Transcript Thermodynamics and kinetics
Thermodynamics and kinetics
• • • • •
Thermodynamic laws Half-cell reactions Kinetics Acid-Base Equilibrium calculations
Speciation calculation from complexation constants
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Provide review of concepts for applications to radiochemistry
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Thermodynamic terms
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Heat Capacity (C p )
Heat required to raise one gram of substance 1 °C Enthalpy (∆H) Energy of a system (heat content)
Internal energy, volume, pressure
Accounts for energy transferred to environment by expansion or heating ∆H = ∆H products - ∆H reactants Exothermic reactions, negative ∆H
Negative ∆H tend to be spontaneous
∆H products ∆H reactants
2 H + + CO 3 2 <--> CO 2 + H 2 O
-393.5 + (-285.8)-(-677.1+2(0)) = -2.2 kj/mol
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Thermodynamic terms
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Endothermic
Reaction requires energy (at 25 °C) 2 HgO + 181.70 kj <--> 2 Hg + O 2 Enthalpy (∆H)
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Energy of a system (heat content)
Internal energy, volume, pressure
Accounts for energy transferred to environment by expansion or heating
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∆H = ∆H products - ∆H reactants Exothermic reactions, negative ∆H
Negative ∆H tend to be spontaneous
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Entropy (∆S) and Gibbs Free Energy (∆G)
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Randomness of a system
increase in ∆S tends to be spontaneous Enthalpy and Entropy can be used for evaluating the free energy of a system Gibbs Free Energy
∆G = ∆H -T∆S
∆G=-RTlnK
K is equilibrium constant
Activity at unity
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Compound ∆G° (kJ/mol) at 298.15 K H 2 O OH (aq) H + (aq)
-237.129
-157.244
0 H 2 O
H + +OH What is the constant for the reaction?
Products-reactants At 298.15 K ∆G(rxn) = 0 + -157.244 - (-273.129) = 79.9 kJ/mol lnK= (79.9E3/(-8.314*298.15))=-32.2; K=1E-14, K w = [H + ][OH ]
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Thermodynamic Laws
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1st law of thermodynamics
Energy is conserved in a system
Can be changed or transferred
Heat and work are energy transfer
∆E = q (heat absorbed) + w (work) 2nd law of thermodynamics
Reactions tend towards equilibrium
Increase in entropy of a system Spontaneous reaction for -∆G
∆G = 0, system at equilibrium 3rd law of thermodynamics
Entropies of pure crystalline solids are zero at 0 K Defines absolute zero
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Half-cell potentials
Cell reaction for
Zn and Fe 3+/2+ at 1.0 M
Write as reduction potentials
Fe 3+ + e <--> Fe 2+
°=0.77 V
Zn 2+ + 2e <-->Zn
°=-0.76 V Reduction potential for Fe 3+ is larger
Fe 3+ is reduced, Zn is oxidized in reaction Overall balanced equation
2Fe 3+ +Zn <--> 2Fe 2+ + Zn 2+
°=0.77+0.76=1.53 V Use standard reduction potential Application of Gibbs Free Energy If work is done by a system
∆G = -
°nF (n= e ) Find ∆G for Zn/Cu cell at 1.0 M
Cu 2+ + Zn <--> Cu + Zn 2+
°=1.10 V
2 moles of electrons (n=2)
∆G =-2(96487C/mole e )(1.10V)
∆G = -212 kJ/mol
2. 30RT nF log [C] c [D] d [A] a [B] b 2-6
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Kinetics and Equilibrium
Kinetics and equilibrium important concepts in examining and describing chemistry
Identify factors which determine rates of reactions
Temperature, pressure, reactants, mixing Describe how to control reactions Explain why reactions fail to go to completion Identify conditions which prevail at equilibrium General factors effecting kinetics
Nature of reactants
Effective concentrations Temperature Presence of catalysts Number of steps
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Rate Law
Concentration of reactant or product per unit time Effect of initial concentration on rate can be examined
rate = k[A] x [B] y
rate order = x + y knowledge of order can help control reaction
rate must be experimentally determined Rate=k[A] n ; A=conc. at time t, A o =initial conc., X=product conc.
