PHASE EQUILIBRIA AND THE PHASE RULE - …live.ispatula.site/1st_Year/2nd_Semester/Physical1/fall_2016/... · phenol and water will be completely miscible and will be one phase. 10/11/2016

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PHASE EQUILIBRIA AND THE

PHASE RULE

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Introduction

• The three primary phases of matter are often defined individually under different conditions.

• In practice we encounter states of matter in coexistence.

• The position of the equilibrium between these states is affected by the environment (temperature, pressure) and composition.

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The phase rule

• The Phase Rule

– A relationship for determining the least number of intensive variables (independent variables that do not depend on the phase volume or mass such as temperature, pressure, density or concentration) that can be changed without changing the equilibrium state of the system.

Or, in other words:

– The least number required to define the state of the system.

F = C – P + 2

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F = C - P + 2

• F: Number of Degrees of Freedom The least number of Intensive variables that must be fixed/ known to describe the system completely. Ex. temperature, concentration, pressure, density, etc.

• P: Number of Phases

A consistent, homogenous, physically distinct segment of a system that is separated from other segments of the system by bounding surfaces.

• C: Number of Components or Chemical Entities

Constituents by which the composition or make-up of each phase in the system can be expressed. Usually the chemical formula or chemical equation is used.

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F = C - P + 2

• The phase rule is used to study and understand the way that temperature, pressure, concentration, etc. effect the phase of a substance.

• Application: – A given mass of a gas, e.g. water vapour confined to a

particular volume.

– Apply phase rule: F=1-1+2=2.

– This means that two intensive variables (temperature and pressure) must be known to duplicate this system exactly.

– Such a system is usually described as bivariant.

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F = C - P + 2

• Application: – A liquid such as water in equilibrium with its vapor ( we have 2 phase

system)

– F=1-2+2=1.

– By stating temperature, the system is completely defined because the pressure under which liquid and vapor can coexist is also fixed.

– If we decide to work under a particular pressure, then the temperature of the system is automatically defined:

– The system is described as univariant.

• Application: – When we have a liquid water, vapor and ice

– Phase rule states that the degrees of freedom = 1-3+2=0

– There are no degrees of freedom, if we attempt to vary the conditions of temperature or pressure necessary to maintain the system, we will lose a phase.

– The combination is fixed and unique.

– The system is invariant.

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F = C - P + 2

• As the number of phases in equilibrium increases, the number

of the required degrees of freedom becomes less.

• As the number of components increases, so do the required

degrees of freedom needed to define the system.

Consequently, as the system becomes more complex, it

becomes necessary to fix more variables to define the system.

• Liquid water+vapor F=1-2+2=1

• Liquid ethanol+vapor F=1-2+2=1

• liquid water+liquid ethanol+vapor F=2-2+2=2

• liquid water+liquid benzyl alcohol+vapor mixture F=2-3+2=1

(benzyl alcohol and water form two separate liquid phases and

one vapor phase).

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One component systems

Figure – Phase diagram for water at moderate pressures

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• The previous figure (phase diagram for water at moderate

pressures), the curve OA is known as the vapor pressure

curve; its upper limit is at the critical temperature, 374ºC

for water, and its lower end terminates at 0.0098ºC, called

the triple point. Along the vapor pressure curve, vapor and

liquid coexist in equilibrium.

• Curve OC is the sublimation curve, and here vapor and

solid exist together in equilibrium.

• Curve OB is the melting point curve at which liquid and

solid are in equilibrium. The negative slope of OB shows

that the freezing point of water decreases with increasing

external pressure.

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• If t1 (above the critical temperature) is held constant, no

matter how much the pressure is raised, the system

can’t be liquefied.

• At t2 below the critical temperature and above the triple

point, water vapor is converted into liquid water by an

increase in pressure.

• At temperature below the triple point e.g. t3, an increase

in pressure on water in the vapor state converts the

vapor first to ice and then at higher pressure into liquid

water.

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• In any one of the three regions in which pure solid, liquid,

or vapor exists, F=1-1+2=2, so two variables (e.g.

temperature and pressure are needed to describe the

system.

• Along any of the three lines, where two phases exist,

F=1-2+2=1, hence, only one condition (either

temperature or pressure) is needed to define the system.

• Finally, at the triple point where the three phases exist,

F= 1-3+2= 0.

The triple point for air-free water is at 0.0098ºC and

4.58mm Hg.

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Condensed Systems:

– These are systems in which the vapor phase is

ignored and only solid and/or liquid phases are

considered.

– The vapor phase is disregarded and the pressure is

considered to be 1atm.

