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Home , Phase separation

5.8 Liquid-Liquid diagrams

Key points 1. Phase separation of partially miscible liquids may occur when T is below UCST or above LCST 2. UCST is the highest T at which phase separation occurs. LCST is the lowest T at which phase separation occurs 3. The outcome of a distillation of a low-boiling azeotrope depends on whether the liquids become fully miscible before they boil or boiling occurs occurs before mixing is complete 4. Consider partially miscible liquids 5.8 Liquid-Liquid phase diagrams

(a) Phase separation • Suppose a small amount of liquid B is added to a sample of another liquid B at T’. Liquid B dissolves completely forming a single phase • As more B is added, a stage comes at which no more dissolves. The sample consists of two phases in equilibrium; A-rich phase (a’’) and B-rich phase (a’) • Lever rule is still applied • As more B is added, a stage comes at so much B is present and it can dissolve all the A forming a single phase 5.8 Liquid-Liquid phase diagrams

(b) Critical temperature • Upper critical solution temperature (UCST) : the highest temperature at which phase separation occurs

• The phase boundary: ∂∆mixG/∂x = 0 x ln +ξ (1− 2x) = 0 1− x • Lower critical solution temperature (LCST) : below LCST they mix in all proportions and above which they form two phases • Some systems have both upper and lower critical solution temperatures (Figure 5.47) 5.8 Liquid-Liquid phase diagrams

(c) The distillation of partially miscible liquids • Consider a pair of liquids that are partially miscible and form a low- boiling azeotrope • This combination is quite common because both properties reflect the tendency of A and B to avoid each other • (1) The liquids become fully miscible before they boil and (2) the boiling occurs before mixing is complete

• Distillation of a1 leads to a vapor of composition b1 • b1 condenses to the completely miscible single-phase solution at b2 • When this distillate is cooled to b3, phase-separation occurs 5.8 Liquid-Liquid phase diagrams

(c) The distillation of partially miscible liquids • Distillate obtained from a liquid of composition of a1 has composition b3 • The distillate is a two-phase of b3’ and b3’’

• A system at e1 forms two phases, which persist up to the boiling point at e2. The vapor of this mixture has the same composition as liquids (azeotrope)

• Condensing a vapor of composition e3 gives a two-phase liquid of the same overall composition 5.9 Liquid-solid phase diagrams

Key points 1. At the eutectic composition the liquid phase solidifies without change of composition 2. The phase equilibria of binary systems in which the components react may also be summarized by a 3. In some cases, a solid compound does not survive melting 5.9 Liquid-solid phase diagrams

(a) Eutectics

• a1 → a2 : The system enters the two-phase region labeled ‘Liquid+B’. Pure solution begins to come out of solution and remaining liquid becomes richer in A

• a2 → a3 : More of the solid B forms • a3 → a4 : At the end of this step, there is less liquid a3, and its composition is given by e2. This liquid now freeze to give a two-phase system of pure B and pure A

• The isopleth e2 corresponds to “the eutectic composition”: a liquid with the eutectic composition freezes at a single temperature, without previously depositing solid A or B • Thermal analysis is a powerful way of detecting eutectics • Eutectic halt 5.9 Liquid-solid phase diagrams

(b) Reacting systems • Many binary react to produce compounds • For example, Ga + As ⇌ GaAs • Although three constituents are present, there are only two components • Consider A + B ⇌ C • Consider a system prepared by mixing an excess of B with A that consists of C and unreacted B • The solid deposited on cooling along the isopleth a is C • Below a4 there are two solid phases: solid B and solid C phases • The pure compound C melts congruently, that is the composition of the liquid it forms is the same as that of the solid 5.9 Liquid-solid phase diagrams (c) Incongruent melting • C is not stable as a liquid (Liquid Na2K is unstable)

• a1 → a2 : A rich in Na is deposited

• a2 → just below a3 : The sample is entirely solid and consists of a solid solution rich in Na and solid Na2K

• b1 → b2 : Solid solution rich in Na begins to deposit • b2 → b3 : A solid solution rich in Na deposits, but a reaction occurs to form Na2K • b3 : Three phases are in mutual equilibrium; liquid + Na2K + a solid solution rich in Na (peritectic line)

• b3 → b4 : The amount of solid compound increases until at b4 the liquid reaches its eutectic composition. It solidifies to give a two- phase solid of a solid solution rich in K and solid Na2K 5.10 The activity

Key points 1. The activity is an effective that preserves the form of the expression for the chemical potential 5.10 The solvent activity

• The general form of the chemical potential of a real or ideal solvent is given by a straightforward modification of µA = µA* + RTln(pA/pA*) • For an , the solvent obeys Raoult’s law and µA = µA* + RTln(xA) • The form of this relation can be preserved when the solution does not obey Raoult’s law ∗ pA µA = µA + RT ln aA aA = ∗ pA

• The quantity aA is the activity of A (an “effective” ) • The activity coefficient, ϒ

aA = γAxA and γA →1 as xA →1 ∗ µA = µA+ RT ln xA + RT ln γA 5.11 The solute activity

Key points 1. The chemical potential of a solute in an ideal- dilute solution is defined on the basis of Henry’s law 2. The activity of a solute takes into account departures from Henry’s law behavior 3. An alternative approach to the definition of the solute activity is based on the of the solute 4. The biological standard state of a species in solution is defined as pH = 7 (and 1 bar) 5.11 The solute activity

