# Rate of particle size change due to condensation or evaporation

Condensation on the surface of an airborne particle results in an increase of size, whereas evaporation results in a decrease of size. The rate of the particle size change can be calculated based on the vapour pressure and temperature at the particle surface and in the air far away from the particle.

 $\frac{dd_{p}}{dt}=\frac{4\cdot D_{v}\cdot M}{R\cdot \rho _{p}\cdot d_{p}}\left (\frac{p_{\infty }}{T_{\infty}} \right )\cdot \Phi$ (1) Particle size change due to condensation or evaporation  $\Phi =\frac{2\cdot \lambda +d_{p}}{d_{p}+5.33\frac{\lambda ^{2}}{d_{p}}+3.42\cdot \lambda }$ (2) Fuchs correction factor $t= \frac{R\cdot \rho _{p}\cdot d_{p}^{2}}{8\cdot D_{v}\cdot M\cdot \left ( \frac{p_{d }}{T_{d}} -\frac{p_{\infty }}{T_{\infty }}\right )}$ (3) Evaporating particle lifetime The change of the particle size due to condensation or evaporation is calculated using equation (1), where Dv is the diffusion coefficient of the vapour molecule, M is the molecular weight of the liquid, R is the gas constant, $\small \rho _{p}$ is the particle density and $\small p_{\infty }$, $\small T_{\infty }$, pd and Td are the vapour pressure and temperature in the air and at the particle surface, respectively.  Usually, $\small T_{\infty }$can be assumed to equal Td. The vapour pressure at the curved particle surface is affected by the Kelvin effect. Vapour will condense onto the surface if the vapour pressure pd at the particle surface is smaller than the vapour pressure in air, whereas particle material will evaporate when it is larger.  $\small \Phi$ is the Fuchs correction factor that takes into account different transport mechanisms of the vapour molecules in the direct vicinity of the particle and far away from it. In equation (2), $\small \lambda$ is the mean free path of gas molecules. Equation (3) describes the lifetime of an evaporating particle, before it is completely evaporated. Equations (1) to (3) are valid only for particles with size $\small d_{p}>\lambda$.

## Execution

 These equations are purely deterministic and can easily be solved using a pocket calculator or spreadsheet software.