Lecture 1. Temperature, Ideal Gas (Ch. 1 )
Outline:
1.
Intro, the structure of the course
2.
Temperature
3.
The Ideal Gas Model
4.
Equipartition Theorem
Physics 351
Professor:
Office:
e-mail:
phone:
office hour:
lectures:
Thermal Physics
Spring 2007
Michael Gershenson
Serin Physics Building Room 122W
gersh@physics.rutgers.edu
732-445-3180 (office)
Mon. 2:30 - 3:30 PM and by appointment
SEC-117, W 10:20-11:40 AM and F 3:20-4:40 PM
Textbook: An Introduction to Thermal Physics,
D.V. Schroeder, Addison-Wesley-Longman, 2000
Web Site Access: http://www.physics.rutgers.edu/ugrad/351
Homework assignments, announcements and notes will be posted
here. The link must be visited regularly.
Lecture
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Date
1/17
1/19
1/24
1/26
1/31
2/2
2/7
2/9
2/14
2/16
2/21
2/23
2/28
3/2
3/7
3/9
3/21
3/23
3/28
3/30
4/4
4/6
4/11
4/13
4/18
4/20
4/25
4/27
5/8
Topic
Introduction, Temperature, Ideal Gas
Energy and Work, the 1st Law of Thermodynamics
Transport Phenomena
Macrostates and Microstates
Entropy and the 2nd Law of Thermodynamics
Entropy and Temperature
Systems with a “limited” Energy Spectrum
Thermodynamic Identities
Overview Ch. 1-3
Midterm
Heat Engines
Toward Absolute Zero, Refrigerators
Thermodynamic Potentials
Equilibrium Between Phases
The van der Waals Gas
Phase Transformations in Binary Mixtures
Dilute Solutions
Chemical Equilibrium
Overview Ch. 4-5
Midterm
Boltzmann Statistics
Canonical Ensembles
Statistics of Ideal Quantum Systems
Degenerate Fermi Gas
Blackbody Radiation
Bose-Einstein Condensation
Debye Theory of Solids
Overview
Final
Chapter
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3
1-3
HW due
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Thermal Physics
- conceptually, the most difficult subject of the undergraduate physics
program.
In the undergraduate mechanics and E&M courses, we consider a few
objects (or fields) whose positions (amplitudes) can be exactly calculated
from the Newton’s (Maxwell’s) equations. This approach works well only for
the simplest situations (even a problem of three interacting bodies cannot
be solved exactly in classical mechanics). It takes us nowhere when we
consider large systems of interacting particles (one needs to solve 1023
differential equations to describe the motion of electrons in 1 cm3 of a
metal).
The solution is to sacrifice the exact knowledge of microscopic
trajectories, and to describe a macro system using some meaningful
macroscopic parameters (e.g., the temperature). Not only is the detailed
dynamic evolution of the system discarded, but the entire dynamical model.
Thermal Physics 
Thermodynamics & Statistical Mechanics
Thermodynamics provides a framework of relating the macroscopic
properties of a system to one another. It is concerned only with macroscopic
quantities and ignores the microscopic variables that characterize individual
molecules (both strength and weakness).
Types of the questions that Thermodynamics addresses:
• How does a refrigerator work? What is its maximum efficiency?
• How much energy do we need to add to a kettle of water to change it to
steam?
Statistical Mechanics is the bridge between the microscopic and
macroscopic worlds: it links the laws of thermodynamics to the statistical
behavior of molecules.
Types of the questions that Statistical Mechanics addresses:
• Why are the properties of water different from those of steam, even though
water and steam consist of the same type of molecules?
• Why does iron lose its magnetism above a certain temperature?
Macroscopic Description is Qualitatively Different!
Why do we need to consider macroscopic bodies as a special class of
physical objects? Because the behavior of macroscopic bodies differs from
that of a single particle in a very fundamental respect:
For a single particle: all equations of classical mechanics, electromagnetism,
and quantum mechanics are time-reversal invariant (Newton’s second law,
F = dp/dt, looks the same if the time t is replaced by –t and the momentum p
by –p).
For macroscopic objects: the processes are often irreversible (a timereversed version of such a process never seems to occur).
Examples: (a) living things grow old and die, but never get younger, (b) if we
drop a basketball onto a floor, it will bounce several times and eventually
come to rest - the arrow of time does exist.
Conservation of energy does not explain why the time-reversed process
does not occur - such a process also would conserve the total energy.
Though the total energy is conserved, it has been transferred from one
degree of freedom (motion of ball’s center of mass) to many degrees of
freedom (associated with the individual molecules of the ball and the floor) –
the distribution of energy changes in an irreversible manner.
The Main Idea of the Course
Statistical description
of a large system
of identical (mostly,
non-interacting) particles
Equation of state
for macrosystems
(how macroparameters of the
system and the temperature
are interrelated)
all microstates of an isolated
system occur with the same
probability, the concepts of
multiplicity (configuration space),
entropy
Irreversibility of
macro processes,
the 2nd Law of Thermodynamics
However, most of the undergraduate courses approach this main line of
attack with caution, moving in circles. Following the same logic, we’ll start
with reviewing some basic facts from the general physics course.
Thermodynamic Systems, Macroscopic Parameters
Open systems can exchange both
matter
and
energy
with
the
environment.
Closed systems exchange energy
but not matter with the environment.
Isolated systems can exchange
neither energy nor matter with the
environment.
Internal and external macroscopic parameters: temperature, volume,
pressure, energy, electromagnetic fields, etc. (average values, fluctuations
are ignored).
No matter what is the initial state of an isolated system, eventually it will
reach the state of thermodynamic equilibrium (no macroscopic
processes, only microscopic motion of molecules).
A very important macro-parameter: Temperature
Temperature is a property associated with chaotic motion of many particles
(it would be absurd to refer to the temperature of a single molecule, or to the
temperature of many molecules moving with the same speed in the same
direction).
Introduction of the concept of temperature in thermodynamics is based on the
the zeroth law of thermodynamics:
A well-defined quantity called temperature exists such that
two systems will be in thermal equilibrium if and only if both
have the same temperature.
The zeroth law implies the existence of some universal property of systems in
thermal equilibrium. This law, in conjunction with the concept of entropy, will
help us to mathematically define temperature in terms of statistical ideas.
Also, if system C is in thermal equilibrium with systems A and B, the latter two
systems, when brought in contact, also will be in thermal equilibrium.
- this allows us to obtain the temperature of a system without a direct
comparison to some standard.
Temperature Measurement
Properties of a thermoscope (any device that quantifies temperature):
1. It should be based on an easily measured macroscopic quantity a
(volume, resistance, etc.) of a common macroscopic system.
2.
3.
The function that relates the chosen parameter with temperature,
T = f(a), should be monotonic.
The quantity should be measurable over as wide a range of T as
possible.
The simplest case – linear dependence T = Aa (e.g., for the ideal gas
thermometer, T = PV/NkB).
the ideal gas thermometer, T = PV/NkB
T
the resistance thermometer with

