ENTROPY
I am teaching Engineering Thermodynamics to an undergraduate class of 75 students.
These slides follow closely my written notes (http://imechanica.org/node/290).
I went through these slides in three 90-minute lectures.
Zhigang Suo, Harvard University
Energy is a loud, pesky, supporting actor
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It is often said that thermodynamics is the science of energy.
This designation steals accomplishments from other sciences, and
diminishes accomplishments of thermodynamics.
Rather, thermodynamics is the science of entropy.
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The play of thermodynamics
ENTROPY
energy
temperature
heat capacity
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space
pressure
compressibility
matter
chemical potential
charge
electrical potential
capacitance
enthalpy, Helmholtz free energy, Gibbs free energy
electrochemical potential
latent heat
thermal expansion, piezoelectric coefficient
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Energy belongs to many sciences
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Energy has many forms.
An isolated system conserves energy.
Convert energy from one form to another.
Transfer energy from one place to another.
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Fire
Spring
Capacitor
Water wheel
Windmill
Steam
Engine
Refrigerator
Turbine
Generator
Battery
Light bulb
Solar cell
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Mechanics
Electrodynamics
Quantum mechanics
Chemistry
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This course: Science of Entropy
• Logic of entropy from first principles
• Intuition of entropy from everyday experience
• Application of entropy in many domains, with
emphasis on heat and motion, engine and refrigerator.
Entropy links theory and experiment.
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Science of Entropy: one subject, many titles
Courses focusing on fundamentals
• Thermodynamics
• Statistical thermodynamics
• Statistical mechanics
Courses emphasizing applications
• Engineering thermodynamics
• Chemical thermodynamics
• Biological thermodynamics
• Thermal physics
Courses using thermodynamics
• Chemical physics
• Electrochemistry
• Solid mechanics
• Fluid mechanics
• Transport of energy and matter
• Electrodynamics
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History of entropy
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Carnot (1824) conceived the Carnot engine, the most efficient engine operating
between thermal reservoirs of two temperatures.
Clausius (1865) named entropy, and connected entropy, energy, and temperature.
Boltzmann (1877) linked entropy to molecular motion.
Gibbs (1873, 1901) formulated the science of entropy in the form used today.
Carnot (1796-1832)
Clausius (1822-1879)
Boltzmann (1844-1906)
Gibbs (1839-1903)
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Dangerous subject
Ludwig Boltzmann, who spent much of his life studying
statistical mechanics, died in 1906, by his own hands. Paul
Ehrenfest, carrying on the work, died similarly in 1933. Now it
is our turn to study statistical mechanics.
Goodstein, States of Matter, 1975
8
Foundation of thermodynamics
1. An isolated system has a certain number of quantum
states. Denote this number by W.
2. The isolated system flips from one quantum state to
another ceaselessly.
3. A system isolated for a long time flips to every
quantum state with equal probability, 1/W.
H2 O
isolated system
molecules move
9
Theory of probability
1. A die has six faces.
2. The die is rolled from one face to another.
3. The die is fair if it reaches every face with
equal probability, 1/6.
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Thermodynamics vs. theory of probability
Thermodynamics
Theory of probability
Setup
Isolated system
Die
Sample space
W quantum states
6 faces
Experiment
Isolate the system for a long time
Roll the die
Probability to get
each sample point
1/W
1/6
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Entropy defined
(entropy) = log (number of sample points)
Entropy of a fair die: S = log 6
Entropy of an isolated system: S = log W
12
Obvious trends of entropy
S = log W
S(U,V,N)
• Warm cheese has more quantum states than cold cheese. An isolated
system of higher energy has more quantum states.
• Molecules in a gas have more quantum states than the same number of
molecules in a liquid. An isolated system of larger volume has more
quantum states.
• Two glasses of wine have more quantum states than one glass of wine. An
isolated system of more molecules has more quantum states.
• Molecules in a liquid have more quantum states than the same number of
molecules in a crystal. Molecules in the liquid can be arranged in more
ways than in a crystal.
13
Questions
• Why do we hide W behind the log?
• How do we count the number of quantum states of
a system in everyday life, such as a glass of wine?
• What is an isolated system?
