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

$p(r)\approx P+{\frac {2\gamma V_{molecule}P}{k_{B}Tr}}=P+{\frac {R_{critical}P}{r}}$ ,

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. Arbitrarily0 ^(talk) 19:35, 9 January 2014 (UTC)