Kelvin equation

Key takeaways

Kelvin 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 Kelvin equation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Kelvin equation from memory before moving on to harder problems.

Reference excerpt

The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex surface is higher than that at a flat surface. The Kelvin equation is dependent upon thermodynamic principles and does not allude to special properties of materials. It is also used for determination of pore size distribution of a porous medium using adsorption porosimetry. The equation is named in honor of William Thomson, also known as Lord Kelvin.

Formulation The original form of the Kelvin equation, published in 1871, is:

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

ρ l i q u i d {\displaystyle \rho _{\rm {liquid}}} = density of liquid

r 1 {\displaystyle r_{1}} , r 2 {\displaystyle r_{2}} = radii of curvature along the principal sections of the curved interface. This may be written in the following form, known as the Ostwald–Freundlich equation:

ln ⁡ p p s a t = 2 γ V m r R T , {\displaystyle \ln {\frac {p}{p_{\rm {sat}}}}={\frac {2\gamma V_{\text{m}}}{rRT}},}

where p {\displaystyle p} is the actual vapour pressure,

p s a t {\displaystyle p_{\rm {sat}}} is the saturated vapour pressure when the surface is flat,

γ {\displaystyle \gamma } is the liquid/vapor surface tension, V m {\displaystyle V_{\text{m}}} is the molar volume of the liquid, R {\displaystyle R} is the universal gas constant, r {\displaystyle r} is the radius of the droplet, and T {\displaystyle T} is temperature. Equilibrium vapor pressure depends on droplet size.

If the curvature is convex, r {\displaystyle r} is positive, then p > p s a t {\displaystyle p>p_{\rm {sat}}}

If the curvature is concave, r {\displaystyle r} is negative, then p < p s a t {\displaystyle p<p_{\rm {sat}}}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Kelvin equation

Start with the simplest possible case. Write down what Kelvin 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 Kelvin 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 Kelvin 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 Kelvin equation

In research: Kelvin 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 Kelvin 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: Kelvin equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Physical chemistry, Surface science, Thought experiments in physics, so understanding it makes those chapters shorter.
In everyday life: Look for Kelvin 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.

How to study Kelvin equation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kelvin 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 Kelvin equation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kelvin equation in simple terms?

The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex surface is higher than that at a flat surface.

Why does Kelvin 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 Kelvin 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 Kelvin equation.