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

Barometric formula 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 in our case U is constant because T is constant Boltzmann:

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

S = k lnW Free energy difference  G =  H - T  S W is the number of micro-states e -1 = 0.37 e -2 = e -3 = 0.05 e -4 = e -5 = HH entropic advantage The energy difference here represents enthalpy H = U + W (internal energy +work) For two global states which can be ensembles of microstates: HH p i /p j pipi pjpj

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

At constant T Helmholtz Free Energy

Helmholtz Free Energy Gibbs Free Energy

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: G bound – G free = RT ln K d

What are all these interactions?

Electrostatic (Coulombic) interactions (in SI) r q1q1 q2q2  charge - charge 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 = 20 o C,  = 80 l B = 7.12 Ǻ

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

r q+q+ q-q- What if there are many ions around as in electrolytes? Poisson eqn 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 dipole moment static with Brownian tumbling d1d1 d2d2 K – orientation factor dependent on angles with Brownian motion static q r

Induced dipoles and Van der Waals (dispersion) forces E - + a - polarizability d r constant dipole induced dipole r I 1,2 – 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

1/r 2 1/r 6 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/r 4 or 1/r 6 ) 1/r 4

Interatomic interaction: Lennard-Jones potential describes both repulsion and attraction r = r 0 ( attraction=minimum ) r = 0.89r 0 r = r 0 steric repulsion Bond stretching is often considered in the harmonic approximation:

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-bonding removal of water from the contact