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

Professor: Michael GershensonOffice: Serin Physics Building Room 122We-mail: gersh@physics.rutgers.eduphone: 732-445-3180 (office)office hour: Mon. 2:30 - 3:30 PM and by appointmentlectures: 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.

Physics 351 Thermal Physics Spring 2007

Lecture Date Topic Chapter HW due1 1/17 Introduction, Temperature, Ideal Gas 12 1/19 Energy and Work, the 1st Law of Thermodynamics 1 3 1/24 Transport Phenomena 1 14 1/26 Macrostates and Microstates 25 1/31 Entropy and the 2nd Law of Thermodynamics 2 26 2/2 Entropy and Temperature 2 7 2/7 Systems with a “limited” Energy Spectrum 3 38 2/9 Thermodynamic Identities 3 9 2/14 Overview Ch. 1-3 1-3

4

410 2/16 Midterm11 2/21 Heat Engines12 2/23 Toward Absolute Zero, Refrigerators 413 2/28 Thermodynamic Potentials 5 514 3/2 Equilibrium Between Phases 5 15 3/7 The van der Waals Gas 5 616 3/9 Phase Transformations in Binary Mixtures 5 17 3/21 Dilute Solutions 5

54-5

7

818 3/23 Chemical Equilibrium19 3/28 Overview Ch. 4-520 3/30 Midterm 6 21 4/4 Boltzmann Statistics 6 22 4/6 Canonical Ensembles 6 23 4/11 Statistics of Ideal Quantum Systems 7 924 4/13 Degenerate Fermi Gas 7 25 4/18 Blackbody Radiation 7 1026 4/20 Bose-Einstein Condensation 7 27 4/25 Debye Theory of Solids 7 1128 4/27 Overview

5/8 Final

Thermal Physics

- conceptually, the most difficult subject of the undergraduate physics program.

In the undergraduate mechanics and E&M courses, we consider a fewobjects (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 microscopictrajectories, 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 macroscopicproperties 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 pby –p).

For macroscopic objects: the processes are often irreversible (a time-reversed 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 descriptionof a large system

of identical (mostly, non-interacting) particles

Irreversibility of macro processes,

the 2nd Law of Thermodynamics

all microstates of an isolated system occur with the same probability, the concepts of

multiplicity (configuration space), entropy

Equation of state for macrosystems

(how macroparameters of the system and the temperature

are interrelated)

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:

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.

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.

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. The function that relates the chosen parameter with temperature,

T = f(a), should be monotonic.3. 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).

Thermometer ≡ a thermoscope calibrated to a standard temp. scale

T

a

the ideal gas thermometer, T = PV/NkB

the resistance thermometer with a semi- conductor sensor, ( )α−∝ TR exp

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).

⎟⎟⎠

⎞⎜⎜⎝

⎛≡

TPPPKT 16.273

- for an ideal gas constant-volume

thermoscope

absolute zero

T,K

PPTP

273.16

0 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

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 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.

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 mThe average speed - 500 m/sThe mean free path - 10-7 m (0.1 micron)The number of collisions in 1 second - 5 ⋅109

The Equation of State of Ideal Gases

P – pressure [Newtons/m2]V – volume [m3]n – number of moles of gasT – the temperature in Kelvins [K]

R – a universal constant

nRTPV =

KmolJR⋅

≈ 31.8

The ideal gas equation of state:

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.

f (P,V,T) = 0

Geometrical representation of the

equation of state:

P

VT

an equilibriumstate

the equation-of-state surface

The Ideal Gas Law

isotherms

In terms of the total number of molecules, N = n⋅NA

TNkPV B=the Boltzmann constant kB = R/NA ≈ 1.38⋅10-23 J/K

(introduced by Planck in 1899)

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.

NA ≈ 6.022045⋅1023

Avogadro’s number

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.

The van der Waals model of real gases

( ) TNkNbVVaNP B=−⎟⎟

⎠

⎞⎜⎜⎝

⎛+ 2

2

For real gases – both quantitative and qualitative deviations from the ideal gas model

r

U(r)

van der Waalsattraction (~ r -7)repulsion

attraction

NbVVeff −=

2

2

VaNPPeff +=

Electric interactions betweenelectro-neutral molecules :

The van der Waals equation of state for real gases(the only model with interactions that we’ll consider)

-3

-2

-1

0

1

2

3

4

1.5 2.0 2.5 3.0 3.5 4.0di t

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 thecontainer.

Strategy: Pressure = Force/Area = [Δp / Δt ]/Area

vxPiston area A

L

Volume = LA

For each (elastic) collision: Δpx = 2 m vx

Intervals between collisions: Δt = 2 L/vx

Momentum

Vmv

Vvp

AvLmvP xxx

x

xi

111/2

2 2===

22x

N

ix vmNmvPV ==∑For N molecules -

?

no-relativisticmotion

Connection between Ktr and T for Ideal Gases (cont.)

