Problems in thermodynamics and statistical physics
1. Let F(x, y) = x · ex2+y2
. Calculate (a) ∂F
∂x
, (b) ∂F
∂y
, (c) ∂2F
∂x2 , (d) ∂2F
∂x∂y
, (e) ∂2F
∂y∂x
, (f)
∂2F
∂y2 .
2. Let dω = A(x, y)dx + B(x, y)dy be any differential form (Pfaffian). Show that if
dω is an exact differential (there exists a function F(x, y) such that dω = dF), then it
must hold
a)
∂A
∂y
=
∂B
∂x
, b) dω = 0,
(b) for each closed integration path.
3. Let dω1 = (x2
− y) dx + x dy. Is it an exact differential, dω2 = dω1/x2
an exact
differential? Calculate the integral dω between the points (1, 1) and (2, 2) along the
lines (1, 1) → (1, 2) → (2, 2) and (1, 1) → (2, 1) → (2, 2).
4. Is dω = p dV + V dp an exact differential? If so, determine the function F whose
exact differentialis dω. Compute the integral dω between the points (V1, p1) and (V2, p2)
along the lines (V1, p1) → (V1, p2) → (V2, p2) a (V1, p1) → (V2, p1) → (V2, p2).
5. Is the fom dQ = c dT + RT
V
dV an exact differential? Calculate the integral dω
between the points (V1, T1) and (V2, T2) long the lines (V1, T1) → (V1, T2) → (V2, T2)
a (V1, T1) → (V2, T1) → (V2, T2). What function f(V, T) must we multiply dQ by to
make the product f dQ an exact differential? Determine the function S for which
dS = f dQ.c and R are constants.
6. Let x, y a z be 3 state variables, connected by the state equation f(x, y, z) = 0.
Show the validity of the relations
∂x
∂y z
=
∂y
∂x
−1
z
,
∂x
∂y z
= −
∂x
∂z y
∂z
∂y x
and
∂x
∂y z
=
∂x
∂y w
+
∂x
∂w y
∂w
∂y z
where the subscript indicates a constant variable and w is another state variable, w =
w(x, y, z).
7. The equation of state pV = NkT connects the variables p, V and T, where N and
k are constants. Verify by direct calculation that
∂p
∂V T
=
∂V
∂p
−1
T
∂p
∂T V
=
∂T
∂p
−1
V
1
∂p
∂V T
= −
∂p
∂T V
∂T
∂V p
∂T
∂V p
= −
∂T
∂p V
∂p
∂V T
8. The equation of state of an ideal gas can be written as
• pV = NkT,
• pV = n1RT,
• p = ρkT
µ
,
• p = nkT,
where p, V, T are pressure, volume and temperature, N is the number of particles, n
is their concentration, k is Boltzmann constant (k = 1, 38 · 10−23
m2
kg s−2
K−1
), R
is the gas constant (R = 8, 31 J mol−1
K−1
), n1 is the amount of substance, ρ is the
gas density and µ is the molecular weight. Verify the units of k and R. What are the
units of n? Show that the individual equations are equivalent rovnice jsou ekvivalentní
NA = 6.022 · 1023
mol−1
.
9. At a constant temperature of 20◦
C an ideal gas expands quasi-statically from a state
with a pressure of 20 atm to a state with a pressure of 1 atm. What work is done by 1
mole of gas?
10. During the quasi-static adiabatic expansion of 6 liters of helium at a temperature
of 350 K, the pressure drops from 40 atm to 1 atm. Calculate the resulting volume and
temperature (assume the ideal gas equation of state). Compare the obtained results
with the values that would be obtained for isothermal expansion (κ = 1, 63). Assume
it is an ideal gas.
11. Calculate the work done by an ideal gas during a quasi-static adiabatic expansion
from a state characterized by p1, V1 to a state p2, V2. Determine the work done by the
gas if it passes from the initial state to the final state first by an isochoric process and
then by an isobaric process, or first by an isobaric process and then by an isochoric
process.
12. During the exchange of air between the lower and upper layers of the troposphere,
expansion occurs, or air compression: rising air expands in an area of lower pressure.
Due to the low thermal conductivity of air, the expansion and compression processes
can be considered adiabatic. Calculate the change in temperature with height due to
these processes. (Consider air as an ideal gas.)