– The variables considered are: temperature and

concentration.

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Two component systems containing

liquid phases:

• Miscible systems such as water and ethanol.

• Immiscible systems such as water and mercury.

• Partial miscibility (or immiscibility) such as phenol and

water.

• Critical solution temperature: The maximum temperature

at which the two – phase region exists.

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liquid phases:

• Phenol and water.

– These liquids are partially miscible in each other.

– At certain ratios the liquids are completely miscible and at others they are immiscible.

– The 2 degrees of freedom of this mixture are temperature and concentration (condensed system).

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Two component systems containing liquid phases:

Figure – Temperature-composition diagram for the system

consisting of water and phenol

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• The curve gbhci shows the limits of temperature and

concentration within which two liquid phases exist in

equilibrium.

• The region outside this curve contain systems with only

one liquid phase.

• The maximum temperature that these two liquids can

exist in a 2 phase system is 66.8 °C.

• This is called the critical temperature or the upper

consolute temperature.

• In this system above 66.8 degrees all combinations of

phenol and water will be completely miscible and will be

one phase.

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At a given temperature, say 50 °C:

• Starting at the point a, a system with 100% water and adding known increments of phenol to a fixed weight of water while maintaining the system at 50oC will result in the formation of a single liquid phase (water with dissolved phenol).

• This continues until point b. At point b, a minute amount of a second phase appears.

• The concentration of phenol and water at which this occurs in 11% by weight of phenol in water.

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• As we prepare mixtures with increasing quantities of phenol (we proceed from point b to point c), the amount of the phenol-rich phase (B) increases and the amount of the water-rich phase (A) decreases.

• Once the total concentration of phenol exceeds 63% at 50oC, a single phenol-rich liquid phase is formed.

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• The line bc drawn across the region containing two phases is

termed the tie line (parallel to the baseline).

• All systems prepared on a tie line, at equilibrium will separate

into phases of constant composition (corresponding to the

solubility of each component in the other), these phases are

termed conjugate phases.

• Any system represented by a point on the line bc at 50oC

separate to give a pair of conjugate phases (A and B) whose

compositions are b and c, respectively.

• The upper one (A) (water rich phase) has a composition of 11% phenol in water and the lower one (B) (phenol rich phase) has a composition of 63% phenol in water.

• Phase B will lie below phase A because it is rich in phenol which has a higher density than water.

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• The relative amounts of the two layers or phases vary.

• The relative weights of the two phases can be calculated using the following formula:

• If we prepared a system containing 24% by weight of phenol and 76% by weight of water (point d), at equilibrium we will have two liquid phases. The relative amounts (A/B) will be:

(63%-24%)/(24%-11%)= 39/13= 3/1

Hence if the total mix was 100 g…Phase A=75g and B= 25g

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If we prepared a system containing 24% by weight of phenol and 76% by weight of water (point d), at equilibrium we will have two liquid phases. The relative amounts (A/B) will be:

(63%-24%)/(24%-11%)= 39/13= 3/1

Hence if the total mix was 100 g…

Phase A=75g and B= 25g

Phase A has 11% phenol, and Phase B has 63% phenol

So, amount of phenol in A= 75 * 0.11= 8.25 g

And amount of phenol in B= 25 * 0.63 = 15.75 g

The total amount of phenol in the system= 8.25 +15.75=24

So 24/100= 24% w/w

Similarly, do the calculations for water in each phase and

in the total system.

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• In the previous example (at point d), if the total amount

of the two liquids was 36 g. Do the calculations.

• What about the points e (37% phenol) and f (50%

phenol)?

• See example discussed page 40 in the fourth edition. The

example is discussed page 51 in the fifth edition.

• See problem 2-21 page 52 in the fourth edition. If you have

the fifth edition, please solve problem 2-21 page 687.

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• Upper consolute temperature: All combinations above this

temperature are completely miscible (one phase).

• Lower consolute temperature: below which the

components are miscible in all proportions.

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Two component systems containing solid and

liquid phases:

• Solid – liquid mixtures in which both components are miscible in the liquid state and completely immiscible as solids.

• This is because the solid phases consist of pure crystalline components.

• These are said to be eutectic mixtures.

• The name (eutectic) comes from the greek 'eutektos', meaning 'easily melted‘.

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Two component systems containing solid

and liquid phases: • The phase diagram of salol-thymol system is an example

of an eutectic mixture.