a semi- conductor sensor, R  exp T
 
a
Thermometer  a thermoscope
calibrated to a standard temp. scale
The Absolute (Kelvin) Temp. Scale
The
absolute
(Kelvin)
temperature scale is based
on fixing T of the triple point
for water (a specific T =
273.16 K and P = 611.73 Pa
where water can coexist in
the solid, liquid, and gas
phases in equilibrium).
T,K
273.16
0
absolute zero
 P  - for an ideal gas
 constant-volume
T  273.16 K 
 PTP 
thermoscope
PTP
P
PTP – the pressure of the gas in a
constant-volume gas thermoscope
at T = 273.16 K
Our first model of a many-particle system: the Ideal Gas
Models of matter:
gas models
(random motion of particles)
lattice models (positions of particles are fixed)
Air at normal conditions:
~ 2.71019 molecules in 1 cm3 of air (Pr. 1.10)
Size of the molecules ~ (2-3)10-10 m, distance
between the molecules ~ 310-9 m
The average speed - 500 m/s
The mean free path - 10-7 m (0.1 micron)
The number of collisions in 1 second - 5 109
The ideal gas model - works well at low densities (diluted gases)
• all the molecules are identical, N is huge;
•
•
•
•
the molecules are tiny compared to their average separation (point masses);
the molecules do not interact with each other;
the molecules obey Newton’s laws of motion, their motion is random;
collisions between the molecules and the container walls are elastic.
The quantum version of the ideal gas model helps to understand the
blackbody radiation, electrons in metals, the low-temperature behavior of
crystalline solids, etc.
The Equation of State of Ideal Gases
An equation of state - an equation that relates macroscopic variables
(e.g., P, V, and T) for a given substance in thermodynamic equilibrium.
In equilibrium ( no macroscopic motion), just a few macroscopic
parameters are required to describe the state of a system.
Geometrical
representation of the
equation of state:
P
an equilibrium
state
the equationof-state surface
f (P,V,T) = 0
V
T
The ideal gas
equation of state:
PV  nRT
P
V
n
T
–
–
–
–
pressure
volume
number of moles of gas
the temperature in Kelvins
R – a universal constant
[Newtons/m2]
[m3]
[K]
R  8.31
J
mol  K
The Ideal Gas Law
In terms of the total number of molecules, N = nNA
Avogadro’s number
NA  6.0220451023
PV  NkBT
the Boltzmann constant kB = R/NA  1.3810-23 J/K
(introduced by Planck in 1899)
The equations of state cannot be derived within the frame of
thermodynamics: they can be either considered as experimental
observations, or “borrowed” from statistical mechanics.
Avogadro’s Law: equal volumes of different
gases at the same P and T contain the same
amount of molecules.
The P-V diagram – the projection of the surface
of the equation of state onto the P-V plane.
isotherms
The van der Waals model of real gases
For real gases – both quantitative and qualitative deviations from the ideal
gas model
4
U(r)
3
Electric interactions between
electro-neutral molecules :
Energy
2
1
0
-1
-2
-3
Veff  V  Nb
repulsion
attraction
aN 2
Peff  P  2
V
1.5
2.0
2.5
r
3.0
3.5
4.0
distance
van der Waals
attraction (~ r -7)

