Aerosol-cloud interaction Anatoli Bogdan Institute of Physical Chemistry, University of Innsbruck Austria and Department of Physics, University of Helsinki.

Aerosol-cloud interaction

Anatoli Bogdan

Institute of Physical Chemistry University of Innsbruck Austria

andDepartment of Physics University of Helsinki

Finland

Contents

- role of aerosol in cloud formation - ideal gas - vapor pressure and partial vapor pressure - Kelvin equation - hygroscopic aerosol particles - Raoultrsquos law - Kohler curves

Cloud condensation nuclei or CCNs or cloud seeds are small particles (typically 02 microm or 1100 th the size of a cloud droplet) about which cloud droplets coalesce Water requires a non-gaseous surface to make the transition from a vapor to a liquid In the atmosphere this surface presents itself as tiny solid or liquid particles called CCNs When no CCNs are present water vapor can be supercooled below 0degC before droplets spontaneously form

At T gt 0 ordmC the air would have to be supersaturated to ~400 before the droplets could form The concept of cloud condensation nuclei has led to the idea of cloud seeding that tries to encourage rainfall by seeding the air with condensation nuclei It has further been suggested that creating such nuclei could be used for marine cloud brightening a geo-engineering technique

Aerosol pollution over Northern India and Bangladesh - NASA

httpenwikipediaorgwikiCloud_condensation_nuclei

httpearthobservatorynasagovFeaturesAerosols

Warm Clouds

Warm clouds consist entirely of droplets which have been formed by condensation onto aerosol particles Aerosol particles respond to changes in humidity in different ways depending on their chemical composition and solubility Particles which are soluble or hydrophilic take on water as humidity increases and increase in size

Above a certain relative humidity soluble particles will deliquesce ndash the solid particle dissolves in the water it has taken on and becomes a tiny liquid drop but not yet a cloud drop For many soluble salts deliquescence happens at relative humidity around 60 ndash 80 These droplets exist in equilibrium with water vapor in the surrounding air

The growth of such particles with increase in relative humidity is expressed by the Koumlhler equation and is a function of the size and chemical composition of the particle

Prior to the consideration of the Kohler aquation we will firstly consider several important notions needed for the understanding of the matter

General about gases and vapors

Perfect gas

Atmosphere is a mixture of gases Prier to the study of the real atmosphere (a mixture of real gases) we firstly will consider a notion of perfect (ideal) gas Gas under very small pressure (le 1 atm) is a very good approximation of perfect gas

In perfect gas i) the distance between molecules is much larger than the length of free

path of molecules and ii) the interaction between molecules is restricted only to their collisions

which are considered to be similar to that of the hard balls

Thus in the perfect gas the molecules possess only kinetic energy whereas potential energy of interaction between the molecules is absent As a result internal energy E of perfect gas is independent on pressure and volume E ne E (p V) Internal energy is determined only by the kinetic motion of molecules and can be easily changed by the addition (or withdrawing) of heat ie by changing its temperature Thus E =E (T)

Vapor pressure of water

Above the surface of liquid water there always exists some amount of vapor and consequently there is a vapor pressure When a container containing water is open then the number of the escaping molecules is larger than the number of molecules coming back from the vaporphase (Fig 1) In this case water vapor pressure is small and far from saturation

The atmosphere is a mixture of gases including water vapor

When the container is closed then the water vapor pressure above the surface increases (concentration of molecules increases) and therefore the number of molecules coming back increases too (Fig2)

After a while the number of molecules escaping the liquid and those coming back becomes equal Such situation is called by dynamic equilibrium between the escaping and returning molecules (Fig 3) In this case the water vapor pressure over the liquid water is called saturated water pressure

Saturated water vapor pressure is a function of temperature only and independent on the presence of other gases

The temperature dependence is exponential In the case of water vapor the semi empirical dependence reads as