• What is a quantum state?
• Why equal probability?
• How long is a long time?
• How does entropy help us understand the
conversion between heat and motion?
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Product rule
W = 12
W1 = 2
W2 = 6
throw a coin and a die
throw a coin
throw a die
W = W 1W 2
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Why do we hide W behind the log?
Entropy is an additive (extensive) property
W, S
W1
S1 = log W1
W2
S2 = log W2
Isolated system 1
Isolated system 2
Isolated system
W = W 1W 2
log W = log W1 + log W2
S = S1 + S2
16
Questions
• Why do we hide W behind the log?
• How do we count the number of quantum states of
a system in everyday life, such as a glass of wine?
• What is an isolated system?
• What is a quantum state?
• Why equal probability?
• How long is a long time?
• How does entropy help us understand the
conversion between heat and motion?
17
Thermodynamic states
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(closed system) = (fixed amount of H2O) = (liquid) + (vapor)
The closed system changes under the fire and the weights.
Isolate the system.
The system isolated for a long time approaches a thermodynamic state of
equilibrium.
• But what is a thermodynamic state of equilibrium?
weights
vapor
liquid
fire
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Thermodynamic properties
• Fixed number of water molecules can be in many thermodynamic
states.
• Name all states using two thermodynamic properties.
• Intensive properties: temperature, pressure.
• Extensive properties: volume, energy, enthalpy, entropy.
• But, in general, what is a thermodynamic property?
weights
P
vapor
state
liquid
fire
V
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Thermodynamic phases
A state of a single phase (ice, water, or vapor)
A state of two coexistent phases (ice-water, water-steam, ice-vapor)
liquid
A state of three coexistent phases (ice-water-vapor)
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Equations of state
• Name all thermodynamic states using two properties (e.g., pressure P
and volume V).
• Any other property is a function of the two independent properties.
T(P,V) and U(P,V), H(P,V), S(P,V).
• Each function is called an equation of state.
• Represent a function of two independent variables by an equation, by
contours on a plane, by a surface in three dimensions, or by a table.
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Count the number of quantum states by
experimental measurement
• For a closed system, entropy is a property, S (U ,V )
• According to calculus,
dS =
(
¶S U ,V
¶U
• In later lectures we will show that
) dU + ¶S (U ,V ) dV
¶V
dS =
1
P
dU + dV
T
T
• Measure entropy incrementally.
weights
No quantum mechanics
No theory of probability
vapor
liquid
fire
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Zero entropy
Absolute entropy
• Recall the definition of entropy, S = log W.
• Zero entropy has physical significance, corresponding to an isolated system.
of a single quantum state.
Relative entropy
• We often choose an arbitrary state of a substance as a reference state.
• Set the entropy at this reference state as zero.
• List the entropy at any other state relative to the reference state.
• Why do we even be bothered with this complication?
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Entropy of a pure substance
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Entropy is a dimensionless number.
Entropy is a thermodynamic property.
Entropy is proportional to the amount of substance.
Report entropy per molecule (or per atom)
at room temperature and atmospheric pressure
Diamond 0.3
Lead
7.8
Water 22.70
O
19.4
O2
24.7
O3
28.6
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Entropy of a pure substance
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Entropy S is an extensive property.
Entropy per unit mass, s = S/m
For a state outside the dome, s(P,T)
For a state inside the dome, s = xsf + (1-x)sg.
P = 0.1 MPa
T
P(T,s)
s
sf
s
sg
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Choices of two independent variables
6 variables (PTvuhs), 15 choices
a
u
P
T
critical
point
liquid
liquid
a
gas
gas
P = 0.1 MPa
gas
liquid
T
a
intensive-intensive
s
extensive-intensive
v
extensive-extensive
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Water
P(T,s)
h(T,s)
r(T,s)
x(T,s)
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Water
P(h,s)
T(h,s)
r(h,s)
x(h,s)
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P
T
liquid
liquid
mixture
gas
gas
T
sf
sg
s
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Three phases
u
gas
liquid
solid
v
intensive-intensive
extensive-intensive
extensive-extensive
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Gibbs’s thermodynamic surface
S(U,V), a function of two variables
S, U, V are extensive properties
Experiment by Andrews (1869).
