Thermodynamic of the Interactions Between
Gas-Solid and Solid-Liquid on Carbonaceous Materials

189
0,0 0,2 0,4 0,6 0,8 1,0
0
5
10
15
20
25
n (mmol/g)
P/Po
COD32
COD48
CUD28
CUD36
N
2

0,00 0,01 0,02 0,03
0
1
2
3
4
n(mmol/g)
P/Po


Thermodynamics – Interaction Studies – Solids, Liquids and Gases

190
The obtained carbon monoliths were tested as potential adsorbents for CO
2
finding a
retention capacity between 88-164 mgCO
2
g
-1
at 273K and atmospheric pressure, in Figure 22
to observe the isotherms of the samples with higher and lower CO
2
adsorption capacity in
each series, the monoliths with a better performance in the retention of this gas were COD32
and CUD28.
The table 10 compiles the characteristics of the carbon monoliths prepared, show the data
obtained for the interaction of three molecules of interest in the characterization of materials.
Additionally, adsorption data were used for the calculation of three parameters: n
oDR
, n
mL,
K
L
which are measures of the adsorption capacity.

Sample
N
2

COD36 1318 14.15 4.91 6.56 0.035 16.80 132 21.33
COD48 975 11.49 4.75 4.75 0.055 18.58 112 22.43
CUD28 1013 12.12 4.93 5.36 0.054 19.12 123 21.47
CUD32 1397 13.35 4.38 6.87 0.028 16.76 130 21.12
CUD36 1711 18.02 2.92 4.53 0.027 16.85 120 14.80
CUD48 1706 18.65 2.36 3.99 0.025 17.63 96 11.48
Table 10. Characteristics of carbon monoliths.
Figure 22 shows the relationship between the number of moles of the monolayer determined
by two different models, n
m
by the Langmuir model and n
o
calculated from Dubinin
Raduskevich, shows that the data are a tendency for both precursors although they are
calculated from models with different considerations. There are two points that fall outside
the general trend CUD28 and COD32 samples, which despite having the highest value of n
o

in each series not have the highest n
m
The Dubinin Raduskevich equation is use to determinate, the characteristic adsorption
energies of N
2
and CO
2
(Eo) for each samples, likewise by the Stoeckli y Krahenbüehl
equation (equation 14) was determined benzene (Eo), in Figure 23 shows the relationship
between the characteristic energies determined by two different characterization techniques
and found two trends in the data which shows the heterogeneity of carbonaceous surfaces
of the prepared samples. The characteristic energy of CO

23456
3
4
5
6
7
8
CUD28
n
m
n
o
COD
CUD
COD32 Fig. 22. Relationship between n
m
and n
o
samples of each series.

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

CUD
Eo (kJ/mol) Calorimetry
Eo (kJ/mol) Adsorption Fig. 23. Relationship between the characteristic immersion energy of benzene and the
characteristic adsorption energy of CO
2
.
Thermodynamic of the Interactions Between
Gas-Solid and Solid-Liquid on Carbonaceous Materials

193
100 120 140 160
10
15
20
25
COD
CUD
Eo Benzene (kJ/mol)


Thermodynamics – Interaction Studies – Solids, Liquids and Gases

194 900 1000 1100 1200 1300 1400
20
22
24
26
C
6
H
6
Eo (kJ/mol)
BET Area (m
2/
g) a) 900 1000 1100 1200 1300 1400
15
16

/g)

c)

800 1200 1600
16
17
18
19
20
CO
2
Eo (kJ/mol)
BET Area (m
2
/g)

d)

Fig. 25. Relationship between the characteristic energy and BET area of the series. a,b) COD.
c,d) CUD.

Thermodynamics – Interaction Studies – Solids, Liquids and Gases


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Iran
1. Introduction

Thermodynamics is the branch of science that is concerned with the principles of energy
transformation in macroscopic systems. Macroscopic properties of matter arise from the
behavior of a very large number of molecules. Thermodynamics is based upon experiment
and observation, summarized and generalized in the Laws of Thermodynamics. These laws are
not derivable from any other principle: they are in fact improvable and therefore can be
regarded as assumptions only; nevertheless their validity is accepted because exceptions
have never been reported. These laws do not involve any postulates about atomic and
molecular structure but are founded upon observation about the universe as it is, in terms of
instrumental measurements. In order to represent the state of a gas or a liquid or a solid
system, input data of average quantities such as temperature (T), pressure (P), volume (V),
and concentration (c) are used. These averages reduce the enormous number of variables
that one needs to start a discussion on the positions and momentums of billions of
molecules. We use the thermodynamic variables to describe the state of a system, by
forming a state function:
P=f (V, T, n) (1)
This simply shows that there is a physical relationship between different quantities that one
can measure in a gas system, so that gas pressure can be expressed as a function of gas
volume, temperature and number of moles, n. In general, some relationships come from the
specific properties of a material and some follow from physical laws that are independent of
the material (such as the laws of thermodynamics). There are two different kinds of
thermodynamic variables: intensive variables (those that do not depend on the size and
amount of the system, like temperature, pressure, density, electrostatic potential, electric
field, magnetic field and molar properties) and extensive variables (those that scale linearly
with the size and amount of the system, like mass, volume, number of molecules, internal
energy, enthalpy and entropy). Extensive variables are additive whereas intensive variables
are not (Adamson, A.W. and Gast, A.P. 1997).
In thermodynamic terms, the object of a study is called the system, and the remainder of the

