Chapter 1 Introducing Thermodynamics

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“In the year nineteen hundred, in the course of purely
theoretical (mathematical) investigation, Max Planck made
a very remarkable discovery: the law of radiation of bodies
as a function of temperature could not be derived solely from
the Laws of Maxwellian electrodynamics . . . This discovery
became the basis of all twentieth-century research in physics
and has almost entirely conditioned its development ever
since.
A. Einstein, “On Quantum Mechanics”, 1945
Chapter 1
Introducing Thermodynamics
1.1
In the Beginning
Perhaps more than other physics subjects of the “classical” era – e.g. mechanics and
electromagnetism – thermodynamics has considerably exceeded a scope and relevance
far beyond its utile industrial revolution origins. Unquestionably, this results from
nearly 100 years of parallel, synergetic development with quantum mechanics, from
which it has gained clarification, enhancement and applicability earning its crucial
role in the development of physics as well as chemistry, biology, engineering, and
even contentious aspects of philosophy.1,2,3,4 Even more surprising is its emergent
role in cosmology.5
1
See, e.g., Ingo Muller, “A History of Thermodynamics”, Springer (2007).
J. Wicken,“Evolution, Thermodynamics and Information,” Oxford University Press, New York
(1987).
3
I. Prigogine, “From Being to Becoming,” Freeman, San Francisco (1980).
4
Entropy, Information and Evolution, Ed. Bruce H. Weber, David J. Depew and James D.
Smith, MIT Press, Cambridge (1988).
5
J. D. Bekenstein,“Black Holes and Entropy”, Phys. Rev. D7 2333 (1973).
2
1
2
CHAPTER 1. INTRODUCING THERMODYNAMICS
1.2
A Mini-History
This subject’s fascinating history is intertwined with seminal characters who contributed much to its present form – some colorful and famous and others lesser
known, ignored or even maligned.
“Atomism” (i.e. molecular models) seems to have had Daniel Bernoulli as its earliest documented proponent when in 1738 he hypothesized a kinetic theory of gases.6
Although science – thermodynamics in particular – is now unthinkable without such
models, Bernoulli’s ideas were largely ignored and for nearly a century little interest was shown in matter as based on microscopic constituents. Despite a continuing atmosphere of suspicion, particle models occasionally reappeared7,8 and evolve,
shedding some of their controversy (Joule 1851) and moving towards firm hypotheses
(Kronig 1856). Adoption of kinetic models by the respected Rudolf Clausius (1857)
further weakened the skeptics position, encouraging the new statistical theories of
Maxwell (1859) and Boltzmann (1872). At about the same time Van der Waals
(1873), theorizing forces between real atoms and molecules, developed an equation
of state for non-ideal gases.9 Nevertheless, as the 20th century dawned controversy
continued only slightly abated. Most likely J. J. Thomson’s discovery of the electron
(1897) put the issue to rest, at last convincing most remaining doubters of matter’s
microscopic essence.
The 20th century paradigm shift towards “quantized” models began with Planck who
in 190010,11 introduced quantized energy into models of electromagnetic radiation,
6
D. Bernoulli,“Hydrodynamica” (1738).
J. Herapath,“On the Causes, Laws and Phenomena of Heat, Gases, Gravitation”, Annals of
Philosophy 9 (1821). Herapath’s was one of the early papers on kinetic theory, but rejected by
the Royal Society, whose reviewer objected to the implication that there was an absolute zero of
temperature at which molecular motion ceased.
8
J.J.Waterston,“Thoughts on the Mental Functions” (1843). This peculiar title was a most
likely cause of its rejection by the Royal Society as “nothing but nonsense.” In recognition of
Waterston’s unfairly maligned achievement, Lord Rayleigh recovered the original manuscript and
had it published as “On the physics of media that are composed of free and perfectly elastic
molecules in a state of motion” Philosophical Transactions of the Royal Society A 183, 1 (1892),
nearly 10 years after Waterston’s death.
9
Van der Waals’ work also implied a molecular basis for critical points and the liquid-vapor
phase transition.
10
Max Planck,“Entropy and Temperature of Radiant Heat,” Ann. der Physik 1, 719 (1900).
11
Max Planck, “On the Law of Distribution of Energy in the Normal Spectrum,” Ann. der Physik
4, 553 (1901).