Described by Gibbs (1873).
A clay model built by Maxwell (1874)
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http://www.sv.vt.edu/classes/ESM4714/methods/Gibbs.html
(entropy) = log (number of sample points)
S = log W
• Coin: S = log 2.
• Die: S = log 6.
• Isolated system: S = log W, where W is the
number of quantum states of the isolated system.
• As a mathematical concept, entropy is not restricted to physical systems,
and is independent of the concepts of energy and temperature.
• Entropy is a dimensionless number.
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Do we need a unit for entropy?
All units are irrational, but some are necessary evils.
Think about meters and inches.
Units for entropy, however, are irrational,
unnecessary, and evil.
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The unit for entropy: (Joule)(Kelvin)-1
• Write S = k log W. The factor k = 1.38 x 10-23 JK-1, known as
the Boltzmann constant, converts entropy without dimension
to entropy with the unit.
• Don’t be intimidated by the great names (Joule, Kelvin and
Boltzmann). This unit for entropy is sillier than any unit for
length (e.g., inch and meter). It is so silly that we need to
explain it in a separate lecture.
• Entropy per unit mass, s = S/m. Unit: JK-1kg-1.
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Thermodynamics vs. theory of probability
Thermodynamics
Theory of probability
Setup
Isolated system
Die
Sample space
W quantum states
6 faces
Experiment
Isolate the system for a long time
Roll the die
Probability to get
each sample point
1/W
1/6
35
Questions
• Why do we hide W behind the log?
• How do we count the number of quantum states of
a system in everyday life, such as a glass of wine?
• What is an isolated system?
• What is a quantum state?
• Why equal probability?
• How long is a long time?
• How does entropy help us understand the
conversion between heat and motion?
36
Isolated system
• We can regard any part of the world as a system.
• A system may interact with the rest of the world in many ways (e.g.,
exchange energy, space, matter, charge).
• A system that does not interact with the rest of the world is called an
isolated system.
• Principles of conservation: an isolated system conserves energy,
space, matter, charge.
system
energy
space
matter
charge
the rest of the world
Isolated
system
the rest of the world
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Quantum Mechanics 1
space, matter, charge, energy
• Name a point in the three-dimensional space by coordinates, (x,y,z).
• A particle has mass M and charge e.
• The particle lives in a field of potential energy, V(x,y,z).
hydrogen atom
r
proton, +e
fixed in space
electron, -e
mass M
electron cloud
V =-
e2
4pe0r
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Quantum Mechanics 2
Particle-wave duality
• Associate the particle with a complex-valued function, Y(x,y,z),
known as the wavefunction.
• The probability for the particle to be in a small volume dxdydz
around a point (x,y,z) is |Y(x,y,z)|2dxdydz (Born 1926).
electron cloud
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Quantum Mechanics 3
The Schrodinger equation (1926)
æ 2
2
2 ö
¶
Y
¶
Y
¶
Y÷
ç
+
+
+V Y = EY
ç
÷
2
2
2
2M è ¶x
¶y
¶z ø
2
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h-bar = 10-34 Js is the Planck constant.
M = mass of the particle. Known.
V(x,y,z) = field of potential energy. Known.
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Eigenvalue problem.
E = energy of the particle, eigenvalue. To be solved.
Y(x,y,z) = wavefunction, eigenfunction. To be solved.
A picture is worth a thousand words.
An equation is worth a thousand pictures.
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Quantum states of a hydrogen atom
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Energy is quantized.
Associated with each energy level are multiple quantum states.
Each quantum state is specified by a wavefunction and a state of spin.
Isolated at the first energy level, the atom has 2 quantum states
(spin up and spin down)
Plot |Y(x,y,z)|2 in some way
Isolated at the second energy level, the atom has 8 quantum states
Atkins and Friedman, Molecular Quantum Mechanics
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A particle in a box
isolated system
z
L
y
x
L
Potential energy V vanishes inside the box.
Potential energy V is infinite outside the box.
