# Ideal Gas Model - Thermal Properties of Matter - Exam, Exams for Thermal Analysis. Allahabad University

## Thermal Analysis

Description: This is the Exam of Thermal Properties of Matter which includes Thermodynamic Quantity, Einstein’s Theory, Ehrenfest Classification, Phase Transitions, Thermodynamic Potential, Clausius-Clapeyron Equation, Molar Volume of Solid etc. Key important points are: Ideal Gas Model, Laws of Thermodynamics, Van-Der-Waals Gas Model, Equipartition Theorem, Lambda Phase Transitions, Specific Heat of Paramagnetic Solids, Schottky Anomaly, Definitions of Entropy
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L A N C A S T E R U N I V E R S I T Y
2010 EXAMINATIONS
Part II
PHYSICS - Paper 2.D
Candidates should answer all those sections identiﬁed with the modules for which
they are registered.
An indication of mark weighting is given by the numbers in square brackets following
each part.
Use a separate answer book for each section.
PHYSICAL CONSTANTS
Planck’s constant h= 6.63 ×1034 J s
¯h= 1.05 ×1034 J s
Boltzmann’s constant kB= 1.38 ×1023 J K1
Mass of electron me= 9.11 ×1031 kg
Mass of proton mp= 1.67 ×1027 kg
Electronic charge e= 1.60 ×1019 C
Speed of light c= 3.00 ×108m s1
Avogadro’s number NA= 6.02 ×1023 mol1
Permittivity of the vacuum ǫ0= 8.85 ×1012 F m1
Permeability of the vacuum µ0= 4π×107H m1
Gravitational constant G= 6.67 ×1011 N m2kg2
Bohr magneton µB= 9.27 ×1024 J T1(or A m2)
Bohr radius a0= 5.29 ×1011 m
Gas constant R= 8.31 J K1mol1
Acceleration due to gravity g= 9.81 m s2
1 standard atmosphere = 1.01 ×105N m2
Mass of Earth = 5.97 ×1024 kg
Radius of Earth = 6.38 ×106m
Density of iron = 7.6×103kg m3
1
Section A: Module 233: Thermal Properties of Matter
(The time allocated for this section is 80 minutes.
Candidates should answer question A1 and
either question A2 or question A3.)
Compulsory question:
A1. (a) State the First, Second and Third Laws of Thermodynamics. [8]
(b) (i) Describe the assumptions of the ideal gas model. Sketch a set of isotherms
for three temperatures, where T1> T2> T3, for an ideal gas on a P-V
diagram.
(i) Describe the assumptions of the Van-der-Waals gas model. Sketch a similar
set of isotherms for the Van-der-Waals gas on a P-Vdiagram. Clearly
identify an isotherm corresponding to temperature below which the Van-
der-Waals gas could be liquiﬁed. [8]
(c) State the equipartition theorem. Using the equipartition theorem, calculate
the internal energy and heat capacity at room temperature for one mole of i)
a monoatomic gas and ii) a diatomic gas. [8]
(d) Explain the distinction between ﬁrst order, second order and lambda phase
transitions. Illustrate your answer with examples (one for each transition) and
sketches showing how the speciﬁc heat varies with temperature. [8]
(e) The speciﬁc heat of paramagnetic solids show a Schottky anomaly at low tem-
perature. Describe brieﬂy what is a Schottky anomaly and how it can be
explained in terms of the microscopic properties of the system. [8]
Answer one of the following two questions:
A2. (a) Give deﬁnitions of entropy in the i) thermodynamical and ii) statistical frame-
works. Explain how the irreversibility of a process is linked to entropy. [16]
(b) (i) Calculate the number of microstates for a 2-level system that consists of
N-elements assuming that the temperature of the system is such that the
probabilities of occupying both energy levels are equal. Using Stirling’s
approximation ln n!nln nnand the statistical framework, calculate
the entropy of the system. [14]
(ii) Calculate the entropy of the same system when only the lowest energy level
is occupied. Discuss the temperature of the system in this case. Explain
whether the system is least ordered in this case or in case (i), when both
levels are occupied equally. [10]
2
A3. (a) Deﬁne two of the four thermodynamic potentials for a hydrostatic system and
any quantities used in their deﬁnition. Describe why each of these potentials
is useful. [16]
(b) Write down the dierential form of the internal energy U, or the central equa-
tion of thermodynamics, for a hydrostatic system.
Hence show that
T= U
S !V
;P= U
V !S
where P,V,Sand Tare pressure, volume, entropy and temperature respec-
tively.
Derive the corresponding Maxwell relation. [14]
(c) The internal energy of the ideal gas is given by
U=U(S, V ) = αN kBN
V2/3
e2S/3NkB
where αis a constant, kBis the Boltzmann’s constant and Nis number of
atoms. Show that the equation of state for the ideal gas follows from this
equation. [10]
3
Section B: Module 235: Nuclei & Particles
(The time allocated for this section is 40 minutes.
Candidates should answer question B1 and
either question B2 or question B3. )
Compulsory question:
B1. (a) The mass of the atom 20
10Ne is 19.9924 u. Find the nuclear binding energy in
MeV. [mn=1.0087 u, MH=1.0078 u, 1 u = 931.494 MeV/c2.] [1]
(b) The Homestake Experiment measured the ﬂux of solar neutrinos by measuring
the rate of the neutrino capture interaction given by νe+37
17Cl37
18Ar++e.
(i) What is the minimum energy in MeV that the neutrinos must have in
order to participate in this interaction?
[M(37
17Cl)=36.965903 u, M(37
18Ar)=36.966776 u, me=0.000549 u,
1 u = 931.494 MeV/c2.]
(ii) Neutrinos produced via the p-p reaction (the ﬁrst step in the proton chain)
in the Sun have a maximum energy of 0.42 MeV. Was Homestake sensitive
to these neutrinos? Justify your answer. [3]
(c) Consider a family of 5 nuclei with even mass number A, and another family of
5 nuclei with odd mass number A.
(i) For each case, sketch the variation of the atomic mass M(Z,A) as predicted
by the Semi-Empirical Mass Formula (SEMF), as a function of Z. Put a
data point for each of the 5 nuclei on your plot.
(ii) Explain why these two sketches are diﬀerent, making reference to the ap-
propriate term in the SEMF.
(iii) On the sketch with even mass number A, indicate which nuclei are likely to
undergo βdecay, β+decay, or electron capture, and which nucleus(nuclei)
is(are) stable against βdecays.
(iv) Explain why some nuclei can undergo electron capture, but not β+decay. [10]
(d) Thorium 228 (228
90Th) is an α-emitter with a half-life of 1.91 years. If the
current activity of a thorium 228 sample is 32.0 Bq, calculate the number of
years needed for the activity to be reduced to 1.00 Bq. [4]
(e) Nuclear reactions may be initiated by bombarding a target with nucleons. For
each of the interactions below, determine the missing nucleon and A,Z for the
resulting unstable nucleus.
(i) 35
17Cl+ ? ?32
16S+4
2He
(ii) 10
5B+ ? ?7
3Li+4
2He [2]
4
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