Transcript V α - Springer

1. Introduction
2. Postulates of thermodynamics
3. Thermodynamic equilibrium in isolated
and isentropic systems
4. Thermodynamic equilibrium in systems
with other constraints
5. Thermodynamic processes and engines
pp. 1–92
6. Thermodynamics of mixtures
(multicomponent systems)
7. Phase equilibria
8. Equilibria of chemical reactions
9. Extension of thermodynamics for
additional interactions (non-simple systems)
10. Elements of
equilibrium statistical thermodynamics
11.Towards equilibrium
– elements of transport phenomena
pp. 92–303
Appendix pp. 305-328
Table of contents
1. Introduction
2. Postulates of thermodynamics
3. Thermodynamic equilibrium in isolated and isentropic systems
4. Thermodynamic equilibrium in systems with other constraints
5. Thermodynamic processes and engines
pp. 1–92
6. Thermodynamics of mixtures (multicomponent systems)
7. Phase equilibria
8. Equilibria of chemical reactions
9. Extension of thermodynamics for additional interactions
(non-simple systems)
10. Elements of equilibrium statistical thermodynamics
11. Towards equilibrium – elements of transport phenomena pp. 92–303
Appendix pp. 305–328
Table of contents
Appendix
F1. Useful relations of multivariate calculus
F2. Changing extensive variables to intensive ones:
Legendre transformation
F3. Classical thermodynamics: the laws
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.
Postulates of thermodynamics
1. 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 .
2. There exists a function (called the entropy, 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.
3. 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.
4. The entropy of any system is non-negative and vanishes in the
state for which the derivative (∂U /∂S )V,n= 0. (I. e., at T = 0.)
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
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
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
Rankine vapor cycle and engines
Qin,1
Qin, 2
C
B
T
Boiler
D
Pump
B’
A
Condenser
A’
E
S
E
Qout
To hot room
Qout
B High pressure
vapor
D
Low pressure
liquid
D’
C
Turbine
High pressure
liquid
D
B
Wout
Wpump
A
heat engine
T
refrigerator
B
Condenser
Throttling
valve
Compressor
Evaporator
A
E
Qin
From cooled room
Low pressure
vapor
Win
D
C
E
A
S
Fugacty and interrelation of activities
Illustration of the
thermodynamic definition
of fugacity
Fugacity and interrelation of activities
Relation of the activities
fi (referenced to infinite dilution)
and
γi (referenced to pure substance
for the same system
Overview of different activities
activity ai
name
relative activity
f i xi
 x ,i xi
(activity
referenced to
Raoult’s law)
rational activity
(activity
referenced to
Henry’s law)
meaning of the standard
condition
i* (T , P )
at any
concentration
0 ≤ xi ≤ 1
(chemical potential of the pure substance)
 p

i,(T )  lim  i (T , pi , i , x )  RT ln i ,i 
p 0
P 

(chemical potential of the hypothetical
pure substance in the state identical
to that at infinite dilution)
i

c,, i (T , P)  lim  i (T , P, ci )  RT ln
ci  0
 m 
 m , i  i 
,
 mi 
molality basis
activity

ci 

ci, 
(chemical potential of the hypothetical ideal
mixture at concentration = 1 mol/kg in
the state identical to that at infinite dilution)

m,,i (T , P)  lim  i (T , P, mi )  RT ln
m 0
i
 c 
 c , i  i 
,
 ci 
 pi 
,
P 
i 
concentration
basis activity
fugacity

mi 

mi, 
(chemical potential of the hypothetical ideal
mixture at concentration = 1 mol/dm3 in
the state identical to that at infinite dilution)
i,
, x (T , P)  lim i (T , P, xi )  RT ln xi 
x 0
(chemical potential of the hypothetical ideal
mixture at a reference pressure φi pi = P,)
i
at any
concentration
in existing
mixtures
in solutions
in solutions
in every
gaseous mixture
Phase diagram of a van der Waals fluid
8Tr
3
Pr 
 2
3Vr  1 Vr
   0 (Tr ,Vr ,0 ) 
 6
24Tr