L
æ 2
2
2 ö
¶
Y
¶
Y
¶
Y÷
ç
+
+
= EY inside the box
ç
÷
2M è ¶x 2 ¶y2 ¶z 2 ø
2
Y = 0 outside the box
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Quantum states of a particle in a box
æ 2
2
2 ö
¶
Y
¶
Y
¶
Y÷
ç
+
+
= EY
Inside the box: ç
÷
2
2
2
2M è ¶x
¶y
¶z ø
2
•
• Outside the box: Y = 0. To make the wavefunction continuous, Y = 0 on the faces
of the box.
• The solution takes the form
• nx, ny, nz being positive integers, known as the quantum numbers.
• The energy is quantized:
• An isolated system has a fixed energy.
• Isolated at the fist energy level, the system has a single quantum state
(nx, ny, nz) = (1,1,1).
• Isolated at the second energy level, the system has three quantum states
(nx, ny, nz) = (2,1,1), (1,2,1), (1,2,2).
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http://scicomp.stackexchange.com/questions/10235/how-do-i-plot-the-surface-of-a-4d-plot
44
Calculation is over-rated: Use quantum states
without knowing much quantum mechanics
• The quantum states of an isolated system are determined by
quantum mechanics.
• In practice, however, computers are too slow to perform
quantum-mechanical calculation for most systems.
• Count the number of quantum states of an isolated system by
experimental measurement.
After all, we have all learned to ride bicycles
without any use of Newton’s mechanics.
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Questions
• Why do we hide W behind the log?
• How do we count the number of quantum states of
a system in everyday life, such as a glass of wine?
• What is an isolated system?
• What is a quantum state?
• Why equal probability?
• How long is a long time?
• How does entropy help us understand the
conversion between heat and motion?
46
Fundamental postulate
A system isolated for a long time flips to every
quantum state with equal probability.
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The fundamental postulate cannot be proved from more elementary facts.
Its predictions have been confirmed without exception by empirical observations.
We regard the fundamental postulate as an empirical fact.
We use the fundamental postulate to build thermodynamics.
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Equilibrium
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Right after isolation, the system flips to some of its quantum states more
often than others. The isolated system is said to be out of equilibrium.
When the system is isolated for a long time, the system flips to every one of
its quantum states with equal probability. The isolated system is then said
to be in equilibrium.
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(out of equilibrium) = (flipping to quantum states with unequal probability).
(in equilibrium) = (flipping to every quantum state with equal probability).
48
Irreversibility
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(out of equilibrium) = (flipping to quantum states with unequal probability).
(in equilibrium) = (flipping to every quantum state with equal probability).
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An isolated system, once in equilibrium, will not get out of equilibrium.
The isolated system out of equilibrium is said to approach equilibrium with
irreversibility.
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All other fundamental laws of nature are time-reversible.
The fundamental postulate of thermodynamics gives time a direction, which
we call the arrow of time.
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Questions
• Why do we hide W behind the log?
• How do we count the number of quantum states of
a system in everyday life, such as a glass of wine?
• What is an isolated system?
• What is a quantum state?
• Why equal probability?
• How long is a long time?
• How does entropy help us understand the
conversion between heat and motion?
50
Kinetics
• Fundamental postulate: A system isolated for a long time flips to every
one of its quantum states with equal probability.
• How long is a long time?
• The fundamental postulate is silent about this question.
• The time to attain equilibrium depends on how fast molecules move.
• The study of how long an isolated system attains equilibrium is a subject
known as kinetics.
51
Thermodynamics is timeless
• This course studies thermodynamics, not kinetics.
• Kinetics studies the evolution of a system in time.
• By contrast, thermodynamics cares about the direction of time, but not the
duration of time.
• Thermodynamics does not use any quantity with dimension of time.
• Time enters thermodynamics merely to distinguish “before” and “after”—that
is, to give a direction of irreversibility.
52
Questions
• Why do we hide W behind the log?
• How do we count the number of quantum states of
a system in everyday life, such as a glass of wine?
• What is an isolated system?
• What is a quantum state?
• Why equal probability?
• How long is a long time?
• How does entropy help us understand the
conversion between heat and motion?
