Aerosol Self Nucleation

1. 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 mm • viruses .002 -.06 mm • if condensationnuclei start asclusters of 3-8 .001- .005 mm molecules

2. 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

3. Free energy and surface tension • What is surface tension • if a liquid has a meniscus surface we could define a force per unit length, gt , that the liquid surface moves from the flat surface of the liquid gt x l = force • force x distance = work • if the distance is dy • work = gtxlxdy • dy xl has the units of area • work/area = gt = surface tension • the free energy of the meniscus moving from position a to b or dy: • DG = DH -TDS ; DH = work + heat • DG = gt xDA + heat -TDS

4. Free energy and surface tension • G = g t x dA + heat -TD S • often the free energy of just the surface is given as: • DGS = g t x DA • for a spherical liquid nuclei or small clusterD GS = 4pr2 xg t • for gas molecules forming a small cluster where Nl gas molecules -> oooo • the change in total free energy is the change in going from a pure vapor to a system that contains particle embryos • D GT = Gembryo system - Ggas vapor

5. Free energy and surface tension • D GT = Gembryo system - Ggas vapor • let mg = chemical potential of the remaining gas, ml the liquid or embryo system; NT will be the tolal number of starting gas molecules and after embryo formation the Ng = # of gas molecules, so, Ng = NT - Nlwhere Nl the number of liquid embryo molecules • D GT = mg x Ng+ mlNl+ 4pr2 xg t - NTmg • Substituting NT= Ng + Nl • D GT = Nl {ml- mg} + 4pr2 xg t

6. Free energy and surface tension • D GT = Nl {ml- mg} + 4pr2 xg t • the number of molecules in a liquid cluster, Nl , is the volume of the cluster divided by the volume of one molecule, vl • where Nl = 4/3 p r 3 / vl • D GT = 4/3 p r 3 / vl{ml- mg} + 4pr2 xg t • the Gibbs Duhem equation describes the change in chemical potential with vapor pressure • dm = v dp; since vg>>> vl • d {ml- mg} = vg dP • {ml- mg} = -kT ln P/Po

7. Free energy and and saturation • {ml- mg} = -kT ln P/Po • defineP/Po as the saturation ratio S • D GT = 4/3 p r 3 / vl{ml- mg} + 4pr2 xg t • D GT = -4/3 p r 3 / vl{kT ln S} +4pr2 g t • A plot of D GT 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 rc) • differentiating and solving for rc • rc= 2gt vl/(kT ln S); • ln S = 2gt Mw/(RTr rc); molar units(Kelvin equation) what happens to vapor pressure over a particle as r decreases and why?? • ln P/Po= 2gt Mw/(RTr rc);

8. An expression for cluster #, Nl • If we go back toDGT = -4/3 p r3c/ vl{kT ln S} +4pr2 g t • and take the derivative with respect tor3c, and set this equal to zero, one gets: • 4p r2c/ vl{kT ln S}=8prg t • mulyiplying both sides by r/3 we get something that looks like the cluster #Nl where Nl = 4/3 p r 3 / vlsincerc= 2gt vl /(kT ln S) • substituting we obtain a valve for Nl , the number of molecules in a cluster with a radius of rcand as function of saturation

9. Estimate cluster rc and the cluster #, Nl • substituting molar values in the Nl expression one obtains: • rc = 2gt Mw/(RTr ln S ); • critical #s (Nl) and rc for 3 organics • saturation ratio 2 3 4 5 • acetone (# Nl) 265 67 33 21(rc in nm) 2.0 1.3 1.0 .8 • benzene (# Nl) 706 177 88 56(rc in nm) 3.0 1.8 1.5 1.3 • styrene (# Nl) 1647 413 202 132(rc in nm) 4.2 2.7 2.1 1.8