22x

N

ix vmNmvPV ==∑

TNkPV B=

22222

23

21

21

xzyxtr vmvvvmvmK =++==

Tkvm Bx =2

TkK Btr 23

= - the temperature of a gas is a direct measure of the average translational kinetic energy of its molecules!

Average kinetic energy of the translational motion of molecules:

TNkKU Btr 23

==The internal energy U of a monatomic ideal gasis independent of its volume, and depends only on T (ΔU =0 for an isothermal process, T=const).

UPV32

= - for an ideal gas of non-relativistic particles, kin. energy ∝ (velocity)2 .

Units for Energy, Temperature

At T = 300K

TkK Btr 23

=- 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

meVeVJ

KKJTkB 26/106.1

300/1038.119

23

≈⋅

×⋅= −

−

Pressure of a photon gas

vpNPV31

= - valid also for relativistic particles (m = f(v))

In particular, it applies to the gas of photons that move randomly within some cavity with mirror walls. For photons,

( )νhcpEc

Ep ph

ph === phEPV31

=

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.

trKNvmNPV32

31 2 == - valid for non-relativistic particles

⇒

Comparison with Experiment

TNkU B23

=- for a point mass with three degrees

of freedom

dU/dT(300K)(J/K·mole)

MonatomicHelium 12.5Argon 12.5Neon 12.7

Krypton 12.3Diatomic

H2 20.4N2 20.8O2 21.1CO 21

PolyatomicH20 27.0CO2 28.5

Testable prediction: if we put a known dU into a sample of gas, and measure the resulting change dT, we expect to get

( )( )moleJ/K5.12

J/K1038.1mole10623

23

231-23

⋅=

⋅⋅=

=

−

BNkdTdU

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.

( ) ( )222222

21

21

zzyyxx IIIzyxMK ϕϕϕ &&&&&& +++++=

Thus, for instance, the degrees of freedom of an ideal gas, in athermodynamic 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 (ϕx, ϕy, ϕz).

Ix – the moment of inertia for rotations around the x-axis, etc.

The degrees of freedom of a system are a collection of independent variables required to characterize the system.

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 )

Polyatomic molecules: 6 transl.+rotat. degrees of freedom

Degrees of Freedom (cont.)

Plus all vibrational degrees of freedom. The one-dimensional vibrational motion counts as two degrees of freedom (kin. + pot. energies):

( ) 22

21

21 xkxmxUK +=+ &

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.

x

U(x)

E1

E2

E3

E4

x

U(x)

-3

-2

-1

0

1

2

3

4

1.5 2.0 2.5 3.0 3.5 4.0di t

Equipartition of Energy

vx

Piston – a mechanical system with one degree of freedom. Thus,

TkuMvm

Bx

21

22

22

==

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”).

Equipartition Theorem: At temperature T, the average energy of any“quadratic” degree of freedom is 1/2kBT.

“Quadratic” degree of freedom – the corresponding energy = f(x2, vx2)

[ translational motion, (classical) rotational and vibrational motion, etc. ]

- holds only for a system of particles whose kinetic energy is a quadratic form of x2, vx

2 (e.g., the equipartition theorem does not work for photons, E = cp)

“Frozen” degrees of freedom

For an ideal gas

PV = NkBT

U = f/2 NkBT

U /kBT

3/2N

5/2N

7/2N

10 100 1000 T, K

Translation

Rotation

Vibration

one mole of H2

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.

Example of H2:

An energy available to a H2 molecule colliding with a wall at T=300 K: 3/2kBT ~ 40meV. If the difference between energy levels is >> kBT, then a typical collision cannot cause transitions to the higher (excited) states and thus cannot transfer energy to this degree of freedom: it is “frozen out”.

x

U(x)

E1

E2

E3

E4

kBT

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” atmosphereConsider 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, ρ. Assume that the temperature of atmosphere is independent of height, the average molecular mass m.

z

z+dzP(z+dz)⋅A

P(z)⋅A Mg

area A AzPMgAdzzP ⋅=+⋅+ )()(

AMgzPdzzP −=−+ )()(

AdzM ρ= gdzdP ρ−=

the density of air: TkPm

TkPVN

VNm

VM

BB

=⎥⎦

⎤⎢⎣

⎡====ρ

⇒

PkTmg

dzdP

−=⇒

⎥⎦⎤

⎢⎣⎡−=

kTmgzPzP exp)0()(

The root-mean-square speed

mTkvv B

rms32 =≡ - not quite the average speed, but close...

For H2 molecules (m ~2⋅1.7⋅10-27 kg ) at 300K: vrms~ 1.84 ⋅103 m/sFor N2 – vrms= 493 m/s (Pr. 1.18), for O2 – vrms= 461 m/s

v

D(v)

vrms

hydrogen moleculesin the “tail” escape the Earth gravitation field

This speed is close to the speed of sound in the gas – the sound wave propagates due to the thermal motion of molecules.