• The phase diagram is made of 4 regions:

i. A single liquid phase

ii. A region containing solid salol and a conjugate liquid phase

iii. A region in which solid thymol is in equilibrium with a conjugate liquid phase

iv. A region in which both components are present as pure solid phases

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liquid phases:

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and liquid phases:

• Example: 60% thymol in salol (w/w): • At 50°C (point x), one liquid phase.

• Upon cooling, the system remains as a single liquid phase until the temperature falls to 29C, at which a small amount of thymol solidifies and separates out, leading to a two phase system (solid thymol + liquid phase (thymol and salol)).

• Further cooling leads to larger amounts of thymol separating out from the liquid phase.

• At 25C (point x1): the system is composed of two phases: a liquid phase and pure solid thymol. The phase ratio and compositions can be figured out from the tie line (a1b1). System x1 (as all mixtures on this tie line) has 2 phases with constant compositions, liquid phase (a1, 53% thymol in salol) and pure solid thymol (b1, 100% thymol). The weight ratios of a1 to b1 is (100-60)/(60-53)= 40/7= 5.7/1. Therefore, if you have a 100 g mix (i.e. 60g thymol and 40 salol), then the system is composed of two phases: a1, liquid phase (85.1g and solid thymol, b1 (14.9 g). The liquid phase, a1 contains 53% thymol (85.1*0.53= 45.1g)……..Total thymol=45.1+ 14.9= 60g

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and liquid phases:

• Cont….example: 60% thymol in salol (w/w): • Further cooling of the system will lead to more solidification of thymol.

So at 20C (point x2): the system has two phases, a liquid phase, a2 (45% thymol in salol) and a pure solid thymol, b1 (100% thymol). The weight ratio a2/b2 is (100-60)/(60-45)= 2.67/1. Therefore, if you have a 100 g mix (i.e. 60g thymol and 40 salol), then the system is composed of two phases: a1, liquid phase (72.76g) and solid thymol, b1 (27.25 g). The liquid phase, a1 contains 45% thymol (72.76*0.45= 32.75g)……..Total thymol=32.75+ 27.25= 60g.

• As the system is cooled, more and more thymol separates out as a solid.

• Below 13C, the liquid phase disappears and the system contains an equilibrium mix of two solid phases, pure solid salol and pure solid thymol.

• For example, at 10C (point x4), the system contains of two solid phases, solid salol a4 and solid thymol b4 with weight ratio a4/b4= (100-60)/(60-0)= 40/60= 0.67/1

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Two component systems containing solid and liquid phases:

• The eutectic temperature is the lowest temperature at which the liquid phase can occur. Above the temperature the components are liquid and below the components are solids. At this temperature, the components in the liquid state pass all at once to the solid phase.

• A eutectic composition is the composition of two (or more) compounds that exhibits a melting temperature lower than that of any other mixture of the compounds.

• In the previous example, it was 13C and this occurs for a mix contains 34% thymol in salol. This point on the phase diagram is known as the eutectic point.

• This is attributed to the fact that the solid phase in the eutectic composition is an intimate mixture of fine crystals of two compounds. This leads to the phenomenon of “contact melting”, resulting in the lowest melting point over a composition range.

• The primary criterion for eutectic formation is the mutual solubility of the components in the liquid or melt phase.

• Systems of two or more components show different interactions (how solutes interact in solution) at different concentrations, with the eutectic point provides the favoured composition for the solutes in solution.

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and liquid phases:

• At the eutectic point, three phases (liquid, solid salol, solid thymol) coexist.

• The eutectic point in this condensed system represent an invariant system (2 components and 3 phases) (pressure is fixed).

• Prilocaine (mp 37-38oC) and lidocaine (mp 68-69oC) are local anesthetics that form an eutectic mixture at 1:1 ratios with a melting point of 18oC, this combination can be used to prepare a liquid dosage form that may be used for topical applications.

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and liquid phases:

Solid dispersions: 1. Eutectic systems: the solid phases constituting the eutectic

each contain only one component and the system may be regarded as an intimate crystalline mixture of one component in the other.

2. Solid solution: each solid phase contains both components, that is, a solid solute is dissolved in a solid solvent to give mixed crystals. Solid solutions are typically not stoichiometric and the minor component (guest) inserts itself into the structure of the host crystal taking advantage of:

– The molecular similarities and/or

– Open spaces in the host lattice

3. Molecular dispersion of one component in another where the overall solid is amorphous.

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solid and liquid phases:

Solid dispersions:

• There is a widespread interest in solid dispersions because they may offer a means of facilitating the dissolution and bioavailability of poorly soluble drugs when combined with freely soluble carriers (hosts) such as urea or polyethylene glycol.