aN 2 
 P  2 V  Nb  NkBT
V 

The van der Waals equation of state for real gases
(the only model with interactions that we’ll consider)
Connection between Ktr and T for Ideal Gases
?
T of an ideal gas  the kinetic energy of molecules
Pressure – the result of collisions between the molecules and walls of the
container.
Momentum
Strategy: Pressure = Force/Area = [p / t ]/Area
px = 2 m vx
Intervals between collisions: t = 2 L/vx
For each (elastic) collision:
Piston
area A
vx
Volume = LA
L
N
For N molecules -
2mvx 1
1
2 1
Pi 
 p x v x  mvx
2 L / vx A
V
V
PV   mvx2  N m v x2
i
no-relativistic
motion
Connection between Ktr and T for Ideal Gases (cont.)
N
PV   mvx2  N m v x2
i
PV  NkBT
Average kinetic energy of
the translational motion of
molecules:
3
K tr  k BT
2
3
U  K tr  Nk BT
2
2
PV  U
3
K tr
m v x2  k BT
1
1
3
2
2
2
2
 m v  m vx  v y  vz  m vx2
2
2
2
- the temperature of a gas is a direct measure of the
average translational kinetic energy of its molecules!
The internal energy U of a monatomic ideal gas
is independent of its volume, and depends only on
T (U =0 for an isothermal process, T=const).
- for an ideal gas of non-relativistic particles,
kin. energy  (velocity)2 .
Units for Energy, Temperature
3
K tr  k BT
2
- the kinetic energy is proportional to the temperature,
and the Boltzmann constant kB is the coefficient of
proportionality that provides one-to-one correspondence
between the units of energy and temperature.
Theorists usually assume that kB = 1 - the same units for energy and
temperature – which makes a lot of sense.
If the temperature is measured in Kelvins, and the energy – in Joules:
kB = 1.3810-23 J/K
In many sub-fields of physics that deal with microscopic particles (including
condensed matter physics, physics of high energies, astrophysics, etc.), a
convenient unit for energy is an electron-Volt (the kinetic energy acquired
by an electron accelerated by the electrostatic potential difference in 1 V).
K = eV  1 eV = 1.6 10-19 J
At T = 300K
1.38 1023 J / K  300 K
k BT 
 26 meV
19
1.6 10 J / eV
Pressure of a photon gas
1
2
2
PV  N m v  N K tr
- valid for non-relativistic particles
3
3
1
PV  N vp
- valid also for relativistic particles (m = f(v))
3
In particular, it applies to the gas of photons that move randomly within
some cavity with mirror walls. For photons,
p
E ph
c
E
ph
 cp  h 

PV 
1
E ph
3
pre-factor 1/3 (instead 2/3) - due to a different relation between
E and p for relativistic particles
We will use this equation later in the course when we consider
the thermal radiation.
Comparison with Experiment
dU/dT(300K)
(J/K·mole)
3
U  Nk BT
2
Monatomic
Helium
12.5
Argon
12.5
Neon
12.7
Krypton
12.3
Diatomic
H2
20.4
N2
20.8
O2
21.1
CO
21
Polyatomic
H20
27.0
CO2
28.5
- for a point mass
with three degrees
of freedom
Testable prediction: if we put a known
dU into a sample of gas, and measure
the resulting change dT, we expect to
get
dU 3
 Nk B
dT 2
3
 6 10 23 mole -1 1.38 10  23 J/K
2
 12.5 J/K  mole



