surface tension 2 sps lectures january 2006 wayne lawton department of mathematics national...
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SURFACE TENSION 2SPS Lectures January 2006
Wayne Lawton
Department of Mathematics National University of Singapore
http://math.nus.edu.sg/~matwml
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 simulationsErik Lindahl and Olle 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.
ENTROPYThermodynamics
Statistical Mechanics
PdVqE 1st Law
2st Law
ofenergy Eby absorbedheat q
todone work PdV
TdSq of ure temperatT
ofentropy S
WkS logBoltzmann’s
tombstone
constant B s'k
smicrostate # W
EQUIPARTITIONEntropy of a System With Two Subsystems
2121211212 loglogloglog SSWkWkWWkWkS
21 EE )()( 2211 ESES therefore
EP Theorem Microstates have equal probability
12'211
'122
'211
'1 )()()()(0 EESEESEESEES
Hence 2nd Law 122
'21
'1
11 )()( TESEST
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 2/kT
A = HELMHOLTZ FREE ENERGYWe 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
))()(()()( ESEESTEEAEEA therefore for every value of
is maximized 0)()( 1 ETESEESE
0)( 1 ETTERemark 0)(1))((1)( 1'' TTESTEA
We consider a system consisting of molecules that can be in states 1 or 2 having respective energies
ENDOTHERMIC REACTIONS
21,EETheorem The fraction p of moleculesin state 1 satisfies )/()(log 121 kTEEp
p
Proof For a system of N molecules the binomial theorem and Stirling approx
)]1log()1(log[)1(/)( 21 ppppkTEppENpA
and the result follows since 0)(' pA
Enthalpy
SURFACE TENSION THERMODYNAMICS
reaAPVEH Guggenheim-Hill [1] incorporate
Gibbs Free Energy TSHG Systems in therm. equil. minimize G
G0 reaATSPV
reaAVPSTE 2nd Law&surf. ten.into
hence into
TSSTAAVPPVE rearea
VEreaVSrea A
ST
A
E
,,
P
rea TAS
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
1. Carry out experiments described inhttp://www.iit.edu/~smile/ch8623.html
RESEARCH PROJECTS
2. Carry out experiments described in [3]
[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.