.
118. Statistical interpretation of entropy is based on:
 Gibbs statement that related the entropy to disorder,
 Statistical aspects that related equilibrium conditions
to the condition of microstates, and
 The quantitisation of energy levels.
Discuss in-detail each of the previous statements.
119. Prove the following relations:
𝑛
∈
 𝑛𝑖 = ( ) exp⁡(− 𝑖 )
𝑃
𝐾𝑇
 𝑈 = 𝐾𝑇𝑙𝑛𝛺 − 𝑛𝐾𝑇⁡𝑙𝑛𝑃
 𝑆 = 𝐾𝑙𝑛𝛺 (Boltzmann’s equation).
120. Define the Helmholtz free energy function, based on
this definition prove that:
 𝑑𝐴 = 𝑆𝑑𝑇 − 𝑝𝑑𝑣
 𝑆 = −(𝜕𝐴/𝜕𝑇)𝑣
 𝑃 = −(𝜕𝐴/𝜕𝑣) 𝑇
𝜕𝑆
 ( ) 𝑇 = (𝜕𝑃/𝜕𝑇)𝑉
𝜕𝑉
𝜕𝐴
𝜕𝑇
𝜕𝑉
𝜕𝑇
𝜕𝑉
𝜕𝐴
 ( )𝑃 ( )𝐴 ( ) = −1
41
121. Define Gibbs free energy function; starting by this
definition, prove that:
 𝑑𝐺 = 𝑓 (𝑃, 𝑇)
 (𝑊 \ ) ≤ −∆𝐺
 𝑑𝐺 = 0.0
 𝑑𝐺 = ∑ 𝜇𝑖 𝑑𝑛𝑖
 𝛿𝑤 = −𝑃𝑑𝑉 + ∑ 𝜇𝑖 𝑛𝑖 ,
state the conditions necessary to apply each of the
previous relations.
122. State the empirical rule that was observed by Dulong
and Petit in 1819 on the values of constant volume
specific heat capacities per gm atom of solid elements
at high temperature.
123. In-detail explain the dependence of Cv on temperature
calculated for Einstein crystal; state the main
assumptions made by Einstein.
124. Why Debye assumed different assumptions to reach
more acceptable Cv –T relation. Discuss in-detail the
Debye theoretical calculations and the assumptions he
made to react Cv to T at very low, normal, and high
temperatures.
42
125. Define the following terms:
 Standard state,
 The reference state,
 Standard enthalpy of transition; give four examples,
 The standard enthalpy of a reaction, conceptual
reactions standard enthalpy of formation.
126. Discuss in-detail:
 Hess’s law, and
 Kirchoff’s law.
127. Prove that:
𝑇
𝑃
 ∆𝐻 = ∫𝑇 2 𝐶𝑝 𝑑𝑇 + ∫𝑃 1 𝑉(1 − 𝛼)𝑑𝑃
1
 ∆𝑆 =
𝑇2
∫𝑇 𝐶𝑝 𝑑 (𝑙𝑛𝑇)
1
128. Prove
that:
conditions).
2
𝑃
− ∫𝑃 2 𝛼𝑉𝑑𝑃
1
𝑇
𝑆𝑇 = 𝑆𝑜 + ∫𝑇 2 𝐶𝑝 𝑑𝑙𝑛𝑇(state
1
129. State the following postulates:
 Nernst Postulates at 1906 (2 postulates).
 Nernst heat theorem.
 Plank’s postulate (third law of thermodynamics).
43
the
What is the incompetence of each of the previous
postulates?
130. Explain in-detail an experiment that verify the third law
of thermodynamics.
131. Complete the following:
°
 Experimentally it was found that ∆𝑆𝑓𝑢𝑠
= ⋯ at
Tfusfor numbers of metals.
 For some metals, experimentally it was found that
°
∆𝑆𝑣𝑎𝑝
= ⋯ at Tvap(Trouton’s law).
 The entropy change of the reaction: 2𝑀𝑐 + 𝑂2 = 2𝑀𝑂𝑐
is given by: ∆𝑆 = − ⋯
132. Define the following terms:





The rate of a chemical reaction,
The extent of the reaction,
First order reaction,
Second order reaction, and
Half-life time of the reaction (t0.5).
133. Relate the extent of the reaction to t for:




First order reaction whose t=0.0, C=𝐶0° ,
Second order reaction for the case:𝐶𝐴° ≠ 𝐶𝐵° and𝐶𝐴° > 𝐶𝐵° ,
Second order reaction for the case: 𝐶𝐴° = 𝐶𝐵° = 𝐶 ° ,
Third order reaction whose 𝐶𝐴° = 𝐶𝐵° = 𝐶𝑐° = 𝐶 ° ,
44
Derive the relation between t0.5 with C° for the previous cases.
134. What is the condition that must exist to apply the relation:
1
1
𝑡0.5 = ( )( 𝑛−1 )
𝐾 𝐶𝑜
How to evaluate the value of n?
45