Due Feb 5

1. A common occurence in the atmosphere is the deposition of water vapor on salt particles at moderate relative humidities, forming a film of water around the salt core. It can be shown that as the humidity rises, there will be a deliquescent point (dependent on the salt but around 70 to 80% RH) where the salt will disolve creating a haze particle.

   Consider the evolution of such a droplet at fixed $T$ and $e$. Let the droplet have density $\rho_{s}$, radius $a$ and surface tension $\sigma_{sv}$. Let the concentration of salt molecules in solution (assume the solution is ideal) be $x=n_{s}/n_{w}$ where $n_{s}$ and $n_{w}$ are the numer of salt and water molecules, respectively. Let $m_{s}$ be the mass of the salt nucleus and $M_{s}$ and $M_{w}$ be the molecular weights of salt and water.

   Assume
   

   $$\mu\left(T,a_{w}\right)=\mu_{water}\left(T\right)+kT\ln a_{w}$$
   
   
   where $a_{w}=1/\left(1+ix\right)$
   1. Show that in a dilute solution $x\sim B/a^{3}$, where $B$ depends on $i$, $a$, $\rho_{s}$, the salt mass and $M_{s}$ and $M_{w}$, and write $\mu_{soln}\left(T,m_{s}\right)$ in terms of $T$, $e_{sat}^{l}\left(T\right)$ and these variables assuming the solution is dilute.
   2. Find $G$, the free energy in a system containing $N-n$ vapor molecules and $n$ molecules of water in a salt solution drop of radius $a$, and activity $a_{w}$.
   3. Find the relationship between $a^{*}$, the equilibrium droplet radius and the supersaturation $S\equiv e/e_{sat}^{l}\left(T\right)-1$.
   4. Find the equilibrium vapor pressure over the drop, $e_{eq}\left(T,m_{s},a^{*}\right)$ in terms of $e_{sat}^{l}\left(T\right)$ and the other parameters of the problem.
   5. Sketch the curves (the Kohler curves) $e_{eq}\left(T,m_{s},a^{*}\right)$ vs. $a^{*}$ for constant $T$ and several values of $m_{s}$.
   6. Find the values of $S=S_{crit}\equiv e_{eq}\left(T,m_{s},a^{*}\right)/e_{sat}^{l}\left(T\right)-1$ and $a^{*}=a_{crit}^{*}$ at the maximum point of the curve, (where $dS/da^{*}=0$). Is the equilibrium at this point stable or unstable? Why?
2. Read and briefly summarize the main findings of Hobbs and Rangno (1985) and Choi et al. (2005) for discussion Wednesday Feb 5.