Dilute System Partial Pressure of Solute 0
Abstract
Raoult's law, published in 1887, is taught in chemistry and chemical engineering fields as a first approximation to the vapor pressure and activity of solutes and solvents in mixtures. In ideal solutions, it is exact but many solutions are known to have substantial deviations from Raoult's law as conventionally interpreted. In 1908, Callendar showed that water hydrated to the solute can explain some of the departures from Raoult's law in aqueous solution. Here, we show that by simply assuming equilibria between the free water in solution and its hydrated forms, Raoult's law and Callendar's extension are valid over the full range of concentrations, while also showing how water and solutes interact in solution. This model of solutions has importance in highly concentrated solutions common in atmosphere aerosols relevant to climate change and air quality, and in numerous industrial processes.
Introduction
Highly concentrated solutions occur in a wide range of industrial and natural processes including brines, carbon capture, atmospheric aerosols, and food drying. These solutions are typically labeled as exhibiting nonideal behavior because the vapor pressures of the solute and solvent and their corresponding activities in solution are not proportional to their mole fraction. Raoult first proposed this linear mole fraction–vapor pressure relationship in 18875 and Raoult's law has become a paradigm in chemistry and chemical engineering. For dilute solutions, this relation holds but deviations grow as the solution becomes more concentrated. In 1908, Callendar6 explained some of the deviations from Raoult's model by positing that in aqueous solutions, solute molecules are hydrated such that some of the water is bound to the solute, so it does not contribute to the vapor pressure. That is, only the so-called "free water" contributes to the vapor pressure. This paradigm extended the validity of Raoult's law to higher concentrations but it still failed at yet higher concentrations. In 1973, Stokes and Robinson7 extended this paradigm further by assuming that for electrolytes there is an equilibrium between the free and bound water and a power law relationship between the equilibrium constants. As the concentration of the solution increases, the water activity decreases and some of the bound water falls off the solute and becomes free water.
Starting in 2011, we published a series of papers that used statistical mechanical techniques to describe solutions over the full range from the infinitely dilute limit to the pure solute limit.8−10 These models were successful at describing solutions of organics and electrolytes in water by assuming that one water was bound to the solute or ion, the next water bound to that water, and so on. Although a huge step forward, this model has a shortcoming of not necessarily capturing hydration the way we understand it. From the prior work of Callendar,6 Stokes and Robinson,7 and many others, we know that multiple waters can bind to solute molecules. They may also bind to each other in a secondary layer, but the single stacked water model of Dutcher and colleagues does not capture what is known from the physics and chemistry of these solutions. For instance, raffinose is thought to have a hydration number (the average number of water molecules bound to the solute molecule in dilute solutions) in excess of 10 but we do not expect 10 waters to be stacked on top of each other to hydrate raffinose.
Results and Discussion
Here, we apply two assumptions to derive equations that describe the "nonideality" of solutions to show that they are actually ideal and fit Raoult's law:
Assumption 1: Raoult's law applies rigorously. That is, the ratio of partial pressure to vapor pressure, the activity of all constituents, a i , be they pure solute, hydrated solute, or free water, is simply their mole fraction given by
1
where n i is the number of moles of constituent i in solution and n t is the total number of moles of all of the constituents in solution. These constituents consist of free water, bare solute, and hydrated solute.
Assumption 2: Each solute may be hydrated and this hydration is governed by the following equilibria
2.1
2.2
2.3
...
2.4
where
a w is the activity of water, a ijw is the activity of solute i hydrated with j water molecules, and K ijw is the equilibrium constant between the solute hydrated with j water molecules and (j – 1) water molecules. All activities are given by eq 1. We will now use eqs 1 and 2.1–2.4 to calculate the water associated with each solute.
First, we multiply all of these equilibria in eqs 2.1–2.4 by each other, so that the activities of the hydrated solutes cancel, except for the most hydrated solute, giving
3
Multiply both sides by n t and applying eq 1, we obtain the number moles of each hydrated solute in terms of the number of moles of free solute, water activity, and the equilibrium constants
4
where the number of moles of each constituent, n ijw, contains one mole of solute i and j moles of water.