7
1.3. THERMODYNAMIC VOCABULARY
3
and Einstein12 who used quantized lattice vibrations in his ground breaking heat
capacity calculation. But it would be another twenty-five years before the inherently
probabilistic microscopic theory of matter –quantum mechanics – with its quantum
probabilities and expectation values – would reshape the scientific landscape and permanently dominate most areas of physics, providing a basis for deeper understanding
of particles, atoms, nuclei and even macroscopic systems. Thermodynamics was being preened for its essential role in modern physics.
Thermodynamics is mainly concerned with mechanical, thermal and electromagnetic
interactions in macroscopic matter – i.e. systems with huge numbers of microscopic
constituents (∼ 1023 particles). Although thermodynamic descriptions are generally
in terms of macroscopic (classical) variables, these macroscopic behaviors are, at
their root, quantum mechanical. Precisely how the classical measurement arises
from quantum behavior has been a subject of some controversy ever since quantum
theory’s introduction. But it now seems clear that macroscopic systems are quantum
systems that are particularly distinguished by always being coupled (however weakly)
to an environment (sometimes referred to as a reservoir) that is also considered a
quantum system. Although environmental coupling may be conceptually simple and
even uninteresting in detail, it has enormous consequences for quantum descriptions
of macroscopic matter, i.e. thermodynamics.13
1.3
Thermodynamic Vocabulary
A few general, large scale terms are used in describing objects and conditions of
interest in thermodynamics.
A. System: A macroscopic unit of particular interest, especially a unit whose thermal properties are under investigation. It may, for example, be a gas confined
within physical boundaries, or a rod of elastic material (metal, rubber or polymer). It can also be electrically polarizable or magnetizable matter coupled to
external electric and/or magnetic fields.
12
A. Einstein, “Planck’s Theory of Radiation and the Theory of Specific Heat,” Ann. der Physik
22, 180 (1907).
13
Macroscopic behavior can be quite different from the behavior of individual constituents (atoms,
molecules, nuclei, etc.) As an example, the appearance of spontaneous bulk magnetism in iron (at
temperatures below some critical temperature Tc ) is not a property of individual iron atoms but
arises from large numbers of interacting iron atoms behaving collectively.
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CHAPTER 1. INTRODUCING THERMODYNAMICS
B. Surroundings: Everything physical that is not the system or not within the
system’s boundaries. Surroundings may be external weights, an external atmosphere or electric and magnetic fields. The system plus surroundings comprise,
somewhat imaginatively, “the universe.”
C. Thermal Variables A set of macroscopic variables that describe the state of
the system. Some variables are intuitive and familiar such as pressure, volume,
elongation, tension, etc. Others may be less intuitive and even abstract – such as
temperature – which nevertheless, also play important roles in thermodynamics.
These will be discussed in detail in this and later chapters.
D. Thermal equilibrium: The final state attained in which thermal state variables
that describe the macroscopic system (pressure, temperature, volume, etc.) no
longer change in time.14 It is only at thermal equilibrium that thermodynamic
variables are well defined. The time elapsed in attaining equilibrium is largely
irrelevant.
1.4
Energy Reallocation
James Joule’s classic contribution on the mechanical equivalent of heat and a theory
of energy reallocation15 (sometimes inappropriately referred to as energy conservation) led Rudolph Clausius16 to the famous generalization:
∆U = Q − W,
(1.1)
called the First Law of Thermodynamics.
Here W is mechanical work done by the system and Q is “Heat” (thermal energy)
added to the system, both of which are classical quantities. In Clausius’ time controversy about atomic theories left U with no definitive interpretation. But being the maximum work which could be theoretically extracted from a substance it
was initially called “intrinsic energy.” As kinetic (atomic) models gained acceptance
14
There are, nevertheless, small departures from equilibrium averages, referred to as fluctuations,
whose averages are also part of any complete thermodynamic description.
15
James P. Joule, “On the Existence of an Equivalent Relation between Heat and the ordinary
Forms of Mechanical Power,” Phil. Mag 27 205 (1850).
16
R. Clausius, “On the Moving Force of Heat, and the Laws regarding the Nature of Heat,” Phil.
Mag. 2 1-21, 102-119 (1851).
1.4.
