Chemical Thermodynamics 2018/2019

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Chemical Thermodynamics2018/2019
11th Lecture Thermodynamics of Simple
Mixtures Valentim M B Nunes, UD de Engenharia
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Introduction
In this lecture we will begin the study of simple
non-reactive mixtures. To do that we will
introduce the concept of partial molar
properties.
At this stage we deal mainly with binary
mixtures, that is
We shall also consider mainly non-electrolyte
solutions, where the solute is not present as
ions.
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Partial Molar Volume

• To understand the concept of partial molar
  quantities, the easiest property to visualize is
  the volume. When we mix two liquids A and B, we
  have three possibilities
• Volume contraction
• Volume dilatation
• No total volume change.

In the last case if we mix, for instance, 20 cm3
of liquid A and 80 cm3 of liquid B, we will
obtain 100 cm3 of solution. This mean that AB
interactions are equal to AA or AB molecular
interactions!
In most cases that is not going to happen! For
instance the molar volume of water is 18
cm3.mol-1. But if we add 1 mol of water to a huge
amount of ethanol the volume will only increase
by 14 cm3. The quantity 14 cm3 is the partial
molar volume of water.
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Partial Molar Volume
The partial molar volumes change with composition
because molecular environment also change. For
water/ethanol system we have
The partial molar volume of a substance is
defined as
Once we know the partial molar volume of two
components of a binary mixture we can calculate
the total volume
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Measuring Partial Molar Volume
One method is measuring the dependence of volume
on composition and determine the slope dV/dn
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Method of Intercepts
Dividing both members of last equation (slide 4)
by the total number of moles, n, we obtain
See next part of lecture!
Differentiating
Substituting
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Method of Intercepts
Finally we obtain
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Partial Molar Gibbs Function
Another partial molar property, already
introduced, is the partial molar Gibbs function
or the chemical potential.
Then, the total Gibbs function of a mixture is
But we have seen before that (recall the
fundamental equations for open systems)
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Gibbs Duhem Equation
Since G is a state function the previous
equations must be equal, and then we obtain
Generalizing
This is the Gibbs Duhem Equation. It tell us
that partial molar quantities cannot change
independently in a binary mixture if one
increases the other should decrease! It can also
be applied to the partial molar volume.
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Thermodynamics of Mixing Perfect Gases
We already now that systems evolves spontaneously
to lower Gibbs energy. If we put together two
gases they will spontaneously mix, then G should
decrease.
Let us consider two perfect gases in two
containers at temperature T and pressure p. The
initial total Gibbs energy is
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Gibbs energy of Mixing
After mixing we have
The difference is the Gibbs energy of mixing and
therefore
Using Daltons Law, yipi/p, we finally obtain
Since molar fractions are always lower than 1,
the Gibbs energy of mixing is always negative!
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Other Thermodynamic mixing functions
Since
we easily obtain
Purely entropic
Now since ?H ?G T?S
And since for perfect gases ?Gmix is independent
of pressure, then
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Summary
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Chemical potential of liquids
In order now to study the equilibrium properties
of liquid solutions we need to calculate the
chemical potential of a liquid.
To do this, we will use the fact that the
chemical potential of a substance present as a
dilute vapor must be equal to the chemical
potential of the liquid, at equilibrium.
Remember also that it is usual, in order to
characterize a given solution, to distingue
between the solvent (usually the substance in
bigger quantity or in the same physical state of
solution) and solute (in lower quantity or in a
different state of aggregation)
Important note We will denote quantities
relating to pure substances by the superscript
so the chemical potential of a pure liquid A
it will be written as µA(l).
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Ideal Solutions
For a pure liquid in equilibrium with the vapor
we can write
If another substance is present (solution!) then
the chemical potential is
Combining the two equations leads to
?
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Ideal Solutions
French chemist François Raoult found that the
ratio pA/pA is equal to mole fraction of A in
the liquid, that is
This equation is known as the Raoults Law.
It follows that for the chemical potential of
liquid
This important equation is the definition of
ideal solutions. Alternatively we may also state
that an ideal solution is the on in which all the
components obey to the Raoults Law.
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Raoults Law
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Examples
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Ideal Dilute Solutions
In ideal solutions the solute, as well as the
solvent, obeys Raoults Law. But in some cases,
in non-ideal solutions, the partial pressure of
solute is proportional to the mole fraction but
the constant of proportionality is not the pure
component vapor pressure but a constant, known as
the Henrys constant, KB
This is known as the Henrys Law. Mixtures
