Y2Exam-January2022 - Isaac Castillo Soto

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A19922 ANY CALCULATOR
.
School of Physics and Astronomy
DEGREE OF B.Sc. & M.Sci. WITH HONOURS
SECOND-YEAR EXAMINATION
03 17296
LI STATISTICAL PHYSICS & ENTROPY
SEMESTER 1 EXAMINATIONS 2021/22
Time Allowed: 1 hour 30 minutes
Answer four questions from Section 1 and two questions from Section 2.
Section 1 consists of four questions and carries 40% of the marks for the examination.
Answer all four questions from this Section.
Section 2 consists of three questions and carries 60% of the marks.
Answer two questions from this Section. If you answer more than two questions,
credit will only be given for the best two answers.
The approximate allocation of marks to each part
of a question is shown in brackets [ ].
All symbols have their usual meanings.
Calculators may be used in this examination but must not be used to store text.
Calculators with the ability to store text should have their memories deleted prior to
the start of the examination.
A formula sheet and a table of physical constants and units that may be required
will be found at the end of this question paper.
Page 1 TURN OVER
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SECTION 1
Answer all four questions from this Section.
1. Calculate the change in entropy of 1 kg of water when it is heated from 20◦C to
100◦C and completely vaporized.
Indicate whether the change in entropy implies any irreversibility in the process.
[At a pressure of 1 atm the specific latent heat of vaporization of water is L =
2.3×106 Jkg−1 and the specific heat capacity of water is cP = 4.2 kJK−1kg−1.] [10]
2. Ten distinguishable, non-interacting atoms undergo random thermal motion inside
a box of volume V . For a sub-volume V/5 inside the box (as illustrated in the
diagram), find the probability that:
(a) all ten atoms are inside the sub-volume;
(b) exactly two atoms are inside the sub-volume.
[10]
Total Volume V
Sub-volume
V/5
3. An atom has discrete energy levels among which are a non-degenerate ground
state and a doubly-degenerate first excited state that is 1 eV above the ground
state. Calculate the ratio of probabilities of finding the atom in the first two energy
levels if it is in thermal equilibrium with a reservoir at 104K. [10]
4. By introducing the free energy F , show that for a gas(
∂T
∂P
)
V
=
(
∂V
∂S
)
T
.
[10]
A19922 Page 2 TURN OVER
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SECTION 2
Answer two questions from this Section. If you answer more than two questions,
credit will only be given for the best two answers.
5. (a) The reversible work done on a purely one-dimensional system can be written
d̄W = ΓdL, where L is the extension of the system and Γ is the tension
exerted on the system. Write down the fundamental thermodynamic relation
for such a system and deduce the corresponding relation for the differential
of the Helmholtz free energy F .
Hence show that the entropy of the system is S = −
(
∂F
∂T
)
L
. [10]
(b) An example of one-dimensional system is the ideal polymer chain. The
tension of the chain at temperature T is given by the equation of state
Γ = αTL, where α is a positive constant. Show that the free energy of
the polymer chain is given by
F = F0 +
1
2
αTL2,
where F0 is the free energy when L = 0. Hence, derive an expression for
the entropy S of the polymer chain as a function of the extension L.
Sketch a graph to show how how S varies with L and explain its physical
significance. [10]
(c) Suppose that the polymer chain is stretched reversibly by a small amount
δL from extension L to extension L+ δL at constant temperature T . Find
the heat absorbed δQ and the work δW done on the polymer. What is
the increase in internal energy δU of the polymer? Explain the physical
significance of your result. [10]
A19922 Page 3 TURN OVER
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6. (a) State how
(
∂S
∂U
)
V
is related to the temperature of a thermodynamic system.
[2]
A system consists of three distinguishable atoms, each of which has three non-
degenerate energy levels, 0, ϵ and 2ϵ.
(b) How many microstates does the system possess? [3]
(c) Make a table showing the statistical weight and the entropy corresponding
to each energy level. [9]
(d) Make a rough sketch of entropy versus energy for this system. [5]
(e) Sketch the temperature of the above system as a function of the energy. [5]
(f) The three atoms are now free to move inside a box and form an ideal gas at
temperature T. Show formally that the fraction of the energy of the gas due
to the internal state of the atoms cannot be larger than 3ϵ. [6]
A19922 Page 4 TURN OVER
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7. (a) A sample of heat capacity C at temperature T1 is brought into contact with
a cold reservoir at temperature T0 < T1 until it reaches thermal equilibrium.
Derive expressions for the changes in entropy of the sample and the reservoir.
[6]
(b) We repeat the experiment in part (a), but, instead of plain thermal contact, a
reversible engine operates between the sample and the cold reservoir.
i. Find the temperature of the sample after a long time. [3]
ii. Over one cycle the engine extracts the heat δQ from the sample at
temperature T and produces the work δW. Derive expressions for
δQ and δW as a function of T, specifying the assumptions you make.
