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

TSUF HelmholtzFreeEnergy

TSPVUG

PVUH

TSHG GibbsFreeEnergy

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

What are all these interactions?

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

desolvdispstericeltot EBAqE

Van der Waals

Here is a typical form in which energy of interactions between two proteins or protein and small molecule can be written Ionic pairs +

H-bondingremoval of waterfrom the contact