1. Physics 414: Introduction to Biophysics Professor Henry Greenside September 07, 2017

2. What happens when valve is opened between
the two balloons and equilibrium is attained?
Insight important for our discussion of osmosis in cells

3. Laplace’s law for sphere with surface tension s
Reference:

4. Time scales for non-sustained equilibrium system to return to equilibrium obey simple scaling law with size of system L
D is diffusivity constant with units of m2/s, associated with random molecular collisions (kT again!)

5. Representative values of some diffusivity parameters
Example: cylindrical chemical reaction chamber 1 cm deep, diameter 25 cm, what is minimum time to wait for equilibrium?
Your turn: approximate time for Earth to reach thermal equilibrium if

6. Small regions of nonequilibrium systems reach thermodynamic equilibrium quickly!
E. coli is cylinder with length 2 microns, radius 0.5 microns: time for thermal equilibrium? Chemical equilibrium?
So can apply equilibrium statistical physics to small regions of biological systems, e.g., to a biomolecule in a blob of cytosol inside a cell.
Also explains why it is meaningful to talk about spatially dependent temperature T(x,y,z) even though T is equilibrium concept: sufficientlly small regions have well-defined equilibrium and temperature.

7. A macroscopic system that is not in thermodynamic equilibrium is said to be “non-equilibrium”
Time dependent macroscopic properties
Temperature not uniform
Pressure not uniform
Chemical gradients
Vast majority of matter in universe is nonequilibrium: stars, atmospheres, oceans, universe itself (since it is expanding over time)
Simple nonequilibrium system: copper bar such that one end is dunked in boiling water, the other end in ice water. Has time-independent properties, pressure is uniform, details not history dependent but nonequilibrium because temperature is not constant throughout.
Biological systems nonequilibrium because of gradients in concentrations of molecules, will discuss this shortly.

8. Two kinds of nonequilibrium systems: transient and sustained
Boiling water that is cooling down (flame turned off) example of transient nonequilibrium system.
Biological systems are sustained nonequilibrium systems. Sustained by consuming energy from light (plants) or from food (animals, fungi, etc)

9. Sustained nonequilibrium systems have remarkable properties, fundamental principles still not yet identified despite big effort over many decades
Rayleigh-Benard convection experiment of pure homogeneous fluid in large horizontal smooth cell with warm floor and cool ceiling, temperatures are held constant during experiment.

10. Class discussion: what pattern of motion do you predict for convection in large cylindrical room?

11. Three most common cases of thermodynamic equilibrium
I. Constant energy E, constant volume V, constant particle # N:
equilibrium when maximum entropy S(E,V,N) = k ln(W)
II. Constant temperature T, V, N: equilibrium when minimum
Helmholtz free energy
F(T,V,N) = E – T S
III. Constant pressure p, T, N: equilibrium when minimum
Gibbs free energy
G(p,T,N) = E – TS + pV

12. Review the “thermodynamic identity” Differential form of energy conservation

13. 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)

14. 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

15. Stirling’s approximation formula for n!
On left: direct comparison of n with Stirling’s approximation, they agree to width of the curves.
On right, plot of the relative error of Stirling’s approximation. For n>7, the relative error is less than one percent (which implies at least two significant digits). As a challenge, can you figure out empirically or analytically the functional form of the relative error as a function of n?

16. One-minute End-of-class Question