By adding all of these together plus the number of moles of bare solute, we obtain the total number of moles of solute i in solution
5
where S i is the total number of moles of solute i per mole of unhydrated solute i and is only a function of water activity. Similarly, the total amount of water hydrated to solute i is given by
6
where W i is the total number of moles of water hydrated to solute i per mole of bare solute i and like S i is only a function of water activity. Dividing eq 6 by eq 5 gives the hydration number as a function of water activity
7
That is, H i (a w) is the total number of water molecules hydrated to all of the hydrated forms of solute i per total number of moles of solute i. The hydration number defined by Callendar (1908) is the value of H i at the limit of infinite dilution, H i (a w = 1). Marcus11 used isothermal compressibility to deduce how hydration number depends on concentration fitting a linear curve through his data.
Equation 7 is the number of bound water molecules per solute molecule. We also need to know the number of free waters in solution. Starting with the definition of water activity from eq 1
8
where ∑ i n iT is the total number of all solutes in solution regardless of their hydration state. Remember that n t is the total number of moles of free water, free solute, and hydrated solute in solution. Solving eq 8 for the number of moles of free water gives
9
The term a w/(1 – a w) appears in other derivations of the thermodynamics of solutions, such as that by Dutcher and co-workers.8−10 Here, we see that it represents the number of moles of free water per mole of solute.
Now, we have the tools that we need to derive the relationship between molality and water activity. The simplest case is an aqueous solution containing a single solute A that does not dissociate or associate. Many alcohols and sugars fit this description. The total amount of water, n wTot, in a solution containing solute A, its hydrated forms, and free water is
10
Combining this with eqs 7 and 9 gives
11
Noting that the molality of solute in solution is given by M w m = n AT /n wTot, where M w is the molar mass of water, yields an equation for the molality as a function of water activity
12
The denominator is the amount of free water per total solute plus the amount of water bound in hydrates per total solute. The solute activity is obtained by taking the total number of constituents in solution
13
Dividing both sides by n t and combining with eq 5 to obtain
14
which can also be obtained using the Gibbs–Duhem equation, eq 12, and noting that W A = a wdS A /da w.
The values of the equilibrium constants, K iqw, govern the performance of eqs 12 and 14 via the functions H A (a w) and S A (a w). For each water hydrated to each solute, there is a K iqw so that say for raffinose that has a hydration number of about 12, at least 12 K iqw values are needed. For organic solutes, we expect that roughly one water molecule will hydrogen bond to each of the OH moieties and that the equilibrium constant will be roughly the same for each hydrogen bond. As a result, the K iqw values should be roughly the same until q (the number of hydrated waters) reaches a value, where all of the bonding locations are taken, then it should precipitously drop in value to zero. The logistic function neatly captures this behavior
15
where K iw x is the equilibrium constant for each water hydrogen bonded to the solute, q 0 is the number of bound waters where the value of K iqw is half K iw x , and Δq governs how rapidly K iqw drops to zero as q increases. Since we expect the value of K iqw to drop precipitously to zero when q becomes larger than the number of hydrogen bonding sites, q 0, we set Δq = 0.1, an arbitrary but small value. We now have only two fit parameters, K iw x and q 0, instead of the 12 or more equilibrium constants needed for say raffinose. Figure 1 shows the fit for raffinose. For this and subsequent solutes, osmotic coefficient defined as φ = −ln(a w)/(M w m) is displayed on the y-axis because it is a very sensitive indicator of deviations between activity data and fits to these data. The square root of the solute mole fraction is displayed on the x-axis to better show the behavior at low concentration, which is not relevant for raffinose but is for other more-soluble solutes.
Osmotic coefficient as a function of solute mole fraction for raffinose in water. Freezing point depression data from ref (2). K iw x was fixed at 2, the fit value for q 0 is 13.8, and the resulting hydration number is 12.3.
Due to scatter in the raffinose data, many combinations of K iw x and q 0 fit the data well, so here we picked K iw x = 2 and just fit q 0. For glycerol and NaCl, the values were simple fits to the data. The fits minimized the sum of the squared error in osmotic coefficient between the model and the data.
