Inputs for Köhler Calculations

The data required for the calculations are described below. These data are: the inorganic composition; the amount of any organic present and its physico-chemical properties (optional); and whether inorganic solids are allowed to form when the particle becomes saturated with respect to one or more of them. See note (1) at the bottom of this page regarding E-AIM Model I.


Enter an absolute temperature (Kelvin) between the specified limits, according to the composition of the solution. These limits differ according to the model chosen:

And for Model IV:

Temperatures below 273.15 K: the particle compositions for which Köhler calculations are carried out include extreme dilution with respect to the dissolved solutes. The formation of ice in the model has been switched off, and no attempt is made to trap input temperatures that are unrealistically low. Users should be aware that: (i) the themodynamic model is poorly constrained by data for dilute solutions at low temperatures, and model calculations for some systems may therefore include extrapolations well beyond the available data. The papers describing the model should be consulted for details. (ii) Ice appears to nucleate spontaneously at about 235 K (even in small particles), so entering temperatures below this is probably unrealistic.

Inorganic Particle Composition

This can be entered in several ways, selected using the buttons in section (1) of the calculation page:

For convenience, the table below lists the amounts (in mole and mass units) present in particles of different radii containing three typical salts. Note that, in the headers, prefix 'n' (nano) means ×10−9, and prefix 'p' (pico) means ×10−12.

Salt Particle
Radius (nm)
      Mass (pg)       Moles (pmol)       Salt Particle
Radius (nm)
      Mass (pg)       Moles (pmol)
NaCl 10       9.09 × 10−6       1.56 × 10−7          NaCl 50       1.14 × 10−3       1.94 × 10−5
(NH4)2SO4 10       7.41 × 10−6       5.61 × 10−8          (NH4)2SO4 50       9.27 × 10−4       7.01 × 10−6
Na2SO4 10       1.13 × 10−5       7.96 × 10−8          Na2SO4 50       1.41 × 10−3       9.95 × 10−6
NaCl 20       7.27 × 10−5       1.24 × 10−6          NaCl 100       9.09 × 10−3       1.56 × 10−4
(NH4)2SO4 20       5.93 × 10−5       4.49 × 10−7          (NH4)2SO4 100       7.41 × 10−3       5.61 × 10−5
Na2SO4 20       9.05 × 10−5       6.37 × 10−7          Na2SO4 100       0.0113       7.96 × 10−5

Organic Particle Composition

The organic component of the particle, if any, is entered as a single additional compound, in the same ways as described for the inorganic composition above. Input boxes for moles, mass, or volume fraction will be displayed if either the "Organic compound" or "Organic acid" button is selected.

For all of the selections in section (1) of the calculation page, except the final one (for κ-Köhler calculations) there are several other input boxes. These enable the physico-chemical properties of the organic compound to be specified, and are needed because the calculations on this page are not linked to the organic compound library used by E-AIM Models I–IV. The meanings and usage of the input boxes are as follows:

The remaining input boxes, starting with "Solubility", describe the formation of the solid organic compound in the particles (the equilibrium between the solid, or solid hydrate, and the dissolved organic compound). If the solubility field is left blank the organic compound will be treated as miscible with water in all proportions. The meanings of the input boxes are as follows:

Finally, when the form of composition input chosen in section (1) of the calculation page is for a κ-Köhler calculation (this requires volume fractions and κ values) there is no input boxes for organic properties because these are all implicit in the κ parameter. Consequently, non-dissociating organic compounds and organic acids are not distinguished from one another.

Thermodynamic Calculations and the Organic Compound

Activity coefficients of the dissolved organic species, and its influence on the water activity (aW) of the particle and the saturation S, are calculated by assuming the mixtures of the uncharged organic species (i.e., the undissociated form if the compound is an acid) and water obey Raoult's Law. The activity coefficients of organic anions, which will also be present if the compound is an acid, are treated as described in section 3 of the chemical systems page. The E-AIM model calculates internally the equilibrium constant for the reaction between the dissolved organic compound and the solid, if a value of the solubility has been specified. In this calculation the generally small influence of dissociation is not taken into account, for simplicity. (Dissociation into organic anion(s) will slightly reduce the concentration of the uncharged form of the compound relative to the value entered in the solubility box.)