V

  V 3 (3V  1) 2
Vr, 0
Vr

8Tr
8T
6
 dV   
 r ln (3Vr  1)
Vr 3 (3Vr  1)
3

Equilibrium condition:  (Tr ,VB )   (Tr ,VE )
A
A
Tr = 0.75
D
-6.0
E
B
-6.5
F
1.2
Cr
1.0
Tr = 1.1
0.8
0.6
C
-7.0
Reduced pressure
-5.5
 –0
and Pr (Tr ,VB )  Pr (Tr ,VE )
0.4
Tr = 1.1
B
-7.5
E
Tr = 0.8
0.2
0
1
2
3
4
reduced volume
0
1
2
3
4
Reduced volume
F
P(V,T ) phase diagram of a pure substance
4
2
1
4
1
2
12
T
V
4
P
4
P
2
1
T
P
Cr
contracting
when freezing
V
T
V
P(V,T ) phase diagram of a pure substance
expanding when
freezing
P
P
T
V
P
T
V
Thermodynamics of phase separation
2.0
2 components, liquid-liquid
g /RT
*
molar Gibbs potential ( g) of
(heterogeneous)
mechanical dispersion
and
(homogeneous) mixture
2
mec
han
ical
disp
ersi
on
1.0
mi
xtu
re
0.5
*
1
0.0
0
0.2
0.4
0.6
0.8
x1
1
molar Gibbs potential
molar Gibbs potential
Common tangents
T < Tcr
Tcr
Tcr
T > Tcr
0
x1
1
0
x1
1
Thermodynamics of phase separation
2 components, solid-liquid
T1
g
T2
g
2
T3
g
3a
L
1
L
2b
xB
0
1
T4

xB
1
T5
xB
0
1
T6

g
g
L
L
L
3b
3c
2a
0

g
6a
4a
5a
4b
6b
4c
0
L
3d
5b
xB
1
0
xB
1
0
xB
1
Thermodynamics of phase separation
2 components, solid-liquid
1
Temperature
2b
liquid
2a
3d 

solid
solution 4c
6b
0
2
3c
3b

phase separation
into + liquid
5b
T1
4b

3a
T2
T3

liquid+  solid
T4
4a solution
phase separation into 
T5
5a
T6
6a
xB
1
Other binary solid-liquid phase diagrams
peritectic reaction
a
compound formation
b
monotectic reaction
a
b
syntectic reaction
a
b
Three-component phase diagrams
a)
3D diagram
dew point
surface
TC
b)
2D projection
boiling point
surface
TA
TB
liquid
C
liquid
A
vapor
projection
of the boiling point
curve at temperature T
vapor
A
B
projection
of the dew point curve
at temperature T
B
a)
3D diagram
b)
2D projection
C
phase
TC
freezing
point
surface
TA
TB
C
melting point
surface of
phase
phase
A
C
B
A
liquid
phase
B
Factors influencing chemical equilibria
G / kJ mol
–1
Example: 1 ½ H2 + ½ N2  NH3
4
reaction
2
0
mixing
–2
–4
–6
0
0,2
 equilibrium
0,4
0,6
0,8
 / mol
1
Extension for additional interactions
surface effects
(elements of surface chemistry)
electrically charged phases
(elements of electrochemistry)
copper wire
copper wire
silver plate
zinc plate
2+
Zn
+
Ag
porous membrane
to avoid mixing
Energy distribution in canonical ensembles
density function of
multiparticle
energy distribution
P(E )
– E
e
E
density function of
single particle
energy distribution
N
p( i)
T1
T1 < T2 < T3
(E )
T2
T3
M(E )
E
i
General interpretation of entropy
Misunderstandings due to the interpretation as “order–disorder”
disordered
ordered
smaller entropy
greater entropy
Viscuous flow as momentum transfer
Lagrange-transformation (Appendix)
y
the envelope of the tangent lines
determines the curve
x
Special terms and notation explained
The words diabatic, adiabatic and diathermal have Greek origin.
The Greek noun διαßασις [diabasis] designates a pass through,
e. g., a river, and its derivative διαßατικος [diabatikos] means the
possibility that something can be passed through. Adding the
prefix α- expressing negation, we get the adjective αδιαßατικος
[adiabatikos] meaning non-passability. In thermodynamic context,
diabatic means the possibility for heat to cross the wall of the
container, while adiabatic has the opposite meaning, i. e. the
impossibility for heat to cross. ….
The name comes from the German freie Energie (free energy). It
also has another name, Helmholtz potential, to honor Hermann
Ludwig Ferdinand von Helmholtz (1821-1894) German physician
and physicist. Apart from F, it is denoted sometimes by A, the
first letter of the German word Arbeit = work, referring to the
available useful work of a system.
Összefoglalás
Summing up
• easy-to-follow basis of thermodynamics
• postulates ready-to-use in equilibrium calculations
• detailed discussion of multicomponent systems
• sound thermodynamic foundations of
phase transitions & related equilibria
chemical reactions (homogeneous & heterogeneous)
surface chemistry
electrochemistry
• exact explanation of statistical thermodynamics
• elements of nonequilibrium thermodynamics (transport)
• Appendix: calculus + laws of classical thermodynamics
Enjoy
your
reading!