5. Desarrollo del trabajo
5.7. Políticas contables Cooperativa COMUNA
In the preceding section, we showed how to calculate the region of U and pH where bulk AxBy is stable against aqueous solvation. Before we proceed, however, it is pru-
dent to reiterate that bulk stability is a precondition for surface stability. At this time, we will extend our treatment of aqueous equilibria to the surfaces of AxBy.
There are many surface chemical reactions that can occur in aqueous solution. These include (1) the dissolution and deposition of A and B, (2) the adsorption and desorp- tion of HaOb species (H, OH, O, and H2O), and (3) the combinations thereof (see Fig.
2.5). For brevity, we will focus on scenarios 1 and 2, that being said, the following approaches are transferable to scenario 3, mutatis mutandis.
Starting with scenario 1, the equilibrium between a surface A atom and aqueous solution is
SA +nwH2OS + [HxAOy] z
+nHH++nee− (2.80)
where SA is the surface plus a surface A atom. The forward and backward reactions here are the dissolution and deposition of A, respectively. Note that it is trivial to rewrite Eq. 2.80 in terms of B. It is convenient to regard surface A dissolution as two consecutive half-reactions, i.e.
SAS + A (s) (2.81)
and the forward reaction in Eq. 2.75. The first reaction (Eq. 2.81) involves the desorption of a surface A atom and its subsequent placement in a reservoir of A(s). It has been shown that the free energy of surface atom desorption (des) can be replaced by the DFT total energy (∆Edes) because the entropy changes for solid-state chemical
dissolution (diss) of a surface A atom is
∆GA,diss(U, pH) = ∆EA,des+ ∆GA,solv(U, pH) (2.82)
where similar expressions exist for the deposition of A, and the dissolution and depo- sition of B. For deposition, however, the first and second terms in Eq. 2.82 correspond to the DFT total energy of adsorption and the free energy of precipitating a neutral atom, respectively.
Moving on to scenario 2, the chemical equation for HaOb adsorption is
S +bH2OSObHa+ (2b−a) H++e−
(2.83)
where SObHa is the surface plus an adsorbed HaOb species. Clearly, the reverse
reaction is the desorption of HaOb. The free energy of HaOb adsorption is given by
∆Gads(U, pH) = ∆Eads+ ∆ZPEads−T∆Sads + ∆IHCads(T)
−α(2.303kBT pH +U)−bkBT lnpH2O(g) (2.84)
where
α= 2b−a (2.85)
∆Eads =ESObHa+αEH2/2−ES−bEH2O (2.86)
∆ZPEads = ZPESObHa +αZPEH2(g)/2−ZPES−bZPEH2O(g) (2.87)
∆Sads(T)≈αSH◦2(g)(T)/2−bS
◦
H2O(g)(T) (2.88)
∆IHCads(T)≈αIHC◦H2(g)(T)/2−bIHC
◦
We have included ZPE, entropy, and IHC changes in Eq. 2.84 because the adsorption of HaOb involves the consumption and production of molecules. EH2 and EH2O are
the DFT total energies of isolated H2 and H2O molecules, respectively. The ZPE
of surfaces with and without adsorbates were calculated using DFPT where only the adsorbates and surface atoms near the binding site were considered. The ZPE, standard entropyS◦, and IHC of H2 and H2O at STP were taken from thermochemical
tables and reproduced in Table 2.1. (133; 143; 144) Note that the final three terms in Eq. 2.84 (i.e. those that depend on pH, U, and the vapor pressure) constitute a modified ∆Gsolv, which is applicable when the dissolution or deposition of surface
atoms does not accompany the adsorption or desorption of HaOb.
Using Eqs. 2.82 and 2.84, one can calculate the free energy changes associated with a plethora of aqueous surface chemical reactions under various conditions (ofU
and pH). These free energy changes can then be used to construct aqueous surface phase diagrams as follows:
1. Choose a reference surface.
2. Generate a catalog of surfaces that offer a wide variety of compositions and structures. This can be achieved by systematically removing atoms from and adding atoms to the reference surface (see Fig. 2.5).
3. Calculate the free energy differences between the surfaces and the reference as a function ofU and pH.
4. Assign a color to each surface and, for each U and pH, plot the color of the surface with the smallest free energy difference.
0
2
4
6
8
10
12
14
pH
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.0
U
(V
vs
. S
HE
)
AB is stable
S
ref-B+H
Sref
S
ref-B
S
ref-B+OH
S
ref-B+O
Figure 2.5: Aqueous surface phase diagram of a hypothetical binary compound AxBy.
Five surface phases are shown, three of which (dark green, purple, and blue) are stable in the region of U and pH where AB is stable. The composition of the surface layer is defined relative to that of some reference (ref) Sref, e.g.Sref-B+H corresponds to a
surface that has one B removed and one H adsorbed. Dotted lines correspond to the region of U and pH where AB is stable.
ZPE T S◦ IHC H2 0.27 0.40 0.09
H2O 0.56 0.58 0.10
Table 2.1: ZPE, standard entropy, and integrated heat capacity of H2 and H2O at