Formation of Solids in the Particles at Different Values of S

At low values of the saturation water vapour pressure, S, the concentrations of the dissolved solutes within the particles are high. Under these conditions, the equilibrium state of the particle may be that one or more solids are present in addition to a liquid phase.

The calculations that use the κ-Köhler theory and equations assume that the particle is fully liquid under all conditions. However, where E-AIM is used to calculate the state of the particles, three choices are offered:

  1. No solids are allowed to form, and the particle is always fully liquid. In this case the particle may be supersaturated with respect to one or more solids at some values of S.
  2. All solids are allowed to form. In this case E-AIM will calculate the equilibrium state of the particle, including the formation of any possible solid(s). This corresponds to the default choice when the models are run from the principal pages ("Simple" or "Comprehensive" calculations, for example).
  3. "Select solids". Here all of the solids are allowed to form by default, but the user is able to exclude individual solids (as many as necessary) using the checkboxes that are displayed. This allows different assumptions to be explored regarding which solid(s) actually form in the particles, and the effect on its size and equilibrium value of S.

If an organic compound is included in the chamical system, then its solid will be allowed to form if a value is entered for its solubility, as described above.

Table 1: Densities of Inorganic Solid Salts and Acidsa (All Models)

Solid ρ (density)   Molar vol. Solid ρ (density)   Molar vol.
H2SO4a1.826 58.39            Na2SO4 2.70 52.61
HNO3a 1.503 53.70            Na2SO4·10H2O 1.46 220.68
HCla 0.8696 41.93            Na3H(SO4)2 2.57 102.01
H2SO4·H2O 1.99 58.28            NaHSO4·H2O 2.10 65.75
H2SO4·2H2O 1.75 76.66            NaHSO4 2.43 49.41
H2SO4·3H2O 1.65 92.40            NaH3(SO4)2·H2O 2.11 111.95
H2SO4·4H2O 1.57 108.10            Na2SO4·(NH4)2SO4·4H2O 2.23 155.57
H2SO4·6.5H2O 1.43 150.37            NaNO3 2.26 37.59
HNO3·H2O 1.80 44.91            NaNO3·Na2SO4·H2O 2.30 106.54
HNO3·3H2O 1.65 70.87            NaCl 2.17 26.93
(NH4)2SO4 1.77 74.66            HCl·3H2O 1.16 77.85
(NH4)3H(SO4)2 1.77 139.32            HNO3·2H2O 1.71 57.89
NH4HSO4 1.78 64.67            NaCl·2H2O 1.77 58.39
NH4NO3 1.72 46.54            
2NH4NO3·(NH4)2SO4 1.74 167.73            
3NH4NO3·(NH4)2SO4 1.74 214.27            
NH4NO3·NH4HSO4 1.75 111.20            
NH4Cl 1.52 35.22            

Units: ρ – g cm−3; Molar volume – cm3 mol−1. aThese are values for the pure liquid acids. See text under the heading "Particle Composition", above.


1.  Köhler calculations using Model I are not included on the website, because it is for mixtures of acids only. With the exception of H2SO4, all of the acids are volatile and might be expected to escape from small particles. However, Model II can be used for calculations that include H2SO4, and HNO3, if required.

2.   The second dissociation constants of di-acids are often tabulated in the literature as "step-wise" values. In such cases the first dissociation is as given in the equation for Kd1 above, but the second dissociation for H2Y, with equilibrium constant K2, is written as: HY = H+ + Y2−. The value of Kd2, which is the quantity that must be entered on the form, is equal to Kd1 × K2.