This document shows how to calculate the equilibrium activity products of
amine (i.e., aminium) salts in saturated aqueous solution. They are needed to
predict the formation of the solid, and should be entered into *E-AIM*
on the data entry page of the amine. We use as an example the nitrate salt of diethylamine
(DEA), which is one of the amines that can be selected from the
Available Compounds
page.

The formation of the solid nitrate salt is described by:

(DEA^{+})NO3(s) = DEA^{+}(aq) +
NO3^{−}(aq) (1)

The activity product *K*S in molality based units is given by:

*K*S =
*m*DEA^{+} ·
γDEA^{+}) ·
*m*NO3^{−} ·
γNO3^{−} (2)

where prefix *m* indicates molality, and γ the molality-based activity coefficient. We require
the values of these quantities for a solution that is saturated with respect to the salt. The
solubility of diethylaminium nitrate – i.e., the molalities in equation (2) above –
are listed in Table 5 of Ge *et al.* (2011) for a series of temperatures from 0 °C to 50 °C. These
molalities, *m*, are given in the second column of the table below.

T | m |
f |
xH_{2}O |
KS |

263.15 | 8.37 | 0.2294 | 0.7683 | 2.176 |

273.15 | 15.27 | 0.2170 | 0.6451 | 4.569 |

293.15 | 26.85 | 0.2462 | 0.5083 | 11.29 |

298.15 | 33.5 | 0.2522 | 0.4531 | 14.65 |

313.15 | 41.67 | 0.2801 | 0.3998 | 21.77 |

323.15 | 66.18 | 0.2899 | 0.2955 | 32.14 |

We need to determine, using
*E-AIM*, the molal activity coefficients γ corresponding to each of these solubilities so that *K*S
can be calculated. This is done in the following steps:

- Select diethylamine on the Available Compounds page so
that it is included in
*E-AIM*calculations. - Go to the Aqueous Solution page for Model II. You will
see that, below the boxes for entering the molalities of the inorganic ions, there is now
a section in which: (i) the
molality of diethylamine can be entered; (ii) the compound can be constrained to exist
in one of two possible liquid phases; (iii) the dissociation reaction of the amine cation
can be switched off; and (iv) the UNIFAC parameter set (for interactions between water and
the neutral amine molecule) can be chosen.
For this calculation select "Aqueous only" under (2), and leave amine dissociation "on" and the standard UNIFAC parameter set selected.

- Carry out a calculation for the saturated aqueous solution at 298.15 K,
in which the molality of (DEA
^{+})NO3 is 33.5 mol kg^{−1}. There is no entry box on the*E-AIM*Model II page for the DEA^{+}ion, only for the uncharged molecule diethylamine. We therefore enter 33.5 mol kg^{−1}of diethylamine, and 33.5 mol kg^{−1}each of H^{+}and NO3^{−}which is the amount of nitric acid required to neutralise the amine to (DEA^{+})NO3.Do the calculation by pressing the "Submit" button, to obtain the following results:

Species Moles Grams Molality Mole Frac. Act. Coeff. Act. Eqn. H(aq) 0.20976E-05 0.2114E-05 0.2098E-05 0.1712E-07 0.2776E+02 PSC DEA+(aq) 0.33500E+02 0.2484E+04 0.3350E+02 0.2734E+00 0.2522E+00 PSC NO3(aq) 0.33500E+02 0.2077E+04 0.3350E+02 0.2735E+00 0.2522E+00 PSC OH(aq) 0.15267E-12 0.2597E-11 0.1527E-12 0.1246E-14 0.3061E+04 PSC H2O(aq) 0.55509E+02 0.1000E+04 0.5551E+02 0.4531E+00 0.1230E+01 DEA(aq) 0.20976E-05 0.1534E-03 0.2098E-05 0.1712E-07 0.2207E+01 UNIFAC

The molalities of the DEA

^{+}and NO3^{−}ions are equal to the 33.5 mol kg^{−1}that was input (because the molality of the uncharged DEA is very small, at 0.2098 × 10^{−5}mol kg^{−1}). The activity coefficients of the ions are both 0.2552. Remember that*E-AIM*lists mole fraction values of activity coefficients (*f*), and these must be converted to a molality basis (γ) for use in equation (2). This is done by multiplying each activity coefficient by the mole fraction of water in the solution (0.4531 from the results above), so that:γDEA

^{+}=*f*DEA^{+}×*x*H2O= 0.2522 × 0.4531 = 0.11427

The analogous relationship applies for NO3

^{−}, also yielding γNO3^{−}= 0.11427.The value of

*K*S at 25 °C can now be calculated from equation (2):*K*S(25 °C) = (33.5 × 0.11427) × (33.3 × 0.11427) = 14.654 mol^{2}kg^{−2} - Repeat this procedure to determine
*K*S from the solubilities at the other temperatures in the table at the top of this page. The mole fraction activity coefficients of both ions (*f*), mole fractions of water (*x*H2O), and*K*S from these calculations are listed there.

The next thing to do is use the values of *K*S we have determined to
obtain best-fit values of *K*S(25 °C), ΔH^{o} and (possibly)
ΔCp for the reaction so that *E-AIM* will be
able to calculate the formation of the salt from aqueous solutions at any temperature. The
equation for *K*S as a function of temperature is:

R · ln(*K*S(T)) = R ·
ln(*K*S(T_{r})) +
ΔH^{o}(T_{r})(1/T_{r} - 1/T) +
ΔC_{p}^{o}(T_{r})
(T_{r}/T - (1 + ln(T_{r}/T))) (3)

where R (8.3144 J mol^{−1} K^{−1}) is the
gas constant and T_{r} is equal to 298.15 K. This equation
should be fitted to the natural logarithms of the tabulated *K*S with
ln(*K*S(T_{r})), ΔH^{o}(T_{r}) and
ΔC_{p}^{o}(T_{r}) as unknowns.

