PHYSICAL CHEMISTRY
ONE-COMPONENT PHASE DIAGRAMS
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Phase Rule
- the thermodynamic criterion for phase
stability
- summarizes the constraints on the
equilibria between phases
- expressed in a general way that can be
applied to systems of more than one
component
- relates the number of variables that may be
changed while the phases of a system
remain in mutual equilibrium
- Josiah Willard Gibbs deduced the phase
rule, which gives the number of parameters
that can be varied independently (at least to
a small extent) while the number of phases
in equilibrium is preserved.
▪ The variance (or number of degrees of
freedom), F, of a system is the number
of intensive variables that can be
changed
independently
without
disturbing the number of phases in
equilibrium.
▪ A constituent of a system is any
chemical species that is present.
▪ A component is a chemically
independent constituent of a system;
minimum number of types of
independent
species
(ions
or
molecules) necessary to define the
composition of all the phases present in
the system.
Phase diagrams
- summarize the behavior of substances
under different conditions, and identify
which phase or phases are the most stable
at a particular temperature and pressure
- such diagrams are important tools for
understanding the behavior of both pure
substances and mixtures
- one of the most succinct ways of presenting
the physical changes of state that a
substance can undergo
- indicates the values of the pressure and
temperature at which a particular phase is
most stable or is in equilibrium with other
phases.
▪ The phase diagram of a pure substance
shows the regions of pressure and
temperature at which its various phases
are thermodynamically stable
▪ The lines separating the regions, which
are called phase boundaries (or
coexistence curves), show the values of
p and T at which two phases coexist in
equilibrium and their chemical
potentials are equal.
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▪ A single phase is represented by an area
on a phase diagram.
A pure substance tends to adopt the phase with
the lowest chemical potential
Gibbs energy is a signpost of spontaneous
change under conditions of constant
temperature and pressure
Phase
- form of matter that is uniform throughout in
chemical composition and physical state
o allotrope: a particular molecular
form of an element (such as O2 and
O3) and may be solid, liquid, or gas
o polymorph: one of a number of
solid phases of an element or
compound
▪ The number of phases in a system is
denoted P. A gas, or a gaseous mixture,
is a single phase (P = 1), a crystal of a
substance is a single phase, and two
fully mixed liquids form a single phase.
▪ Two metals form a two-phase system
(P = 2) if they are immiscible, but a
single-phase system (P = 1), an alloy, if
they are miscible (and actually mixed).
▪ A solution of solid B in solid A—a
homogeneous mixture of the two
miscible substances— is uniform on a
molecular scale.
▪ In a solution, atoms of A are
surrounded by atoms of A and B, and
any sample cut from the sample, even
microscopically
small,
is
representative of the composition of the
whole. It is therefore a single phase.
▪ A dispersion is uniform on a
macroscopic scale but not on a
microscopic scale, because it consists
of grains or droplets of one substance
in a matrix of the other.
- A small sample could come
entirely from one of the minute
grains of pure A and would not
be representative of the whole.
A dispersion therefore consists
of two phases.
Phase Transition
- the spontaneous conversion of one phase
into another phase, occurs at a characteristic
transition temperature 𝑇𝑡𝑟𝑠 for a given
pressure.
- At the transition temperature, the two phases
are in equilibrium and the Gibbs energy of
the system is a minimum at the prevailing
pressure.
▪ The detection of a phase transition is
not always straightforward as there may
be nothing to see, especially if the two
phases are both solids.
▪ The thermodynamic analysis of phases
is based on the fact that at equilibrium,
the chemical potential of a substance is
the same throughout a sample.
Characteristic properties related to phase
transitions
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The pressure of a vapor in equilibrium with the
liquid is its vapor pressure. Therefore, the
liquid–vapor phase boundary in a phase
diagram shows how the vapor pressure of the
liquid varies with temperature.
The solid–vapor phase boundary shows the
temperature variation of the sublimation vapor
pressure, the vapor pressure of the solid phase.
The vapor pressure of a substance increases
with temperature because at higher
temperatures more molecules have sufficient
energy to escape from their neighbors.
The condition of free vaporization throughout
the liquid is called boiling.
The temperature at which the vapor pressure of
a liquid is equal to the external pressure is called
the boiling temperature at that pressure.
For the special case of an external pressure of 1
atm, the boiling temperature is called the
normal boiling point, 𝑻𝒃 .
With the replacement of 1 atm by 1 bar as
standard pressure, there is some advantage in
using the standard boiling point instead: this
is the temperature at which the vapor pressure
reaches 1 bar.
o Because 1 bar is slightly less than 1 atm
(1.00 bar = 0.987 atm), the standard
boiling point of a liquid is slightly
lower than its normal boiling point.