(a) Ideal-dilute

• A solute B that satisfies Henry’s law has a vapor pressure pB = KBxB

∗ pB ∗ KB µB = µB + RT ln ∗ = µB + RT ln ∗ + RT ln xB pB pB

• KB and pB* are characteristics of solute Θ ∗ KB µ B = µB + RT ln ∗ pB Θ µB = µ + RT ln xB

Θ ∗ • If the solution is ideal, KB = pB* and µB = µB 5.11 The solute activity

(b) Real solutes

• We now permit deviations from ideal-dilute, Henry’s law behavior

Θ pB µ B = µ + RT ln aB aB = K B • An activity coefficient : all the deviations from ideality are captured in the activity coefficient ϒB.

aB = γ BxB

aB → xB and γ B →1 as xB → 0 5.11 The solute activity

(c) Activities in terms of

• The selection of a standard state is entirely arbitrary • In chemistry, compositions are often expressed as molalities, b, in place of mole fractions

Θ µB = µB + RT ln bB • We incorporate the deviations from ideality by introducing an activity and a dimensionless activity coefficient b a = γ B where γ →1 as b → 0 B B bΘ B B µ = µΘ + RT ln a 5.11 The solute activity

(d) The biological standard state

• The conventional standard state of hydrogen ions : unit activity corresponding to pH = 0 • In biochemistry, it is common to adopt the biological standard state, in which pH = 7 • G⊕, H⊕, µ⊕, and S⊕ for the biological standard state µ (H+ ) = µΘ (H+ ) + RT ln a(H+ ) = µΘ (H+ ) −(RT ln10)× pH µ ⊕ (H+ ) = µΘ (H+ ) − 7RT ln10 • 7RTln10 = 39.96 kJ/mol. The two standard values differ by about 40 kJ/mol 5.12 The activities of regular solutions

Key points 1. The Margules equations relate the activities of the components of a model regular solution to its composition 2. The Margules equations lead to expressions for the vapor pressure of the components of a regular solution 5.12 The activities of regular solutions

• Starting point is the expression for the Gibbs energy of mixing for a regular solution

ΔmixG = nRT (xA ln xA + xB ln xB +ξ xA xB ) • Margules equations

ΔmixG = nRT(xA lnaA + xB lnaB ) 2 2 lnγ A = ξ xB lnγ B = ξ xA 2 2 ξxB ξ(1−xA ) aA = γ A xA = xAe = xAe 2 p = x eξ(1−xA ) p∗ A { A } A ξ ∗ pA = xAe pA for x << 1 K 5.13 The activities of ions in solutions

Key points 1. Mean activity coefficients apportion deviations from ideality equally to the cations and anions in an ionic solution 2. The Debye-Hückel theory ascribes deviations from ideality to the Coulombic interaction of an ion with the ionic atmosphere that assembles around it 3. The Debye-Hückel theory limiting law is extended by including two further empirical constants 5.13 The activities of ions in solutions

(a) Mean activity coefficients

• Interactions between ions are so strong that the approximation of replacing activities by molalities is valid only in very dilute solutions • In precise work activities themselves must be used + • µ+ : the chemical potential of M - • µ- : the chemical potential of X • The molar Gibbs energy of an ideal solution of such ions : Gideal = µ ideal + µ ideal m + - • Real solution of such ions G = µ + µ = µ ideal + µ ideal + RT lnγ + RT lnγ = Gideal + RT lnγ γ m + − + - + − m + − • All deviations from the ideality are contained in the last term 1 2 γ± = (γ+γ− ) : the mean activity coefficient µ = µ ideal + RT lnγ µ = µ ideal + RT lnγ +− + ± −− - ± 5.13 The activities of ions in solutions

(a) Mean activity coefficients

• Generalize the previous approach to the case of a compound MpXq

ideal Gm = pµ+ + qµ− = Gm + pRT lnγ+ + qRT lnγ−

p q 1 s γ± = (γ +γ − ) s = p + q

• The chemical potential of eqch ion

ideal µi = µi + RT lnγ± 5.13 The activities of ions in solutions

(b) The Debye-Hückel limiting law

• The long range and strength of the Coulomb interaction means that it is likely to be primarily responsible for the departures from ideality in ionic solutions and to dominate all the other contribution to non-ideality • Oppositely charged ions attract one another • Overall, the solution is electrically neutral, but near any given ion there is an excess of counter ions : ionic atmosphere • The chemical potential of any given central ion is lowered as a result of its electrostatic interaction with its ionic atmosphere by RTlnϒ± 1 2 ln γ± = − z+z− AI :the Debye-Huckel limiting law

1 2 Θ I = ∑zi (bi b ) :the ionic strength 2 i • For a solution consisting of two types of ions, 1 I = (b z2 + b z2 ) bΘ 2 + + − − 5.13 The activities of ions in solutions

(c) The extended Debye-Hückel law

• The name ‘limiting law’ was applied because ionic solutions of moderate molalities may have activity coefficients that differe from the values given by Debye-Hückel limitingn law • When the ionic strength is high enough, A z z I1 2 lnγ = − + − +CI ± 1+ BI1 2 • B and C are dimensionless constants and are best regarded as an adjustable empirical parameters