obeying to the Henrys Law are called ideal
dilute solutions.
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Example
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Ideal Mixture of Liquids
When two liquids are separated we have
When they are mixed we have
As a consequence
This equation is the same as that for two perfect
gases! All the other conclusions are equally
valid. The driving force for mixing is the
increase of entropy.
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Excess Functions
Real solutions have different molecular
interactions between AA, AB and BB particles.
As a consequence ?H?0 and ?V?0. For instance if
?H is positive (endothermic) and ?S negative
(clustering) ?G is positive and the liquids are
immiscible. Alternatively the liquids can be only
partially miscible.
Thermodynamic properties of real solutions can be
expressed in terms of excess functions (GE, SE,
etc.). For example for entropy
In this context the concept of regular solutions
is very important. In this solutions HE?0 but the
SE0.
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Examples
These figure shows experimental excess functions
at 25 ºC. HE values for benzene/ cyclohexane show
that the mixing is endothermic (since ?H0 for an
ideal solution).
VE for tetrachloroethene/cyclopentane shows that
there is a contraction at lower mole fractions of
tetrachloroethene but expansion at higher mole
fractions (because ?V0 for an ideal solution)
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Solutions containing non-volatile solutes
Let us now consider a solution in which the
solute is non-volatile, so it does not contribute
to the vapor pressure, and it does not dissolve
in solid sovent. Recalling that the chemical
potential of liquid A in solution is
We can conclude that the chemical potential in
the liquid solvent is lower that the chemical
potential of pure liquid (remember that xAlt1).
These give origin to a set of properties of this
solutions that are called colligative
properties. For instance the vapor pressure
lowering
or
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Colligative Properties
All colligative properties have in common that
they only depend on the amount of solute but not
on the particular substance. The other
colligative properties are Boiling point
elevation, Freezing point depression and Osmotic
pressure.
Graphically we can observe the first two
properties
Effect of the solute
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The elevation of boiling point
The heterogeneous equilibrium that maters when
considering boiling is the solvent vapor and the
solvent in the solution.
The equilibrium condition is
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The elevation of boiling point
This rearranges to
When xB0 , the boiling point is that of pure
liquid, Tb, and
The difference between the two equations is
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The elevation of boiling point
Assuming now dilute solutions, that is, xBltlt1,
then ln(1-xB) -xB, so
Since T Tb we can also write
Which gives
This means that the elevation of boiling point,
?T, depends on xB (no reference to the identity
of solute, the other properties are related to
the solvent!)
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The elevation of boiling point
We can also calculate the elevation of boiling
point in terms of the molality of the solute, mB.
Since the mole fraction of solute is small nA gtgt
nB, so
Therefore we obtain
Boling point elevation can now be calculated by
or
Kb is the molal ebullioscopic constant.
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The depression of freezing point
The equilibrium of interest now is between the
pure solid solvent and the solution with
dissolved solute
At freezing point the equilibrium condition is
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The depression of freezing point
All the previous calculations are equal, so we
can write directly
or
Kf is the molal cryoscopic constant.
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Constants of Common Liquids
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Osmosis
We have shown that boiling point and freezing
point depends on the equilibrium between the
solvent in solution and in the solid or vapor
state. The last possibility is to have an
equilibrium between the solvent in solution and
the pure solvent. This is the basis of the
phenomenon of osmosis. This consists in the
passage of a pure solvent into the solution
separated from it by a semipermeable membrane.
? ?hg is the osmotic pressure
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Osmotic pressure
The equilibrium condition is
From these equation we can easily obtain
For dilute solutions ln xA ln (1-xB) -xB, so,
assuming that Vm is constant
Now, since xB nB/nA and nAVm V, we finally
obtain
This is the vant Hoff equation for osmotic
pressure. Note the remarkable similitude with
the perfect gas equation!
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Applications
Vant Hoff equation can be rewritten as
This is a very useful equation for the
determination of molar mass of polymers and other
macromolecules, assuming dilute solutions and
incompressible solutions.
Other application is the reverse or inverse
osmosis, If we apply from the solution side a
pressure bigger than the osmotic pressure we will
reverse the passage of water molecules. This can
be used to obtain fresh water from sea water ( p
30 to 50 atm)
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Ideal solubility of solids
Although is not strictly a colligative property
the solubility of a solid can be estimated by the
same techniques of previous slides. In contact
with a liquid the solid will dissolve until the
solution is saturated. The equilibrium condition
is
From this starting point and using similar
deduction we will obtain
Being the properties related only to the solute.