13. Assume that the atmosphere of the planet Venus contains k1 = 96.5% molecules
CO2 and k2 = 3.5% molecules N2. We can neglect the other components. The temperature
of the atmosphere is t = 464◦
C and the atmospheric pressure on the surface of
Venus contains p0 = 9.1MPa. The mass of the planet is M = 4.87 · 1024
kg and the
radius R = 6052 km.
2
(a) Determine the density ρ0 of the atmosphere and the gravitational acceleration gv
at the surface of Venus.
(b) To research the planet’s atmosphere, we will use an open "hot air balloon"(filled,
of course, with the planet’s atmosphere) with a volume of V = 50 m3
. The weight of
the structure is m = 100 kg. To what temperature t1 must we heat the gas in the
balloon to start rising above the surface of the planet? At what temperature t2 inside
the balloon will we reach a height of 1 km?
Neglect the rotation of Venus and the drop in gravitational acceleration as the
balloon exits. Consider the temperature of the atmosphere up to a height of 1 km as
constant. The molar masses of the two main components of the Venus atmosphere are
Mm (CO2) = 44.0 · 10−3
kg · mol−1
, Mm (N2) = 28.0 · 10−3
kg · mol−1
.
14. A gas is described by the equation of state p = p(V, T). Show by direct calculation
that δQ is not an exact differential.
15. The energy of a particle enclosed in an infinitely high potential well with dimensions
Lx × Ly × Lz is given by the relation
E =
2
2m
π2 n2
x
L2
x
+
n2
y
L2
y
+
n2
y
L2
y
Assume that Lx = Ly = Lz = L. Assume that the system as a whole is characterized
by the energy of the system. How is the microstate determined, how is the macrostate
determined? Calculate the force that the particle exerts on the walls of the container.
Determine the relationship between the energy of the system and the pressure.
16. Derive from the existence of the equation of state f(p, V, T) = 0 the relation
α = p · β · κ
between the thermal coefficient of expansion α := 1
V
∂V
∂T p
, isochoric expansion coefficeint
β := 1
p
∂p
∂T V
and the coefficient of isothermal compressibility κ := − 1
V
∂V
∂p
T
.
17. The state equation has the form p = f(V ) · T. Prove::
(a) ∂E
∂V T
= 0
(b) if a holds, then ∂E
∂p
T
= 0.
18. For an ideal gas, calculate ∂p
∂V ad
, ∂p
∂V T
.
19. Show that for a gas described by the equation of state f(p, V, T) = 0 holds
∂p
∂V ad
=
∂p
∂V T
cp
cV
20. Show the validity of the relation cp − cV = R between the isobaric and isochoric
specific heats of one mole of an ideal gas. The internal energy of an ideal gas does not
depend on its volume.
3
21. Calculate the entropy of an ideal gas when cp = konst., cV = konst. Show that δQ
is not an exact differential.
22. Prove the validity of the relation pV κ
= konst. ( κ = cp/cV is the adiabatic
exponent) for quasi-static adiabatic process of an ideal gas. Calculate κ assuming that
cV = 3
2
R.
23. For a gas, it has been found experimentally that the product of pressure and volume
is a function of temperature only, teploty, pV = f(T) and that the internal energy also
depends only on temperature. What shape does f(T) have?
24. For a photon gas, the energy density is only a function of temperature and the
pressure is given by p = 1
3
u(T), where u(T) = E/V . Calculate
(a) Function u(T),
(b) entropy„
(c) isotherm and adiabat equation.
25. A rod is twisted by a moment of force M through an angle ϕ. (a) Prove that the
first theorem of thermodynamics in this case is of the form
dE = δQ + M dϕ.
(b) Derive from the definition of heat capacity (and the first theorem of thermodynamics)
expressions for cM a cϕ.
(c) Find the relationship between ∂M
∂ϕ
adiab
a ∂M
∂ϕ
izoterm
.
26. (a) 1 kg of water at 0◦
C is brought into thermal contact with a large reservoir at
100◦
C. Calculate the entropy change of the water, the reservoir, and the entire system
after equilibrium is established.
(b) Calculate the change in entropy of the entire system if the water was first in contact
with a reservoir at a temperature of 50◦
and then with a reservoir at a temperature
100◦
C.
(c) How to ensure that the entropy of the system does not change when the water is
heated?
