Table Of Content1
Thermodynamics of Adsorption
from Solution
H.-H.KOHLER
InstituteofAnalyticalChemistry,Chemo-andBiosensors,Universityof
Regensburg,D-93040Regensburg,Germany
I. Introduction.................................................................... 2
II. Fundamental Relations.................................................. 3
A. The System.............................................................. 3
B. The Interface........................................................... 4
C. Gibbs’ Adsorption Equation, Adsorption
Isotherm, and Interfacial State Equation.............. 7
D. Sustained Equilibrium and Enthalpy of
Adsorption............................................................. 10
E. Nonequilibrium Interfaces and the Gibbs
Free Energy of Adsorption.................................... 12
III. One-Component Adsorption......................................... 14
A. General Relations.................................................. 14
B. Langmuir Adsorption............................................ 16
1
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2 ThermodynamicsofAdsorptionfromSolution
C. Temkin Adsorption................................................ 19
D. van der Waals Adsorption.................................... 24
E. Surface Tension of Micellar Solutions.................. 26
IV. Solution Containing Two Nonionic Adsorbates.......... 27
A. Differential Adsorption Enthalpy......................... 27
B. Competitive Langmuir Adsorption....................... 29
C. Dissociation from a Solid Surface with
Independent Sites................................................. 30
V. Adsorption from Electrolyte Solutions........................ 32
A. Symmetrical Electrolyte: General Relations........ 32
B. Symmetrical Electrolyte:
Gouy–Chapman–Stern Theory............................. 34
C. Adsorption from Solutions with High Total
Electrolyte Concentration..................................... 37
1. General Thermodynamic Relations................ 37
2. Gouy–Chapman–Stern Adsorption................ 38
References.............................................................................. 41
I. INTRODUCTION
Adsorption processesoften dominate processesof aggregation
and flocculation in solution. The aim of this chapter is to give
a concise description of the thermodynamic background of
adsorption including its relation to interfacial tension.
In the author’s experience, the implications and limita-
tions of a theoretical result often remain obscure, unless the
theoretical line of arguments is traced back to the funda-
mental equations and assumptions. Therefore, an attempt is
made to establish a reasonable balance between theoretical
foundations and derivations, on the one hand, and results, on
the other hand. The fundamental thermodynamic relations
will be presented in Section II. Subsequent sections are de-
voted to special types of single-layer adsorption advancing
fromone-componentadsorptiontotwo-componentandtoelec-
trolyte adsorption. A similar approach worth reading is pre-
sented in Aveyard and Haydon [1].
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CoagulationandFlocculation 3
II. FUNDAMENTAL RELATIONS
A. The System
Adsorptionistheexchangeofmatterbetweenavolumephase
and an interface. Therefore, the thermodynamics of adsorp-
tion are closely related to the thermodynamic description of
interfaces. We assume that adsorption takes place between a
liquid solutionand amacroscopically homogeneousandplane
phase boundary between the solution and another solid or
fluid phase. Assuming, for convenience, that the composition
ofthesecondphaseispracticallyfixedandthatallchangesof
the interfacial properties are due to exchange with the solu-
tion phase, we will (mostly) refer to a simplified thermo-
dynamic system consisting of the solution and the interfacial
phase only. We further assume that temperature Tand pres-
sure p are uniform all over the system.
Thethermodynamicdescriptionofthesystemstartsfrom
the first law of thermodynamics.
