Talk:Ostwald–Freundlich equation
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OSTWALD–FREUNDLICH equation
[edit]"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)
[edit]According to Lord Kelvin's equation of 1871,[1][2]
where
- = vapor pressure at a curved interface of radius
- = vapor pressure at flat interface () =
- = surface tension
- = density of vapor
- = density of liquid
- , = radii of curvature along the principal sections of the curved interface.
Note: Kelvin defined the surface tension as the work that was performed per unit area by the interface rather than on the interface; hence the term containing has a minus sign. In what follows, the surface tension will be defined so that the term containing has a plus sign.
If the particle is assumed to be spherical, then .
Since , then .
Hence
.
Assuming that the vapor obeys the ideal gas law, then
where
- = mass of a volume of vapor
- = molecular weight of vapor
- = number of moles of vapor in volume of vapor
- = ideal gas constant =
- = Avogadro’s number
- = Boltzmann's constant
- = absolute temperature.
Since mass of one molecule of vapor or liquid, then
volume of one molecule .
Hence
,
where .
Thus
.
Since , then .
Since , then .
If , then .
Hence
.
Therefore
,
which is the Ostwald-Freundlich equation.
- ^ 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.
Cwkmail (talk) 10:50, 20 February 2013 (UTC)
- I have added this to the article. Arbitrarily0 (talk) 19:35, 9 January 2014 (UTC)