• This is usually achieved in solid dispersions by reducing the particle size, reduced aggregation and agglomeration and enhanced wetting.

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liquid phases:

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Phase equilibrium in three component

systems:

• As one proceeds from binary systems to systems with three or more components, the phase diagrams become more complex.

• Each component adds another dimension to the representation of the phase equilibrium.

• Thus, for three components, two dimensions are required to represent the phase diagrams for a single temperature and pressure; these are conveniently depicted by a triangular diagram in which each vertex represents a pure component.

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Phase equilibrium in three component

systems:

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Phase equilibrium in three component systems:

Rules relating to Triangular diagrams:

• Each corner or apex (A, B, or C).

• Each line joining the corner points.

• The area within the triangle (e.g. point x).

• If a line is drawn through any apex to a point on the opposite side (e.g. line CD).

• If a line is drawn parallel to one side of the triangle (e.g. line HI).

Refer back to the book for more details.

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Ternary Systems with one pair of partially miscible

liquids

• A ternary system is composed of 3 liquids (A, B and C).

• Water (liquid A) and benzene (liquid C) are partially miscible, so a mixture of the two usually produces a two-phase system (i.e. line AC). “Water rich phase” in the bottom and the “benzene rich phase” on the top.

• Alcohol (liquid B) is completely miscible with both water and benzene. So it works as a cosolvent.

• Hence, it is expected that the addition of sufficient amount of alcohol (B) to a two-phase system (A and C) would produce a single-liquid phase (see bd and je lines). Alcohol in this case is acting in a manner comparable to that of temperature in the binary phenol-water system. See figure on the next slide.

• The curve afdeic, frequently termed the binodal curve, marks the extent of the two-phase region.

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Water Benzene

Alcohol

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Ternary Systems with one pair of partially

miscible liquids

• The tie lines in this case are not necessarily parallel to one another or to the baseline. The directions of the tie lines are related to the shape of the binodal curve which in turn depends on the relative solubility of the added component (B) in the other two components (A and C). It seems in this case that liquid B is more soluble in liquid C than in liquid A.

• Would the tie lines run parallel to the base line?

• The properties of tie lines discussed earlier still apply.

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Ternary Systems with one pair of partially

miscible liquids

For example: Tie line (fi):

Concentrations are presented as % w/w (A,B, C)

f (90, 5, 5) and i (10, 20, 70).

It seems that along the tie line, an increase of C% necessitates the addition of increasing amounts of B to keep the conjugate phases with constant compositions. This is attributed to the higher solubility of B in C than that of B in A. Therefore, the tie lines are not parallel to the base line in this case.

Now, all systems prepared on this tie line (e.g. g and h) will give rise to two phases having compositions denoted by the points f and i. Again, the relative amounts of the two conjugate phases can be figured out from the position of this system on the tie line.

For example, system h will separate to two conjugate phases, water rich phase (f: 90,5,5) and benzene rich phase (i: 10,20,70). The weight ratios f:i equals hi: hf. As the point h is on the halfway along the tile line, equal amounts of the two phases will be produced at equilibrium.

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Phase equilibrium in three component systems:

• In a system containing three components, but only one phase there are 4 degrees of freedom. These are temperature, pressure, and the concentrations of 2 of the 3 components.

• If we hold the temperature constant and the system is condensed where pressure is held at 1atm then the degrees of freedom of the system are only 2 (the concentrations of 2 components).

• In the two-phase region, F= 1, so the concentration of one component is enough.

• In the previous figure (a system of three liquids, one pair of which is partially miscible), F=2 in a single-phase region, so two concentrations have to fix the particular region. Within the binodal curve, afdeic, F=1 so we need to know one concentration term, since this will allow the composition of one phase to be fixed on the binodal curve. From the tie line, we can then obtain the composition of the conjugate phase.

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Q1) A 100 g mix contains:

35 g Chloroform

30 g Acetic acid

35 g Water

How many phases?

Relative weights of phases?

Compositions of each phase?

HOMEWORK

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Q2) A ternary mix contains:

15 g Chloroform

10 g Acetic acid

25 g Water

How many phases?

Calculate the amount of acetic

acid distributed in each phase.

HOMEWORK

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Q3) A binary mix composed of:

10 g Chloroform and 10 g Water

What’s the lowest amount of

acetic acid required to get one

liquid phase system?

HOMEWORK

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PHASE EQUILIBRIA AND THE PHASE RULE - …live.ispatula.site/1st_Year/2nd_Semester/Physical1/fall_2016/... · phenol and water will be completely miscible and will be one phase. 10/11/2016