Lecture 5
Barometric formula and the Boltzmann equation(continued)
Notions on Entropy and Free Energy
Intermolecular interactions: Electrostatics
kTmgh
o
ep
p /Barometric formula
kTE
o
potec
c /
kTmgh
o
en
n / n = number of particles per unit volume
c = concentration (which is probability)
because pressure is proportional to the number of
particles p ~ n
normalizing to the volume c = n/V
potEUH kTH
o
ec
c /
in our case U is constant because T is constant
Boltzmann:
Boltzmann equation uses probabilities
kTEE
j
i jiep
p /)(
the relative populations of particles in states i and j separated by an energy gap
t
j
kTE
kTE
jj
j
e
ep
1
/
/
t
j
kTE je1
/- partition function
the fraction of particles in each state:
E2-1
E3-2
1
3
2
S = k lnW
Free energy difference G = H - TS
W is the number of micro-states
e-1 = 0.37
e-2 = 0.135
e-3 = 0.05
e-4 = 0.018
e-5 = 0.007
H
entropic advantage
The energy difference here represents enthalpyH = U + W (internal energy +work)
kTG
j
i ep
p /
For two global states which can be ensembles of microstates:
H
pi/pj
pi
pj
kTH
j
i ep
p /
Carnot cycle and Entropy
V
p
T1
T2
Q1 - Q2 = W (reversible work)
2
2
1
1
T
Q
T
Q
0 Ti
Qi
0T
dQ T
dTC
T
dQdS
prev
B
A
revT
dQASBS )()( S = k lnW
W = number of accessible configurations
Q1
Q2
B
A
revdQASBST )]()([
At constant T
QUW
)]()([)()( ASBSTBUAUW
)()( BFAFW
)()()( ATSAUAF
)()()( BTSBUBF
STUFW
HelmholtzFreeEnergy
What determines affinity and specificity?
Tight stereochemical fitand Van der Waals forces Electrostatic interactionsHydrogen bondingHydrophobic effect
All forces add up giving the total energy of binding:
Gbound– Gfree= RT
lnKd
Electrostatic (Coulombic) interactions
20
21
4 r
qqF
r
qqFdrU
r 0
21
4
(in SI)
r
q1 q2
charge - charge
dielectric constant of the medium that attenuates the field≥
kTelB 02 4/
The Bjerrum length is the distance between two charges at which the
energy of their interactions is equal to kT
When T = 20oC, = 80 lB = 7.12 Ǻ
rq
Electrostatic self-energy, effects of size and dielectric constant
r
qqGel
04
q
r
qqdqG
q
el 0
2
00 84
1
brought from infinity
rq
?
Consider effects of 1. charge2. size3. value of 2 relative to 1
on the partitioning between the two phases
r
q+q-
What if there are many ions around as in electrolytes?
02 / Poisson eqn
)exp()( 1 KrArr Solution in the Debye approximation:
The radial distribution function shows the probabilities of finding counter-ions and similar ions in the vicinity of a particular charge
Point charge and radial symmetry predict a decay that is steeper than exponential
K – Debye length, a function of ion concentration
same charge ions
counter-ions
Charge-Dipole and Dipole-Dipole interactions
+ q’
- q’
a
charge - dipole
r
204
cos
r
qdU
qad
dipole moment
static
420
22
3)4( kTr
dqU
with Brownian tumbling
30
21
4 r
KddU
d1 d2
K – orientation factor dependent on angles
620
22
21
)4(32
rkT
ddU
with Brownian motion
static
q
r
Induced dipoles and Van der Waals (dispersion) forcesE
Ed -
+ a - polarizability
dr 6
02
2
)4( r
dU
constant dipole
induced dipole
r64
21
2121
3)( rnII
IIU
I1,2 – ionization energies
1,2 – polarizabilities
n – refractive index of the medium
induced dipoles(all polarizable molecules
are attracted by dispersion forces)
neutral molecule in the field
d – dipole moment
Large planar assemblies of dipoles are capable of generating long-range interactions
0 2 4 6 8 100
0.2
0.4
0.6
0.8
U1 r( )
U2 r( )
U4 r( )
U6 r( )
r
1/r2
1/r6
1/r
Long-range and short-range interactions
Even without NET CHARGES on the molecules, attractive interactions always exist. In the presence of random thermal forces all charge-dipole or dipole-dipole interactions decay steeply (as 1/r4 or 1/r6)
1/r4
Interatomic interaction: Lennard-Jones potential describes both repulsion and attraction
Uo 1
U x( ) Uo x12
2x6
0.6 0.8 1 1.2 1.4 1.6 1.8
1
0
1
2
U x( )
x
600
1200 )/(2)/()( rrErrErEp
r = r0 (attraction=minimum)
r = 0.89r0
r = r0
steric repulsion
Bond stretching is often considered in the harmonic approximation:
202
1 )()( xxxU