DTTCT

BA

sw ep

ln

where temperature is in Kelvin and A = 7734 B = -7235 C = - 82 D = 0005711

132 Air humidity Amount of water vapor in the air can be expressed by several different ways

Specific humidity Mass of water vapor per unit mass of humid air air

OH

m

m 2

Absolute humidity Mass of water vapor per unit volume of humid air (kgm3) Relative humidity (RH) Ratio of water vapor pressure pw to the saturated water vapor pressure at that temperature multiplied by 100

RH = 100)(

Tp

ppure

ws

w (153)

Saturation ratio S Ratio

S =)( Tp

ppure

ws

w (154)

From the last two definitions we see that RH = S100 ie their physical meanings are almost the same Supersaturation S - 1 gt 0

For the perfect gas a following (experimental) equation of state p = f (T V n) is true

pV= nNAkbT (11) where p is pressure V volume n number of moles NA Avogadro number kb

Boltzmann constant and T temperature in Kelvin Since kb NA = R (gas constant) for the amount of gas of one mole (n = 1) the equation of state (11) can be written as

pV = RT (12) Using simple kinetic theory of gases and Newtonrsquos laws of motion one can show (see for example Understanding Physics (1998) p 249 or Hinds (1982) pp 15-16) that

pV = RT = KE = 3

2Nmmol (13)

where KE is kinetic energy of the molecules composing the perfect gas mmol mass of

molecule N number of molecules and 2 average of square of molecular velocities (see below) The expression (13) means that the kinetic energy KE which is also the internal energy of the perfect gas E is independent on pressure volume or molecular weight and depends only on temperature ie KE = E = E (T)

Equations derived for the perfect gas work also in the case of mixture of gases provided that the individual gases (components of the gaseous mixture) and the mixture itself behave perfectly Mathematically this means that are valid both the equations of state for each component of the gaseous mixture i

piV = niRT (17) and the equation of state for the mixture itself

pV = nRT (18) Daltonrsquos law for a mixture of perfect gases The pressure p exerted by a mixture of the perfect gases is the sum of the partial pressures pi of the gases Mathematically it reads p = sum pi (19)

Physical meaning of partial pressure According to the Daltonrsquos law for the perfect gases the pressure p exerted by a gaseous mixture is the sum of the partial pressures pi of the gases In the case of the perfect gases the partial pressure pi is simply the pressure which the gas would exert if it occupied the container alone at the same temperature But in the case of the real gases at moderate pressures the interaction between molecules of different gases makes the total pressure being slightly different (smaller) from the simple sum of the pressures of the gas components if they were alone Therefore for the real gas the partial pressure pi of the gas component i in the gaseous mixture is defined as pi = Xi p (115) where p is the total pressure of the mixture(Compare (115) with (110)) It follows from Eqs (19) and (115) that p = (X1 + X2 + X3 + ) p = p1 + p2 + p3 + = sum pi (116) ie similar for the mixture of the perfect gases the sum of the partial pressures of real gases is also equal to the total pressure of the mixture (But total pressure of the mixture of the perfect gases is not equal to that of the real gases)

Kelvin equation

ps(r) ps(infin) = exp (2 σwρwRvTr) = exp (ar)r = droplet radius

ps(r) = the actual vapour pressure of droplet of radius r

ps(infin)= the saturation vapour pressure over bulk water

σw = surface tension

ρw= water density

Rv - the universal gas constant T - temperature

ExampleSaturation ratio Critical radius 1 012 μm11 00126 μm2 173 nm

What is going on with soluble aerosol particles for example such which are composed of NaCl sea salt ammonium sulfate etc

Hygroscopy is the ability of a substance to attract water molecules from the atmosphere through either absorption or adsorption

Hygroscopic substances include sugar honey glycerol ethanol methanol sulfuric acid many salts and many other substances

Deliquescent materials (mostly salts) have a strong affinity for water Such materials absorb a large amount of moisture (water vapor) from the atmosphere until it dissolves in the absorbed water and forms an aqueous solution