ENERGY REALLOCATION
5
(Clausius having played an influential role) U became interpreted as the mean kinetic energy of the system’s microscopic constituents or, more generally, as internal
energy. Although the change in internal energy, ∆U, is a consequence of mechanical and thermal, i.e. “classical”, interactions (see Eq. 1.1), quantum mechanics is
able to give a clear and specific meaning to ∆U as an average change in energy of
the macroscopic system’s microscopic constituents. The precise meaning of this and
similar averages lies at the heart of thermodynamics.17
1.4.1
Defining Thermodyanmic Variables
Some thermodynamic concepts and macroscopic variables are familiar from classical
physics while others arise simply from operational experience. But the first law
arrays Q and W (both classical) against ∆U (quantum mechanical), suggesting an
unavoidable partitioning of macroscopic variables into “classical”(C ) vs. quantum
(Q).
As will be discussed in Chapter 3, quantum mechanics provides definitions and context not only for internal energy U, but for other macroscopic (thermodynamic)
variables. According to Schrödinger’s quantum postulates, these variables acquire
representations as linear quantum mechanical operators which provide the mathematical basis for quantum models and, ultimately, thermodynamic (macroscopic)
variables.
Q-variables.
1. energy Hop (the dynamical hamiltonian operator)
∂Hop
2. pressure p = −
∂V
3. electric polarization Pop ,
4. magnetization Mop ,
5. elongation (length) Xop ,
6. particle number Nop .18
17
Details about macroscopic “averages” will be discussed in Chapter 3.
The particle number operator is not a part of Schrödinger’s fixed particle number theory. It
makes its appearance quite naturally in quantum field theories. Variable particle number is essential for thermodynamic descriptions of phase transitions, chemical reactions and inhomogeneous
18
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CHAPTER 1. INTRODUCING THERMODYNAMICS
C-variables – Classical (macroscopic) variables that exist apart from quantum mechanics, i.e. there are no quantum operators to define them.
1. temperature T ,
2. volume V,
3. static magnetic fields B or H,
4. static electric fields E or D,19
5. elastic tension τ ,
6. chemical potential µ.20
Static electromagnetic fields usually appear in interaction hamiltonians that
supplement Hop . These field quantities are said to be “conjugate” to the quantum
operators with which they appear. For example,
a. A static magnetic induction field B contributes an interaction hamiltonian
(energy)
M
Hop
= −mop · B
(1.2)
where B is “conjugate” to the magnetic moment operator mop of an elementary particle or the magnetic moment of a composite particle.21
b. A static electric field E contributes an interaction hamiltonian (energy)
P
Hop
= −pop · E
(1.3)
where E is “conjugate” to the electric polarization operator pop of, for example, some polarizable ion.
systems. Implementing variable particle number requires:
a. operators to create and destroy particles,
b. operators to count particles.
Number operator applications will be discussed in Chapters 13 and 14.
19
Electromagnetic radiation fields are, on the other hand, representable by quantum field operators Bop and Eop obtained from a quantum Electromagnetic Vector Potential Operator Aop . This
will be discussed in Chapter 15, the chapter on radiation theory.
20
Essentially, µ is an energy per particle and is associated with processes having varying particle
number.
21
This provides an opportunity to clarify ambiguites in the thermodynamic roles of static fields,
e.g. Maxwell’s local magnetic average B vs. an external B0 and Maxwell’s local electric average E vs.
external E0 . This is not a trivial matter and misunderstanding has led to serious misinterpretation
of experimental results as will be extensively discussed in Chapter 12.
1.4.
ENERGY REALLOCATION
7
Variable particle number is associated with open systems in which particles enter or
leave a system (as characterizes chemical reactions and phase transitions.) For this
case there is an “open system” interaction hamiltonian
N
= −µNop
Hop
(1.4)
where the chemical potential µ (energy per particle) is “conjugate” to a particle
number operator Nop .22
1.4.2
Closing Comment
So far this might seem an unusual and unexpected thermodynamic introduction.
Nevertheless, it is the author’s view that these are appropriate points for an approach
that ultimately brings unity, structure and range of application, i.e. modernity, to
thermodynamics.
22
Treating systems of identical particles as having variable number of particles (called open
systems) achieves enormous mathematical and conceptual simplification. See Chapter 14.
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