There comes a stage when the density of the
vapor is equal to that of the remaining liquid
and the surface between the two phases
disappears. The temperature at which the
surface disappears is the critical temperature,
𝑻𝒄 , of the substance.
The vapor pressure at the critical temperature is
called the critical pressure, 𝒑𝒄 .
At and above the critical temperature, a single
uniform phase called a supercritical fluid fills
the container and an interface no longer exists.
That is, above the critical temperature, the
liquid phase of the substance does not exist.
The temperature at which, under a specified
pressure, the liquid and solid phases of a
substance coexist in equilibrium is called the
melting temperature.
Because a substance melts at exactly the same
temperature as it freezes, the melting
temperature of a substance is the same as its
freezing temperature.
The freezing temperature when the pressure is
1 atm is called the normal freezing point, 𝑇𝑓 ,
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and its freezing point when the pressure is 1bar
is called the standard freezing point.
The normal and standard freezing points are
negligibly different for most purposes. The
normal freezing point is also called the normal
melting point.
There is a set of conditions under which three
different phases of a substance (typically solid,
liquid, and vapor) all simultaneously coexist in
equilibrium. These conditions are represented
by the triple point, a point at which the three
phase boundaries meet.
o The temperature at the triple point is
denoted 𝑇3 .
o The triple point of a pure substance
cannot be changed: it occurs at a single
definite pressure and temperature
characteristic of the substance.
o Triple point marks the lowest pressure
at which a liquid phase of a substance
can exist.
o If the slope of the solid–liquid phase
boundary is as shown in the diagram,
then the triple point also marks the
lowest temperature at which the liquid
can exist.
o The triple point of water lies at
273.16K and 611Pa (6.11mbar,
4.58Torr), and the three phases of water
(ice, liquid water, and water vapor)
coexist in equilibrium at no other
combination
of
pressure
and
temperature.
o This invariance of the triple point was
the basis of its use in the now
superseded definition of the Kelvin
scale of temperature
Ideal Solution
Quantities relating to pure substances are
denoted by a superscript *
François Raoult found that the ratio of the
partial vapor pressure of each component to its
vapor pressure when present as the pure liquid,
pA/pA*, is approximately equal to the mole
fraction of A in the liquid mixture. That is, he
established what is now called Raoult’s law.
o Raoult’s law provides a relation
between the vapor pressure of a
substance and its mole fraction in a
mixture.
o Some mixtures obey Raoult’s law very
well, especially when the components
are structurally similar.
o Mixtures that obey the law throughout
the composition range from pure A to
pure B are called ideal solutions.
o An ideal solution is a solution that
obeys Raoult’s law over its entire range
of compositions; for real solutions it is
a limiting law valid as the mole fraction
of the species approaches 1.
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PHASE DIAGRAMS OF BINARY SYSTEMS:
LIQUIDS
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The phase diagram of a liquid mixture can be
understood in terms of the variation with
temperature and pressure of the composition of
the liquid and vapor in mutual equilibrium.
The partial vapor pressures of the components
of an ideal solution of two volatile liquids are
related to the composition of the liquid mixture
by Raoult’s law:
𝑝𝐴 = 𝑥𝐴 𝑝𝐴∗
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Points that lie between the two lines do provide
additional information if the horizontal axis
denotes the overall composition of the
mixture 𝒛𝑨 in equilibrium at a given
temperature rather than the liquid or vapor
composition separately.
The overall composition does not change
regardless of how much liquid vaporizes, so the
system moves up the vertical line at a.
Such a vertical line is called an isopleth (from
the Greek words for ‘equal abundance’).
𝑝𝐵 = 𝑥𝐵 𝑝𝐵∗
𝑝 = 𝑝𝐴 + 𝑝𝐵
𝑇𝑜𝑡𝑎𝑙 𝑣𝑎𝑝𝑜𝑟 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒: 𝑝 = 𝑝𝐵∗ + (𝑝𝐴∗ − 𝑝𝐵∗ )𝑥𝐴
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This expression shows that the total vapor
pressure (at some fixed temperature) changes
linearly with the composition from 𝑝𝐵∗ to 𝑝𝐴∗ as
𝑥𝐴 changes from 0 to 1.
The compositions of the liquid and vapor that
are in mutual equilibrium are not necessarily
the same. Common sense suggests that the
vapor should be richer in the more volatile
component.
