EAS 6140 - Thermodynamics of Atmospheres and Oceans
Phase Nucleation
1. Nucleation of a pure phase of one component is referred to as
a) homogeneous
b) heterogeneous
2. Most of the nucleation processes in the atmosphere and ocean occur via
a) homogeneous nucleation
b) heterogeneous nucleation
Surface tension work
3. Write an expression for surface tension work
dWst   dA
4. What are the units for surface tension?
J / m2
5a. The effect of surface tension on the internal energy of a droplet is
(greater than, less than, the same) for a smaller drop.
5b. At the critical point, the surface tension is equal to zero (true, false)
6a How much work is required to break a 1 cm cube of water into drops with radius 10
m? (use surface tension 0.076 N m-1.)
dW=9.55e-11J for one water drop. For entire volume of 2.387x108 drops, dW=.0228J
6b. An issue of considerable environmental concern is contaminants in the ocean. The spreading
of a drop of insoluble oil on a water surface involves the buoyancy of oil on water as well as the
surface interaction between the three substances: oil, water, and air.
The rim of an oil lens is pulled outward if the air-water surface tension is larger than the sum of
the air-oil and oil-water surface tensions. The spreading of an oil slick thus depends on the sign
of the spreading coefficient, Sc
Sc = (air-water) – (air-oil) – (oil-water)
If Sc is positive, the oil will spread until it becomes distributed as a thin film on the water. If Sc
is negative, then the density difference between oil and water may still push the rim of the oil
patch outward, but the surface tension pulls it inward, and the oil will tend to form lens-shaped
globules. Compare the spreading of olive oil and paraffin oil on seawater at a temperature of
20°C. The surface tension between seawater and air can be evaluated using the expression in 1.
The surface tensions of the two types of oil at 20°C are
Surface tension (x 10-3 N m-1)
air-oil
oil-water
Olive Oil
Paraffin Oil
32.0
26.5
20.6
48.4
Scoo  80.485 103 N / m  32.0 103 N / m  20.6 10 3 N / m
 27.885  103 N / m
Sc po  80.485 103 N / m  26.5 10 3 N / m  48.4 10 3 N / m
 5.585  103 N / m
7. Surface tension work on a droplet is proportional to the
a) surface area of the drop
b) volume of the drop
8. The latent heat released in the growth of a droplet is proportional to the
a) surface area of the drop
b) volume of the drop
9. The critical radius for homogeneous nucleation of a drop occurs when
a) the surface tension work exceeds the latent heat release
b) the surface tension work and latent heat release are in balance
c) the latent heat release exceeds the surface tension work
10 Write an expression for the surface area of a drop.
A  4 r 2
11. Combine #10 with #3 to write a differential expression for surface tension work in
terms of dr.
dWst   lv 8 rdr
Nucleation of cloud drops
12. Write the first and second laws combined in Gibbs function form.
dG   dT  Vdp  pdV
13. Augment the Gibbs function equation in #12 by adding a term for surface tension
work.
dG   dT  Vdp   dA
14. Augment the Gibbs function equation in #13 by adding chemical potential terms
associated with a change of phase between liquid and vapor (see equation 4.7; you shoul
obtain (5.8)
dG   dT  Vdp   lv dA  l dnl  v dnv
15. After numerous steps and integration of #14, the following expression is obtained for
nucleation of a liquid water drop from the vapor:
G = 4  r 2  lv– 4  r 3  l Rv T ln S
3
(5.13)
where S = es(r)/es is the saturation ratio and es is the saturated vapor pressure over a plane
surface (4.21). This relationship is plotted in Fig. 5.2. With increasing values of the saturation
ratio, the critical radius r* of the embryo drop (increases, decreases, remains the same).
16. Refer to Fig. 5.3, which is a graph of Kelvin’s equation
r* =
2  lv
l R v T ln S
(5.14a)
a) Has a drop with radius 1 x 10-3 m and S=1.5 been activated (i.e. will it grow
spontaneously)? No
a) Has a drop with radius 1 x 10-3 m and S=4 been activated (i.e. will it grow
spontaneously)? Yes
17. The formation of pure water droplets requires a vapor pressure that is (less than,
equal to, greater than) the saturation vapor pressure over a plane surface of pure water
18 .If the relative humidity of the air is 100%, droplets of pure water will
a) evaporate b) grow further by condensation c) remain the same size
19. Are values of S=1.5 and S=4 observed in the atmosphere? What is a realistic
maximum value of S that is observed in the atmosphere? No, a realistic value is about
1.01 or 1%
20. Aerosol particles that attract water are called
a) hydrophobic
b) hydrophilic
21. Cloud condensation nuclei (CCN) are aerosol particles that nucleate water drops at
supersaturations less than
a) 100%
b) 10%
c) 1%
22. Which of these aerosol particles are likely to act as cloud condensation nuclei (circle
all that apply)
a) clay
b) NaCl
c) (NH4)2SO4
d) AgI
e) pollen
f) sand
23. The saturation vapor pressure over a solution is (less than, equal to, greater than) the
vapor pressure over pure water.
24. The saturation vapor pressure over an electrolytic solution is (less than, equal to,
greater than) the saturation vapor pressure over a non-electrolytic solution.
