Címlap
Postulatory
Thermodynamics
Ernő Keszei
Loránd Eötvös University
Budapest, Hungary
http://keszei.chem.elte.hu/
•
Introduction
•
Survey of the laws of classical thermodynamics
•
Postulates of thermodynamics
•
Fundamental equations and equations of state
•
Equilibrium calculations based on postulates
Outline
Avant
propos
Thermodynamics is a funny subject. The first
time you go through it,
you don’t understand it
at all
. The second time you go through it,
you
think you understand it, except for one or two
small points
. The third time you go through it,
you know you don’t understand it, but by that
time you are so used to it, it doesn’t bother you
anymore
.
Arnold Sommerfeld
Problem with teaching thermodynamics
Let’s take an example:
probability theory
Postulates
of probability theory:
1.
The probability of event A is P(A) > 0
2.
If A and B are disjoint events,
i. e.
A
B=0,
瑨en(A
B)=(A)+(B)
3⸠
Forallposs楢ieeven瑳(瑨een瑩esamplespaceS)
the equality P(S) = 1 holds
Using these postulates,
„all possible theorems” can be proved,
i. e.,
all probability theory problems can be solved.
Important definition:
random experiment and its outcome
;
a (random) event
Fundamentals of (classical) thermodynamics:
the laws
Two popular textbooks of
physical chemistry
Atkins
P, de Paula J (2009) Physical Chemistry
,
9
th
edn., Oxford University Press, Oxford
Silbey L J,
Alberty
R A, Moungi G B (2004)
Physical Chemistry,
4
th
edn., Wiley, New York
(Traditional textbook of MIT
;
t
ypically, a new co

author replaces an old one at each new edition.)
Definition
of a
thermodynamic system
Atkins:
The
system
is the part of the world, in which
we have special interest.
The
surroundings
are where
we make our measurements.
Alberty
:
A
thermodynamic system
is that part
of the physical universe that is under consideration.
A system is separated from the rest of the universe by a real or
imaginary boundary. The part of the universe outside the boundary
is referred to as
surroundings
.
(
Introduction:
Thermodynamics is concerned with
equilibrium states
of matter and
has nothing to do with time
.)
The
Zeroth
Law
of thermodynamics
Atkins:
If A is in thermal equilibrium with B, and B is
in thermal equilibrium with C, then C is also
in thermal equilibrium with A.
Preceeding statement:
Thermal equilibrium
is established if
no change of state occurs when two objects A and B are
in contact through a diathermic boundary.
Alberty
:
It is an experimental fact that if system A is in
thermal equilibrium with system C, and system B
is also in thermal equilibrium with system C,
then A and B are in thermal equilibrium
with each other.
Preceeding
statement:
If two
closed systems with fixed volume
are brought together so that they are in thermal contact,
changes may take
place in the properties of both. Eventually,
a state is reached in
which there is no further change, and this is
the state of
thermal equilibrium
.
The
First Law
of thermodynamics
Atkins:
If we write
w
for the work
done on a system,
q
for the
energy transferred as
heat
to a system,
and Δ
U
for the resulting change in internal energy,
then it follows that
Δ
U
=
q
+
w
Alberty:
If both
heat
and
work
are added to the system,
Δ
U
=
q
+
w
For an infinitesimal change in state
d
U
= đ
q
+ đ
w
The đ indicates that
q
and
w
are not exact differentials
.
The
Second Law
of thermodynamics
Atkins:
No process is possible, in which the sole result
is the absorption of heat from a reservoir and
its complete conversion into work.
(In terms of the entropy:)
The entropy of an
isolated system
increases in the course of
spontaneous change:
Δ
S
tot
> 0
where
S
tot
is the total entropy of the system
and its surroundings
.
Later (!!):
The
thermodynamic definition
of entropy is based on the expression:
Further on:
proof of the entropy being a state function
,
making use of a Carnot cycle.
The
Second
Law
of thermodynamics
Alberty
:
The second law
in the form we will find most useful:
In this form, the second law provides a criterion
for a spontaneous process, that is, one that can
occur
,
and can only be reversed by work
from outside the system.
Previously
:
(
Anal
y
zing three coupled Carnot

cycles, it is stated that…
)
… there is a state function
S
defined by
The
Third
Law
of thermodynamics
Atkins:
If the entropy of every element in its most stable
state at
T
= 0 is taken as zero, then every substance
has a positive entropy which at
T
= 0 may become
zero
,
and which does become zero for all perfect
crystalline substances, including compounds.
Afterwards:
It should also be noted that the Third Law does not state that
entropies
are
zero at
T
= 0: it merely implies that all perfect
materials have the same entropy at that temperature. As far as
thermodynamics is concerned, choosing this common value as zero
is then a matter of convenience. The molecular interpretation
of entropy, however, implies that
S
= 0 at
T
= 0.
… The choice
S
(0) = 0 for perfect crystals will be made from now on.
The
Third
Law
of thermodynamics
(Péter Esterházy in a novel on communism)
The
Kádár

