mathematics

Ostwald–Freundlich equation

Ostwald–Freundlich equation is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ostwald–Freundlich equation rather than just read about it. In short: The Ostwald–Freundlich equation governs boundaries between two phases; specifically, it relates the surface tension of the boundary to its curvature, the ambient temperature, and the vapor pressure or chemical potential in the two phases. The Ostwald–Freundlich equation for a droplet or particle with radius R {\displaystyle R} is: p p e q = exp ⁡ ( R c r i t i c a l R ) {\displaystyle {\frac {p}{p_{\rm {eq}}}}=\exp…

Key takeaways

Ostwald–Freundlich equation belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ostwald–Freundlich equation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ostwald–Freundlich equation from memory before moving on to harder problems.

Reference excerpt

The Ostwald–Freundlich equation governs boundaries between two phases; specifically, it relates the surface tension of the boundary to its curvature, the ambient temperature, and the vapor pressure or chemical potential in the two phases. The Ostwald–Freundlich equation for a droplet or particle with radius R {\displaystyle R} is:

p p e q = exp ⁡ ( R c r i t i c a l R ) {\displaystyle {\frac {p}{p_{\rm {eq}}}}=\exp {\left({\frac {R_{\rm {critical}}}{R}}\right)}}

R c r i t i c a l = 2 ⋅ γ ⋅ V a t o m k B ⋅ T {\displaystyle R_{critical}={\frac {2\cdot \gamma \cdot V_{\rm {atom}}}{k_{\rm {B}}\cdot T}}}

V a t o m {\displaystyle V_{\rm {atom}}} = atomic volume

k B {\displaystyle k_{\rm {B}}} = Boltzmann constant

γ {\displaystyle \gamma } = surface tension (J ⋅ {\displaystyle \cdot } m−2)

p e q {\displaystyle p_{\rm {eq}}} = equilibrium partial pressure (or chemical potential or concentration)

p {\displaystyle p} = partial pressure (or chemical potential or concentration)

T {\displaystyle T} = absolute temperature One consequence of this relation is that small liquid droplets (i.e., particles with a high surface curvature) exhibit a higher effective vapor pressure, since the surface is larger in comparison to the volume. Another notable example of this relation is Ostwald ripening, in which surface tension causes small precipitates to dissolve and larger ones to grow. Ostwald ripening is thought to occur in the formation of orthoclase megacrysts in granites as a consequence of subsolidus growth. See rock microstructure for more.

History In 1871, Lord Kelvin (William Thomson) obtained the following relation governing a liquid-vapor interface:

p ( r 1 , r 2 ) = P − γ ρ v a p o r ( ρ l i q u i d − ρ v a p o r ) ( 1 r 1 + 1 r 2 ) , {\displaystyle p(r_{1},r_{2})=P-{\frac {\gamma \,\rho \,_{\rm {vapor}}}{(\rho \,_{\rm {liquid}}-\rho \,_{\rm {vapor}})}}\left({\frac {1}{r_{1}}}+{\frac {1}{r_{2}}}\right),}

where:

p ( r ) {\displaystyle p(r)} = vapor pressure at a curved interface of radius r {\displaystyle r}

P {\displaystyle P} = vapor pressure at flat interface ( r = ∞ {\displaystyle r=\infty } ) = p e q {\displaystyle p_{eq}}

γ {\displaystyle \gamma } = surface tension

ρ v a p o r {\displaystyle \rho \,_{\rm {vapor}}} = density of vapor

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Ostwald–Freundlich equation

Start with the simplest possible case. Write down what Ostwald–Freundlich equation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ostwald–Freundlich equation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ostwald–Freundlich equation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ostwald–Freundlich equation

In research: Ostwald–Freundlich equation appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ostwald–Freundlich equation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ostwald–Freundlich equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Petrology, Surface science, Thermodynamic equations, so understanding it makes those chapters shorter.
In everyday life: Look for Ostwald–Freundlich equation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Ostwald–Freundlich equation” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Ostwald–Freundlich equation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ostwald–Freundlich equation means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ostwald–Freundlich equation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ostwald–Freundlich equation in simple terms?

Why does Ostwald–Freundlich equation matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ostwald–Freundlich equation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ostwald–Freundlich equation.