Derive expressions for the total work extracted from the machine, and
the entropy changes of the sample and the cold reservoir. [11]
(c) Assume the same engine now executes a cycle composed of two isotherms
and two isochores. Without derivation, explain how the total work extracted
from the machine and the entropy changes of the sample and cold reservoir
compare to those found in parts (a) and (b). [10]
A19922 Page 5 TURN OVER
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Formula Sheet
Classical thermodynamics
• Carnot theorem:
Q1
Q2
=
T1
T2
• Carnot efficiency: η = 1− T1
T2
• Clausius inequality:
∮
d̄Q
Text
⩽ 0
• First law: dU = d̄Q+ d̄W , ∆U = Q+W
• Entropy definition: T dS = d̄Qrev, ∆S =
∫
d̄Qrev
T
• Second law: Text dS ⩾ d̄Q, ∆S ⩾
∫
d̄Q
Text
• Fundamental thermodynamics identity (gas): dU = T dS − P dV
• Thermodynamics potentials (for a gas)
– Free energy: F = U − TS
– Enthalpy: H = U + PV
– Gibbs free energy: G = U − TS + PV
Ideal gas (monoatomic)
• PV = nRT = NkT
• E = 32NkT =
3
2nRT = CV T
• CP − CV = nR
• isentropic transformation: PV γ constant with γ = CP/CV
Statistical physics
• Boltzmann entropy: S = k lnΩ
• Boltzmann probability (state): pi =
1
Z
e−
Ei
kT with Z =
∑
states i
e−
Ei
kT
• Boltzmann probability (energy level): p(En) =
gn
Z
e−
En
kT with Z =
∑
levels n
gn e
−EnkT
A19922 Page 6 TURN OVER
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Mathematical results
• For functions x(y, z), y(x, z) and z(x, y),
� dz =
(
∂z
∂x
)
y
dx+
(
∂z
∂y
)
x
dy and
∂2z
∂x∂y
=
∂2z
∂y∂x
�
(
∂x
∂y
)
z
=
[(
∂y
∂x
)
z
]−1
and
(
∂x
∂y
)
z
(
∂y
∂z
)
x
(
∂z
∂x
)
y
= −1
• Stirling formula: lnN ! ≃ N lnN −N
• 1 + x+ x2 + . . . =
1
1− x for x < 1
•
∫ +∞
−∞
e−αx
2
dx =
√
π
α
•
∫ +∞
−∞
x2e−αx
2
dx =
1
2
√
π
α3
• sinhx =
ex − e−x
2
, coshx =
ex + e−x
2
•
d
dx
coshx = sinhx
A19922 Page 7 TURN OVER
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Physical Constants and Units
Acceleration due to gravity g 9.81m s−2
Gravitational constant G 6.674× 10−11 N m2 kg−2
Ice point Tice 273.15K
Avogadro constant NA 6.022× 1023 mol−1
[N.B. 1 mole ≡ 1 gram-molecule]
Gas constant R 8.314 J K−1 mol−1
Boltzmann constant k, kB 1.381× 10−23 J K−1 ≡ 8.62× 10−5 eV K−1
Stefan constant σ 5.670× 10−8 W m−2 K−4
Rydberg constant R∞ 1.097× 107 m−1
R∞hc 13.606eV
Planck constant h 6.626× 10−34 J s ≡ 4.136× 10−15 eV s
h/2π h̄ 1.055× 10−34 J s ≡ 6.582× 10−16 eV s
Speed of light in vacuo c 2.998× 108 m s−1
h̄c 197.3MeV fm
Charge of proton e 1.602× 10−19 C
Mass of electron me 9.109× 10−31 kg
Rest energy of electron 0.511MeV
Mass of proton mp 1.673× 10−27 kg
Rest energy of proton 938.3MeV
One atomic mass unit u 1.66× 10−27 kg
Atomic mass unit energy equivalent 931.5MeV
Electric constant �0 8.854× 10−12 F m−1
Magnetic constant µ0 4π × 10−7 H m−1
Bohr magneton µB 9.274× 10−24 A m2 (J T−1)
Nuclear magneton µN 5.051×10−27 A m2 (J T−1)
Fine-structure constant α = e2/4π�0h̄c 7.297× 10−3= 1/137.0
Compton wavelength of electron λc = h/mec 2.426× 10−12 m
Bohr radius a0 5.2918× 10−11 m
angstrom Å 10−10 m
barn b 10−28 m2
torr (mm Hg at 0 ◦C) torr 133.32Pa (N m−2)
1
A19922 Page 8 END OF PAPER