53
Thermodynamics vs. theory of probability
Thermodynamics
Theory of probability
Experimental setup
Isolated system
Die
Sample space
W quantum states
6 faces
Experiment
Isolate the system for a
long time
Roll the die
Probability to get each
sample point
1/W
1/6
Subset of sample points
Configuration
Macrostate
Thermodynamic state
Event
(e.g., a throw that obtains
an even number)
Probability to realize a
subset
(Number of quantum
states in the subset)/W
3/6
A map from sample space
to a real variable
Internal variable
Thermodynamic property
Random variable
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Event of a die
(sample space) = (six faces).
{1,2,3,4,5,6}
A subset of the sample space is called an event.
•Event “getting an even” = {2,4,6}
•Event “getting a 5 or a 6” = {5,6}
55
Thermodynamic state of an isolated system
H2O
(sample space) = (a set of quantum states)
A subset of quantum states is called
• a configuration,
• a macrostate, or
• a thermodynamic state.
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So many quantum states!
So many subsets of quantum states!
• An isolated system has W quantum states.
• This sample space has a total of 2W subsets.
Identify a subset through an experiment
57
A long-chained molecule
• So many quantum states.
• How do we select a subset of quantum states?
water
58
Two subsets
chain
Subset A: chains
Subset B: loops
loop
light
water
59
Thermodynamics vs. theory of probability
Thermodynamics
Theory of probability
Experimental setup
Isolated system
Die
Sample space
W quantum states
6 faces
Experiment
Isolate the system for a
long time
Roll the die
Probability to get each
sample point
1/W
1/6
Subset of sample points
Configuration
Macrostate
Thermodynamic state
Event
(e.g., a throw to obtain an
even number)
Probability to realize a
subset
(Number of quantum
states in the subset)/W
3/6
A map from sample space
to a real variable
Internal variable
Thermodynamic property
Random variable
60
Ink disperses in water
http://qcpages.qc.cuny.edu/~instr%5Cjgao104/diffusion.html
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•
•
What makes ink particles move?
Why is dispersion irreversible?
How fast will ink disperse?
61
Brownian motion
Discovery. Observed in a microscope, particles move
randomly in water (Brown, 1828).
Interpretation. A particle observed in the microscope is much
larger than individual water molecules. Water molecules kick
the particle.
particle
water
62
Why is dispersion irreversible?
(a thermodynamic state) = (all ink particles are in volume V)
(number of quantum states in this thermodynamic state ) µV N
probability for N particlesin volume V
=
probability for N particlesin volume V/7
a drop of ink
VN
(
V /7
)
N
= 7N
dispersed ink
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How fast does ink disperse?
Two kinetic processes to disperse ink:
• Convection
• Diffusion
64
Diffusion (Einstein 1905)
D=
kT
6p ah
1.38´10 J/K ) ( 300K )
(
D=
= 2 ´10
6p (10 m) (10 Pa s)
D = diffusivity
h = viscosity
a = radius of the ink particle
k = Boltzmann constant
T = temperature
-23
-13
-6
-3
m2 /s
t = time
L = average distance traveled
L = 2Dt
In a year
(
)(
)
L = 2 2´10-13 m2 /s 365 ´24 ´ 3600s = 3.6mm
65
Thermodynamics vs. theory of probability
Thermodynamics
Theory of probability
Experimental setup
Isolated system
Die
Sample space
W quantum states
6 faces
Experiment
Isolate the system for a
long time
Roll the die
Probability to get each
sample point
1/W
1/6
Subset of sample points
Configuration
Macrostate
Thermodynamic state
Event
(e.g., a throw to obtain an
even number)
Probability to realize a
subset
(Number of quantum
states in the subset)/W
3/6
A map from sample space
to a real variable
Internal variable
Thermodynamic property
Random variable
66
High-school math: map (or function)
map
random variable
internal variable
set A: domain of the map
sample space
x
set B: range of the map
real variable, x
67
Random variable
a map (i.e., function) from a sample space to a real variable
domain of the map:
sample space
{a,b,c,d,e, f }
range of the map:
amount of winning
map
a ® $200
b ® $600
c ® $100
d ® $400
e ® $700
f ® $0
68
Internal variable
a map (i.e., function) from a sample space to a real variable
domain of the map:
sample space,
All quantum states
range of the map:
volume V in which to find the particle
V
The volume V is an internal variable of the isolated system.
