Engineering & Science
Kelvin Equation: Nanodroplet Vapor Pressure & Capillary Condensation RH
Calculate convex aerosol nanodroplet supersaturation ratio P / P_sat = EXP(+2 * γ * V_m / (r * R * T)) and concave silica-mesopore capillary condensation relative humidity RH_cond (%) = 100 * EXP(-2 * γ * V_m / (r * R * T)).
Why 2 nm cloud droplets evaporate instantly at 100% RH—while 2 nm silica pores flood at 59% RH!
Lord Kelvin (William Thomson, 1871) proved that surface curvature shifts chemical potential and equilibrium vapor pressure exponentially: ln(P / P_sat) = ±(2 * γ * V_m) / (r * R * T)! Over a CONVEX liquid aerosol nanodroplet (+ sign), surface molecules are weakly bound so P > P_sat—explaining why clean atmosphere requires supersaturation or cloud condensation nuclei to nucleate rain! Inside a CONCAVE hydrophilic mesopore (- sign), meniscus tension stabilizes liquid so vapor condenses into liquid FAR below 100% relative humidity!
Check water at T = 298.15 K (γ = 0.0720 N/m, molar volume V_m = 18.068 cm^3/mol, R = 8.3144626 J/(mol·K)) at radius r = 2.0 nm: a convex 2 nm nanodroplet demands P / P_sat = 1.6901 (169.01% relative humidity!) just to avoid evaporating, whereas a 2 nm hydrophilic silica gel desiccant pore spontaneously condenses water vapor at RH_cond = 59.17% (= 100 / 1.6901)!
| Meniscus Curvature Radius r (Water at 298.15 K) | Convex Droplet P / P_sat | Concave Pore Condensation RH (%) | Physical Consequence |
|---|---|---|---|
| Ultrafine r = 2.0 nm Meniscus | 1.6901 (169.0% RH) | 59.17% RH | Strong Silica Gel Desiccant Pickup! |
| Mesoporous r = 10.0 nm Meniscus | 1.1107 (111.1% RH) | 90.04% RH | Moderate Curvature Shift |
| Flat Bulk Surface (r -> ∞) | 1.0000 (100.0% RH) | 100.00% RH | Standard Saturation Equilibrium |
Convex Nanodroplet Equilibrium Supersaturation Ratio P / P_sat
Converts molar volume B3 from cm^3/mol (* 1E-6) and droplet radius B4 from nm (* 1E-9) inside EXP(+2 * γ * V_m / (r * R * T)).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Kelvin Inputs", ROUND(EXP((2 * B2 * (B3 * 0.000001)) / ((B4 * 1E-9) * 8.3144626 * B5)), 4))Concave Hydrophilic Mesopore Capillary Condensation RH (%)
Evaluates 100 * EXP(-2 * γ * V_m / (r * R * T)) assuming complete wetting (cos θ = 1) inside hydrophilic pores.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Kelvin Inputs", ROUND(100 * EXP(-(2 * B2 * (B3 * 0.000001)) / ((B4 * 1E-9) * 8.3144626 * B5)), 2))Try the example
Use the original values or change them to test your understanding.