era: filthy land, lying to the marrow, shit as is,
in which, aside
from this, one could live, aside from the fact that one couldn't put it aside,
even though we did put it aside
.
Atkins:
If the entropy of every element in its most stable
state at
T
= 0 is taken as zero, then every substance
has a positive entropy which at
T
= 0 may become
zero
,
and which does become zero for all perfect
crystalline substances, including compounds.
Afterwards:
It should also be noted that
the Third Law does not state that
entropies
are
zero at
T
= 0: it merely implies that all perfect
materials have the same entropy at that temperature. As far as
thermodynamics is concerned, choosing this common value as zero
is then a matter of convenience. The molecular interpretation
of entropy, however, implies that
S
= 0 at
T
= 0.
… The choice
S
(0) = 0 for perfect crystals will be made from now on.
The
Third
Law
of thermodynamics
Alberty
:
The entropy of each pure element or substance
in a perfect crystalline form is zero at absolute zero.
Afterwards :
We will see later that statistical mechanics gives a reason
to pick this value
.
Atkins:
If the entropy of every element in its most stable
state
at
T
= 0 is taken as zero, then every substance
has a positive entropy which
at
T
= 0 may become
zero
,
and which does become zero for all perfect
crystalline substances, including compounds.
Afterwards
: It should also be noted that the Third Law does not state that
entropies
are
zero at
T
= 0: it merely implies that all perfect
materials have the same entropy at that temperature. As far as
thermodynamics is concerned, choosing this common value as zero
is then a matter of convenience. The molecular interpretation
of entropy, however, implies that
S
= 0 at
T
= 0.
… The choice
S
(0) = 0 for perfect crystals will be made from now on.
Avant
propos
After all it seems that Sommerfeld was right…
Thermodynamics is a funny subject. The first
time you go through it, you don’t understand it
at all. The second time you go through it, you
think you understand it, except for one or two
small points. The third time you go through it,
you know you don’t understand it, but by that
time you are so used to it, it doesn’t bother you
anymore
.
But thermodynamics is an exact science…
Development of axiomatic thermodynamics
1878
Josiah Willard
Gibbs
Suggestion to axiomatize chemical thermodinamics
1909
Konstantinos
Karathéodori
(greek matematician)
The first system of postulates (axioms)
(heat is
not
a basic quantity)
1966
László
Tisza
Generalized Thermodynamics, MIT Press
(Collected papers, with some added text)
1985
Herbert B.
Callen
Thermodynamics and an Introduction
to Thermostatistics, John Wiley and Sons, New York
1997
Elliott H.
Lieb
and Jacob
Yngvason
The Physics and Mathematics of the Second Law of Thermodynamics
(15 mathematically sound but simple axioms)
Fundamentals of
postulatory
thermodynamics
An important definition:
the
thermodynamic
system
The objects described by thermodynamics are called
thermodynamic systems
. These are not simply
“the part
of the physical universe that is under consideration
” (or
in which
we have special interest
), rather
material bodies
having a
special property; they are
in equilibrium
.
The condition of equilibrium can also be formulated so that
thermodynamics is valid for those bodies at rest for which
the predictions based on thermodynamic relations coincide
with reality (
i. e.
with experimental results). This is an
a posteriori
definition; the validity of thermodynamic
description can be verified after its actual application.
However,
thermodynamics offers a valid description for an
astonishingly wide variety of matter and phenomena.
Postulatory
thermodynamics
A practical simplification:
the
simple system
Simple systems
are pieces of matter that
are
macroscopically
homogeneous
and
isotropic
, electrically
uncharged
, chemically
inert
, large enough so that
surface
effects can be neglected
, and they are
not acted on by
electric, magnetic or gravitational
fields
.
Postulates will thus be more compact, and these
restrictions largely facilitate thermodynamic description
without limitations to apply it later to more complicated
systems
where these limitations are not obeyed.
Postulates will be formulated for physical bodies that are
homogeneous and isotropic, and their
only
possibility to
interact with the surroundings is
mechanical work
exerted
by volume change, plus
thermal
and
chemical interactions
.
(Implicitely assumed in the classical treatment as well.)
Postulate 1
of thermodynamics
There exist particular states (called
equilibrium states
) of
simple systems that, macroscopically,
are characterized
completely
by the internal energy
U
, the volume
V
, and the
amounts of the
K
chemical components
n
1
,
n
2
,…,
n
K
.
1.
There
exist
equilibrium states
2. The equilibrium state is
unique
3. The equilibrium state has
K
+
2 degrees of freedom
(in simple systems!)
2.
The equilibrium state
cannot depend on the ”past history”
of the system
3.
State variables
U
,
V
and
n
1
,
n
2
,…,
n
K
determine the state;
their functions
f(
U
,
V
,
n
1
,
n
2
,…
n
K
) are
state functions
.
Postulate 2
of thermodynamics
There exists a function (called the entropy and denoted by
S
)
of the extensive parameters of any composite system, defined
for all equilibrium states and having the following property:
The values assumed by the extensive parameters in the
absence of an internal constraint are those that maximize the
entropy over the manifold of constrained equilibrium states.
1.
Entropy is defined
only for equilibrium states.
2.
T
he
equilibrium state in an
isolated
composite
system
will be the one which
has the
maximum of entropy.
Definition:
composite system:
contains at least two subsystems
the two subsystems are sepatated by a wall (
constraint
)
Over
what variables
is entropy maximal?
isolated cylinder
fixed, impermeable,
thermally insulating piston
U
α
,
V
α
,
n
α
U
β
,
V
β
,
n
β
In the absence
of an internal constraint
, a manifold of different
systems can be
imagined
;
all of them could be realized by
re