Number of quantum states of the isolated system when the particle is confined in volume V is
( )
W V µV
69
Half bottle of water
seal
vacuum2O
2
2O
H2 O
H2 O
O
• In equilibrium, how many water molecules will be in the vapor?
• Why is the approach to equilibrium irreversible?
70
A constraint internal to an isolated system
N=0
subset. W(0)
N=1
subset. W(1)
N=2
subset. W(2)
No constraint
sample space. W
2O
2O
2O
2
2O
H2 O
H2 O
H2 O
H2 O
O
•
Make the half bottle of water an isolated system.
•
Without the seal, the isolated system flips to every quantum state in the sample space.
The sample space has a total of W number of quantum states.
•
The seal provides a constraint internal to the isolated system. The number of molecules
in the vapor, N, is called an internal variable.
•
When the seal constrains the number of molecules in the vapor to N, the isolated system
flips among a subset of quantum states. The subset has W(N) number of quantum states.
•
When the seal is punctured, the internal variable N changes, and the probability for the
vapor to have N molecules equals W(N)/W.
71
Use an internal variable to dissect
the sample space into subsets
x
map
internal variable
set A: domain of the map
sample space
set B: range of the map
real variable, x
72
Probability in general terms
An isolated system without constraint flips to every quantum state in the sample space.
The sample space has a total of W number of quantum states.
1. Construct an isolated system with an internal variable, x.
2. When the internal variable is constrained at x, the isolated
system flips among a subset of its quantum states. This subset
has W(x) number of quantum states.
3. When the constraint is lifted, x changes, and the probability for
the internal variable to take value x is W(x)/W.
73
From probability to (almost) certainty
•
•
•
•
•
16 dies together produce a sample space of 616 sample points.
(a random variable) = (the number of even faces, N = 0, 1, 2, …, 16).
W(N) = (number of sample points that give N even faces).
W(N) maximizes at N = 8.
When the sample space is large, the maximum is sharp.
W(N)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
N
74
From probability to (almost) certainty
• For an isolated system with a large number of quantum states, the most
probable value of the internal variable N is much more probable than
other values of N.
• We therefore often focus our attention exclusively on N that maximizes
the function, W(N), and not be bothered with less probable values of N.
2
2O
W(N)
O
H2 O
N
75
Basic algorithm
1. Construct an isolated system with an internal variable, x.
2. When the internal variable is constrained at x, the isolated
system flips among a subset of its quantum states. Obtain the
number of quantum states in this subset, W(x).
3. After the constraint is lifted, x changes to maximize W(x).
•
•
•
•
No change in x decreases W(x).
A change in x is irreversible if W(x) increases.
A change in x is reversible if W(x) does not change.
The isolated system is in equilibrium when W(x) maximizes.
76
summary
Fundamental postulate
1. An isolated system has a set of quantum states.
2. The isolated system flips from one quantum state to another ceaselessly.
3. A system isolated for a long time flips to every quantum state with equal probability.
Immediate consequences
• (in equilibrium) = (flipping to every quantum state with equal probability)
• Arrow of time. An isolated system approaches equilibrium with Irreversibility.
• (Thermodynamic state) = (a subset of quantum states)
• (Thermodynamic property) = (a map from a sample space to a real variable)
Basic algorithm
1. Construct an isolated system with an internal variable x.
2. When the internal variable is constrained at x, the isolated system flips among a subset
of its quantum states. Obtain the number of quantum states in this subset, W(x).
3. After the constraint is lifted, x changes to maximize W(x).
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The second law
How does entropy steal the show?
Recall (entropy) = log (number of sample points).
Call S(x) = log W(x) the entropy when the internal variable is constrained at x
1. Construct an isolated system with an internal variable, x.
2. When the internal variable is constrained at x, the isolated
system has entropy S(x).
3. After the constraint is lifted, x changes to maximize S(x).
Cryptic statements of the second law
• Entropy always increases.
• An isolated system maximizes entropy.
• After a constraint internal to an isolated system is removed, the
internal variable changes to increase entropy.
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