Combination of Kelvin’s Law (#16, pure droplets with surface tension effects) with
Raoult’s Law (4.48) for solutions yields Kohler’s equation:
e s r,m solt
= 1  b3 exp a r
es
r
(5.17)
where a = 2lv /(l Rv T). Use the Kohler curve below to estimate:
25. The radius of the droplet that will be in equilibrium on a NaCl particle of mass 10-15
g in air which is 0.1% supersaturated.  .5 m or 5x10-5cm
To get this result, follow curve for 10-15 gm and intersect this with a saturation ratio of
1.001. Follow this point directly down to determine the radius ~ 5x10-5 cm
26. The relative humidity of the air adjacent to a droplet of 0.3 microns which contains
10-15 g NaCl.  99.85%
27. The critical supersaturation required for a NaCl particle of mass 10-16 g to grow
beyond the haze state.  1.0035 or 0.35%
28. Consider the droplet with radius r with a CCN consisting of 10-15 g NaCl in an
environment with saturation ratio S, as indicated by point A. Is the droplet growing,
evaporating, or in equilibrium?
29. Consider the droplet with radius r with a CCN consisting of 10-15 g NaCl in an
environment with saturation ratio S, as indicated by point B. Is the droplet growing,
evaporating, or in equilibrium?
Nucleation of the ice phase
30. At approximately what temperature will a droplet of radius 10 m freeze via
homogeneous nucleation? a) 0oC b) -20oC c) -40oC d) -60oC
31. Which of these aerosol particles are likely to act as ice forming nuclei (circle all that
apply)
a) clay
b) NaCl
c) (NH4)2SO4
d) AgI
e) pollen
f) sand
The last two are plausible but not probable
Diffusional growth of droplets
Diffusional growth of water drops
1. Once a drop is activated (has reached its critical radius), it will grow spontaneously by
diffusion. This growth is associated with (circle all that are correct)
a) diffusion of heat towards the drop
b) diffusion of heat away from the drop
c) diffusion of water vapor towards the drop
d) diffusion of water vapor away from the drop
2. The equation for the diffusion of water vapor is
2  v
 v
=
D
t
x 2i
Doing dimensional analysis, determine the units of D.
m2 / s
4
3. The volume of a sphere is _____  r 3 ________________
3
4. The surface area of a sphere is ___ 4 r 2 _______________
5. The droplet growth rate as influenced by both diffusion of water vapor and heat is
dr
r  =
dt
S1
2
L lv  l
 R vT
2
+
l R v T
1
= S
K+D
es T D
Are curvature and solute effects included in this equation? Why or why not?
No, their impact is negligible once the drop size has increased past a few microns
6. Assume that S, K, D in #5 are constant. Integrate #5 from an initial drop size ro to
find r(t).
r
1 2 (t )  S  1 
r

t
2 r0  K  D 
1
 S 1  1 2
 S 1 
2
2
 r 2 (t )  
 t  r0  r (t )  2 
 t  r0
2
K D 2
K D
7. Large cloud droplets will increase their radius (faster, slower, at the same rate) by
diffusion than small cloud drops
8. Consider Table 5.5. How long does it typically take to grow a drop to 50 microns
radius by diffusion? Is it realistic to expect rain drops to form by diffusional growth?
Between 41,500 and 44,500 seconds, this is not realistic for rain drops
9. Consider a cloud with a concentration of 500 drops per cm3 volume of air that are
growing by diffusion, in an air parcel initially with S=1.01. As a result of the diffusional
growth, the saturation ratio will (increase, decrease, remain the same).
10. What processes determine how supersaturation varies with time.
the production of supersaturation (via cooling) and condensation
11. What are some physical mechanisms that would cause relative humidity (and
supersaturation) to increase? Cooling in adiabatic ascent, rising motion
12. Consider the following equation (5.28).
dwl
dS
= A1 u z  A2
dt
dt
In the derivation of the expression of A given in (5.29)- (5.37), circle all of the
thermodynamical “laws” or equations that are used directly in this derivation:
a) ideal gas law
b) Dalton’s law of partial pressures
c) hydrostatic equation’
d) first law of thermodynamics
e) adiabatic lapse rate
f) second law of thermodynamics
g) Clausius-Clapeyron equation
13. Consider Fig. 5.6, which was derived from a simple model that included both
diffusional growth and the saturation ratio equation, in the presence of a steady updraft
and a prescribed spectra of CCN. Above a height of about 10 above cloud base, why
does supersaturation start to decrease?
Diffusional growth leads to decreasing supersaturation
14. Discuss why saturation ratios in clouds are rarely observed to exceed 1.01.
Supersaturation comes into balance with diffusion growth
15. Assuming that droplets are activated at different values of r* associated with
different sizes of aerosol particles, a spectrum of drop of drop sizes will result. As a
result of diffusional growth of this spectrum of drops, the spread in drop sizes will
(increase, decrease, remain the same).
17. The spread of observed drop size spectra are observed to (increase, decrease, remain
the same).
18. List two processes that may contribute to the broadening of drop size spectra.
Giant particles acting as embyos for large drops, small-scale turbulence,
entrainment of dry air
Diffusional growth of ice crystals
19. What ice crystal habit would you expect to have at -20oC and a relative humidity
(with respect to water) of 95%? Solid to Skeleton Thin Plates
20. Ice crystals of non-spherical shape grow by diffusion (faster than, slower than, at
the same rate as) spherical ice crystals
21. Refer to Table 4.4. At a temperature of –20oC and RH=100%, what is the saturation
ratio
a) with respect to water
1
b) with respect to ice
1.22
22. At relative humidity of 101% with respect to liquid water, which particle will grow
faster, a water drop or an ice crystal? Ice crystal
23. If water drops and ice crystals exist in the same air parcel and the temperature is
slightly below freezing, what will be the final equilibrium state? The water drops will
change into ice crystals