1. SURFACE TENSION 2 SPS Lectures January 2006 Wayne Lawton Department of Mathematics National University of Singapore http://math.nus.edu.sg/~matwml matwml@nus.edu.sg

2. ABSTRACT The Journal of Chemical Physics -- September 1, 2000 -- Volume 113, Issue 9, pp. 3882-3893 Spatial and energetic-entropic decomposition of surface tension in lipid bilayers from molecular dynamics simulations Erik LindahlErik Lindahl and Olle EdholmOlle Edholm Theoretical Physics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden We explain molecular cause of surface tension using thermodynamic concepts that explain the role of both energy and entropy – cutting edge concepts in biochemistry and life sciences.

3. ENTROPY Thermodynamics Statistical Mechanics 1 st Law 2 st Law Boltzmann’s tombstone

4. EQUIPARTITION Entropy of a System With Two Subsystems therefore EP Theorem Microstates have equal probability Hence 2 nd Law  Corollary Two subsystems in thermal equilibrium with constant total energy will maximize EP Theorem (Boltzmann) Each translational or rotational component of the random thermal motion of a molecule has an average kinetic energy

5. A = HELMHOLTZ FREE ENERGY We consider a constant volume system whose entropy S = S(E) that is in thermal equilibrium with an infinite reservoir that has temperature T Theorem Energy will flow into / out of the system so as to minimize A(E) = E – TS(E) Proof At thermal equilibrium the total entropy is thereforefor every value of is maximized  Remark

6. We consider a system consisting of molecules that can be in states 1 or 2 having respective energies ENDOTHERMIC REACTIONS Theorem The fraction p of molecules in state 1 satisfies Proof For a system of N molecules the binomial theorem and Stirling approx  and the result follows since

7. Enthalpy SURFACE TENSION THERMODYNAMICS Guggenheim-Hill [1] incorporate Gibbs Free Energy Systems in therm. equil. minimize G  2 nd Law&surf. ten.  into hence into

8. TUTORIAL PROBLEMS 3. Study the role of entropy in the chemical equilibrium formula in http://en.wikipedia.org/wiki/Chemical_equilibrium 1. Boltzmann’s formula uses the natural log and log W gives information in nats. How many bits of information = 1 nat ? 2. Derive A(p) endothermic in reactions

9. 1.Carry out experiments described in http://www.iit.edu/~smile/ch8623.html RESEARCH PROJECTS 2.Carry out experiments described in [3]

10. [1] Chemistry of Interfaces, M. J. Jaycock and G. D. Parfitt, Ellis Horwood, Chichester, 1986. REFERENCES [2] Dynamics of Surface Phenomena, P. Joos, Ridderprint, Utrecht, 1999. [3] Science with Soap Films, D. Lovett, Institute of Physics Publishing, Bristol, 1994.