27. Show that for small deviations δρ, δp from the equilibrium values of density ρ0 and
pressure p0 the propagation of sound waves can be described by a wave equation
∂2
δp
∂t2
= c2 ∂2
δp
∂x2
,
where the speed of sound is given by the relation c = (∂p/∂ρ)ad if we assume that
events are on- ad fast that there is no heat exchange between the individual elements
of the air. Show that the speed of sound can also be calculated as c = κad/ρ0, where
the adiabatic compressibility κad := −V ∂p
∂V ad
Calculate the speed of sound in air
assuming that air is made up of only N2 molecules and that κ = cp/cV = 7/5.
28. An ideal gas expands adiabatically from volume V1 into vacuum. Calculate the
increase in entropy if the gas in the final state has volume V2 and prove that the
expansion process is irreversible.
4
29. The van der Waals equation of state for 1 mole of gas has the form
p +
a
V 2
(V − b) = RT
where a, b are constants. For a given T he curve can have two extremes given by
equation
∂p
∂V T
= 0
In addition, at the critical point determined by the parameters Tc, pc a Vc
∂2
p
∂V 2
T
= 0
Calculate the values of Tc, pc a Vc. Write the state equation using the variables T′
=
T/Tc, p′
= p/pc a V ′
= V/Vc.
30. Determine::
(a) the internal energy and entropy of a van der Waals gas,
(b) the work of a van der Waals gas during reversible isothermal expansion,
(c) the temperature change of a van der Waals gas during adiabatic expansion into a
vacuum.
31. The Joule-Thomson coefficient is defined using a parameter
λ = −
∂T
∂p H
(a) Show that
dH = T dS + V dp
and
λ =
V
Cp
(1 − Tαp)
αp := 1
V
∂V
∂T p
is the coefficient of isobaric expansion.
(b) Show that
λ =
T ∂p
∂T V
+ V ∂p
∂V T
Cp · ∂p
∂V T
(c) Verify that λ = 0 for a classical ideal gas.
(d) Show that it holds for a van der Waals gas
λ =
bp + 3ab
V 2 − 2a
V
p − a
V 2 + 2ab
V 3 · Cp
(e) Express the equation of the inverse curve that represents the interface between the
region λ > 0 and λ < 0 in the p − V diagram for the case of a van der Waals gas.
5
32. Show that the thermal coefficient of expansion
α :=
1
V
∂V
∂T p
meets the relation
∂S
∂p T
= −
∂V
∂T p
= −V α
33. Show that the specific heat at constant pressure, cp, and at constant volume, cV ,
satisfy the relation
cp − cV = T
∂p
∂T V
∂V
∂T p
= −T
∂S
∂V T
∂S
∂p T
.
34. Free energy of the system F(V, T) = −1
3
· const ·V T4
. Determine its pressure,
internal energy, 3 enthalpy and Gibbs potential.
35. Calculate the efficiency of the Carnot cycle (1. isothermal expansion, T2 = konst,
2. adiabatic expansion„ S = konst, 3. isothermal compression, T1 = konst, 4. adiabatic
compression, S = konst) for an ideal gas using its equations of state.
36. Calculate the efficiency of the following ideal gas cycle. Can this process be carried
out reversibly?
1. isothermal expansion T2 = konst
2. isochoric cooling V2 = konst
3. isothermal compression T1 = konst
4. isochoric heating V1 = konst.
37. Determine the efficiency coefficient of an (idealized) Otto engine that works with
an ideal gas 5 of specific heat cV = 5
2
R/mol at a compression ratio of 10:1.
1. adiabatic compression„
2. isochoric heating (=burning of fuel),
3. adiabatic expansion (doing work),
4. cooling (=hot gas exhaust, new, cold gas is drawn in).
38. The Diesel cycle consists of the following parts:
1. adiabatic compression of atmospheric air,
2. combustion of the injected mixture and isobaric expansion,
3. adiabatic expansion
4. and isochoric cooling.
Determine the cycle efficiency as a function of compression ratio for an ideal gas.
39. What is the total entropy change if we mix 2 kg of water at a temperature of 363
K adiabatically and at constant pressure with 3 kg of water at a temperature of 283
K? (cp = 4184 J/Kkg)
40. A refrigerator can turn 10 liters of water at 0◦
C into ice at the same temperature
in one hour. For this, the thermal energy Q = 800kcal(= 800 × 1, 163Wh) must be
6
transferred to the air (27, 3◦
C). What is the minimum power the refrigerator must
have?
41. Prove that for T → 0 there is no system described by pV = const ·T.
42. A closed system consists of two simple subsystems that are separated by a movable
wall that allows
(a) only heat exchange,
(b) both heat and mass exchange,
(c) neither heat nor mass exchange.