This expectation can be confirmed as follows:
𝑦𝐴 =
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𝑝𝐴
𝑝
𝑦𝐵 =
𝑝𝐵
𝑝
Provided the mixture is ideal, the partial
pressures and the total pressure may be
expressed in terms of the mole fractions in the
liquid by using Raoult’s Law and the equation
for the total vapor pressure. The result of
combining these relations is:
𝑦𝐴 =
𝑝𝐵∗ +
𝑥𝐴 𝑝𝐴∗
(𝑝𝐴∗ −
𝑦𝐵 = 1 − 𝑦𝐴
𝑝𝐵∗ )𝑥𝐴
→ 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑎𝑝𝑜𝑟
Temperature–Composition Diagram
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A temperature–composition diagram is a phase
diagram in which the boundaries show the
composition of the phases that are in
equilibrium at various temperatures (and a
given pressure, typically 1 atm).
Temperature–composition diagrams are central
to the discussion of distillation.
They are normally constructed from empirical
data on the composition of the phases in
equilibrium at each temperature.
The horizontal axis of the diagram denotes the
value of the mole fraction xA when interpreting
the ‘Liquid’ line and the mole fraction yA when
interpreting the ‘Vapour’ line
Tie line: horizontal line at a temperature
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Lever Rule: The distances ℓ𝛼 and ℓ𝛽 are used
to find the proportions of the amounts of the
vapor and liquid present at equilibrium. The
lever rule is so called because a similar rule
relates the masses at two ends of a lever to their
distances from a pivot.
The lever rule applies to any phase diagram, not
only to liquid−vapor equilibria.
𝑛𝛼 ℓ𝛼 = 𝑛𝛽 ℓ𝛽
Distillation
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In a simple distillation, the vapor is withdrawn
and condensed. This technique is used to
separate a volatile liquid from a non-volatile
solute or solid.
In fractional distillation, the boiling and
condensation cycle is repeated successively.
This technique is used to separate volatile
liquids.
The efficiency of a fractionating column is
expressed in terms of the number of theoretical
plates, the number of effective vaporization
and condensation steps that are required to
achieve a condensate of given composition
from a given distillate.
Azeotropes: When so much A has been
evaporated that the liquid has reached the
composition b, the vapor has the same
composition as the liquid. Evaporation then
occurs without change of composition. The
mixture is said to form an azeotrope.
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When the azeotropic composition has
been reached, distillation cannot
separate the two liquids because the
condensate has the same composition
as the azeotropic liquid.
PHASE DIAGRAMS OF BINARY SYSTEMS:
SOLIDS
MULTICOMPONENT SYSTEMS
Liquid-liquid Phase Diagrams
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partially miscible liquids: liquids that do not
mix in all proportions at all temperatures
The upper critical solution temperature, 𝑻𝒖𝒄
(or upper consolute temperature), is the highest
temperature at which phase separation occurs.
Above the upper critical temperature, the two
components are fully miscible. This
temperature exists because the greater thermal
motion overcomes any potential energy
advantage in molecules of one type being close
together.
The thermodynamic interpretation of the upper
critical solution temperature focuses on the
Gibbs energy of mixing and its variation with
temperature.
Some systems show a lower critical solution
temperature, 𝑻𝒍𝒄 (or lower consolute
temperature), below which they mix in all
proportions and above which they form two
phases.
Additional Notes:
1. Raoult’s law is used to calculate the total vapor
pressure of a binary system of two volatile
liquids.
2. A temperature–composition diagram is a
phase diagram in which the boundaries show
the composition of the phases that are in
equilibrium at various temperatures.
3. The composition of the vapor and the liquid
phase in equilibrium are located at each end of
a tie line.
4. The lever rule is used to deduce the relative
abundances of each phase in equilibrium.
5. Separation of a liquid mixture by fractional
distillation involves repeated cycles of boiling
and condensation.
6. An azeotrope is a liquid mixture that
evaporates without change of composition.
7. Phase separation of partially miscible liquids
may occur when the temperature is below the
upper critical solution temperature or above
the lower critical solution temperature; the
process may be discussed in terms of the model
of a regular solution.
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The isopleth (constant-composition line) at 𝑒2
in Fig. 5D.1 corresponds to the eutectic
composition, the mixture with the lowest
melting point.
A liquid with the eutectic composition freezes
at a single temperature, without previously
depositing solid A or B.
A solid with the eutectic composition melts,
without change of composition, at the lowest
temperature of any mixture.
Solutions of composition to the right of 𝑒2
deposit B as they cool, and solutions to the left
deposit A: only the eutectic mixture (apart from
pure A or pure B) solidifies at a single definite
temperature without gradually unloading one or
other of the components from the liquid.
Additional Notes:
1. At the eutectic composition the liquid phase
solidifies without change of composition.
2. A peritectic line in a phase diagram represents
an equilibrium between three phases.
3. In congruent melting the composition of the
liquid a compound forms is the same as that of
the solid compound.
4. During incongruent melting, a compound
melts into its components and does not itself
form a liquid phase.