Thermodynamics: (pure) chemical equilibrium

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Thermodynamics(pure) chemical equilibrium
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An example - 1
A (completely gaseous) system is made of various
amounts of CO and CO2, let us say n(CO) moles of
CO and n(CO2) moles of CO2 and is placed in
variable - volume vessel. Temperature and
pressure inside the vessel are controlled by
appropriate devices and are fixed at given
values. If this T is high enough, we know that
a chemical reaction can take place The chemical
reaction modifies the amounts of CO and CO2, and
therefore modifies the free energy of the system.
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An example - 2
A further quantity n(O2) is seemingly required
to precise the state of the system. However, if
we restrict the discussion the thermodynamic
equilibrium, the quantities n(CO), n(CO2) and
n(O2) are not independent of each other but are
constrained by stoichiometry (and by initial
conditions). Using now n0(CO) and n0(CO2) to
indicate the initial values of the substances, we
have
? ? 0
Only one variable (?) is needed to describe the
transformation.
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An example - 3

• Three chemical variables are required if we look
  at the general case. These can be
• n(CO), n(CO2) and n(O2) or
• two of the above (depending on how we have
  prepared the system) plus the ?

Two chemical variables are then required if we
consider thermodynamic equilibrium (the
constraint), and the second choice is the right
choice we use two of the n (to describe how we
have prepared the system) plus the ?, and we look
for the equilibrium value of the ?. It is now
clear that this x is not a state variable but
merely describes how the system approaches
equilibrium from its initial conditions.
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An example - 4
If we know (from experiments or from theory) how
to calculate the free energy as a function of ?
at the given values of T and P (and the two n),
we can calculate the ? corresponding to minimum
G, that is we obtain the equilibrium state of the
system.
This is a particular case of the general case (G
is the appropriate thermodynamic potential) here
x is the advancement degree ? of the chemical
reaction.
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Using G
Using
This holds for a generic ?.
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Using G
At thermodynamic equilibrium there are only 2
independent chemical potentials. We say that
there are only two INDEPENDENT CHEMICAL
COMPONENTS.
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At thermodynamic equilibrium, the relation
between chemical potentials is exactly the same
as the relation between chemical components in
the stoichiometry of the reaction.

• In both relations
• gt 0 for products and
• lt 0 for reagents

Chemical equation (stoichiometry)
Relation between chemical potentials
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Towards useful results
For a mixture of ideal gases, the general
relation takes the form Then (at
equilibrium)
Partial pressure
Standard pressure
P/P0 replaced by p.
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Towards useful results
Then (at equilibrium)
Depends only from Temperature Therefore also the
lhs depends only from temperature
Depends only from temperature
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The equilibrium constant
Then (at equilibrium) The ratio of the
intensive chemical variables here partial
pressures corresponding to the various chemical
components is a constant (at a given T) it is
named the equilibrium chemical constant of the
reaction (K).
The knowledge of this constant makes possible to
solve the equilibrium, that is to say how much
CO, how much CO2 and how much O2 the system
contains at the given T and P (total pressure).
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Solving the equilibrium

• A closer look would show that solving the
  equilibrium means solving a 3d degree equation
  in ?.
• Instead of working out the full solution, let us
  consider the case where
• the rhs of the equation is very low (let us say
  below 10-10)
• or, (in an equivalent way) ? ltlt of the two n0
  (very small amount of CO).
• Then, the equation can be simplified

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Solving the equilibrium
In terms of partial pressures the final results
is
These initial values are the partial pressures at
? 0, that is before starting of the reaction
We can control with a chemical buffer the partial
pressure of oxygen. The pressure value that can
be realized depends on K and on the reagent
ratio. For instance if K 10-15, it is not
difficult to ACHIEVE and CONTROL partial
pressures 4 orders of magnitude above or below
10-15.
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The equilibrium constant K
K is related to fundamental thermodynamic
quantities by
In the case of ideal solutions, K is related to a
product of partial pressures of reaction products
and reagents.
?i is the stoichiometric coefficient of component
i in the reaction.
To unify the notation, let us write ?i gt 0 for a
product, and ?i lt 0 for a reagent.
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Chemical equilibrium between ideal gases
Conventionally writing for a reaction product ?i
gt 0 for a reagent ?i lt 0
Chemical reaction
Always true
Chemical potentials at equilibrium
A mixture of ideal gases is an ideal solution
Equilibrium constant
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General case of chemical equilibrium
Chemical reaction
Always true
Chemical potentials at equilibrium
Only for ideal solutions
Equilibrium constant
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Components and independent components
As seen, a chemical reaction reduces the number
of independent variables of the system. In
particular, it reduces the number of intensive
chemical variables (concentrations, molar
fractions, partial pressures, ) needed to
specify the state of the system. For that reason,
we also say that it reduces the number of
INDEPENDENT components of the system. This number
(nindip) is then ntot nindip nreact
Number of independent components
Number of independent chemical reaction between
these components
Total number of components