Order rate equation 0 [A 0 ]-[A]=kt, [X]=kt k mole/L sec 1 1/sec 2 3 1 [A] ln[A 0 ]-ln[A]=kt, ln[A 0 ]-ln([A o ]-[X])=kt 1 [A]
1 [A o ]
kt 1 [A o ]
[X]
1 [A o ]
kt 2
1 [A o ] 2
kt 2 1 ([A o ]
[X]) 2
1 [A o ] 2
kt 2 L/mole sec L 2 /mole 2
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sec
Acid-Base Equilibria
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Brønsted Theory of Acids and Bases
Acid
Substance which donates a proton
Base
Accepts proton from another substance
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NH 3 + HCl <--> NH 4 + H 2 O + HCl <--> H 3 O + NH 3 + H 2 O <--> NH 4 + + Cl + Cl + OH Remainder of acid is base Complete reaction is proton exchange between sets Extent of exchange based on strength Water can act as solvent and reactant
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Dissociation Constants
Equilibrium expression for the behavior of acid HA + H 2 O <--> A + H 3 O + Water concentration is constant
K [A ][H 3 O ] [HA][H 2 O] •
pK a =-logK a
K a K[H 2 O]
Can also be measured for base
[A ][H 3 O ] [HA]
Constants are characteristic of the particular acid or base Acid Acetic Carbonic Phosphoric Oxalic Formula HC 2 H 3 O 2 H 2 CO 3 HCO 3 H 3 PO 4 H 2 PO 4 HPO 4 2 H 2 C 2 O 4 HC 2 O 4 K a 1.8E-5 3.5E-7 5E-11 7.5E-3 6.2E-8 4.8E-13 5.9E-2 6.4E-5
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Buffer Solutions
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Buffers can be made over a large pH range Can be useful in controlling reactions and separations
Buffer range
Effective range of buffer
Determined by pK a HA + H 2 O <--> A -
K a [A ][H 3 O ] [HA]
of acid or pK b + H 3 O -
[H 3 O ] K a [HA] [A ]
of base Write as pH
pH pK a log [HA] [A ]
The best buffer is when [HA]=[A ]
largest buffer range for the conditions
pH = pK a - log1 For a buffer the range is determined by [HA]/[A ]
[HA]/[A ] from 0.1 to 10 Buffer pH range = pK a ± 1 Higher buffer concentration increase durability
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Hydrolysis Constants
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Reaction of water with metal ion
Common reaction
Environmentally important
Strength dependent upon metal ion oxidation state 2 H 2 O <--> H 3 O +
water: + OH Water concentration remains constant, so for
K w = [H 3 O + ][OH ]= 1E-14 at 25°C Metal ions can form hydroxide complexes with water M z+ + H 2 O <--> MOH z-1+ + H + Constants are listed for many metal ion with different hydroxide amounts
Database at: http://www.escholarship.org/uc/item/9427347g
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Equilibrium
Le Châtelier’s Principle At equilibrium, no further change as long as external conditions are constant Change in external conditions can change equilibrium
A stressed system at equilibrium will shift to reduce stress
concentration, pressure, temperature N 2
+ 3 H 2 <--> 2 NH 3 + 22 kcal What is the shift due to
Increased temperature?
Increased N 2 ?
Reduction of reactor vessel volume?
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Equilibrium Constants
For a reaction
aA + bB <--> cC + dD At equilibrium the ratio of the product to reactants is a constant
The constant can change with conditions
By convention, constants are expressed as products over reactants
K [C] c [D] d [A] a [B] b •
Conditions under which the constant is measured should be listed
Temperature, ionic strength
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Activities
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Strictly speaking, activities, not concentrations should be used
K C [C] c D [D] d A [A] a B [B] b
At low concentration, activities are assumed to be 1 The constant can be evaluated at a number of ionic strengths and the overall activities fit to equations Debye-Hückel (Physik Z., 24, 185 (1923))
log A 1 0.5085Z
a 2 0.3281R
A
Z µ = molal ionic strength R A A = charge of species A = hydrated ionic radius in Å (from 3 to 11) First estimation of activity
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Activities
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Debye-Hückel term can be written as:
• D 0.5107
1 1.5
Specific ion interaction theory
Uses and extends Debye-Hückel
long range Debye-Hückel
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ij
Short range ion interaction term = specific ion interaction term
log i
Pitzer
log ß( ) logß(0) Z i 2 D ij Z 2 D ij
Binary (3) and Ternary (2) interaction parameters
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6.6
6.5
6.4
6.3
Experimental Data Cm-Humate shows change in stability constant with ionic strength Ion Specific Interaction Theory used
6.2
6.1
6.0
0.0
K +
0.5
1.0
1.5
sqrt Im
2.0
2.5
3.0
Ca 2+ Al 3+ Fe(CN) 6 4-
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Constants
Constants can be listed by different names
Equilibrium constants (K)
Reactions involving bond breaking
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2 HY <--> H 2 + 2Y Stability constants (ß), Formation constants (K)
Metal-ligand complexation
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Pu 4+ + CO 3 2 <--> PuCO 3 2+ Ligand is written in deprotonated form Conditional Constants
An experimental condition is written into equation
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Pu 4+ + H 2 CO 3 <--> PuCO 3 2+ +2H +
Constant can vary with concentration, pH
Must look at equation!
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Realistic Case
Consider uranium in an aquifer
Species to consider include
free metal ion: UO 2 2+
hydroxides: (UO 2 ) x (OH) y
carbonates: UO 2 CO 3
humates: UO 2 HA(II), UO 2 OHHA(I) Need to get stability constants for all species
UO 2 2+ + CO 3 2 <--> UO 2 CO 3 Know or find conditions
Total uranium, total carbonate, pH, total humic concentration
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Stability constants for selected uranium species at 0.1 M ionic strength Species logß UO 2 OH + UO 2 (OH) 2 UO 2 (OH) 3 UO 2 (OH) 4 2 (UO 2 ) 2 OH 3+ (UO 2 ) 2 (OH) 2+ UO 2 CO 3 UO 2 (CO 3 ) 2 2 UO 2 (CO 3 ) 3 4 UO 2 HA(II) UO 2 (OH)HA(I) 8.5
17.3
22.6
23.1
11.0
22.0
8.87
16.07
21.60
6.16
14.7±0.5
Other species may need to be considered. If total uranium concentration is low enough, binary or tertiary species can be excluded.