installing the constraint (“virtual states”).
Completely releasing the internal constraint(s) results in a well
determined state which
–
over
the manifold of virtual states
–
has the maximum of entropy.
Postulate
3
of thermodynamics
The entropy of a composite system is
additive
over the
constituent subsystems. The entropy is
continuous
and
differentiable
and is a strictly
increasing
function of the
internal energy.
1.
S
(
U
,
V
,
n
1
,
n
2
,…
n
K
)
is an
extensive function,
i
. e.
, a
homogeneous first order
function
of its extensive variables
.
2.
There exist the derivatives
of the entropy function.
3. The entropy function
can be inverted
with respect to energy
:
there exists
the function
U
(
S
,
V
,
n
1
,
n
2
,…
n
K
)
, which
can be calculated
knowing the entropy function.
4.
Knowing the entropy function, any equilibrium state
(after any change) can be determined:
S = S
(
U
,
V
,
n
1
,
n
2
,…
n
K
)
is a
fundamental equation
of the system
.
5. Consequently, its inverse,
U = U
(
S
,
V
,
n
1
,
n
2
,…
n
K
)
contains
equivalent information, thus it is
also
a fundamental equation
.
Postulate
4
of thermodynamics
The entropy of any system is non

negative and vanishes in the
state for which the derivative
is zero.
As
, this also means that
the
entropy is exactly zero at zero temperature
.
The
scale of entropy
–
contrarily to the energy scale
–
is well determined
.
(This makes calculation of chemical equilibrium constants possible.)
(“Residual entropy”: no equilibrium
!!)
Summary of the postulates
(Simple) thermodynamic systems
can be described by
K
+
2 extensive variables
.
Extensive quantities are their
homogeneous linear functions
.
Derivatives of these functions are
homogeneous zero order
.
Solving thermodynamic problems
can be done using
differential

and integral calculus of multivariate functions
.
Equilibrium calculations
–
knowing the fundamental equations
–
can be reduced to
extremum calculations
.
Postulates
together with fundamental equations
can be used directly
to solve any thermod
y
namical problems.
Relations of the functions
S
and
U
S
(
U
,
V
,
n
1
,
n
2
,…
n
K
)
is
concave
,
and
a
strictly
monotonous
function of
U
In
at constant energy
U
,
S
is maximal
;
at constant entropy
S
,
U
is minimal
.
Fundamental equations in
U
and
S
Equilibrium
at constant energy (in an
isolated system
):
at the
maximum of the function
S
(
U
,
V
,
n
1
,
n
2
,…
n
K
)
Equilibrium
at constant entropy (in an
isentropic system
):
at the
m
in
imum
of the function
U
(
S
,
V
,
n
1
,
n
2
,…
n
K
)
(In simple systems:
isentropic
=
adiabatic
)
To find extrema of the relevant functions, we search for
the zero values of the
first order differentials
:
Identifying (first order) derivatives
We know:
at constant
S
and
n
(in
closed, adiabatic systems
):
(This is the
volume work
.)
Similarly:
at constant
V
and
n
(in
closed, rigid wall systems
):
(This is the
absorbed heat
.)
Properties of the derivative confirm:
at constant
S
and
V
(in
rigid, adiabatic systems
):
(This is
energy change due to material transport
)
The relevant derivative is called
chemical potential
:
Identifying (first order) derivatives
is negative
pressure
,
is
temperature
,
is
chemical potential
.
The total differential
can thus be written (in a simpler notation) as:
Fundamental equations and equations of state
Equations of state:
Equations of state:
Energy

based
fundamental equation:
Entropy

based
fundamental equation :
U
=
U
(
S
,
V
,
n
)
S
=
S
(
U
,
V
,
n
)
Its differential form:
Its differential form:
Some formal relations
U
=
U
(
S
,
V
,
n
)
is a
homogeneous linear function.
According to Euler’s theorem:
Euler equation
Gibbs