Reversible differential changes of the internal energy
U are given by
dU ¼ dQrþdWr, (1)
where dQr and dWr are heat and work reversibly exchanged
with the surroundings (the greek d is used to denote differen-
tial changes of path-dependent variables). We assume that
reversible work can be exchanged by variation of the volume
V, giving rise to volume work dQr ¼(cid:1)pdV, by variation of
vol
theamountn ofcomponenti,givingrisetothechemicalwork
i
dWr ¼P m dn, where m is the chemical potential of com-
chem i i i
ponentiandbyvariationoftheinterfacialareaAofthephase
boundary, giving rise to interfacial work dWr ¼sdA, where
surf
s is the interfacial tension. Hence, with dQr¼T dS, Equation
(1) transforms into Gibbs’ fundamental equation:
X
dU ¼ TdS(cid:1)pdV þ m dn þsdA: (2)
i i
It is convenient to introduce the Gibbs free energy
G ¼ U þpV (cid:1)TS ¼ H(cid:1)TS: (3)
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4 ThermodynamicsofAdsorptionfromSolution
Inserting Equation (2), dG¼dUþpdVþVdp(cid:1) TdS(cid:1)SdT
canbeexpressedby
X
dG ¼ (cid:1)SdTþVdpþ m dn þsdA: (4)
i i
We use c to denote the molar concentration of component i in
i
the bulk solution. Component 1 is always the solvent. Since
equilibrium is assumed between the interface and the solu-
tion, the interfacial tension s is determined by the intensive
state of the bulk solution, which in turn is a function of T, p
and the solute concentrations c ,..., c . Thus
2 n
s ¼ s(T,p,c , ...,c ): (5)
2 n
Thesystemmaybebuiltupcontinuouslybyincreasingthen0 s
i
and the interfacial area A in such a way that the intensive
state of the system keeps constant (which implies dT, dp¼0
andm,s¼constant)Then,accordingtoEquation(4),Gcanbe
i
expressed by
nði ðA
X X
G ¼ m dn þs dA or G ¼ m n þsA (6)
i i i i
0 0
From the last equation
X X
dG ¼ m dn þ n dm þsdAþAds, (7)
i i i i
whichifequatedwithEquation(4),leadstotheGibbs–Duhem
equation
X
SdT(cid:1)Vdpþ n dm þAds ¼ 0: (8)
i i
B. The Interface
We define the amount of substance i contained in the bulk
phase by
nb ¼ c V: (9)
i i
The amount of substance ns contained in the interfacial
i
phase, often called the interfacial excess of i, is n (cid:1)nb.
i i
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CoagulationandFlocculation 5
Theinterfacialconcentration(ortheinterfacialexcessconcen-
tration) G is defined by G ¼ns/A. Thus
i i i
n ¼ nbþns ¼ c V þG A: (10)
i i i i i
The definition of nb in Equation (9) is not complete as long as
i
the volume V is not defined precisely. In our thermodynamic
description, an interface is a two-dimensional plane separat-
ing homogeneous bulk phases. In physical reality interfacial
effects are spatially distributed and extend into the neighbor-
ingvolumephases.Thereforethepreciselocationoftheinter-
face and, consequently, the precise value of the volume of the
bulk phase is a matter of definition.
Taking into account the special role of substance 1, we
define the position of the dividing interface by
n
1
V ¼ : (11)
c
1
This implies nb¼n and ns¼0 or G ¼0. (See Defay [2] for
1 1 1 1
moredetails,includingthedistinctionbetweentheinterfaceof
tension and the dividing interface.) With volume V given by
Equation (11), the bulk value of any extensive variable X,
related to the volume, is given by
Xb ¼ xbV, (12a)
where xb is the bulk concentration. As a generalization of
Equation (10), the interfacial contribution to the extensive
variable X, Xs, and the interfacial concentration, xs, are
A
given by
Xs
Xs ¼ X (cid:1)Xb, xs ¼ : (12b)
A A
According to Equation (11), we have V¼Vb. Hence from
Equation (12b)
Vs ¼ 0: (12c)
With Equation (12b), dG and G now can be written as
dG ¼ dGbþdGs, G ¼ GbþGs, (13a)
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6 ThermodynamicsofAdsorptionfromSolution
where Gb is the Gibbs free energy of an ordinary bulk phase.
