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
Belleville roller coaster (1817) !
modern roller coaster!
In thermodynamics (T constant) : valid only at T = 0 K
Equilibrium : E - TS minimum
Interpretation
Sunivers maximum
S univers = S syst + S ext
$ ext
∂S ext
E
ext
ext
ext
& S = S (Etot − E) ≈ S (Etot ) +
(−E ) = S0 −
with %
∂E
T
& S syst = S
'
External environment!
Thermostat!
system!
E ⤻!
gives T!
S
univers
E
= S − (+S0ext )
T
minimization of the energy = maximization of
the entropy of the external environement
Competition between S and E
Equilibria : minimisation of F = E – TS
!
V(r) =
potential
energy
between two
molecules!
(= -TSuniverse)!
V(r)!
repulsion !
attraction !
!! σ $12 ! σ $6 $
typically, V (r) = 4ε ### & − # & &&
"" r % " r % %
r!
distance between the molecules!
ε!
σ!
Solid!
U predominant!
atoms at fixed distance!
Low T!
Gas!
S predominant!
desorder!
High T!
ε: depth of the attraction!
σ : size of the molecules!
Liquid!
U ≈ TS ? ? ?!
Intermediate T !
Hard spheres and entropy
a (crystalline) solid phase can have a higher entropy than a liquid phase!
solid!
liquid (or dense fluid)!
Hard spheres and entropy
a (crystalline) solid phase can have a higher entropy than a liquid phase!
solid!
liquid (or dense fluid)!
Hard spheres and entropy
a (crystalline) solid phase can have a higher entropy than a liquid phase!
solid!
liquid (or dense fluid)!
Hard spheres and entropy
a (crystalline) solid phase can have a higher entropy than a liquid phase!
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 and entropy
a (crystalline) solid phase can have a higher entropy than a liquid phase!
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!
Solid phases entropically
favourable at high pressure
T!
Hard
spheres
phase
diagram
F!
S
+
F
S
(fcc)!
0.494! 0.545! 0.740!
Φ (packing fraction)!
(exception:
water)!
Liquid crystals
Liquid crystal:
long rigid molecules!
Isotropic
liquid phase!
rot : L
trans: L!
Ex:
MPPA!
Nematic
!
Smectic
top view!
top view!
rot : S
trans: L!
Crystal
solid phase!
rot : S
trans: 1/3S+2/3L!
Competition of translational and rotational entropy
top view!
rot : S
trans: S!
Liquid crystals
Liquid crystal:
Ex:
MPPA!
long rigid molecules!
Isotropic
liquid phase!
2
P / kbar!
Nematic
!
Smectic
solid phase!
So!
Octylcyanobiphényl
(8CB)!
Shashidhar et
Venkatesh!
J. Phys. Coll. (1979)!
N!
1
top view!
Sm!
rot : L
trans: L!
rot : S
trans: L!
Crystal
L!
top view!
rot : S
trans: 1/3S+2/3L!
T / oC!
80!
40!
Competition of translational and rotational entropy
top view!
rot : S
trans: S!
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)…
EntropyN ooft e the
solvation
t e c hsolvent
n i q u e D Rfor
CP/S
CPS/2008/21
Page 37/50
Born model for solvation
εr!
!
1 molal
Ze! 2RBorn!
2 molal
Δ solvG 0
Z 2e2 # 1 &
=
% −1(
8πε 0 RBorn $ εr '
≤0
3 molal
solvation
entropy
stantanées Corresponding
représentant les deux
premières
sphères de coordination des chlorures
% T ∂εr (
∂Δ solvG
Z 2e2
TΔ solv S = −T
=
×'
*
8πε 0εr Rsphère
εr coordina∂T )
&de
cules d’eau et des chlorures dans la première∂T
et la deuxième
Born
s pour chaque
23).
Forconcentration
water at 25o(Tableau
C
T ∂εr
−1.36
Δ solv Saqueuse
< 0 à 300 K
étés dynamiques des molécules d’eau et=des
ions Cl-⇒
en solution
εr ∂T
s de UO2Cl2.
résidence Solvation
(TMR) des: molécules
d’eau
dans
la première
sphère
(TMR(1)WAT
) etless
la translation
decrease
of
the
solvent
entropy
(less
rotation
and
(2)
2+
TMR WAT) de coordination de UO2 (en ps).
of solvent molecules)
résidence (TMR) des Cl- dans la première sphère (TMR(1)Cl-) et la deuxième sphère
ination de UO22+ (en ps).
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
p12 p22
V (r) = −
3(4πε 0 )2 kBTr 6
< kT (weak) but water
p1!
p2!
p 2Q 2
V (r) = −
6(4πε 0 )2 kBTr 4
< kT (weak) but water
Debye!
Dipole mobile /
Ion
Q!
p!
Potential : free energy of the pair of particles
f (r)
V (r) =
T
∂F
E
TS = T
= −F ⇒ F = E − TS = E − F =
∂T
2
The entropic contribution is unfavourable (lost rotation)
and is it is half of the energetic one!
Depletion
" "" " " " " "
"" " " " "
"" "" " "
"
"
"
"
"
"" " " " 0" "
"
" " " "" " " " "
Mixture of big and small hard spheres
"
""
"
"
"
""
""
"
""
"
-  Big hard spheres (ex: colloid)
- Small hard spheres (ex: polymer)
Purely repulsive system… but an attraction is measured !
Possible explanation
P!
""
"
" " " ""
" ""
"
"
"
"
"
""""
"
"
"
"
" " "
" " "
"
"
" " "
"
"
" " "
" "
"
" ""
"
"
" 0"" "
"
"
"
"
""
"
depletion
domain!
P!
Depletion
Entropic interpretation
Asakura et Oosawa
-  excluded volume of the small spheres
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
Asakura et Oosawa
Z =
N − βV
dr
∫ e
= (V −Vexclu ) N
F(x )=−kT ln Z = F id +Veff (r)
r!
Hence
πD3 % 3r
r3 (
Veff (r) /kT = ρ
+
'1−
*
6 & 2D 2D3 )
€
Effective potential
€
potential averaged over the
solvent configurations
2R < r < 2R + σ = D
Veff (r )
attraction
Depletion
Phase separations
Nature, 416, 801 (2002)!
Direct measurement
Phys. Rev. Lett., 81, 4004 (1998)!
Interfacial depletion
Big solutes are attracted to an interface just because they are
big
Entropy of the solvant
Similar to Asakura et Oosawa
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 + …
!