Oxygen deficiency and fugacity

seventy-five years after its invention, transmission electron microscopy TEM has taken a great step forward with the introduction of aberration-corrected electron optics....... The energy resolution is about 100 milli–electron volts, and the accuracy of spatial measurements has reached a few picometers. However, understanding the results is generally not straightforward and only possible with extensive quantum-mechanical computer calculations.

..........In principle, two types of aberrations are always involved: geometrical aberrations, such as spherical aberration, and chromatic aberrations, which arise from the electron-energy dependence of the refraction properties of magnetic fields. Current lens designs can only correct for the former. To reduce the effect of chromatic aberrations, field-emission electron sources and, in some cases, energy filters (monochromators) are used in the electron-beam–forming system of the new generation of 100-to-300-keV instruments.






{∑}3{111} twin boundary in BaTiO3. All atomic species, including oxygen, can be identified. The atomic resolution is confirmed by image simulations. These indicate that, because of negligible point spread, neighboring atoms have no effect on the intensity measured in a given atomic position. This provides the basis for the quantitative site-occupation measurements. The local intensity values indicate that, in the individual oxygen-atom columns seen end-on, only between about 40 and 70% of the sites are occupied. This provides evidence for oxygen deficiency, which is presumed to have a detrimental influence on electronic properties.



http://www.sciencemag.org/cgi/content/full/321/5888/506

POWER POINT FOR TEACHING



Fugacity: definitions, concepts, and geology

FUGACITY (and activity).

First, some symbols:

f(i) = fugacity of component i in a mixture.

P = total pressure Pi = partial pressure of component i e.g., P(CO2) = partial pressure of CO2

j = activity coefficient of i (subscripts left off for simplicity) J = fugacity coefficient of pure i "

a(i) = activity of component i in a A SINGLE PHASE mixture X(i) = mole fraction " "

µ(i) = chemical potential of component i

ss = standard state K = equilbrium constant

Fugacity

Formal definition: whenever we are dealing with the chemical potential of a component in a gas phase, or a component that *MAY* be in a gas phase (see footnote 1), then we can use the fugacity to account for the difference between the chemical potential of interest µ(P,T), and the chemical potential of the pure substance at T,1 bar:

f(i) = exp{[µ(T,P) - µ(T,1 bar, pure)]/RT}

qualitative concept: Note that at a given T,there are only two variables in the above equation: f(i) and µ(T,P). Therefore, as, fi increases, so does µ. A high fugacity of water or oxygen means a high chemical potential of water or oxygen, respectively. A high chemical potential of water or oxygen indicates a "wet" or "oxidized" system, respectively.

BUT what does it MEAN??? well, look at the most general equation for fugacity:

f(i) = j X(i) J P.

If a gas is ideal AND the mixture is ideal, then, f(i) = X(i) P, and fi is also equal to Pi .

So, if you ask "what is the fugacity, and how do I think about it", just think of it as a partial pressure: it is a strong function of the mole fraction of the component in the gas phase, and of the total pressure of the gas phase, just like a partial pressure; more precicely, remembering that chemical potential is a quantitative measure of the reactivity of a component in a phase, we can think of fiugacity as a measure of how much the chemical potential of the component in the gas deviates from the chemical potential of some reference, namely, the standard state, due to changes in P and/or the mole fraction of the component i.

How is it different then, from partial pressure or total pressure? Well, taking a pure gas first, we know that fi deviates from P by:

f(i) = J P.

WHY is this so, you ask? Well, IT GOES BACK TO the fact that the rate of change of the chemical potential of a pure gas (i) with respect to changes in P (at a given T) is equal to the volume of the gas, V. For an ideal gas, V = RT/P; dµ = VdP, then becomes dµ = RT(dP/P), and integration yields µ(P) = µ(1 bar) + RtlnP, where the RtlnP term corrects the one-bar chemical potential up to the pressure of interest, P. BUT, for a non-ideal gas, V has a different value from RT/P, so something other than RTlnP must be used as the "pressure correction" term. The solution to this problem, is, on the surface, a rather "unsatisfying" one: we use Rtlnf(i), instead of RTlnP. Now the pressure has been multiplied by the fugacity coefficient, which are tabulated, or are available as output from computer models, but which ultimately are based on experimental determinations made by by hard working experimental geochemists and others scientists by integrating the *difference* between the observed molar volume of the gas and the ideal molar volume (RT/P), with respect to pressure (dP) from one bar to the pressure of interest (see footnote 2).

