Explain in detail, using thermodynamic principles, why the mixing of pure chemicals to form a homogeneous mixture is a so called spontaneous process, while the separation of that mixture into its pure...

Mixing is several initially separate systems of different composition, each in a
thermodynamic state of internal equilibrium, are mixed without chemical reaction by the
thermodynamic operation of removal of impermeable partition between them, followed by a
time for establishment of a new thermodynamic state of internal equilibrium in the new
unpartitioned closed system.
In general, the mixing may be constrained to occur under various prescribed conditions.
In the customarily prescribed conditions, the materials are each initially at a common
temperature and pressure, and the new system may change its volume, while being
maintained at that same constant temperature, pressure, and chemical component masses.
The volume available for each material to explore is increased, from that of its initially
separate compartment, to the total common final volume.
The final volume need not be the sum of the initially separate volumes, so that work can be
done on or by the new closed system during the process of mixing, as well as heat being
transferred to or from the surroundings, because of the maintenance of constant pressure and
temperature.
The internal energy of the new closed system is equal to the sum of the internal energies of
the initially separate systems.
The reference values for the internal energies should be specified in a way that is
constrained to make this so, maintaining also that the internal energies are respectively
proportional to the masses of the system
For concision in this article, the term 'ideal material' is used to refer to an ideal gas (mixture)
or an ideal solution.
In a process of mixing of ideal materials, the final common volume is the sum of the initial
separate compartment volumes.
There is no heat transfer and no work is done.
The entropy of mixing is entirely accounted for by the diffusive expansion of each material
into a final volume not initially accessible to it.
On the mixing of non-ideal materials, the total final common volume may be different from
the sum of the separate initial volumes, and there may occur transfer of work or heat, to or
from the surroundings; also there may be a departure of the entropy of mixing from that of
the corresponding ideal case.
That departure is the main reason for interest in entropy of mixing.
https://en.wikipedia.org/wiki/Thermodynamic_operation
https://en.wikipedia.org/wiki/Ideal_gas
https://en.wikipedia.org/wiki/Ideal_solution
These energy and entropy variables and their temperature dependences provide valuable
information about the properties of the materials.
On a molecular level, the entropy of mixing is of interest because it is a macroscopic variable
that provides information about constitutive molecular properties.
In ideal materials, intermolecular forces are the same between every pair of molecular
kinds, so that a molecule feels no difference between other molecules of its own kind and of
those of the other kind.
In non-ideal materials, there may be differences of intermolecular forces or specific
molecular effects between different species, even though they are chemically non-reacting.
The entropy of mixing provides information about constitutive differences of intermolecular
forces or specific molecular effects in the materials.
The statistical concept of randomness is used for statistical mechanical explanation of the
entropy of mixing. Mixing of ideal materials is regarded as random at a molecular level, and,
correspondingly, mix1 Mixing of ideal materials at constant temperature and pressure
Mixing of ideal materials at constant temperature and
pressure
In ideal materials, intermolecular forces are the same between every pair of molecular kinds,
so that a molecule "feels" no difference between itself and its molecular neighbours. This is
the reference case for examining corresponding mixings of non-ideal materials.
For example, two ideal gases, at the same temperature and pressure, are initially separated by
a dividing partition.
Upon removal of the dividing partition, they expand into a final common volume (the sum of
the two initial volumes), and the entropy of mixing is given by
.
where is the gas constant, the total number of moles and the mole fraction of component ,
which initially occupies volume . After the removal of the partition, the moles of component
may explore the combined volume , which causes an entropy increase equal to for each
component gas.
In this case, the increase in entropy is due entirely to the irreversible processes of expansion
of the two gases, and involves no heat or work flow between the system and its surroundings.
Gibbs free energy of mixing
The Gibbs free energy change determines whether mixing at constant (absolute) temperature
and pressure is a spontaneous process. This quantity combines two physical effects—the
enthalpy of mixing, which is a measure of the energy change, and the entropy of mixing
considered here.
https://en.wikipedia.org/wiki/Constitutive_equation
https://en.wikipedia.org/wiki/Gas_constant
https://en.wikipedia.org/wiki/Mole_%28unit%29
https://en.wikipedia.org/wiki/Mole_fraction
https://en.wikipedia.org/wiki/Gibbs_free_energy
https://en.wikipedia.org/wiki/Spontaneous_process
https://en.wikipedia.org/wiki/Enthalpy
For an ideal gas mixture or an ideal solution, there is no enthalpy of mixing (), so that the
Gibbs free energy of mixing is given by the entropy term only:
For an ideal solution, the Gibbs free energy of mixing is always negative, meaning that
mixing of ideal solutions is always spontaneous. The lowest value is when the mole fraction
is 0.5 for a mixture of two components, or 1/n for a mixture of n components.
Solutions and temperature dependence of miscibility
Ideal and regular solutions
The above equation for the entropy of mixing of ideal gases is valid also for certain liquid (or
solid) solutions—those formed by completely random mixing so that the components move
independently in the total volume. Such random mixing of solutions occurs if the interaction
energies between unlike molecules are similar to the average interaction energies between
like molecules

.
The value of the entropy corresponds exactly to random mixing for ideal solutions and for
regular solutions, and approximately so for many real solution
For binary mixtures the entropy of random mixing can be considered as a function of the
mole fraction of one component.
For all possible mixtures, , so that and are both negative and the entropy of mixing is positive
and favors mixing of the pure components.
Also the curvature of as a function of is given by the second derivative
This curvature is negative for all possible mixtures , so that mixing two solutions to form a
solution of intermediate composition also increases the entropy of the system. Random
mixing therefore always favors miscibility and opposes phase separation.
For ideal solutions, the enthalpy of mixing is zero so that the components are miscible in all
proportions.
For regular solutions a positive enthalpy of mixing may cause incomplete miscibility (phase
separation for some compositions) at temperatures below the upper critical solution
temperature (UCST)
.

This is the minimum temperature at which the term in the Gibbs energy of mixing is
sufficient to produce miscibility in all proportions.
Systems with a lower critical solution temperature
https://en.wikipedia.org/wiki/Ideal_gas
https://en.wikipedia.org/wiki/Ideal_solution
https://en.wikipedia.org/wiki/Ideal_solution
https://en.wikipedia.org/wiki/Regular_solution
https://en.wikipedia.org/wiki/Upper_critical_solution_temperature
https://en.wikipedia.org/wiki/Upper_critical_solution_temperature
Nonrandom mixing with a lower entropy of mixing can occur when the attractive interactions
between unlike molecules are significantly stronger (or weaker) than the mean interactions
between like molecules. For some systems this can lead to a lower critical solution
temperature (LCST) or lower limiting temperature for phase separation.
For example, triethylamine and water are miscible in all...