Kelvin Effect: Physical Cartoon

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Kelvin Effect Physical Cartoon
Equilibrium vapor pressure is higher over a
curved surface than a flat one. Important for
nucleation of new particles, lifetime of small
droplets.
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Kelvin Effect Energy of Droplet Formation
Energy of converting vapor molecules to bulk
liquid molecules
Energy required to maintain a liquid-gas surface
boundary
Xv ? Xl
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S ratio of eq. vapor pressure around a droplet
relative to to above a flat surface
Slt1
Sgt1
?G
For a droplet to exist, Sgt1. pcurved gt pflat
always
Rp
Radius at which ?G maximizes and beyond which
droplet formation becomes possible
How does S vary with Rp?
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Kelvin Equation
Relates molecular properties (molecular weight,
surface tension, density) to the degree to which
v.p. over curved surface is enhanced
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Questions

1. Some organic compounds are highly surface active.
   That is, they prefer to reside at the gas-liquid
   interface, and lead to a lower surface tension.
   By how much would S change if the surface tension
   of a droplet changed from 75 dynes (pure water)
   to 35 dynes (surfactant coated water)?
2. Do you have a physical explanation to the above
   answer?
3. What is the surface tension of a cluster of 10
   H2SO4 molecules and 10 H2O? Is it the same as the
   surface tension for a 50 wt H2SO4 bulk solution?

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Continuum versus Free Molecular Dynamics
Transition Regime
Rp
Rp
Free Molecular or Non-continuum Regime
Continuum Regime
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Mean Free Path
?
In 1 second red has swept through a volume
For N molecules per cm3, red-blue collisions per
sec
The distance traveled between collisions
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Mean Free Path Increases with Altitude
For 10 nm particle Kn ltlt1
Free Molecular
For 1 um particle Kn gt1
Continuum
For 0.2 um particle depends on altitude
Continuum to transition
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Question
What is a physical explanation for the mean free
path being related to diffusivity and gas
viscosity? Does the atmospheres viscosity
depend on pressure? Temperature? Does the
diffusivity of air depend on pressure?
Temperature?
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Terminal Settling Velocity
After induction time ?, drag will balance
gravity Velocity that results from this balance
is the settling (or terminal) velocity
v mpgCc/3pimuDp

• lt 0.01 sec for all
• atmospheric aerosols

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Particle Mobility
We can generalize the gravitational settling
problem.
A particle experiencing an external force will
accelerate until its velocity leads to a drag
force opposing the external force. A
steady-state terminal velocity can be found by
balancing the external forces with the drag force
Fext 3pimuDpv
V Fext/3pimuDp
Fext gravity, electric field, random collisions
by fluid
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Brownian Motion
Collisions between particle and randomly moving
molecules leads to irregular-jerky particle
motion
As Kn?inf random walks are more the norm
Mean square displacement proportional to time
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Particle Diffusivity
Mean square displacement in 1-D, by diffusion
ltx2gt 2Dt
D kBTCc/3pimuDp
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Gravitational Settling vs. Diffusion
In 1 second, how far does a 1 micron particle
move due to gravity? How far does it move by
diffusion? What about for a 0.1 micron
particle? What do you conclude about the
relative importance of gravitation settling for
big and small particles?