Lecture 5 Barometric formula and the Boltzmann equation (continued)

1. Lecture 5 Barometric formula and the Boltzmann equation (continued) Notions on Entropy and Free Energy Intermolecular interactions: Electrostatics

2. Barometric formula because pressure is proportional to the number of particlesp ~ n n = number of particles per unit volume normalizing to the volumec = n/V c = concentration (which is probability) Boltzmann: in our case U is constant because T is constant

3. Boltzmann equation uses probabilities the relative populations of particles in states i and j separated by an energy gap 3 2 DE3-2 the fraction of particles in each state: DE2-1 1 - partition function

4. S = k lnW The energy difference here represents enthalpy H = U + W (internal energy +work) W is thenumber of micro-states pi pj DH DH Free energy difference DG = DH - TDS pi/pj e-1 = 0.37 e-2 = 0.135 e-3 = 0.05 e-4 = 0.018 e-5 = 0.007 entropic advantage For two global states which can be ensembles of microstates:

5. Q1 - Q2 = W (reversible work) Carnot cycle and Entropy p Q1 T1 T2 Q2 V S = k lnW W = number of accessible configurations

6. At constant T Helmholtz Free Energy

7. Helmholtz Free Energy Gibbs Free Energy

9. What determines affinity and specificity? Tight stereochemical fit and Van der Waals forces Electrostatic interactions Hydrogen bonding Hydrophobic effect All forces add up giving the total energy of binding: Gbound– Gfree= RT lnKd

10. What are all these interactions?

11. Electrostatic (Coulombic) interactions q1 q2 e (in SI) charge - charge r e≥1 dielectric constant of the medium that attenuates the field The Bjerrum length is the distance between two charges at which the energy of their interactions is equal to kT When T = 20oC, e = 80 lB = 7.12 Ǻ

12. Electrostatic self-energy, effects of size and dielectric constant e r q brought from infinity e1 e2 Consider effects of 1. charge 2. size 3. value of e2 relative to e1 on the partitioning between the two phases q r ?

13. What if there are many ions around as in electrolytes? The radial distribution function shows the probabilities of finding counter-ions and similar ions in the vicinity of a particular charge q- q+ r Poisson eqn counter-ions Solution in the Debye approximation: K – Debye length, a function of ion concentration Point charge and radial symmetry predict a decay that is steeper than exponential same charge ions

14. Charge-Dipole and Dipole-Dipole interactions static charge - dipole dipole moment + q’ a q q r with Brownian tumbling - q’ K – orientation factor dependent on angles static d1 d2 r with Brownian motion

15. Induced dipoles and Van der Waals (dispersion) forces E - d d – dipole moment r a - polarizability + constant dipole induced dipole neutral molecule in the field r I1,2 – ionization energies a1,2 – polarizabilities induced dipoles (all polarizable molecules are attracted by dispersion forces) n – refractive index of the medium Large planar assemblies of dipoles are capable of generating long-range interactions

16. 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) Long-range and short-range interactions 1/r 1/r2 1/r6 1/r4

17. Interatomic interaction: Lennard-Jones potential describes both repulsion and attraction r = r0 steric repulsion r = 0.89r0 Bond stretching is often considered in the harmonic approximation: r = r0 (attraction=minimum)

18. Here is a typical form in which energy of interactions between two proteins or protein and small molecule can be written Ionic pairs + H-bonding removal of water from the contact Van der Waals