Basic Thermodynamic Relations: Isolated System: This Is A System That Does Not

Basic Thermodynamic Relations
Isolated system: this is a system that does not exchange energy with the surrounding media. First Postulate (equilibrium theorem) : Isolated system always reaches the equilibrium state and never leaves it spontaneously. Second Postulate (temperature) Every equilibrium system is completely determined by the set of external variables (volume, pressure, magnetic field, etc.) plus one internal variable TEMPERATURE. At least one additional internal parameter is needed to describe a non-equilibrium system.
Equilibrium Process: This is a process that proceeds so slowly that the system is always in equilibrium state
The First Law of Thermodynamics

The First Law of Thermodynamics is a statement of conservation of energy in which the equivalence of work and heat flow is taken into account.
dU = dQ + dW
The internal energy of the system U depends only on the actual state of the system and not on the way the system is driven to it, i.e. it is a function of state.
The Second Law of Thermodynamics

Efficiency. This is a measure of how well the heat flow from the hotter thermal reservoir is converted to work. If the work done in one complete cycle is W, then efficiency is defined as the ratio of the work done to the total heat flow Q
W Q
One of the main consequences of the Second Law of Thermodynamics is the existence of another function of state called entropy
dQ dS T
Processes in closed systems are always connected with an increase of entropy. In equilibrium the system has maximum of entropy.
TdS dU + dW
for dW
= pdV
TdS = dU + pdV
Free energy For open systems
dW (dU TdS)
F = U TS G = H TS
W (F1 F2 )T
dF=dU-TdS-SdT=-pdV-SdT (for T=const)
dF = SdT pdV
Basic Thermodynamic Relations

First derivatives of Thermodynamic Potentials:
F S = T V
F p = V T G V = p T
G S = T p
F G = = O V ,T O p ,T
F G U H = = = = n V ,T n p ,T n V ,T n p ,S
Second derivatives of Thermodynamic Potentials: Heat capacity 2F cv = T 2 T V compressibility
k=
1 V = Vo P T
1 2F Vo V 2 V
2G S c p = T 2 = T T T p p
k = 1 Vo V = P T 1 2G Vo 2 p T
Thermal expansion coefficient
1 = Vo
V T P
Thermodynamics of Phase Transitions

Every homogeneous part of a heterogeneous system is called PHASE. Phase is a macroscopic, homogeneous, quasi closed part of a system, separated from the other parts of the system by a separating surface.
According to the Classification of Ehrenfest the order of the phase transition is determined by the order of the derivative of thermodynamic potential that jumps at this point.
Thermal equilibrium
dQ
dS+dS 0
dQ dQ + 0 T T
T T
Mechanical equilibrium (at T =T )

P P dV
dF+dF 0 dF = -pdV SdT (SdT=0)
-pdV+pdV 0
p p
Chemical equilibrium
dn
At T,P = const the two phases exchange material
dG + dG 0 dG= vdp SdT + idni -idni + i dni 0 i i
For the first order phase transitions: In the transition point the chemical potentials of the two phases are equal 1 = 2
Tg
P = const
Tm T
It is seen that I order phase transitions are accompanied by a heat transfer Q=T (S2 S1) . The heat of phase transition is the heat required to transfer substance from state 1 to state 2. The maximal work is
Wmax= (F)T = Q - (U)T Q C = p T p
The Kirhoff equation gives the temperature dependen of the heat of ce phase transition in differential form
Equation of Clausius-Clapeiron 1 ( T , p ) = 2 ( T , p ) . p=p(T) d (p,T) = d1 (p,T) 1 or 1 dT + 1 dp = 2 dT + 2 p p T p T p T So that
dp T
dp s2 s1 Q = = dT v2 v1 T (v 2 v1 )
Equation of Clausius-Clapeiron
For the second order phase transitions: In the transition point the molar enthalpies, the second order derivatives of thermodynamic potentials of the two phases are equal
dp s2 s1 0 = = =? dT v 2 v1 0 LHopital ds2 ds1 dT p dT p c p dp = = dv2 dv1 dT dv T dT p dT p dT p
ds2 ds1 dv dp dp T dp T dT p = = dT dv 2 dv1 dv dp dp dp T T T
v 2 s = = because T TP p
Ehrenfest Equation
kC p
TV
=1
SUPERSATURATION
G S = T p
S =
0 T
Tm
=
Tm
S (T )dT
T
C p T
dT = S m
C p T
dT
= Sm T dT
T T
Tm
Tm
C p T
dT
note:
Tm
C p T
dT = S m S (T )
Therefore
=
Tm
S (T )dT
T
A truncated Taylor expansion of S(T) in the vicinity of Tm gives:

S T T S (T ) = Sm + S (T Tm ) = m C p ,m m Tm T Tm
So that supersaturation becomes

= S m T C p ,m T 2 2Tm
= H m
T C p ,m T 1 Tm 2 S m Tm example: for Cp=const

T T T + C pTm ln Tm Tm Tm
= (H m C pTm )
Nature of Glass Transition: experimental evidence
glass
liquid (undercuuled melt) crystal
Structure
18 16
PMMA
14 12
lg [Pa.s.]
10 8 6 4 2 0 20 22 24 26 28 30
4
1 0 /T
63
Heat flow [mW]
60
57
54 360 390 420

o
450
Temperature [ C]
3,5 3,0 2,5 2,0
Tf Tg
ln(q)
1,5 1,0 0,5 0,0 -0,5 1,51 1,51 1,51 1,52 1,52 1,53 1,53 1,54 1,54 1,55 1,55
1000/T

Basic Thermodynamic Relations: Isolated System: This Is A System That Does Not

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Basic Thermodynamic Relations

The First Law of Thermodynamics

The Second Law of Thermodynamics

Basic Thermodynamic Relations

Second derivatives of Thermodynamic Potentials: Heat capacity 2F cv = T 2 T V compressibility

Thermal expansion coefficient

Thermodynamics of Phase Transitions

Mechanical equilibrium (at T =T )

dF+dF 0 dF = -pdV SdT (SdT=0)

At T,P = const the two phases exchange material

dG + dG 0 dG= vdp SdT + idni -idni + i dni 0 i i

Equation of Clausius-Clapeiron 1 ( T , p ) = 2 ( T , p ) . p=p(T) d (p,T) = d1 (p,T) 1 or 1 dT + 1 dp = 2 dT + 2 p p T p T p T So that

dp s2 s1 0 = = =? dT v 2 v1 0 LHopital ds2 ds1 dT p dT p c p dp = = dv2 dv1 dT dv T dT p dT p dT p

ds2 ds1 dv dp dp T dp T dT p = = dT dv 2 dv1 dv dp dp dp T T T

A truncated Taylor expansion of S(T) in the vicinity of Tm gives:

So that supersaturation becomes

T C p ,m T 1 Tm 2 S m Tm example: for Cp=const

Nature of Glass Transition: experimental evidence

liquid (undercuuled melt) crystal

Heat flow [mW]

54 360 390 420

3,5 3,0 2,5 2,0

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