Conclusion: diatomic and polyatomic
gases can store thermal energy in forms
other than the translational kinetic
energy of the molecules.
Degrees of Freedom
Indeed, the polyatomic molecules have more than just three degrees of
freedom – they rotate and vibrate.
The degrees of freedom of a system are a collection of independent
variables required to characterize the system.
Thus, for instance, the degrees of freedom of an ideal gas, in a
thermodynamic description, are P and V (or a pair of other variables).
The independent coordinates that describe the position of a mechanical
system in space - e.g., for a rigid body, it is sufficient to specify the
coordinates of its center of mass (x, y, z) and three angles (x, y, z).
K

 

Ix – the moment of inertia
for rotations around the
x-axis, etc.
Polyatomic molecules: 6 transl.+rotat. degrees of freedom
1
1
M x 2  y 2  z 2  I x x2  I y y2  I z z2
2
2
Diatomic molecules: 3 + 2 = 5 transl.+rotat. degrees of
freedom (QM: degrees of freedom corresponding to rotations
that leave the molecule completely unchanged do not count )
Degrees of Freedom (cont.)
Plus all vibrational degrees of freedom. The one-dimensional vibrational
motion counts as two degrees of freedom (kin. + pot. energies):
K  U x  
1
1
m x 2  k x 2
2
2
For a diatomic molecule (e.g., H2), 5 transl.+rotat. degrees of freedom
plus 2 vibrational degrees of freedom = total 7 degrees of freedom
Among 7 degrees of freedom, only 3 (translational) degrees correspond to a
continuous energy spectrum, the other 5 – to a discrete energy spectrum.
U(x)
U(x)
4
E4
3
Energy
2
1
E3
0
E2
-1
-2
E1
-3
1.5
2.0
2.5
3.0
x
distance
3.5
4.0
x
Equipartition of Energy
“Quadratic” degree of freedom – the corresponding energy = f(x2, vx2)
[ translational motion, (classical) rotational and vibrational motion, etc. ]
Equipartition Theorem:
At temperature T, the average energy of any
“quadratic” degree of freedom is 1/2kBT.
- holds only for a system of particles whose kinetic energy is a quadratic
form of x2, vx2 (e.g., the equipartition theorem does not work for photons, E =
cp)
Piston – a mechanical system with one degree of
freedom. Thus,
vx
m v 2x
2

M u2
2

1
k BT
2
M – the mass of a piston, u2 the average u2,
where u is the piston’s speed along the x-axis.
Thus, the energy that corresponds to the one-dimensional translational
motion of a macroscopic system is the same as for a molecule (in this
respect, a macrosystem behaves as a giant “molecule”).
“Frozen”
degrees of
freedom
U /kBT
one mole of H2
7/2N
Vibration
5/2N
Rotation
For an ideal gas
3/2N
PV = NkBT
U = f/2 NkBT
Example of H2:
Translation
10
100
1000
T, K
An energy available to a H2 molecule colliding U(x)
E4
with a wall at T=300 K: 3/2kBT ~ 40meV. If the
E3
difference between energy levels is >> kBT,
E2
then a typical collision cannot cause transitions kBT
E1 x
to the higher (excited) states and thus cannot
transfer energy to this degree of freedom: it is
“frozen out”.
The rotational energy levels are ~15 meV apart, the difference between
vibrational energy levels ~270 meV. Thus, the rotational degrees start
contributing to U at T > 200 K, the vibrational degrees of freedom - at T >
3000 K.
For the next lecture:
Recall the ideal gas laws, look at the ideal gas problems (some problems
are posted on our Web page).
A useful book (covers ~ ½ of this course): T.A. Moore, Six Ideas that
Shaped Physics” Unit T: Some processes are irreversible.
Problem 1.16
The “exponential” atmosphere
Consider a horizontal slab of air whose thickness is dz. If this slab is at rest,
the pressure holding it up from below must balance both the pressure from
above and the weight of the slab. Use this fact to find an expression for the
variation of pressure with altitude, in terms of the density of air, . Assume
that the temperature of atmosphere is independent of height, the average
molecular mass m.
P(z+dz)A
area A
z+dz
z
P(z)A
Mg
P( z  dz )  A  Mg  P( z )  A
Mg
P( z  dz )  P( z )  
A
dP
M  Adz 
  g
dz
M Nm 
PV  Pm

 N 
the density of air:  

V
V
k BT  k BT

 mgz 
P( z )  P(0) exp 

 kT 
dP
mg


P

dz
kT
The root-mean-square speed
vrms 
v
2
3k BT

m
- not quite the average speed, but close...
For H2 molecules (m ~21.710-27 kg ) at 300K: vrms~ 1.84 103 m/s
For N2 – vrms= 493 m/s (Pr. 1.18), for O2 – vrms= 461 m/s
This speed is close to the speed of sound in the gas – the sound wave
propagates due to the thermal motion of molecules.
D(v)
hydrogen molecules
in the “tail” escape the
Earth gravitation field
vrms
v