Postulatory Thermodynamics - Springer

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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⸠
Forallposs楢ieeven瑳(瑨een瑩esamplespaceS)



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

!