Chem 300 - Ch 16/#3 Today’s To Do List

More on IM Forces A bit on Potential Energy Lennard-Jones & Other Potentials What is a London Dispersion

force?? The return of Van der Waals

2 Interacting Linear Molecules

B2V & Potential Energy [u(r)]

2 0

22 1

V AB N rT u r kT dr( ) ex p ( ) /

A sim p le ex p ress io n fo r u (r) is n eed ed .

Compare with “Real” Equations of State

Attractive and repulsive (excluded volume) forces:

Attractive forces • are influential at far distances • and are (-)

Excluded volume effects • are short-range • and are (+)

Reasonable forms for u(r)

u r cr

u r crnn

( )

( )

66 (when r large)

(r small, n = 9 -12)

Lennard-Jones Potential

u r r r( )

41 2 6

= m easu re o f in te rm o lec a ttrac tio n

= m easu re o f m o lecu la r s iz e

To Evaluate B2V

Find & for selected gas Substitute & into Lennard-Jones

equation Substitute L-J into expression for B Integrate With B2V gas pressure can be calculated

from virial eq.

Lennard-Jones Potential

Significance of &

Compare gases at molecular level Examples: /k

• He 10.2 256• Ar 120 341• Xe 229 406• CO 100 376• CO2 189 449

What are these Attractive forces?

Dipole-Dipole Dipole-Induced dipole London Dispersion

All are C/r6 dependent

They differ in the size of the coefficient, C

Permanent & Induced Dipoles

(1) Dipole-Dipole udd(r)

ddu rr

kT( )

( )( )

2

0 3

11

2

2

2

2 6

4

(2) Induced Dipole Moment

Proportional to the electric field strength of the neighboring dipole:

induced = E = polarizability (dimensions of

volume)• E = electric field strength

inducedu ( )( / )

( )r

r

2 4

4

2

0

0

6

(3) London Dispersion Force

Exists even without perm dipole QM effect Electronic Wave function of one

molecule is momentarily distorted by passing molecule

Creates an instant induced dipole Feeds back to other molecule

dispuI II I r

r( ) ( )

3

20

1 2

1 2

1 22 64

I = Ionization Energy

Compare 2 HCl molecules

Dipole-dipole -1.82 x 10-78 J -m6

Dipole-induceddipole

-0.56 x 10-78 J -m6

Londondispersion

-10.6 x 10-78 J -m6

Other Potential Functions (1)

Hard-sphere Potential• Mimic molecules as hard spheres of

diameter • For r< u(r) = for r> u(r) = 0• H-SP assumes no attractive force• Pretty good at high T

Substitution gives:B2V = (2 2N)/3 (T-indep)

Hard Sphere(a) & Square-Well(b) Potentials

Other Potential Functions (2)

Square-Well Potential• For r< u(r) = • for < r< u(r) = -• for r > u(r) = 0• S-WP assumes crude attractive

potential

• Gives T-dependent B2V

• Reduces to HSP when = 1 or =0

Van der Waals Again

P = [RT/(Vm-b)] - a/V2m

Compare with• Z = 1 + B2V/Vm + ...

Next Time

Start Chapter 19: 1st Law• P-V work• State Functions• 1st Law• Adiabatic Processes