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:
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.
This can be entered in several ways, selected using the buttons in section (1) of the calculation page:
The acids H2SO4(liq.), HNO3(liq.), and HCl(liq.) have been added as possible "solids", so that the user has the flexibility to specify acid particles. The densities assigned to these components are those of the pure liquids at 298.15 K, e.g., 1.8263 g cm−3 (H2SO4), and 1.5029 g cm−3 (HNO3). Very acidic particles may be volatile (losing HNO3(g) if nitrate is also present, for example). However, in the E-AIM Köhler calculations no partitioning to the gas phase is calculated. The particle always has the same composition as entered by the user.
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 |
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:
HX(aq) = H+(aq) + X−(aq) (Kd1)
and if organic compound is a di-acid, H2Y, then:H2Y(aq) = H+(aq) + HY−(aq) (Kd1)
The equation for the second dissociation of the di-acid, is:H2Y(aq) = 2H+(aq) + Y2−(aq) (Kd2)
See note (2) at the bottom of the page regarding the relationship between Kd2 and the "stepwise" dissociation constants commonly listed in the literature. It is also explained in section 6b of the help page for organic compound properties.
For mono-acids, enter a value for Kd1 only, and leave the box for Kd2 blank. (If both boxes are left blank the compound will be treated as non-dissociating, as if the "Organic compound" button had been selected.)
In the output of the model the undissociated organic compound is named "Org" (for non-acids), "HOrg" (for mono-acids), and "H2Org" (for di-acids). Organic anions, if any, are named "Org-" (mono-acid), and "HOrg-" and "Org2-" (from a di-acid).
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.
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.)
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:
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.
Solid | ρ (density) | Molar vol. | Solid | ρ (density) | Molar vol. | ||
H2SO4a | 1.826 | 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.