Thermodynamics of Small Systems

1
Thermodynamics of Small Systems
dUTdS-pdVmdNgdA
dAd(4pr2)8prdr
N4pr3/(3Vm)
dN4pr2dr/(Vm)rdA/(2Vm)
dA ((2Vm)/r)dN
dUTdS-pdV(m g (2Vm)/r) dN
m m bulk g (2Vm)/r)
2
Thermodynamics of Small Systems
(4pr2)g
m m bulk g (2Vm)/r)
mkmk0RTlnakm bulk g (2Vm)/r)
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Thermodynamics of Small Systems
mkmk0RTlnakm bulk g (2Vm)/r)
RTlnak(m bulk-mk0) g (2Vm)/r)
DGsol(mk0- m bulk)
RTlnak(-DGsol g (2Vm)/r)
ak( r)exp((-DGsol)/RT)exp((g (2Vm)/r)/RT)
ak( r)ak(large)exp((g (2Vm)/r)/RT)
4
Thermodynamics of Small Systems
Gibbs-Thompson or Ostwald-Freundlich
ak( r)ak(large)exp((g (2Vm)/r)/RT)
r goes down solubility increases -- convex
r
r
concave r goes down solubility decreases
5
Thermodynamics of Small Systems
Ostwald ripening
diffusion
mkhigher akhigher Particles dissolve
mklower aklower Particles get bigger
6
Thermodynamics of Small Systems
Equilibrium, large particle
msolid(p,Tm) mliquid(p,Tm)
Equilibrium, small particle melting is higher
due to greater m
msolid(p,TmDT) m bulk (p,TmDT) g (2Vm)/r)
mliquid(p,TmDT)
DHfusion ln(1DT/Tm) g (2Vm)/r) 0
Tm( r)Tm(large)1- g (2Vm)/(r
DHfusion)Tm(large)1- r/r
r0.2-1.7 nm
g0.59R(Tm )/(aNavo)
7
Thermodynamics of Small Systems