Such a fit yields the following result: ln(*K*S(T_{r}))
= 2.655 ± 0.045, ΔH^{o}(T_{r}) = 28.41 ± 1.4 kJ
mol^{−1}, and
ΔC_{p}^{o}(T_{r}) = -320 ± 119
J mol^{−1} K^{−1}. Clearly the uncertainty
in ΔC_{p}^{o}(T_{r}), which is
related to the second differential of ln(*K*S(T)) with respect to T, is
large. The absolute value is also large compared to the -101.2 J mol^{−1} K^{−1}
obtained by Clegg *et al.* (1998) for
NH4NO3(s) (see their Table 2).

Further test fits of the data show that if the measurement for 263.15 K is omitted then
the level of significance of the fitted ΔC_{p}^{o}(T_{r})
is so low that it can be set to zero. Thus, it appears that the data do not extend over a wide enough temperature
range, and/or are too imprecise, to reliably determine ΔC_{p}^{o}(T_{r}).
Such a fit yields
ln(*K*S(T_{r})) = 2.598 ± 0.030,
and ΔH^{o}(T_{r}) = 28.13 ± 1.3 kJ mol^{−1}.
One further fit, fixing
ΔC_{p}^{o}(T_{r}) to -101.2
J mol^{−1} K^{−1} and using all the data, yields
ln(*K*S(T_{r})) = 2.595 ± 0.039 and
ΔH^{o}(T_{r}) = 30.07 ± 1.3 kJ mol^{−1}.
Values of ln(*K*S(T_{r}))
and ΔH^{o}(T_{r})
from both of these fits agree to within the uncertainties in the fitted parameters
and either set of results could be used in the model.

How are the equilibrium constant and thermal properties included in *E-AIM*? On the data entry page for diethylamine, the last
part of section 6 has the heading "Formation of inorganic aminium salts". If "nitrate"
is checked then input boxes for *K*S(T_{r}),
ΔH^{o}(T_{r}) and
ΔC_{p}^{o}(T_{r}) will appear.
Enter the values that have been fitted. There is also a box for the molar volume of the
salt. This can be entered, if it is known, otherwise it can be left blank.

Next, save the data using the button in section 10, selecting "Overwrite existing data" from
the browser page that appears. If you return to one of the *E-AIM* problem pages,
such as the
one used to calculate *K*S, it will be possible to calculate the
formation of the salt. The ability to switch off the formation of the salt will also
have been added (there will be a checkbox for (DEA^{+})NO3(s)
in the Solids section of the page).

The procedure for determining *K*S and the associated thermal properties for chloride
salts of singly dissociating amines is exactly the same as described above for the
nitrate salt, except that HCl is used as the neutralising acid instead of HNO3. For
the sulphate, H2SO4 is used, remembering that the stoichiometry
of the reaction is different:

(DEA^{+})2SO4(s) = 2DEA^{+}(aq) +
SO4^{2−}(aq) (4)

*K*S =
(*m*DEA^{+} ·
γDEA^{+})^{2} ·
*m*SO4^{2−} ·
γSO4^{2−} (5)

Thus, if the solubility of the diethylaminium sulphate salt were 1.0 mol kg^{−1} then
the molalities entered for the calculation of the activity coefficients would be:
2.0 mol kg^{−1} H^{+},
1.0 mol kg^{−1} SO4^{2−},
and 1.0 mol kg^{−1} diethylamine. The conversion
of the activity coefficients from mole fraction to a molality basis, and equation (3),
are exactly the same as for the nitrate and chloride salts.

Diamines form doubly charged cations, so that the stoichiometries of the solubility reactions differ from those given above. For example, for a diaminium nitrate salt we would have:

H3NRNH3(NO3)2(s) =
^{+}H3NRNH3^{+}(aq) +
2NO3^{−}(aq) (6)

*K*S = *a*(^{+}H3NRNH3^{+}) ·
(*a*NO3^{−})^{2}
= *m*(^{+}H3NRNH3^{+}) ·
(*m*NO3^{−})^{2} ·
γ(^{+}H3NRNH3^{+}) ·
(γNO3^{−})^{2} (7)

If the solubility of the salt in water were 10 mol kg^{−1} then,
to calculate *K*S, we
would enter 20.0 mol kg^{−} of H^{+},
20.0 mol kg^{−1} of NO3^{−}
and 10.0 mol kg^{−1} of the
diaminium salt on the *E-AIM* problem page.

- Conversions of activity coefficients and concentrations between different scales: these are described
in Chapter 2 of
*Electrolyte Solutions*(2nd Revised Edition) by R. A. Robinson and R. H. Stokes, Butterworth & Co. (Publishers), London, 1970. - The variation of equilibrium constants with temperature: expressions for this, and the
inter-relationships between equilibrium constants,
ΔH
^{o}(T_{r}), and ΔC_{p}^{o}(T_{r}) are given in most thermodynamic textbooks. For example, see*Chemical and Engineering Thermodynamics*, (2nd Edition), by S. I. Sandler, John Wiley and Sons, 1989. See also equation (19) of Carslaw*et al.*(1995)

References

K. S. Carslaw, S. L. Clegg, and P. Brimblecombe (1995) *J. Phys. Chem.* **99**, 11557-11574.

X. Ge, A. S. Wexler and S. L. Clegg (2011) *Atmos. Environ.* **45**, 561-577.