Concepts in Environmental Behavior of Trace Metals

1. CHEMICAL KINETICS AND SORPTION:

A REVIEW

LAW OF MASS ACTION

dcreverse

baforward

dcreverse

baforward

]D[]C[k]B[]A[kkRateOverall

reactionreverseofRate]D[]C[k

reactionforwardofRate]B[]A[k

dDcCaBaA

ba

dc

reverse

forwardeq

]B[]A[

]D[]C[

k

kK

At equilibrium, the rate of forward reaction becomes equal to the rate of the reverse reaction and k = 0.

kforward*[A]a[B]b = kreverse*[C]c[D]d

ELEMENTARY REACTIONS

•They occur in a single step. For these reactions, the law of mass action holds.

•Bimolecular elementary reactions are common in the environment:

•Tri-molecular elementary rxns are less common but do exist. Complicated stoichiometric equations than tri-molecular do not occur (it is highly improbable that more than 3 molecules collide simultaneously to effect a reaction)

]A[kdt

]B[dor]A[k

dt

]A[d

BA

]B][A[kdt

]A[dor]A[k

dt

]A[d

CBAorBA2

2

RATE CONSTANT AND TEMPERATURE

•Arrhenius equation relates reaction rate k and temperature (T)

RT

Eact

Aek

2121

actTT

RT

E

TRT

E

T

TTTRT

Eexpkk

AekandAek

21

2

act

21

act

1

T

1

RT

EAlnkln act

Biological reactions work on the basis of enzyme and have a surprisingly small range of activation energies in natural systems.Q10 rule in biology = The rate of a reaction will approximately double as T increases by

100C increments

•Linearization

•conversion of a rate constant from a known or reference temperature to a second temperature

REACTION ORDERS AND TESTING REACTION ORDERS AND TESTING REACTION RATE EXPRESSIONSREACTION RATE EXPRESSIONS

.........................cbanorderactionRe

.......]C[]B[]A[k)k(rateactionRe

oductsPr......cDaBaAcba

forward

Most elementary reactions are either zero, first, or second order. Fractional order reactions (e.g. 0.5, 1.5, etc. ) are also observed.

Zero-Order Reactions

Heterogeneous complex reactions that occur in many steps in natural waters, methane production and release of hydrolysis products (e.g. NH3, PO4

3-) from anaerobic sediments are examples zero-order reactions

0kdt

]A[d

[A]

Time

Linearization and plot

k0 unitstime

unitsionConcentratunitsk0

First-Order Reactions

0.11

0.11

k

]A[kdt

]B[d

]A[kdt

]A[d

BA 1

Alntk)BAln(

eABA

eAAB

eAAofonsubstutuitbyionLinearizat

)e1(AB

dteAkdB

methodfirstbysolvableequationaldifferentiogeneoushomnonordinaryLinear

eAkdt

dB

]A[kdt

]B[d

___________________________

AlntkAln:ionLinearizat

eAA

tkA

Aln

]A[kdt

]A[d

10

tk0

tk00

tk0

tk0

B

0

t

0

tk01

tk01

1

01

tk0

10

1

1

1

1

1

1

1

1

First-Order Reactions (Cont’d)

___

AlntkAln:ionLinearizat

eAA

tkA

Aln

]A[kdt

]A[d

A2A

01

tk0

10

1

1

Example first order reactions: Radioisotope decay, BOD, re-areation and gas transfer, log growth phase of microorganisms, sedimentation of non-coagulating solids, death and respiration rates for bacteria and algae

Second-Order Reactions

___

]

R2RA:reactionicAutocatlyt

DBA:tstanreac2

BAA:ttanreac1

1. Zooplankton death rate, atmospheric gas phase reactions2. Microbial kinetics (substrate-cells), sorption rxns, aquatic

redox rxns3. Autocatalytic rxns = nucleation/crystal growth, some photo

redox reactions, microbial division, etc.

Second-Order Reactions (Cont’d)

0

0002

02

B

Alnt)AB(k

B

Aln.2

A

1tk

A

1.1

Linearization

For autocatalytic reactions, one needs to know the stoichiometry or yield and linearization would lead to a plot similar to #2 above.

