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GG325 L5, F
Lecture 5
Chemical Thermodynamics
- Mixtures
- Chemical Potential
- Non ideality in gasses and aqueous mixtures
GG325 L5, F
Thermodynamics of Chemical Mixtures
Partial Molar Quantities and Chemical Potential
Many thermodynamic properties like G depend on composition, which can be expressed in a multi-component system with a partial molar value.
The partial molar value expresses how that property (volume, pressure, enthalpy, entropy) depends on changes in amount of one component
For example, we can define the partial molar volume of component a in phase p as:
Vap^ = dV/dna (at constant T, P) such that Vp^ = Σ Va /na
i
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http://commons.wikimedia.org/wiki/File:Forsterite-Olivine-4jg54a.jpg
Thermodynamics of Chemical Mixtures
Partial Molar Volume is useful for understanding the properties of mixtures.
Let’s say we wanted to know how much space is occupied by 1 mole of calcite , CaCO 3 , versus 1 mole of dolomite , CaMg[CO 3 ] 2 , in an outcrop of limestone.
Or we might want to know how volume in a magma chamber might change if 1 mole of pure forsterite Mg 2 SiO 4 was turned into 1 mole of forsterite mixed with 20% fayalite, often abbreviated as Fo 80 = (Mg0.8 Fe0.2 ) 2 SiO 4.
The partial molar volumes of the components in these systems would give us the answer.
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Thermodynamics of Chemical Mixtures
Chemical Potential is Partial Molar Gibbs Free Energy. It is another useful property for understanding mixtures.
Chemical Potential, written as μ, is change in Gibbs free energy relative to change in a particular component. It is a measure of Free energy change with compositional change.
For all components “i”, the finite change version of μ is:
Σμi ·∆ni = ∆G at constant T, P
for one component “a” and at nother = constant, this becomes
μa ·∆na = ∆G or μa = ∆G/∆na
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Thermodynamics of Chemical Mixtures
μi can also be used to look at the exchange of one component between multiple phases of a closed system.
Example 1: small amounts of Ar in the atmosphere and oceans...
μAratm^ ·∆nAratm^ + μArocean^ ·∆nArocean^ = 0 (at equilibrium)
Example 2: Rb in plagioclase and clinopyroxene - two minerals typically found in a basalt
μRbplag^ ·∆nRbplag^ + μRbcpx·∆nRbcpx^ = 0 which rearranges to...
μRbplag^ ·∆nRbplag^ = -μRbcpx·∆nRbcpx
at equilibrium, ∆G = ∆GRbplag^ + ∆GRbcpx^ =0 and chemical exchange between these two phases should be balanced.
Thus, ∆nRbplag^ = -∆nRbcpx^ which leads to μRbplag^ = μRbcpx
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Thermodynamics of Chemical Mixtures
μi can also be used to examine multiple components of a single phase (i.e., Rb, Sr, Ba and K in a basaltic plagioclase). Initially, we’d try to treat this system as an ideal mixture…
ideal mixtures are ones where the components behave in fractional proportions to how they would in pure substances
- in other words, there is no energy effect of the interaction of the ideal components.
In general, for systems of “i” ideal components... μi = μi * + RT lnXi
where **μi *** is the chemical potential at pure i (Xi = 1) and Xi is the mole fraction of component i (e.g., Xi = ni /ntot )
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Non-Ideality in Chemical Mixtures
What if materials in mixtures do not behave ideally?
Non-Ideality in Solutions of Gasses Ideal gasses are those whose individual molecules or atoms do not interact with each other in the gas phase.
In this case, the partial pressure of any species i can be related to the total pressure:
Pi = Xi PTotal
Chemical potential for an ideal gas can be expressed as
μi = μi * + RT lnPi
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Non-Ideality in Chemical Mixtures
Gasses do not behave ideally in most geologically relevant systems. Instead, they have an effective pressure, known as the Fugacity , F Typically F (^) i < Pi
Fugacity and pressure are related by a non ideality factor known as a fugacity coefficient γi
Fi = γi Pi
Typically, γi gets lower as Ptotal increases (note: γi = 1 in an ideal gas; Many gasses behave close to ideally at low P).
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Non-Ideality in Chemical Mixtures
What do we really mean by “activity”, or “apparent concentration”?
This example should help to illustrate the point…
Activity = Effective Concentration
Instructor
Students
Yah, yah, yah
4 students (all alert) activity = concentration (ideal case) (^) γ=
Instructor
Students
Blah, blah,blah
4 students (1 alert, 3 sleeping) activity (1) ≠ concentration (4) (non ideal case) (^) γ=1/4 = 0.
Zzzzzzzzzzz
CASE 2 Non-ideal
CASE 1 ideal
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Non-Ideality in Chemical Mixtures
In aqueous systems, it is relatively easy to develop a theory of non-ideality for ionic species, where non-ideality usually results from charge (couloumbic) interactions between different solutes and with the solvent.
We use something called the Ionic Strength of the solution to help determine the level of couloumbic interactions.
Ionic Strength, I , is defined as I = ½ Σ m (^) i zi^2 , where
m (^) i = ion molality of species i zi = ion charge of species i
e.g., Isea water ~ 0.7 Iaverage rivers ~ 0.
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Non-Ideality in Chemical Mixtures
Ion interaction at high I causes the effective molality or molarity to differ from that expected of the true number of moles of solute in solution.
This has a very real effect of important mineral solubilities in natural waters, such as that of CaCO 3.
At low I Ksp = m (^) Ca2+ · m (^) CO3 2- At high I Ksp < m (^) Ca2+ · m (^) CO3 2- because...
Ksp = aCa2+ · aCO3 2- and ai < m (^) i
In fact, non-ideality allows there to actually be about 3 times more Ca 2+^ and CO 3 2-^ ions in solution than predicted by K (^) sp and the ideal solute molalities.
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Non-Ideality in Chemical Mixtures
Just what is γi? It is a quantification of the non-ideality of a solute in a solvent.
In aqueous solutions, non-ideality is a result of:
☼ charge interactions/ion charge ☼ ion diameter ☼ charge stabilizing capability of the solvent (the dielectric constant) ☼ other ionic solutes present
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Non-Ideality in Chemical Mixtures
γi is not predicted from thermodynamic theory of ideal solutions.
Instead, γi is parameterized empirically by matching ion behaviors in known non-ideal solutions.
The simplest parameterization is known as Debye-Hückel theory and is based on the simplest Coulombic electric field arguments. -log γi = A zi^2 I½
where z = ion charge, I = Ionic Strength and A is a temperature dependent constant related to the solvent dielectric constant. A ~ 0.5 for water at 25°C.
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Non-Ideality in Chemical Mixtures
For ionic solutes in water, activity is lower than ideal as a result of ion-ion interactions. At most natural ionic strength, these ions are more soluble than the ideal case.
For neutral (non-charged) molecules, log γi = k (^) i I, where k is the dielectric constant. At most natural ionic strength, these molecules are less soluble than the ideal case.
River to seawater range