So, pressure is still pressure, and fugacity is not a "corrected pressure". Fugacity has meaning only when related to chemical potential, and the correction factor term RtlnF(i) corrects the chemical potential of gas for the fact that tabulated free energies of the gas are at a stated total pressure (usually one bar), and for the pure gas; because the chemical potentail is central to thermodynamics, fugacity becomes so also.

As µ(T,P, pure) = µ(T,1,pure) +Rtlnf(pure )i, we can, at a given T, use f(i)i as a proxy for the chemical potential, when we are thinking about how pressure will affect a reaction where we have a gas phase present (e.g., "ah yes, at T = 200 C, as we increase the fugacity of water in the gyp-anh-gas assemblage, we will definitely favor the stability of gypsum relative to anhydrite" ).

When we have a mixture, fi does double duty, because

fi = j X(i) J P,

which is of course equal to: fi = a(i) f (pure i). Then, for a gas,

mu(T,P, mix) = mu (T,1,pure) +Rtlnf(i),

and we may leave it in this form in our "condition for equilibrium" statement (e.g., when subbing for the FIRST TERM in the equilibrium condition:

mu-CO2(T,P, vapor,mix) = mu-CO2(T,P, liquid,mix)

so that it ends up in the equilibrium constant as f(i). IN THIS CASE, the "delta mu - zero, of Rxn" HAD BETTER contain mu (T,1,pure). Remember, K is calculated from "delta mu - zero, of Rxn" = -RTlnK.

NOW, *if you want* you CAN have the activity of (i) in the gas in the equilibrium constant. then,

mu(T,P, mix) = mu (T,1,pure) + Rtlnf (pure i) + RTlna(i) , GAS EQ 1.

AND, mu(T,P, mix) = mu (T,1,pure) +Rtlnf (pure i) is put into the "delta mu - zero, of Rxn"; because "delta mu - zero, of Rxn" = -RTlnK, and the form of K (whether you will use fi or a(i) )for gases HAS TO BE CONSISTENT with what you put into the "delta mu - zero, of Rxn"!!!!!!

EQ 1 is DIRECTLY analogous to what we do for non-gases (liquids and solids, see footnote 3), where

µ(T,P, mix) = µ(T,1,pure) + (P-1)delta V + Rtlna(i) LIQ/SOLID EQ 2

PLEASE COMPARE EQ 1 AND EQ 2.

When you use a(i) for liquid or solid in an equil constant, K, and all you have from a table, etc, is µ(T,1,pure), then you MUST include µ(T,1,pure) + (P-1)delta V into the "delta µ - zero, of Rxn" = -RTlnK.

footnote 1. like when we are dealing with:

1. the vapor pressure of liquid water at 25 C, and **1 bar total pressure without a gas present**; or

2. a system where oxygen or sulfur is a system component, but there is no gas phase (see below).

footnote 2. (These result from many hundreds of measurements of actual molar volumes of gasses, and in the case of gas mixtures, measurements of the molar volumes of the gas mixture,so that once we have the fugacity coefficient, we can also estimate the activity coefficient).

footnote 3. we call liquids and solids "condensed" phases, because of their relative incompressibility. THIS DOES NOT MEAN THAT THEIR CHEMICAL POTENTIAL's DO NOT HAVE TO BE CORRECTED FOR PRESSURE!!!! IT ONLY MEANS WE CAN HOPE TO INTEGRATE VdP to (P-1)delta V, without V changing too much with pressure. We cannot EVEN PRETEND to integrate VdP for gases with V constant, HENCE, fugacity!!!!!!!!!!!

Philip A. Candela, 1997, Maryland Univ.