Duhem equation
We know:
Equilibrium calculations
isentropic, rigid, closed system
impermeable,
initially fixed,
thermally isolated piston,
then
freely moving, diathermal
S
α
,
V
α
,
n
α
S
β
,
V
β
,
n
β
S
α
+
S
β
= constant;
–
dS
α
=
dS
β
Consequences of impermeability (piston):
n
α
= constant;
n
β
= constant
→
d
n
α
= 0;
d
n
β
= 0
V
α
+
V
β
= constant;
–
dV
α
=
dV
β
Equilibrium condition:
dU= dU
α
+
dU
β
=
0
U
α
U
β
Equilibrium:
T
α
=
T
β
and
P
α
=
P
β
Equilibrium calculations
isentropic, rigid, closed system
S
α
,
V
α
,
n
α
S
β
,
V
β
,
n
β
Condition of
thermal
and
mechanical
equilibrium
in the composit system:
U
α
U
β
T
α
=
T
β
and
P
α
=
P
β
4 v
ariables
S
α
,
V
α
,
S
β
and
V
β
are to be known at equilibrium.
They can be
calculated by solving the
4
equations:
T
α
(
S
α
,
V
α
,
n
α
) =
T
β
(
S
β
,
V
β
,
n
β
)
P
α
(
S
α
,
V
α
,
n
α
) =
P
β
(
S
β
,
V
β
,
n
β
)
S
α
+
S
β
=
S
(
constant
)
V
α
+
V
β
=
V
(
constant
)
Equilibrium at constant temperature and pressure
isentropic, rigid, closed system
T
=
T
r
and
P
=
P
r
(constants)
S
r
,
V
r
,
n
r
T
r
,
P
r
S
,
V
,
n
T
,
P
equilibrium condition:
the „internal system” is closed
n
r
= constant
and
n
= constant
d
(
U
+U
r
)
= d
U
+
T
r
dS
r
–
P
r
dV
r
=
0
S
r
+
S
= constant;
–
dS
r
=
dS
V
r
+
V
= constant;
–
dV
r
=
dV
d
(
U
+U
r
)
= d
U
+
T
r
dS
r
–
P
r
dV
r
= d
U
+
T
r
dS
–
P
r
dV
=
0
T
=
T
r
and
P
=
P
r
d
(
U
+U
r
)
=
d
U
–
TdS
+
PdV
= d
(
U
–
TS
+
PV
)
=
0
minimizing
U
+
U
r
is equivalent to minimizing
U
–
TS
+
PV
Equilibrium condition at constant temperature and pressure:
minimum of the Gibbs potential
G = U
–
TS
+
PV
Summary of equilibrium conditions
Via
intensive variables
:
identity
of these variables in all phases
φ
Thermal equilibrium:
T
φ
T
,
Mechanical equilibrium:
P
φ
P
,
Chemical equilibrium :
μ
φ
μ
i
,
i
Extension is simple for
variables characterizing
other interactions
:
E. g
. electrostatic equilibrium:
Ψ
φ
Ψ
,
(
Ψ
φ
: electric potential of phase
φ
)
For chemical equilibrium, there is a condition for individual components
;
for all components that can freely move
between the subsystems (phases) of a composite system.
Summary of equilibrium conditions
Other (entropy

like) potential functions
can also be applied if needed.
Constraints
Condition of
equilibrium
Mathematical condition
Condition of
stability
U
and
V
constant
maximum
of
S
(
U
,
V
,
n
)
S
and
V
constant
maximum
of
U
(
S
,
V
,
n
)
S
and
P
constant
maximum
of
H
(
S
,
P
,
n
)
T
and
V
constant
maximum
of
F
(
T
,
V
,
n
)
T
and
P
constant
maximum
of
G
(
T
,
P
,
n
)
Via
extensive variables
:
extrema
of these variables in the system
•
Postulatory
thermodynamics is easy to understand
•
Postulates are based on quantities characteristic
of the system only
•
Relevant quantities (as internal energy and entropy)
are defined in
the postulates
•
Postulates are ready to use in equilibrium calculations
•
Derivation of auxiliary thermodynamic functions
(as free energy and Gibbs potential)
is straightforward
•
Exact mathematical treatment of
equilibria
is easy
Conclusions
Thus,
it is worth
both
teaching
and
learning
postulatory
thermodynamics
!
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