Assignment 2, Due Jan 25.

1. Obtain a value for $d_{0}$ consistent with an approximate expression for the critical radius for nucleation of an embryo
   
   $$r^{*}=\frac{d_{0}}{s}$$
   
   
   where $s$ is the supersaturation and $s\ll1$. Derive a corresponding expression for $\Delta G^{*}$ in terms of $d_{0}$. Assume a temperature of 0 $^{\circ}$C.
2. Plot the equilibrium saturation ratio $S=e/e_{sat}$ over pure ice and pure water as a function of particle radius at -20 $\textdegree$C, on a logarithmic plot between 0.001 and 1 $\mu$m radius. Why are the two different?
3. Read the paper by Facchini et al. ``Cloud albedo enhancement by surface-active albedo enhancement in growing droplets'', and briefly summarize its findings in the context of what we have covered in class.

Useful parameters

• $\sigma_{solid-liquid}=33$ mJ/m$^{2}$
• $\sigma_{liquid-vapor}=76$ mJ/m$^{2}$
• $\sigma_{solid-vapor}=\sigma_{solid-liquid}+\sigma_{liquid-vapor}$
• $e{}_{water}$(T) = [2.8262$\times10^{9}$ - 1.0897$\times10^{6}$T - 94934T$^{2}$ + 582.2T$^{3}$]exp(-5450/T$_{K}$)
• $e{}_{ice}$(T) = [3.6646$\times10^{10}$ - 1.3086$\times10^{6}$T - 33793T$^{2}$]exp(-6150/T$_{K}$)
  where vapor pressures $e$ are in mbar, the temperature T is in degrees Celsius, and T$_{K}$ is in degrees Kelvin. These approximate expressions are accurate to better than 0.1 percent from -50C to 50C.