cours3 entropy temperature · External environment! Thermostat! ... favourable at high pressure...
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Jean-François Dufrêche Entropy and Temperature A fifth force in the nature ? E ⬌ TS Rudolf Clausius (Koszalin, 1822 - Bonn, 1888) Entropy = η τροπη = the transformation
Transcript of cours3 entropy temperature · External environment! Thermostat! ... favourable at high pressure...
Jean-François Dufrêche!
Entropy and Temperature
A fifth force
in the nature ?
E ⬌ TS
Rudolf Clausius (Koszalin, 1822 - Bonn, 1888)
Entropy = η τροπη = the transformation
Thermodynamics
In mechanics
Equilibrium : E minimum
E = potential energy
In thermodynamics (T constant) : valid only at T = 0 K
Equilibrium : E - TS minimum
Interpretation
Sunivers maximum
Belleville roller coaster (1817) ! modern roller coaster!
Sunivers = Ssyst + Sext with Sext = Sext (Etot −E) ≈ Sext (Etot )+
∂Sext
∂E−E( ) = S0
ext −ET
Ssyst = S
$
%&
'&
External environment!Thermostat!system!
E ⤻ ! gives T!
Sunivers = S − ET
+S0ext( )
minimization of the energy = maximization of the entropy of the external environement
Competition between S and E
Equilibria : minimisation of F = E – TS (= -TSuniverse)!
Solid!U predominant!
atoms at fixed distance!Low T!
Gas!S predominant!
desorder!High T!
Liquid! U ≈ TS ? ? ?!Intermediate T !
r! distance between the molecules!
typically, V (r) = 4ε σr
!
"#
$
%&
12
−σr
!
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$
%&
6!
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$
%&&
repulsion !attraction !
!
V(r) = potential energy
between two molecules! ε: depth of the attraction!
σ : size of the molecules!ε!
σ!
V(r)!
Hard spheres and entropy
solid! liquid (or dense fluid)!
a (crystalline) solid phase can have a higher entropy than a liquid phase!
Hard spheres and entropy
solid! liquid (or dense fluid)!
a (crystalline) solid phase can have a higher entropy than a liquid phase!
Hard spheres and entropy
solid! liquid (or dense fluid)!
a (crystalline) solid phase can have a higher entropy than a liquid phase!
Hard spheres and entropy
solid! liquid (or dense fluid)!A high density, an organized phase (solid) gives more space for every molecules, because it can move in relatively large site!
a (crystalline) solid phase can have a higher entropy than a liquid phase!
Hard spheres and entropy
solid! liquid (or dense fluid)!A high density, an organized phase (solid) gives more space for every molecules, because it can move in relatively large site!
Hard spheres phase diagram
Φ (packing fraction)!
T!
S (fcc)!
S+F
F!
0.494! 0.545! 0.740!
Solid phases entropically favourable at high pressure
(exception: water)!
a (crystalline) solid phase can have a higher entropy than a liquid phase!
Liquid crystals
Liquid crystal:
long rigid molecules!Ex: MPPA!
Isotropic liquid phase!
Crystal solid phase!
top view!
Smectic
top view!
Nematic !
top view!
rot : L trans: L!
rot : S trans: L!
rot : S trans: 1/3S+2/3L!
rot : S trans: S!
Competition of translational and rotational entropy
Liquid crystals
Liquid crystal:
long rigid molecules!Ex: MPPA!
Isotropic liquid phase!
Crystal solid phase!
top view!
Smectic
top view!
Nematic !
top view!
rot : L trans: L!
rot : S trans: L!
rot : S trans: 1/3S+2/3L!
rot : S trans: S!
Competition of translational and rotational entropy
2
1
40! 80!T / oC!
P / kbar! So!
L!
N!
Sm!
Octylcyanobiphényl (8CB)!Shashidhar et Venkatesh!J. Phys. Coll. (1979)!
Micelles theory for dummies
Amphiphilic molecule
hydrophobic tail / hydrophilic head Law of attraction : like attracks like
Micelles theory for dummies
Amphiphilic molecule
hydrophobic tail / hydrophilic head Law of attraction : like attracks like
-!-!
-!-!-!-!-!-!-!-!-!
-!-! -!-!-!-!Completly wrong with electrostaic forces ! one interaction is missing… Entropy
! chains ! solvent (solvation)…
Born model for solvation Corresponding solvation entropy For water at 25oC Solvation : decrease of the solvent entropy (less rotation and less translation of solvent molecules)
Entropy of the solvent for solvation
Note technique DRCP/SCPS/2008/21
Page 37/50
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0,5 molal
1 molal
2 molal
3 molal
Figure 19 : Vues instantanées représentant les deux premières sphères de coordination des chlorures d’uranyle.
Les TMR des molécules d’eau et des chlorures dans la première et la deuxième sphère de coordina-
tion ont été calculés pour chaque concentration (Tableau 23).