Figure 2a shows the fit for glycerol that is miscible in water, so data exist for the full range of solute mole fractions. For both raffinose and glycerol, the hydration number deduced from these fits agrees well with the values reported in the literature. Figure 2b shows the change in hydration as a function of water activity for glycerol. The corresponding figure for raffinose and NaCl is not shown because the change over the water activity range is extremely small.
(a) Osmotic coefficient as a function of solute mole fraction for glycerol in water. Data from triangle,3 diamond.4 Parameter values are K iw x = 0.553 and q 0 = 5.1; the resulting hydration number is 1.0. (b) Predicted hydration as a function of water activity for glycerol using the same parameters as in (a).
For two or more solutes in solutions A and B, what kind of mixing rule applies? The total amount of water in solution is now
16
Dividing both sides by n wTot and rearranging gives
17
Comparing this to eq 12, we see that the square-bracketed quantities in the denominator are simply the single-solute molalities; that is, the numerator in each term is the molality in the solution mixture, whereas the denominator is the molality in the single-solute solution. This is the ZSR mixing rule posited independently by Zdanovski12 and Stokes and Robinson,13 which has been shown to be valid for a wide range of solutes that do not associate in solution. It also shows the role that free water a w/(1 – a w) and hydrated water H A play. In the work of Dutcher and colleagues,9 free water was assumed to be associated with each solute. Here, we obtain the same result without assumption.
Electrolytes can be thought of as multicomponent solutions, wherein the salt dissociates into its ionic components and each of these ions comprises a solute in solution. In contrast to mixtures of nonionic solutes, the ions project their electrostatic fields beyond the vicinity of the ion altering the thermodynamics of the solution. This effect was first recognized by Debye and Hückel in 1923.14,15 Pitzer16 evaluated a number of forms for representing these long range forces and found that it is well represented by
18
where I is the ionic strength, Ax = 2.917 at 298 K and ρ is a fit parameter related to the ion size and charge. The factor K w DH modifies the water activity in the equations derived here. Employing the stoichiometry of the electrolyte, the equations derived above and this Debye–Huckel term yield an expression for the molality as a function of water activity for electrolytes
19
where v A and v B are the number of cations and anions in the electrolyte, respectively, assuming that the electrolyte is only comprised of one type of cation and one type of anion. More complex electrolytes are an easy extension. Figure 3 shows the fit for NaCl to saturation.
Osmotic coefficient as a function of solute mole fraction for sodium chloride in water. Data from ref (1). Parameter values are given in the table.
Conclusions
In summary, the theory presented here shows that Raoult's original idea, that the activity of solutes in solution is proportional to the mole fraction, needs no correction if we consider the hydrated forms of the solute in solution as Callendar suggested and employ simple equilibria between the hydrated forms.
Most models of solutes in solution use one of the forms derived by Pitzer17 because they accurately model the molality–activity relationship over a wide range of conditions. For instance, the Clegg and Pitzer formulation is employed by E-AIM, our popular model of aerosol thermodynamics that performs about 100 000 calculations annually for scientists worldwide working in the fields of air pollution and climate change. The shortcoming of the Pitzer family of models is that they do not extrapolate well to concentrations higher than the available data because they are semiempirical. In fact, beyond the physical data, they can give unphysical results. The formulation by Dutcher and co-workers8−10 extrapolates well at concentrations beyond the available data but does not capture conceptually how we understand that solutes hydrate in solution. The model presented here, based on the work published more than 100 years ago, extrapolates well to concentrations higher than the available data, represents hydration of the solute realistically, covers both organic and electrolyte solutes in water, and due to its simplicity shows the role of free and bound water in solution.
Acknowledgments
I would like to thank Ahmad Ikram, a student at Franklin High School, for his help with the calculations and figures and Simon Clegg, University of East Anglia, for his valuable guidance.
Notes
The author declares no competing financial interest.
Dilute System Partial Pressure of Solute 0
Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6682120/