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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3
244p
l v pl
RG g g R
Energy of converting vapor molecules to bulk liquid molecules
Energy required to maintain a liquid-gas surface boundary
Xv Xl
Kelvin Effect: Energy of Droplet Formation
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Rp*
Radius at which G maximizes and beyond which droplet formation becomes possible
S<1
S>1
G
S = ratio of eq. vapor pressure around a droplet relative to to above a flat surface
For a droplet to exist, S>1.pcurved > pflat
alwaysHow does S vary with Rp?
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2 2exp expcurved
lflat
B p p
vp MSp k TR RTR
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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Questions1. 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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Rp
Rp
Free Molecular orNon-continuum Regime Continuum Regime
Transition Regime
Continuum versus Free Molecular Dynamics
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Mean Free Path
In 1 second: red has swept through a volume 2redr
For N molecules per cm3, red-blue collisions per sec ~ redN
The distance traveled between collisions:1
redN
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0.05 0.1 0.15 0.2 0.25 0.30
2
4
6
8
10
12
Mean Free Path (microns)
Alti
tude
(km
)
Mean Free Path Increases with Altitude
For 10 nm particle: Kn <<1
For 1 um particle: Kn >1
For 0.2 um particledepends on altitude…
Free Molecular
Continuum
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 atmosphere’s viscosity depend on pressure? Temperature?
Does the diffusivity of air depend on pressure? Temperature?
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Terminal Settling VelocityAfter induction time , drag will balance gravity
Velocity that results from this balance is the settling (or terminal) velocity
v = mpgCc/3pimuDp < 0.01 sec for all atmospheric aerosols
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Particle MobilityWe 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
V = Fext/3pimuDp
Fext = 3pimuDpv
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 Kninf: random walks are more the norm
Mean square displacement proportional to time
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Particle Diffusivity
D = kBTCc/3pimuDp
Mean square displacement in 1-D, by diffusion: <x2> = 2Dt
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Gravitational Settling vs. DiffusionIn 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?