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Thermodynamics of Solution Crystallization

Peter VekilovUniversity of Houston

World market $72 billion/year

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World market $304 billion/year

World market $350 billion/year

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Flint and Quarts Crystals

Kremuk

Hematite Drawings

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Synthetic Crystals

The history of salt-making in salt-pans, from sea water or salt springs, goes further back than human records.

G.AgricolaDe re metalica (1556)

Insulin BiosynthesisG. Dodson, D. Steiner,Curr. Op.Struct. Biol. 8 (1998) 189

Single crystal per vesicle

Fast crystal growth

Ready response to fluctuations in conversion rate

Crystals exclude proinsulin present in islet cells

Slow dissolution at undersaturation

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0th, I, II, and III laws

Equations of stateWeighted averaging of molecular states

Predictions of thermodynamic properties Sta

tistic

al A

ppro

ach

Cla

ssic

al A

ppro

ach

Classical and Statistical Thermodynamics

The Four Laws of Thermodynamics

0th Law

A B C

A B C

B AIf

and B C

then A C

1st Law

U Q W Ek EpU Q W

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Entropy

2nd Law

Suniverse 0

Sisolated system 0

3rd Law

at T = 0 K

S = 0

IsolatedSystem

dV =0,dS =0

Environment

T=const

dT =0,dp =0

Other Formulations of the Second Law

Sisolated system 0

US,V 0

Gp,T 0

dG=VdpSdT+

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0Atp =const,

0

0 0

Other Formulations of the Second Law

T=const

dT =0,dp =0

Homogenous and Heterogeneous Systems

Homogeneous: properties do not change or change smoothly

Heterogeneous: properties change abruptly

Phase: homogeneous part of a heterogeneous system

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x

Fluctuations

The only way forward for activated processes

E

L

S1

S2

Gibbss Phase Rule

Equilibrium between phases: N components

T T T.p p p

f=N +2

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C

p

T

Ce(p,T)

Solubility of a Crystal

N = 2 (solvent and solute) = 2 (crystal and solution)

f = N + 2 = 2

Ce = Ce(p, T)

Ce = Ce(T)p = 1 atm

Solvent and SoluteSolution: a mixture of two or more components

majority component: solventpresent at lower concentration: solutes

At p = const and T = const

dG =

ln ln

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Molecular-level View of the Solution

n1 n2 n1 +n2

Ideal Solutions

ln

1ln 1 2ln 2

1 2 ln 1 2 2ln 2

n1 n2 n1 +n2

0

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Regular Solution Model

0

n1 n2 n1 +n2

2 2 2

m12=znx2(1x2)

2

H/nkBT= x2(1x2)

G/nkBT=(1x2)ln(1x2)+x2lnx2+12x2(1x2)

Regular Solution Modeln1 n2 n1 +n2

, , 2 ln 1

ln lnln = 1

: properties of pure soluteRTlnC2: entropy of dissolution

RTln2: interactions between solute and solvent

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The Crystallization Driving Force

dG =dGcryst +dGs =

dG=

0onlyif

If thendG

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The Crystallization Driving Force

ln ln ,

ln / ,ln , ln ,

ln / ,

Multi-component Systems

Ca2+(s) + CO32-(s) CaCO2(cryst).

, ,

ln

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Thermodynamic Effects of the Solvent

Na+(gas) + Cl-(gas) NaCl(solid)

Na+(gas) + nH2O Na+(aq)

Cl-(gas) + nH2O Cl-(aq)

Na+(aq) + Cl-(aq) NaCl(solid)

H0 =788kJmol1

H0 =19kJmol1

H0 =406kJmol1.

H0 =366kJmol1

The Effect of Temperature

,

ln ,

ln ,,

C

T

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Thermodynamics of Hematin Crystallization

0 20 40

0.15

0.30

0.45

BulkAFM

c e(m

M)

T (oC)

1/T (1/K)

lnc e

0.0032 0.0036-3-2-1

The crystallization enthalpy = -378 kJ mol-1The crystallization entropy = -497 J mol-1K-1

suggests a process leading to higher disorder accompanies hematin crystallization

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T

T1T2

The Effect of Temperature

ln ln

ln / ,

ln / ,

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The Phase diagram of Ionic Solutes

100

2040

200 40 60 80 100

200

406080

Brine Brine+NaClcryst

Brine+NaCl.2H2Ocryst NaCl.2H2Ocryst+NaClcrystIce+NaCl.2H2Ocryst

Ice+Brine

NaClConcentration[%wt.]

Temperature

[o C]

Theoretical phase diagrams ofsmall-molecule solutes

T

C2

Solidus

Tcr

Ttr

C2,cr

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Theoretical phase diagrams ofof protein and colloid solutions

T

C2

Tcr

C2,cr

Solidus

Macroscopic Methods of Solubility Determination

t

C2

t

C2

t

T,I

Te

C2,e

C2,e

(a)

(b)

(c)

Crystal growth

Crystal dissolution

Step wise dissolution

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Microscopic Methods of Solubility Determination

SurfaceCoordinate mmTim

es

0

250

50100150200

1

22

0 2 4 6 108

12

3Times

050100150200

0.0

1.0

2.0

3.0

0 5 10 15 20 25 30Time [days]

Prot

ein

Con

cent

ratio

n [m

g/m

l]

4 C25 C

C2 =2.5mgml1

C2 =2.3mgml1

Microscopic Methods of Solubility Determination

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RudolfClausius(1822 1888)

JosiahWillardGibbs(1839 1903)