Tableau 23 : Propriétés dynamiques des molécules d’eau et des ions Cl- en solution aqueuse à 300 K pour les simulations de UO2Cl2. a Temps moyen de résidence (TMR) des molécules d’eau dans la première sphère (TMR(1)
WAT) et la deuxième sphère (TMR(2)
WAT) de coordination de UO22+ (en ps).
b Temps moyen de résidence (TMR) des Cl- dans la première sphère (TMR(1)Cl-) et la deuxième sphère
(TMR(2)Cl-) de coordination de UO2
2+ (en ps).
TMR(1)
WATa TMR
(2)WAT
a TMR
(1)Cl-
b TMR
(2)Cl-
b
Sim(D1) > 2000 11 - -
Sim(D2)nd > 2000 9 > 2000 -
Sim(D2)d > 2000 11 - -
Sim(0,5) > 2000 11 - 31
Sim(1) > 2000 12 - 30
Sim(2) > 2000 13 - 31
Sim(3) > 2000 20 - 53
Aucun échange n'a été observé entre les molécules de la première et de la deuxième sphère de coor-
dination de l’uranyle, quelques soient la concentration et les points de départ simulés. Cette observa-
tion rejoint celle faite pour l’hydratation de l’uranyle dans l’eau pure. Dans ces conditions, il semble
difficile d’observer les ions chlorure passer de la deuxième à la première sphère de coordination.
Concernant les molécules d’eau de la deuxième sphère, les TMR sont du même ordre de grandeur
jusqu’à 2 molal (d’une dizaine de picosecondes) et augmentent à 3 molal (20 ps). Cette variation des
TMR est également observée pour les chlorures en deuxième sphère de coordination de l’uranyle.
Notons tout de même que les TMR des chlorures dans cette sphère de coordination sont supérieurs à
ceux calculés pour les molécules d’eau, comme cela a été observé pour les sels de lanthanides.
4.2.2. Sels d’uranyle avec les ions nitrate (UO2(NO3)2)
Les propriétés structurales d’hydratation des nitrates d’uranyle sont déduites des fonctions de distribu-
tion radiale U-OW et U-N (Tableau 24).
2RBorn!
εr!
� ! Ze!ΔsolvG
0 =Z 2e2
8πε0RBorn1εr−1
#
$%
&
'(
≤ 0
TΔsolvS = −T∂ΔsolvG∂T
=Z 2e2
8πε0εrRBorn×Tεr
∂εr∂T
%
&'
(
)*
Tεr
∂εr∂T
= −1.36 ⇒ ΔsolvS < 0
Micelles theory and solvent entropy
-!-!
-!-!-!-!-!-!-!-!-!
-!-! -!-!-!-!
Dissociated molecules A lot of solvating solvent molecules Low solvent entropy
-!
-!Associated molecules
Less solvating solvent molecules Higher solvent entropy
For the solute entropy, it is the contrary (see lesson 1)!
Van der Waals forces with rotation
Keesom!Dipole mobile / Dipole mobile
V (r) = − p12p2
2
3(4πε0 )2kBTr
6
< kT (weak) but water
Debye!Dipole mobile /
Ion
V (r) = f (r)T
< kT (weak) but water
p!
p1! p2!
Q!
V (r) = − p2Q2
6(4πε0 )2kBTr
4
Potential : free energy of the pair of particles The entropic contribution is unfavourable (lost rotation)
and is it is half of the energetic one!
TS = T ∂F∂T
= −F ⇒ F = E −TS = E −F = E2
Depletion
Mixture of big and small hard spheres Purely repulsive system… but an attraction is measured ! Possible explanation
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- Big hard spheres (ex: colloid)
- Small hard spheres (ex: polymer)
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P! P!
depletion domain!
Entropic interpretation
- excluded volume of the small spheres
Asakura et Oosawa
When the colloids come together
- overlapping of the excluded volumes
- more volume for the small spheres
- more important entropy
- favourable configuration "attraction
Depletion
Entropic interpretation
Z = drNe−βV∫= (V −Vexclu )
N
Hence
€
F(x )=−kT ln Z=F id +Veff (r)
€
Veff (r )
attraction
Effective potential
potential averaged over the
solvent configurations
r!
€
Veff (r) /kT = ρπD3
61− 3r2D
+r3
2D3
%
& '
(
) * 2R < r < 2R +σ = D
Asakura et Oosawa
Depletion
Phase separations
Nature, 416, 801 (2002)!
Direct measurement
Phys. Rev. Lett., 81, 4004 (1998)!
Depletion
Big solutes are attracted to an interface just because they are big
Entropy of the solvant Similar to Asakura et Oosawa
Interfacial depletion
Forbidden domain for the solvent!
Smaller forbidden domain if the particle is at the surface ! Bigger entropy ! stability!
Brazilian Nuts effect!
Conclusion
Entropy = a force in the nature
Important parameter: temperature Driving phenomenon: excluded volumes + … !
