Transcript címlap - Springer

Címlap
Postulatory
Thermodynamics
Ernő Keszei
Loránd Eötvös University
Budapest, Hungary
http://keszei.chem.elte.hu/
Outline
• Introduction
• Survey of the laws of classical thermodynamics
• Postulates of thermodynamics
• Fundamental equations and equations of state
• Equilibrium calculations based on postulates
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
Important definition: random experiment and its outcome; a (random) event
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,
then P(AB) = P(A) + P(B)
3. For all possible events (the entire 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.
Fundamentals of (classical) thermodynamics:
the laws
Two popular textbooks of physical chemistry
Atkins P, de Paula J (2009) Physical Chemistry,
9th edn., Oxford University Press, Oxford
Silbey L J, Alberty R A, Moungi G B (2004)
Physical Chemistry, 4th edn., Wiley, New York
(Traditional textbook of MIT;
typically, 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 ΔU for the resulting change in internal energy,
then it follows that
ΔU = q + w
Alberty: If both heat and work are added to the system,
ΔU = q + w
For an infinitesimal change in state
dU = đq + đw
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:
ΔStot > 0
where Stot is the total entropy of the system
and its surroundings.
Later (!!): The thermodynamic definition of entropy is based on the expression:
dS 
đqrev
T
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:
đq
dS 
T
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: (Analyzing three coupled Carnot-cycles, it is stated that…)
… there is a state function S defined by
dq
dS  rev
T
The Third Law of thermodynamics
Atkins: If the entropy of every element in its most stable
state atT = 0 is taken as zero, then every substance
has a positive entropy which atT = 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
Atkins: If the entropy of every element in its most stable
state atT = 0 is taken as zero, then every substance
has a positive entropy which atT = 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 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.
(Péter Esterházy in a novel on communism)
The Third Law of thermodynamics
Atkins: If the entropy of every element in its most stable
state atT = 0 is taken as zero, then every substance
has a positive entropy which atT = 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.
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.
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 n1, n2,…, nK .
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 n1, n2,…, nK determine the state;
their functions f(U, V, n1, n2,… nK) are state functions.
Postulate 2 of thermodynamics
Definition: composite system: contains at least two subsystems
the two subsystems are sepatated by a wall (constraint)
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. The equilibrium state in an isolated composite system
will be the one which has the maximum of entropy.
Over what variables is entropy maximal?
isolated cylinder
U α, V α, n α
U β, V β, n β
fixed, impermeable,
thermally insulating piston
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,n1,n2,…nK) 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,n1,n2,…nK), 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, n1, n2,… nK )
is a fundamental equation of the system.
5. Consequently, its inverse, U = U(S,V,n1,n2,…nK ) 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
 U
state for which the derivative 
 S
 U
As 
 S


is zero.
V , n1 , n2 ,... nK


 T , this also means that
V , n1 , n2 ,... nK
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 thermodynamical problems.
Relations of the functions S and U
S(U,V,n1,n2,…nK) is concave,
and a strictly monotonous
function of U
S
S = S0 plane
U
In
at constant energy U,
S is maximal;
at constant entropy S,
U is minimal.
U = U0 plane
Xi
Fundamental equations in U and S
Equilibrium at constant energy (in an isolated system):
at the maximum of the function S(U,V,n1,n2,…nK)
Equilibrium at constant entropy (in an isentropic system):
at the minimum of the function U(S,V,n1,n2,…nK)
(In simple systems: isentropic = adiabatic)
To find extrema of the relevant functions, we search for
the zero values of the first order differentials:
K
 U 
 S 
 S 

dS  
dni
 dU  
 dV   
 U V , n
 V U , n
i 1  ni U ,V , n
j i
K
 U 
 U 
 U 

dU  
dni
 dS  
 dV   
 S V , n
 V  S , n
i 1  ni  S ,V , n
j i
Identifying (first order) derivatives
K
 U 
 U 
 U 

dU  
dni
 dS  
 dV   
 S V , n
 V  S , n
i 1  ni  S ,V , n
j i
We know:
at constantS and n (in closed, adiabatic systems):
(This is the volume work.)
Similarly:
at constantV and n (in closed, rigid wall systems):
(This is the absorbed heat.)
Properties of the derivative confirm:
at constant S andV (in rigid, adiabatic systems):
(This is energy change due to material transport)
The relevant derivative is called chemical potential:
dU  PdV
 U 

