Aerosol Self Nucleation Why are we interested?  Contribute to natural aerosol concentrations  global warming implications  health implications  serve as sites for the sorption of other gas phase compounds-toxic Usually they are very small  pyrene (gas).0007  m  viruses  m  if condensation nuclei start as clusters of  m molecules

If gases are coming together to form particles or clusters level of gas saturation amount of cluster growth free energy of the system surface tension vapor pressure of the gas molecules

Free energy and surface tension What is surface tension  if a liquid has a meniscus surface we could define a force per unit length,  t, that the liquid surface moves from the flat surface of the liquid  t x l = force  force x distance = work  if the distance is dy  work =  t x l x dy  dy x l has the units of area  work/area =  t = surface tension  the free energy of the meniscus moving from position a to b or dy:   G =  H -T  S ;  H = work + heat   G =  t x  A + heat -T  S

Free energy and surface tension G =  t x dA + heat -T  S often the free energy of just the surface is given as:   G S =  t x  A for a spherical liquid nuclei or small cluster  G S = 4  r 2 x  t for gas molecules forming a small cluster where N l gas molecules -> o o o o the change in total free energy is the change in going from a pure vapor to a system that contains particle embryos  G T = G embryo system - G gas vapor

Free energy and surface tension  G T = G embryo system - G gas vapor let  g = chemical potential of the remaining gas,  l the liquid or embryo system; N T will be the tolal number of starting gas molecules and after embryo formation the N g = # of gas molecules, so, N g = N T - N l where N l the number of liquid embryo molecules   G T =  g x N g +  l N l + 4  r 2 x  t - N T  g  Substituting N T = N g + N l   G T = N l {  l -  g } + 4  r 2 x  t

Free energy and surface tension  G T = N l {  l -  g } + 4  r 2 x  t the number of molecules in a liquid cluster, N l, is the volume of the cluster divided by the volume of one molecule, v l where N l = 4/3  r 3 / v l   G T = 4/3  r 3 / v l {  l -  g } + 4  r 2 x  t  the Gibbs Duhem equation describes the change in chemical potential with vapor pressure  d  = v dp ; since v g >>> v l  d {  l -  g } = v g d P  {  l -  g } = - kT ln P/P o

Free energy and and saturation  {  l -  g } = - kT ln P/P o  define P/P o as the saturation ratio S   G T = 4/3  r 3 / v l {  l -  g } + 4  r 2 x  t   G T = - 4/3  r 3 / v l { kT ln S } + 4  r 2  t  A plot of  G T vs particle diameter for different saturation ratios >1,shows it to go thru a maximum and then fall; this max is called the critical diameter (or radius r c )  differentiating and solving for r c  r c = 2  t v l /(kT ln S);  ln S = 2  t Mw/(RT  r c ); molar units (Kelvin equation) what happens to vapor pressure over a particle as r decreases and why??  ln P/P o = 2  t Mw/(RT  r c );

An expression for cluster #, N l  If we go back to  G T = - 4/3  r 3 c / v l { kT ln S } + 4  r 2  t  and take the derivative with respect to r 3 c, and set this equal to zero, one gets:  4  r 2 c / v l { kT ln S }= 8  r  t  mulyiplying both sides by r/3 we get something that looks like the cluster # N l where N l = 4/3  r 3 / v l since r c = 2  t v l /(kT ln S)  substituting we obtain a valve for N l, the number of molecules in a cluster with a radius of r c and as function of saturation

Estimate cluster r c and the cluster #, N l  substituting molar values in the N l expression one obtains:  r c = 2  t Mw/(RT  ln S );  critical #s (N l ) and r c for 3 organics  saturation ratio 2345  acetone (# N l ) (r c in nm)  benzene (# N l ) (r c in nm)  styrene (# N l ) (r c in nm)