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CHEM F111 General Chemistry Lecture 13

Phase equilibria of pure substances
 Review of lecture 12

• Thermodynamic potentials
• Maxwell relations
• Multi-component systems and chemical potential
• Chemical equilibrium, van't Hoff equation,
feasibility of chemical reactions

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Equilibrium and spontaneity
criteria
In terms of intensive properties, the criteria of equilibrium are:
Thermal equilibrium: Uniformity of temperature T
Heat transfer occurs spontaneously in the direction of decreasing
temperature
Mechanical equilibrium: Uniformity of pressure
Displacement of ‘wall’ occurs spontaneously in the direction of decreasing
pressure
Material equilibrium (transfer of substance j): Uniformity of the chemical
potential of j
Net transfer of substance j occurs in the direction of decreasing chemical
potential μj

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Spontaneity and equilibrium
In the chemistry laboratory, one is most commonly concerned with
systems at constant T and P. At constant pressure, q = ΔH = qsurr so
that
(ΔS)tot = ΔSsys + ΔSsurr = ΔSsys – (ΔH)/T ≥ 0
('>' for irreversible and '=' for reversible process).
Or ΔH –TΔS ≤ 0
If T is constant, this may be written as
Δ(H – TS) ≤ 0 ; that is, ΔG≤ 0
('<' for irreversible and '=' for reversible process)
The dependence of Gibbs' energy on temperature and pressure is
given by:

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Equilibrium between two
phases of a pure substance
Consider two phases (1) and (2) of a pure substance in equilibrium. The
dependence of molar Gibbs energy of each phase on temperature and
pressure can be written as

At equilibrium, . It follows:

It follows:

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Clapeyron equation

(From the previous slide), we have:

The transition entropy is defined as

Thus, it follows:

This equation is known as Clapeyron (or Clausius Clapeyron) equation.

This form is exact and is applicable for transition between any two phases
of a pure substance.
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Liquid-vapour/solid-vapour
phase transitions
Consider the Clapeyron equation

The molar volume of liquid (or solid) phase is negligible as

compared to the molar volume of vapour phase. Therefore, during
vapourisation (or sublimation), ΔtrsV can be approximated as molar
volume of the vapour phase (Vm). Using ideal gas approximation and
substituting Vm = RT/P; and integrating,it follows:

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Solid-liquid phase transition

Consider the Clapeyron equation

In case of solid-liquid phase transition, where the transition

enthalpy and volume difference between the phases are fairly
constant over a range of temperatures and a range of pressures, the
equation can be integrated to obtain

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Thermodynamic stability of
a phase
At given T and P, the stable phase of a pure substance is the one with
the minimum molar Gibbs energy Gm.
For example, at a pressure of 1 atm, ice is the most stable phase of
water below 0ºC, while the steam is the most stable phase above
100ºC. For temperatures above 0ºC and below 100ºC, the liquid
water is the most stable phase.

At this pressure, ice and liquid water have the same value of Gm at
0ºC, and they coexist in equilibrium, while at 100ºC, liquid water and
steam have the same Gm, and they coexist in equilibrium.

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Variation of Gibbs energy
with pressure
At constant temperature,
Over a range of pressures, the molar volumes of solids and liquids
change only marginally. Thus, the Gibbs energy for solids and liquids
increases almost linearly with pressure.
However, the molar volume of vapour phase is strongly dependent on
pressure. Under the ideal gas approximation, the molar volume will
be inversely proportional to the pressure. Integrating the above
equation over a range of pressures (from Pi to Pf ), it follows:

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Variation of Gibbs energy
with pressure

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Variation of Gibbs energy
with pressure
The pressure at which the Gibbs energy curves of two phases intersect is the
transition pressure for the given temperature.
At this point, the corresponding two phases are in equilibrium. Clearly, the
transition pressure will change with change in temperature.
In the current example, the solid-liquid transition pressure is higher than the
liquid-vapour transition pressure. The liquid phase is the most stable between
these two transition pressures as it has the minimum Gibbs energy in this
range.
However, if the solid-liquid transition pressure is lower than the liquid-
vapour transition pressure, then the Gibbs energy of the liquid phase will be
minimum at no pressure which is the case with sublime materials.

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Variation of Gibbs energy
with temperature
At constant pressure,
The Gibbs energy of each phase decreases with increase in
temperature. The entropy is minimum for a solid phase and is
maximum for a vapour phase at given pressure. This is reflected in
the the slopes of the Gibbs energy curves for solid, liquid and
vapour phases.
At constant pressure, the temperature corresponding to intersection
of two curves is a transition temperature.
At a standard pressure of 1 atm, the transition temperatures refer to
melting/freezing point (solid-liquid), boiling point (liquid-vapour)
and sublimation point (solid-vapour)
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Variation of Gibbs energy
with temperature

In most cases, the

sublimation temperature
lies between the melting
and the boiling points
ensuring lowest Gibbs
energy for liquid phase over
a range of temperature.

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Variation of Gibbs energy
with temperature
In some cases, the
sublimation temperature is
higher than the boiling
temperature. In such cases,
liquid phase cannot have
lowest Gibbs energy at any
temperature for the given
pressure and the substance
is found to be sublime.

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Phase diagram analysis

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Stable phases and phase
equilibria
We see that for a given phase can be stable over a range of pressures and
temperatures for which its Gibbs energy will be the lower than that of any
other phase. Thus, to determine the thermodynamic state of a single-phase
one-component system (pure substance), we require two parameters to be
specified – two degrees of freedom.
On the other hand, if equilibrium between two phases of a pure substance
is to be studied, one needs to specify only one parameter, that is, either
pressure or temperature, and the other parameter gets fixed – one degree of
freedom.
In a system of a pure substance, at the max, three phases can coexist in
equilibrium. However, this is possible only for unique temperature and
pressure (they cannot be altered) – no degrees of freedom.
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Gibbs phase rule

In case of multi-component systems, the amounts (concentration or number of

moles, etc.) of all the components (except one) are the additional parameters to be
specified (in addition to the pressure and/or temperature or none) in order to
determine the thermodynamic state of the system. The number of degrees of
freedom increases accordingly.
These observations were studied by J.W. Gibbs. Published by him during
1875-1878, in a research article, entitled “On the Equilibrium of
Heterogeneous Substances”, the phase rule (now popularly known as Gibbs
phase rule) is a mathematical equality: f = c – p + 2
where, f = number of degrees of freedom, c = number of components, p =
number of phases (in equilibrium)

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Phase diagram of a pure
substance

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