Accordingly
X
dGb ¼ (cid:1)SbdTþVdpþ m dnb,
i i
(13b)
X
Gb ¼ m nb:
i i
OnsubtractingtheseequationsfromEquations(4)and(6),we
find
dGs ¼ (cid:1)SsdTþsdAþX0m dns, Gs ¼ X0m ns þsA:
i i i i
(13c)
Because of ns¼0, summation over i here can be restricted to
1
the range i¼2 to i¼n, as indicated by P0. For the Helmholtz
free energy, F¼U(cid:1)TS, we have, correspondingly
dF ¼ dFbþdFs, F ¼ FbþFs (14a)
with
X X
dFb¼(cid:1)SbdT(cid:1)pdVþ m dnb, Fb¼ m nb(cid:1)pV,
i i i i
(14b)
and
dFs ¼ (cid:1)SsdTþsdAþX0m dns, Fs ¼ X0m ns þsA:
i i i i
(14c)
Note that, according to Equations (13c) and (14c), there is no
differencebetweenGibbsandHelmholtzinterfacialfreeener-
gies, i.e.
dGs ¼ dFs, Gs ¼ Fs: (14d)
This also implies
Hs ¼ Us (14e)
The Gibbs–Duhem equation of the bulk phase is
X
SbdT(cid:1)Vdpþ nbdm ¼ 0: (15)
i i
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CoagulationandFlocculation 7
We subtract this from Equation (8) and obtain
SsdT(cid:1)AdsþX0nsdm ¼ 0: (16)
i i
This is the Gibbs–Duhem equation of the interface.
C. Gibbs’ Adsorption Equation, Adsorption
Isotherm, and Interfacial State Equation
Dividing Equation (16) by A gives
ds ¼ (cid:1)ss dT(cid:1)X0G dm , (17a)
A i i
which is known as the Gibbs’ adsorption equation [3]. Note
that s does not explicitly depend on p. At constant tempera-
ture we obtain
X0
ds ¼ (cid:1) G dm : (17b)
i i
For an ideal solution under constant pressure (by ideal solu-
tion we always mean an ideally diluted solution) this simpli-
fies to
X0
ds ¼ (cid:1)RT G dlnc : (17c)
i i
Hence, the dependence of the interfacial tension on the bulk
concentrations c is regulated by the interfacial concentration
i
G. If G is positive, substance i is called surface-active (or a
i i
surfactant), otherwise it is surface-inactive. In view of Equa-
tion (17a), s can be written as
s ¼ s(T,m , ...,m ): (18)
2 n
According to Equation (17a) G is the partial derivative of s
i
withrespecttom.Thereforeitdependsonthesameindepend-
i
ent variables as s. Thus
G ¼ G (T,m , ...,m ), i ¼ 2,3, ... (19a)
i i 2 n
The set of equations given by Equation (19a) can be inverted
to
m ¼ m (T,G , ...,G ), i ¼ 2,3, ... (19b)
i i 2 n
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8 ThermodynamicsofAdsorptionfromSolution
This result shows that m, taken as a function of interfacial
i
variables, can be written as a function of Tand G through G
2 n
alone and does not depend on p. In bulk solution thermo-
dynamics m is written as
i
m ¼ m (T,p,c , ...,c ): (19c)
i i 2 n
According to Equations (19b) and (19c), chemical equilibrium
between the interface and the bulk solution now can be ex-
pressed by
m (T,G , ...,G ) ¼ m (T,p,c , ...,c ): (19d)
i 2 n i 2 n
On the left-hand side there are n independent variables, on
the right-hand side, however, nþ1. Obviously, something
is wrong. Recall that the interface is part of a two-phase
system.Although,inourcontext,we(tryto)ignorethesecond
phase, it is still there. Gibbs’ phase rule states that, at equi-
librium,thenumberofindependentintensivevariablesofthe
two-phase system is smaller by one than the number of
the one-phase system. So actually one of the variables on the
right-hand side of Equation (19d) is a function of the remain-
ing ones and therefore should be removed from the list of
independent intensive bulk variables of our system. Arbit-
rarily, we omit the pressure. Equations (19c) and (19d) now
become
m ¼ m (T,c , ...,c ), (19e)
i i 2 n
m (T,G , ...,G ) ¼ m (T,c , ...,c ): (19f)
i 2 n i 2 n
In a shorter notation we write the last equation as
ms ¼ mb, (19g)
i i
where ms and mb are introduced to denote the chemical
i i
potential as a function of the intensive variables of the inter-
facialphaseandthebulkphase,respectively.DuetoEquation
(19e), Equations (19a) and (19b) can be transformed into
G ¼ G (T,c , ...,c ), i ¼ 2,3, ... (20a)
i i 2 n
c ¼ c (T,G , ...,G ), i ¼ 2,3, ... (20b)
i i 2 n
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CoagulationandFlocculation 9
These equations relate bulk concentrations to interfacial con-
centrationsandaregeneralformsoftheadsorptionisotherms
of the system, the term isotherm reflecting that, in practical
use,theG sandc’sareusuallytreatedasvariables(whileTis
i i
used as a parameter).