What are the corresponding equilibrium conditions?
43. Two equal quantities of ideal gas with the same temperature T and different pressures
p1, p2 are separated from each other by a partition. Determine the change in
entropy resulting from the mixing of the two gases.
44. Determine the maximum work that can be obtained by combining equal quantities
of the same ideal gas with the same temperature T0 (and different volumes or pressures).
45. Molar volume of water v(2)
= 18 cm3
/mol, the molar volume of ice is 9.1% larger (at
a pressure of 105
Pa ), The molar mass of water is 18 g/mol. The latent heat of melting
of ice is 330 kJ/kg. Calculate the change in melting point as the pressure changes.
46. When the magnetization M changes by dM the system performs work dW =
−H dM, where H is the intensity of the magnetic field. (This is work done per unit
volume; volume V = konst. = 1.) Determine the difference in heat capacities cH − cM
at a constant field H and at a constant magnetization.
47. Determine the equation of the adiabat of an isotropic magnetic.
48. Show that it holds cH
cM
=
χT
χS
,
where
χT =
∂M
∂H T
and
χS =
∂M
∂H S
49. The gamma function is defined by an integral
Γ(n) :=
∞
0
dt exp(−t)tn−1
.
1. Prove the relationship
Γ(n + 1) = nΓ(n),
2. calculate Γ(n), n ∈ N,
7
3. calculate
Γ n +
1
2
, n ∈ N.
50. Using the Gamma function, calculate the approximate expression ln(n!) for large
values of n (Stirling’s formula).
51. A hydrogen atom is in the level n = 3. Assuming that the occupation of the energy
levels is given by the microcanonical distribution, calculate the probability that the
atom is in states with the same secondary quantum number l.
52. Entropy for an isolated system is given by the relation S = kB ln Γ, where Γ is
the number of microstates. For a closed system S = −kB n wn ln wn. Show that both
relations are not contradictory.
53. Show that the heat capacity cV is given by the energy fluctuation, i.e.
cV =
1
kBT2
∆E2
.
54. Calculate the thermodynamicproperties of the system of N distinguishableclassical
harmonic oscillators with frequency ω.
55. Consider a gas with diatomic molecules. Calculate the molar heat capacity of the
given gas. Consider only the vibrational motion of molecules, when the energy is given
by the relation
En = ω n +
1
2
. (1)
First, calculate the statistical sum from which you will determine the free energy and
from the free energy you can already determine the required heat capacity. You can
write the resulting heat capacity in the approximation of low and high temperatures.
56. Derive the form of the Maxwell-Boltzmann distribution of the speeds of gas molecules.
Proceed only from the assumption that the space is isotropic and that the
movement of gas molecules in individual directions is independent.
57. Derive the Maxwell-Boltzmann distribution of atomic momentum using the canonical
distribution.
58. Assuming the validity of the Maxwell-Boltzmann distribution of the speeds of
gas molecules, calculate (a) px , (e)
√
∆E2, (b) p , (c) p2
, (f) the most probable
magnitude of momentum, (d) v2
, (g) probability that pz > 0.
59. Calculate the density distribution in a column of gas with base A under the
influence of a homogeneous gravitational field (in the atmosphere). Assume that the
gas is made up of indistinguishable particles, each of mass m.
60. Show that pressure and energy density have the same unit.
61. Calculate the density of states for relativistic particles and find the limit relations
for classical and ultra relativistic particles.
8
62. Show that in the classical case it is possible to derive the Maxwell-Boltzmann
velocity distribution law from the grand canonical distribution of a single particle.
63. Let’s define functions
Bn(y) =
1
Γ(n)
∞
0
dx
xn−1
exp(x − y) − 1
, (2)
and
Fn(y) =
1
Γ(n)
∞
0
dx
xn−1
exp(x − y) + 1
. (3)
For these functions prove
dBn+1(y)
dy
= Bn(y),
and
dFn+1(y)
dy
= Fn(y).
64. From a relationship
Ω = −kBT
gV
(2π )3
(2πmkBT)
3
2 B5
2
µ
kBT
, (4)
valid for a non-relativistic ideal boson gas, count the number of particles N and derive
the relation for the chemical potential within the classical limit.
65. Calculate cV of a non-relativistic fermion gas and verify the validity of the classical
limit for cV /N.
9