Talk:Ostwald–Freundlich equation

OSTWALD–FREUNDLICH equation
"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))

The usual version of the Gibbs-Thomson equation is different - see Melting-point depression and a page I have in preparation : User:Dr.BeauWebber/Gibbs-Thomson Equation / Effect - Kelvin equation is also different - but they are all related, and derived from / special cases of the generalised Gibbs Equations.

What are peoples views on if and how my page should be merged with this page ? Do we want to list the (many) permutations of the G-T equation, or just the main two variants for different applications (isolated particles / crystals in pores) ? cheers, Beau - Dr.BeauWebber (talk) 22:44, 21 February 2011 (UTC)

Derivation of Ostwald-Freundlich equation from Kelvin's equation (1871)
According to Lord Kelvin's equation of 1871,

$$ p(r_1, r_2) = P - \frac { \gamma\, \rho\, _{vapor} } {(\rho\,_{liquid} - \rho\,_{vapor})}\left ( \frac {1}{r_1} + \frac {1}{r_2}\right ) $$

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)
 * I have added this to the article.   A rbitrarily 0    ( talk ) 19:35, 9 January 2014 (UTC)