Kelvin Equation Calculator

Calculator Inputs

Set your scenario, choose units, then compute or solve inversely.

Formula Used

The Kelvin equation connects curvature with equilibrium vapor pressure. It is commonly written using a pressure ratio:

Here, γ is surface tension, Vm is molar volume, r is radius, R is the gas constant, T is temperature, and θ is contact angle.

How to Use This Calculator

1. Select what you want to solve for: ratio, radius, or temperature.
2. Choose the geometry that matches your physical situation.
3. Enter γ, Vm, and the remaining known values with units.
4. For concave pores, set θ or directly provide cos(θ).
5. Click Calculate to view results above the form.
6. Use CSV or PDF buttons to export your computed report.

Example Data Table

Example values (water-like, 25°C, γ=0.0728 N/m, Vm=18.07 cm³/mol, convex).

Professional Notes on the Kelvin Equation

1) What the Kelvin Equation Describes

The Kelvin equation quantifies how surface curvature shifts equilibrium vapor pressure relative to a flat interface. For a convex droplet, molecules at the surface experience a higher chemical potential, so the ratio p/p0 becomes greater than 1 as radius decreases. This explains why very small droplets evaporate more readily.

2) Convex vs Concave Curvature

Geometry changes the sign. Convex curvature yields ln(p/p0) = +2γVm/(rRT). Concave curvature, typical of a meniscus in a pore, produces ln(p/p0) = −2γVmcos(θ)/(rRT). A wetting liquid (cosθ near 1) lowers p/p0 and promotes condensation in narrow pores.

3) Sensitivity to Radius at the Nanoscale

Because ln(p/p0) scales with 1/r, nanoscale radii dominate the result. With water-like parameters near 298 K, reducing radius from 20 nm to 2 nm increases ln(p/p0) by an order of magnitude. This scaling matters in aerosols, pores, and condensation studies.

4) Temperature and the RT Term

Temperature appears in the denominator as RT. Higher temperature reduces ln(p/p0) for the same γ, Vm, and r. In practice, γ and Vm may also vary with temperature, so careful work uses temperature-consistent property data rather than a single constant.

5) Role of Surface Tension (γ)

Surface tension sets the energetic cost of creating curved interface area. A larger γ increases |ln(p/p0)| linearly. Surfactants reduce γ and can dramatically weaken curvature effects, changing equilibrium humidity thresholds for condensation and altering droplet stability in sprays and emulsions.

6) Choosing a Molar Volume (Vm)

Vm links pressure work to molecular volume in the condensed phase. For liquids, Vm can be approximated from density: Vm ≈ M/ρ. Small errors in Vm produce proportional errors in the exponent, so selecting a credible density at the working temperature improves reliability.

7) Contact Angle in Pores

In concave mode, cos(θ) accounts for wetting. θ below 90° gives positive cos(θ), amplifying capillary condensation (p/p0 < 1). θ above 90° yields negative cos(θ) and can invert expectations. When θ is uncertain, it is useful to sweep values to bound the result.

8) Practical Interpretation for Humidity

Many workflows interpret p/p0 as equilibrium relative humidity fraction for water vapor at fixed temperature. The calculator reports RH (%) = 100·(p/p0) for convenience. For inverse problems, solving for radius or temperature helps design pore sizes, predict aerosol activation, and estimate where condensation begins in environmental control, drying, and nanoporous material design.

FAQs

1) What does p/p0 represent here?

It is the equilibrium vapor pressure over a curved surface divided by the flat-surface vapor pressure at the same temperature. For water vapor, it often corresponds to the equilibrium relative humidity fraction.

2) When should I use concave geometry?

Use concave for a meniscus inside a pore or capillary where curvature points inward. This case commonly predicts lower p/p0 and therefore capillary condensation at humidity below 100%.

3) Why is contact angle included?

Contact angle captures wetting. The factor cos(θ) modifies the curvature effect for pores. Better wetting (smaller θ) increases cos(θ) and strengthens condensation predictions.

4) Can p/p0 be greater than 1?

Yes. In convex mode, small droplets typically yield p/p0 > 1, meaning higher equilibrium vapor pressure than a flat surface. This supports faster evaporation of tiny droplets.

5) What inputs most affect the result?

Radius has the strongest influence because the exponent scales as 1/r. Surface tension and molar volume scale linearly, while temperature reduces the exponent magnitude through the RT term.

6) How accurate are the outputs?

Accuracy depends on property data (γ, Vm, θ) and whether Kelvin theory is valid for your scale. At extremely small radii, molecular effects and non-idealities can require more advanced models.

7) How should I pick Vm for non-water liquids?

Use Vm from reliable thermophysical tables at your temperature, or estimate Vm ≈ M/ρ using molar mass and liquid density. Consistent units are essential for meaningful results.