Inserting Equation (19b) in Equation (18) we obtain
s ¼ s(T,G , ..., G ) (21)
2 n
which is the general form of the interfacial equation of state.
Changes of the interfacial tension due to adsorption can
be expressed in terms of the interfacial pressure p defined by
p ¼ s (cid:1)s, (22)
0
where s is the interfacial tension of the ‘‘clean interface’’
0
(moreprecisely,atG ,...,G ¼0).Ifinterfacialconcentrations
2 n
are proportional to bulk concentrations, so that
G ¼ a (T)c (23a)
i i i
then Equation (17c) gives
X0
p ¼ RT G (23b)
i
which is the two-dimensional equivalent of the ideal gas
equation.
Note that, as a result of Equations (10) and (11), G can
i
also be expressed as
(cid:1) (cid:1)
@ni(cid:1) @ni(cid:1)
Gi ¼ (cid:1) ¼ (cid:1) (24a)
@A(cid:1)ci,V @A(cid:1)ci;c1;n1
But c will be constant if Tand c ,..., c are constant (again
1 2 n
we omit the pressure as an independent variable, see discus-
sion following Equation (19b)). Therefore, Equation (24a) can
be rewritten as
(cid:1)
@ni(cid:1)
Gi ¼ (cid:1) : (24b)
@A(cid:1)
T;n1;cj6¼1
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10 ThermodynamicsofAdsorptionfromSolution
This shows that G is the amount of substance i (i 6¼ 1) to be
i
addedto thesystemperchangeof interfacialareato preserve
the intensive state of the system at constant n .
1
D. Sustained Equilibrium and Enthalpy of
Adsorption
Because of Gs¼Hs(cid:1) TSs, Equation (13c) yields
ms ¼ @Gs(cid:1)(cid:1)(cid:1) ¼ hs (cid:1)Tss, (25a)
i @ns(cid:1) i i
i T;A;ns
j6¼i
where (cf. Equation (14e))
@Hs @Us @Ss
hs ¼ ¼ , ss ¼ : (25b)
i @ns @ns i @ns
i i i
The partial derivatives are taken under the conditions speci-
fied in Equation (25a). Accordingly
(cid:1)
@Gb(cid:1)
mb ¼ (cid:1) ¼ hb(cid:1)Tsb: (26)
i @nb(cid:1) i i
i (cid:1)T;p;nb
j6¼1
Introducing the partial molar adsorption enthalpy
Dhad ¼ hs (cid:1)hb, (27a)
i i i
and the partial molar adsorption entropy
Dsad ¼ ss (cid:1)sb, (27b)
i i i
and using Equations (25a), (26), (27a), and (27b), the equilib-
rium condition of Equation (19g) can be written as
Dmad ¼ ms (cid:1)mb ¼ Dhad(cid:1)TDsad ¼ 0: (28)
i i i i i
Reversible adsorption processes therefore require d(Dmad)¼0
i
or
dmb ¼ dms: (29a)
i i
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