  P
 V  S , n
dU  TdS
 U 

 T
 S V , n
dU  i dni
 U 


 i
 ni  S ,V , n ji
Identifying (first order) derivatives
 U 

   P is negative pressure,
 V  S , n
 U 


 i
 ni  S ,V , n ji
 U 

  T is temperature,
 S V , n
is chemical potential.
The total differential
K
 U 
 U 
 U 

dU  
dni
 dS  
 dV   
 S V , n
 V  S , n
i 1  ni  S ,V , n
j i
can thus be written (in a simpler notation) as:
K
dU  TdS  PdV   i dni
i 1
Fundamental equations and equations of state
Energy-based fundamental equation:
Entropy-based fundamental equation :
U = U(S, V, n)
Its differential form:
S = S(U, V, n)
Its differential form:
K
dU  TdS  PdV   i dni
i 1
K
i
1
P
dS  dU  dV   dni
T
T
i 1 T
Equations of state:
Equations of state:
T  T ( S , V , n)
1 1
 (U , V , n)
T T
P  P ( S , V , n)
1
1

(U , V , n)
P
P
i  i (S , V , n)
1
i

1
i
(U , V , n)
Some formal relations
U = U(S, V, n) is a homogeneous linear function.
According to Euler’s theorem:
K
 U 
 U 
 U 

U 
ni
 S 
 V   
 S V , n
 V  S , n
i 1  ni  S ,V , n
j i
K
U  TS  PV   i ni
i 1
Euler equation
We know:
K
dU  TdS  PdV   i dni
i 1
K
SdT  VdP   ni di  0
i 1
Gibbs-Duhem equation
Equilibrium calculations
isentropic, rigid, closed system
S α, V α, n α
S β, V β, n β
Uα
Uβ
Equilibrium condition:
dU= dUα + dU β = 0
S α + S β = constant; – dSα = dS β
V α + V β = constant; – dV α = dV β
impermeable, initially fixed,
thermally isolated piston,
then freely moving, diathermal
 U 
dU  

 S

 U 


dS  

 V
V  ,n
Consequences of impermeability (piston):
n α = constant; n β = constant →

 U 


dV  

 S
 S  , n
dn α = 0; dn β = 0

 U 


dS  

 V
V  ,n 
dU  T  dS   T  dS   P dV   P  dV   0
dU   T   T   dS    P  P   dV   0
Equilibrium: Tα = T β and Pα = P β


dV   0
 S  ,n 
Equilibrium calculations
isentropic, rigid, closed system
S α, V α, n α
S β, V β, n β
Uα
Uβ
Condition of thermal and
mechanical equilibrium
in the composit system:
Tα = T β and Pα = P β
4 variables Sα , Vα , 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
equilibrium condition:
the „internal system” is closed
n r = constant and n = constant
d(U+U r ) = dU + T r dS r – P r dV r = 0
S, V, n
T, P
S r + S = constant; – dS r = dS
V r + V = constant; – dV r = dV
d(U+U r ) = dU + T r dS r – P r dV r = dU + T r dS – P r dV= 0
T = T r and P = P r
d(U+U r ) = dU – 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 :
μi φ  μi , 
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.
Extension is simple for variables characterizing other interactions:
E. g. electrostatic equilibrium: Ψ φ  Ψ , 
(Ψ φ: electric potential of phase φ)
Summary of equilibrium conditions
Via extensive variables: extrema of these variables in the system
Constraints
Condition of
equilibrium
Mathematical condition
U and V
constant
maximum of
S (U, V, n)
K

1
P
dS  dU 
dV   i dni  0
T
T
i 1 T
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)
K
dU  TdS  PdV    i dni  0
Condition of
stability
d 2S  0
d 2U  0
i 1
K
dH  TdS  VdP    i dni  0
d 2H  0
i 1
K
dF   SdT  PdV    i dni  0
d 2F  0
i 1
K
dG   SdT  VdP    i dni  0
d 2G  0
i 1
Other (entropy-like) potential functions can also be applied if needed.
Conclusions
• 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
Thus,
it is worth both
teaching and
learning
postulatory
thermodynamics!