Other Reaction OrdersMany times a reaction is not elementary (i.e. single step), and the rate expression is not simply zero, first or second order. In this case, the change in reaction rate can be plotted against concentration (on a log-log plot) to get an estimate of the order of reaction (also known as Van’t Hoff Plot)

•It may be fractional•Fractional order reactions occur in precipitation and dissolution reactions

Log[C]Logk

Log(-dC/dt)]

n = 1n = 2n =3

1nforexcept,nofvalueany

forvalidisequationaboveThe

kt)1n(]C[

1

]C[

1

asrewrittenbecan

kt]C[

1

]C[

1

)1n(

1

givesegrationint

]C[kdt

]C[dRate

1n0

1nt

1n0

1nt

n

Van’t Hoff Plot[C]

Time (t)

n=2

n=3n=3

SORPTION / ADSORPTION

A. SORPTION:

•This term denotes the distribution of a compound into/onto a sorbent without considering a specific mechanism.

•Therefore, sorption is a more inclusive and less definitive term and is a more appropriate designator for the distribution of chemicals between solid and aqueous phases

•Soils/sediments are so complex that several mechanisms are usually active in sorption

B. ADSORPTION:

•A substance is said to adsorb if the concentration in a boundary region is higher than in the interior of the contiguous phase.

•Adsorbent: adsorbing surface

•Asorbate: compound that adsorbs from solution

•Physio-sorption (e.g. London vdW forces)

•Chemi-sorption (chemical bonds)

SORPTION ISOTHERMS

An isotherm describes the relationship of the concentrations of a solute between two separate phases at equilibrium at constant temperature

An adsorption isotherm expresses the relationship between amount of vapor or solute adsorbed as a function of the equilibrium concentration of of the vapor or solute in solution

A sorption isotherm describes the same process without reference to the mechanisms

DEFINITIONS

Langmuir Isotherm

Primarily for gases and based on the following assumptions

1. The energy of adsorption is constant and independent of the extent of surface coverage

2. The adsorption is on localized sites and there is no interaction between the adsorbed molecules

3. The maximum adsorption possible is that of a complete monolayer

C

1).

bX

1(

X

1

X

1

formLinearized

bC1

C*b*XX

mm

m

X = mole of solute adsorbed per gram of adsorbentC = equilibrium conc. of solute in solutionXm = Number of moles adsorbed/g of adsorbent to give a complete monolayerb = constant related to the energy of adsorption

Freundlich Isotherm

Empirical relation used to express the relation between the amount of compound sorbed and Keq

e

ne

ClognKlogm

xlog

formLinearized

KCm

x

Where x/m = amount of compound sorbed

And Ce = equilibrium concentration

The use of this relation to evaluate experimental data is essentially “curve fitting” and has no mechanistic base. 1. K provides an index of the extent of sorption and is often listed w/o units2. n = indicates whether the relation between x/m and Ce is linear (n=1) or concave down (n<1) or concave up (n>1) – and when n = 1, K equals the distribution coefficient

Isotherm types that are commonly observed in environmental sciences

q

q

Ce Ce

L-curve isotherm•Solute has a relatively high affinity for the sorbent surface at low surface coverage - Affinity sorbate-sorbent decreases with increasing coverage

H-curve isotherm•Very high affinity sorbate-sorbent.•Probability of inner sphere complexes formation

C-curve isotherm•Characteristics of non-ionic and hydrophobic compounds•Constant partitioning

S-curve isotherm•Affinity of sorbent for sorbate is less than of solution at low solute concentration. •As the concentration of solute increases and exceeds the retention capacity of the solution, sorption peaks up.

q

SORPTION KINETICS

Batch Studies Column Studies

•Distribution defined by sorption - not desorption•Common 24-hr equilibration batch experiments•Rapid removal from solution: >99% in 2 to 3 hrs•True equilibrium usually not attained

•Movement of pollutant monitored as a function of sorbent type (e.g. soil), column dimensions, flow rates, and ionic strength. Water used as eluent•Principles of chromatography can be used•Data used to analyze sorption kinetics by 2-site Freundlich relation and where

•S = amount sorbed at respective site (f=fraction of S1 and 1-f=fraction S2)•C= concentration in solution•k= rate constant•K and n are Freundlich parameters from batch experimentsand K1=Kf and K2=K(1-f)

)SCK(kdt

dS

and

)SCK(kdt

dS

21n

222

1n

111