Talk:Ostwald–Freundlich equation

message: "Gibbs-Thomson equation" used interchangeably with "Ostwald–Freundlich equation" in ALLAN S. MYERSON (Ed.);  Handbook of industrial crystallization, Second Edition;  Butterworth–Heinemann, Boston;  2002;  313 pp.  
Comments?
— DIV (128.250.204.118 09:01, 12 March 2007 (UTC)
thread_title: OSTWALD–FREUNDLICH equation

message: According to Lord Kelvin's equation of 1871,Sir William Thomson (1871) "On the equilibrium of vapour at a curved surface of liquid," Philosophical Magazine, series 4, 42 (282) :  448-452.  See equation (2) on page 450.The derivation here is based on pages 522-525 of:  Robert von Helmholtz (1886) "Untersuchungen über Dämpfe und Nebel, besonders über solche von Lösungen" (Investigations of vapor and mists, and especially of such things from solutions), Annalen der Physik, 263 (4)  :  508-543.
 p(r_1 , r_2) = P - \frac { \gamma\,  \rho\, _{vapor} } {(\rho\,_{liquid} - \rho\,_{vapor})}\left ( \frac {1}{r_1} + \frac {1}{r_2}\right ) 
where 
:  p(r)   =  vapor pressure at a curved interface of radius  r 
:  P   =  vapor pressure at flat interface ( r = \infty ) =  p_{eq} 
:  \gamma   =  surface tension
:  \rho\, _{vapor}   = density of vapor
:  \rho\, _{liquid}  = density of liquid
:  r_1  ,   r_2  = radii of curvature along the principal sections of the curved interface.
Note:  Kelvin defined the surface tension  \gamma  as the work that was performed per unit area by the interface rather than on the interface; hence the term containing  \gamma  has a minus sign.  In what follows, the surface tension will be defined so that the term containing  \gamma  has a plus sign.
If the particle is assumed to be spherical, then   r =  r_1 = r_2  .
Since  \rho\, _{liquid} \gg \rho\, _{vapor} , 
then  \rho\, _{liquid} -  \rho\, _{vapor} \approx \rho\, _{liquid} .
Hence
p(r) \approx P + \frac {2  \gamma\,  \rho\, _{vapor} } {\rho\,_{liquid} \cdot r}.
Assuming that the vapor obeys the ideal gas law, then
 \rho\, _{vapor} = \frac {m_{vapor}} {V} = \frac {MW \cdot n} {V} = \frac {MW \cdot P} {RT} = \frac {MW \cdot P} {N_0 k_B T}  
where
: m_{vapor}  = mass of a volume  V  of vapor
: MW = molecular weight of vapor 
: n  = number of moles of vapor in volume  V  of vapor
: R  = ideal gas constant =  N_0 k_B
: N_0  = Avogadro’s number
: k_B  = Boltzmann's constant
: T  = absolute temperature.
Since  \frac {MW} {N_0} =  mass of one molecule of vapor or liquid, then
 \frac {\left ( \frac {MW} {N_0} \right )} {\rho\, _{liquid}} =  volume of one molecule  = V_{molecule}  .
Hence
 p(r) \approx P + \frac {2 \gamma V_{molecule} P} {k_B T r} = P + \frac {R_{critical} P} {r}  ,
where  R_{critical} = \frac {2 \gamma V_{molecule}} {k_B T}  .
Thus
 \frac {p(r) - P} {P} \approx \frac {R_{critical}} {r}  .
Since  \frac {p(r)} {P} = 1 - \frac {P - p(r)} {P}  , then  \log \left ( \frac {p(r)} {P} \right ) = \log \left (1 - \frac {P - p(r)} {P} \right )  .
Since   p(r) \approx P , then  \frac {P - p(r)} {P} \ll 1  .
If  x \ll 1, then \log \left (1 - x \right ) \approx -x  .
Hence 
 \log \left ( \frac {p(r)} {P} \right ) \approx \frac {p(r) - P} {P}  .
Therefore
 \log \left ( \frac {p(r)} {P} \right ) \approx \frac {R_{critical}} {r}  ,
which is the Ostwald-Freundlich equation.
Cwkmail (talk) 10:50, 20 February 2013 (UTC)
replies: text: I have added this to the article. Arbitrarily0 (talk) 19:35, 9 January 2014 (UTC)
thread_title: Derivation of Ostwald-Freundlich equation from Kelvin's equation (1871)