Physics 414: Introduction to Biophysics Professor Henry Greenside September 12, 2017

Physics colloquium by Michael Rubenstein Wednesday, 3:30pm Physics 128 "The Story of Love and Hate in Charged Polymers"

Case I: constant E, V, N Determine equilibriums state by calculating entropy S(E,V,N) then maximize S Not experimentally relevant case, most experiments involve constant T, not constant E. It is hard technically to calculate entropy for most systems. In stat mech course, only a few systems are considered for which S can be explicitly calculate: adsorption of molecules into discrete sites ideal gas Einstein solid (periodic lattice of identical quantum oscillators) system of spin-1/2 magnetic dipoles (paramagnetism)

See Phillips et al pages 219-222 Entropy S of lattice model of identical protein molecules binding to a DNA molecule See Phillips et al pages 219-222

Entropy S(E,V,N) of ideal gas of N >> 1 identical atoms of mass m given by Sackur-Tetrode equation

Sackur-Tetrode equation implies the ideal gas law PV = N k T

Sackur-Tetrode equation implies the ideal gas law P = (N/V) k T = c k T But Sackur-Tetrode depends only on non-interaction of atoms so holds for “ideal” solution, which means a dilute solution such that solute atoms do not interact most of the time. So we expect “idea gas law” to hold for dilute solutions provided we interpret the equation correctly in terms of physics.

Osmotic Pressure Experiment Membrane is permeable to water, impermeable to solute (say sugar in solution)

van’t Hoff relation for osmosis If membrane is impermeable to two solutes, then pressure difference across membrane is generalization of the van’t Hoff relation:

Consequences, Applications of Osmosis Cell shape and volume (how to kill a slug with salt...) Rehydration of dried fruit, dried vegetables. Photosynthesis by plants (stomata), roots extracting water Cholera: bacterial-induced release of water into the small intestine Reverse osmosis: desalination of seawater by pushing sea water through semi-permeable membrane with high pressure.

Osmosis hugely important for cells

Stomata of a plant cell open and close via osmotic pressure, enabling photosynthesis

Wrinkled fingers

Case II: constant V, T, N Probability for small system to have energy Ei when in contact with equilibrium reservoir with temperature T Much easier in general to use Boltzmann factor than calculate S. Free energy F of small system minimized when Boltzmann distribution holds. Will discuss case III later, to include variable number of particles and constant pressure. This is where “independent of history” assumption is important: system has to be able to repeatedly make transitions between all different microstates with different energies or formula doesn’t work

Boltzmann factor used to calculate average properties, other thermodynamic quantities

First step in applying statistical mechanics: identify all microstates and their energies E_i consistent with macroscopic constraints (conserved E, V, ...) What is microstate of molecule in tall column of water that is in equilibrium with temperature T?

Example: molecules of mass m in tall water column of height H, temperature T: how does concentration vary with height? Mean height <h> of molecules? Given thermal energy, gravitational potential energy, guess magnitude of average height.

Some numbers for myoglobin, large globular protein For large molecules, need to include buoyance correction because volume of molecule excludes surrounding water: For myoglobin, m ~ 17,000 g/mole, m_eff = 0.25 m So in 5 cm tall test tube in equilibrium, concentration at top compared to bottom is So myoglobins never precipitate out, get so-called colloidal suspension.

One-minute End-of-class Question