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Equations
Write concentrations in terms of species
[UO 2 ] tot = UO [CO 3 2 ] free =f(pH) [OH ] = f(pH) 2free +U-carb+U-hydroxide+U-humate
[HA] tot = UO 2 HA + UO 2 OHHA+ HA free Write the species in terms of metal, ligands, and constants
[(UO 2 ) x A a B b ] = 10 -(xpUO2+apA+bpB-log(UO2)xAaBb) pX = -log[X] free
[(UO 2 ) 2 (OH) 2 2+ ]=10 -(2pUO2+2pOH-22.0) Set up equations and solve for known terms
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U speciation with different CO
2
partial pressure
0% CO 2 1% CO 2
1.0
1.0
UO 2 HA(II) UO 2 (OH) UO 2 2+ 3 UO 2 HA(II) UO 2 OHHA(I)
0.8
UO 2 2+ UO 2 (CO 3 ) 3 4 UO 2 OHHA(I)
0.8
UO 2 (OH) 2
0.6
0.6
0.4
0.4
UO 2 (CO 3 ) 2 2-
0.2
0.2
0.0
2.0
4.0
pH
6.0
1.0
8.0
10.0
0.0
2.0
UO 2 2+ UO 2 HA(II) UO 2 OHHA(I) UO 2 (CO 3 ) 3 4-
0.8
4.0
pH
6.0
8.0
10.0
0.6
0.4
0.2
0.0
UO 2 (CO 3 ) 2 2-
2-22 2.0
4.0
pH
6.0
8.0
10.0
10% CO 2
Comparison of measured and calculated uranyl organic colloid
1.0
0.8
0.6
0.4
0.2
0.0
100% 10% 1% 0% 0.035%
2.0
4.0
pH
6.0
8.0
10.0
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Excel spreadsheets CHESS Program
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Energy terms
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Constants can be used to evaluate energetic of reaction
From Nernst equation
∆G=-RTlnK
76 74 72 70
∆G=∆H-T∆S
-RTlnK = ∆H-T∆S
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RlnK= - ∆H/T + ∆S
64 0.003
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Plot RlnK vs 1/T
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Temperature effect on Np-Humate stability
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T emp (°C)
32 24 0.0031
0.0032
1/T (K)
0.0033
²H = -22.2 ± 2.8 kJ/mol ²G 298 =-21.7 kJ/mol ²S=1.2±1.4 J/molK 0.0034
16 0.0035
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Solubility Products
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Equilibrium involving a solid phase
AgCl(s) <--> Ag + + Cl -
K [Cl ][Ag ] [AgCl ]
AgCl concentration is constant
Solid activity and concentration is treated as constant
By convention, reaction goes from solid to ionic phase in solution Can use K sp solution for calculating concentrations in
K sp K[AgCl] [Cl ][Ag ] 2-26
Solubility calculations
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K sp of UO 2 = 10 -52 . What is the expected U 4+ concentration at pH 6. Generalize equation for any pH
Solubility reaction:
UO 2 + 2 H 2 O
U(OH) 4
U 4+ + 4 OH -
K sp = [U 4+ [U 4+ ]= K sp ][OH /[OH ] 4 ] 4
pOH + pH =14
At pH 6, pOH = 8, [OH ]=10 -8 [U 4+ ]= 10 -52 /[10 -8 ] 4 = 10 -52 /10 -32 = 10 -20 M For any pH
[U 4+ ]= 10 -52 /[10 -(14-pH)*4 ]
Log [U 4+ ]= -52+((14-pH)*4)
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Limitations of K
sp
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Solid phase formation limited by concentration
below ≈1E-5/mL no visible precipitate forms
colloids formation of supersaturated solutions
slow kinetics Competitive reactions may lower free ion concentration Large excess of ligand may form soluble species
AgCl(s) + Cl <--> AgCl 2 (aq)
K sp really best for slightly soluble salts 2-28
Overview
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Understand heats of reactions
Enthalpy, entropy, Gibbs free energy Reaction data from constituents
Understand half-cell reactions
Nernst Equation
Kinetics
Influence of reaction conditions
Equilibrium and constants
Use to develop a speciation spreadsheet
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Questions
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What is the difference between 1 st kinetics?
and 2 nd order What can impact reaction rates?
How can a compound act as a base and acid? Provide an example.
What does the dissociation constant of an acid provide?
Provide the speciation of acetic acid at pH 3.5, 4.5, and 5.5.
What are the species from carbonic acid at pH 4.0, 6.0, and 8.0?
Set up the equations to describe the speciation of uranyl, the uranyl monocarbonate, and the uranyl dicarbonate.
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Pop Quiz
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Show the relationship between Gibbs free energy, enthalpy, and entropy
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