Deliquescence occurs when the vapour pressure of aqueous solution is smaller than the partial pressure of water in the atmosphere

All soluble salts will deliquesce if the air is sufficiently humid

141 Raoultrsquos law a) Ideal solution The expression (158) shows that the chemical potential of the liquid A in the solution increases with increasing the partial vapor pressure pA and becomes equal to the chemical potential of pure liquid when pA = pA The French chemist F Raoult experimentally found that the ratio of the partial vapor pressure of the component A to its vapor pressure as a pure liquid is approximately equal to the mole fraction XA (see 113) in the solution ie

AA

A Xp

p or pA = XA pA (160)

The Eq (160) is known as the Raoultrsquos law The Raoultrsquos law is valid also for the component B Linear dependence of the partial vapor pressures pA and pB are shown by the dashed lines in Fig 6

Fig 6 Equilibrium partial pressures of the components of ideal and non-ideal binary solution as a function of the mole fraction XA For the real solution relationships between the pA pB and the mole fractions XA XB are not linear as it is in the case of ideal solution where the linear relationships are shown by the straight dashed lines LR and KQ When XA rarr 1 then we have a dilute solution of B in A In this region the Raoultrsquos law pA = XA pA is applied for the component A For the component B the Henryrsquos law pB = KB XB is applied In the region where XA rarr 0 we have the Raoultrsquos law pB = XB pB for B component and the Henryrsquos law pA = KA XA for A component The straight lines LM and KN depict the Henryrsquos law

The solutions which obey the Raoultrsquos law throughout the whole composition range from pure A to pure B are called ideal solutions

The solutions which components are structurally similar obey the Raoultrsquos law very well In Fig 6 the equilibrium partial pressures of the components A and B in the ideal solution are shown by dashed lines of LR and KQ Solution is only ideal if is satisfied for each component

Dissimilar liquids (large difference in the liquid structures) significantly depart from the Raoultrsquos law Nevertheless even for these mixtures the Raoultrsquos law is obeyed closely for the component in excess as it approaches purity ie when XA rarr 1 or XA rarr 0 (Fig 6)

Raoultrsquos law Mathematically for a plane water surface the reduction in vapour pressure due to the presence of a non-volatile solute is expressed

p (infin)ps(infin) = 1 ndash (3νmsMw)(4 πMsρwr3) = 1 - br3

wherep (infin) - the saturation vapour pressure of pure water ps (infin) - the saturation vapour pressure of bulk solution Ms- molecular weight of the solute Ms- mass of the solute Ν- degree of dissociation

Combining the Kelvin equation and the expression from the Raoultrsquos law we can obtain so called Kohler equation

Koumlhler Curve = Kelvin equation + Raoultrsquos law

p(r)ps(infin) = (1 - br3)middotexp(ar) asymp 1 + ar - br3

wherea ~ 33 10-7T [m]b ~ 43 10-6i Msms [m3mol]Ms= molecular mass of salt [kgmol]ms= mass of salt [kg]

The critical radius rc and critical supersaturation Sc are calculated as

rc= (3ba)12 and Sc= (4 a3[27 b])12

httpenwikipediaorgwikiFileKohler_curvespng

Kohler curves show how the critical diameter and criticalsupersaturation are dependent upon the amount of solute

As humidity increases aerosol continues to swell even after vapor saturation is reached Once a critical supersaturation is reached corresponding to the peak of the Koumlhler curve for that particle a particle becomes activated as a cloud droplet Activated particles are no longer in stable equilibrium with the vapor phase but are able to continue to grow by vapor deposition provided that conditions remain supersaturated

Droplet size is determined by the number of particles activated and the amount of water vapor available for condensation which is generally determined by the vertical height of an adiabatic ascent If droplets are able to grow to a sufficient size and the cloud exists for a sufficient length of time droplets will coalesce as they collide with each other through random motion gravitational settling or motion within the dynamics of the cloud system