(AEROSOL) AND CLOUD PHYSICS Exercise Sheet 2

Problem 1: Homogeneous Nucleation Homogeneous nucleation is based on the idea that random fluctu- ations that one finds at a given humidity, and which are described by statistical mechanics of the water-vapor, will produce super-critical cluster, or embryos, that then grow because es(a) decreases as the droplet em- bryos grow in size. The nucleation rate of a cluster of size a is given as ! 4πa2 4πa2σ J = √ eZn exp − (1) 2πmkT 3kT where k is the Boltzmann constant, e the vapor pressure, Z ≈ 0.05 the Zeldovich, or non-equilibrium factor, and n the number density of vapor molecules, m the molecular mass.

1. Show that the units of J take the form of particles per volume per unit time.

2. By rewriting Kelvin’s relation in terms of rc the size of a drop in equilibrium at a given supersaturation, derive an expression for the nucleation rate as a function of S. For this calculation note that the surface tension of water is σ = 75.6 − 0.14(T − 273.15) mN m−1

3. If we define the critical supersaturation as that which produces a droplet nucleation rate of 1 cm s−1 calculate the critical supersaturation ratio as a function of Temperature. For this you need the su

Problem 1: Kohler¨ Equation The Kohler¨ equation described the equilibrium supersaturation of a solu- tion drop, with a certain mass and type of solute as a function of the solution drop radius.

Species ν aw (Molality = 0.3) aw (Molality = 3) NaCl 2 0.9901 08932 (NH4)2SO4. 3 0.9886 0.9022 Table 1: Disassociativity and water activity for common CCN at different molality

1. From the table above calculate the practical osmotic coefficient given the water activity for a solution drop with the stated molality.

2. From the Kohler¨ equation calculate, and plot, the supersaturation ratio S as a function of the droplet radius a for a solution that forms (i) an ammonium sulfate nucleus of mass 10−16 g; (ii) a sea-salt (NaCl) nucleus of mass 10−16 g; (iii) a sea-salt (NaCl) nucleus of mass 10−18 g. For this you need to know the practical Osmotic Coefficient, and disassociation factors, for NaCl and (NH4)2SO4. Take appropriate values given your answer to the first question.

3. Using the Kohler¨ equation calculate the equilibrium size each of these particles would have at 80 % and 90% relative humidity. This increase in the aerosol size as it takes up water from the humid environment is called deliquescence. Deliquescence of the aerosol is why humid skies are often hazier, as the light scatters more efficiently as the particulate matter in the atmosphere swells with relative humidity.

1 4. The critical supersaturation, S∗, and the critical radius, r∗ are the maximum equilibrium supersatu- ration, and the radius at which it occurs, as given by the Kohler¨ equation for a solute of a given type ∗ ∗ and mass. Derive an analytic formula for S and r as a function of A(T ) and Bs and solve for the critical supersaturations and critical radii of the particles in the first part of this question.

5. The equilibrium supersaturation defined by the Kohler¨ curve is unstable for a > r∗ and stable for a < r∗. Prove this fact and explain what is meant by it.

Problem 3: Droplet Growth

1. Given the simplified equation of droplet growth presented in lecture, solve for the growth of a droplet (with time) assuming that the supersaturation field is held fixed at a value 10% larger than the critical supersaturation for a sea-salt particle of mass 10−16 (i.e., S = 1.1S∗). Repeat this calculation (but do not change the value of the supersaturation field, for a sea-salt particle whose initial mass is 100 times larger, and for a sea-salt particle particle whose initial mass is 100 times smaller.

2. Prove that, for large enough drops so that solution and solute effects can be neglected, that the drop q 2 diameter increases in time as D(t) = D0 + 2ζt. Where ζ is a thermodynamic factor. 3. From the above show that the difference between the surface area of two droplets growing in the same environment is constant; but that their difference in mass increases with time

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