system can be defined as a given thermodynamic state. It should be noted that a
thermodynamic state is completely different from a molecular state because only after the
precise spatial distributions and velocities of all molecules present in a system are known
can we define a molecular state of this system. An extremely large number of molecular
states correspond to one thermodynamic state, and the application of statistical
thermodynamics can form the link between them (Lyklema, J. 2005), (Dabrowski A., 2001).
2. Energy, work and heat
2.1 The first law of thermodynamics
Generally, when a system passes through a process it exchanges energy U with its
environment. The energy change in the system ΔU may result from performing work w on
the system or letting the system perform work, and from exchanging heat q between the
system and the environment
Uqw

 (2)
The heat and the work supplied to a system are withdrawn from the environment, such
that, according to the first law of thermodynamics
0
system environment
UU

  (3)
The First Law of thermodynamics states that the energy content of the universe (or any
other isolated system) is constant. In other words, energy can neither be created nor
annihilated. It implies the impossibility of designing a perpetuum mobile, a machine that
performs work without the input of energy from the environment. The First Law also
implies that for a system passing from initial state 1 to final state 2 the energy change

Thermodynamics of Interfaces


of component i, and the number of moles n
i
of component i. As a
rule, X varies with Y but for an infinitesimal small change of Y, X is approximately constant.
Hence, we may write

ii
i
dU q pdV dA dQ dn
 
    

(5)
The terms of type XdY in Eq. above represent mechanical (volume), interfacial, electric, and
chemical works, respectively.
i

implies summation over all components in the system. It
is obvious that for homogeneous systems the γdA term is not relevant.
2.2 The second law of thermodynamics: entropy
According to the First Law of thermodynamics the energy content of the universe is
constant. It follows that any change in the energy of a system is accompanied by an equal,
but opposite, change in the energy of the environment. At first sight, this law of energy
conservation seems to present good news: if the total amount of energy is kept constant why
then should we be frugal in using it? The bad news is that all processes always go in a
certain direction, a direction in which the energy that is available for performing work
continuously decreases.
Entropy, S, is the central notion in the Second Law. The entropy of a system is a measure of
the number of ways the energy can be stored in that system. In view of the foregoing, any
spontaneous process goes along with an entropy increase in the universe that is, ΔS > 0. If as

occurs(sraelachvili, J. 1991):

2
1
q
S
T



(7)
Because the temperature may change during the heat transfer is written in differential form
(Pitzer, K.S. and Brewer L. 1961).
2.3 Reversible processes
In contrast to the entropy, heat is not a function of state. For the heat change it matters
whether a process 1 2 is carried out reversibly or irreversibly. For a reversible process,
that is, a process in which the system is always fully relaxed

2
.
1
rev
q
S
T



(8)
Infinitesimal small changes imply infinitesimal small deviations from equilibrium and,

electric charge, and composition. The required conditions make this definition very
impractical, if not in operational. If the interface is extended it is very difficult to keep
variables such as entropy and volume constant.
The other intensive variables may be expressed similarly as the change in energy per unit
extensive property, under the appropriate conditions (Tempkin M. I. and Pyzhev V., 1940).
2.4 Maxwell relations
At equilibrium, implying that the intensive variables are constant throughout the system, (9)
may be integrated, which yields

iii
U TdS pV A Q n


 (11)

Thermodynamics of Interfaces

205
To avoid impractical conditions when expressing intensive variables as differential
quotients as, for example auxiliary functions are introduced. These are the enthalpy H,
defined as

HUpV

 (12)
the Helmholtz energy
AUTS

 (13)
and the Gibbs energy

heating or cooling a system at constant p, the heat exchange between the system and its
environment is equal to the enthalpy exchange. Hence, for the heat capacity, at constant p,

()( )
p
p
p
q
dH
C
dT dT


(18)
In general, for a function of state f that is completely determined by variables x and y, df =
Adx + Bdy. Cross-differentiation in df gives( / ) ( / )
xy
Ay Bx

, known as a Maxwell
relation. Similarly, cross-differentiation in dU, dH, dA, and dG yields a wide variety of
Maxwell relations between differential quotients. For instance, by cross-differentiation in dG
we find, (Lyklema, L. 1991), (Pitzer, K.S. and Brewer L. 1961).

,,, ,,,
,,
() ()
pAQn TpQn
is is
S

i
is defined as the differential quotient (Prausnitz,
J.M., and et al. 1999).

, , ,
1
()
i
Tp n
i
j
Y
y
n




(20)
The partial molar quantities are operational; that is, they can be measured. Now
,,
,
Tp n
is
Y can
be obtained as
iii
ny A partial molar quantity often encountered is the partial molar Gibbs
energy (Aveyard, R. and Haydon, D.A., 1973),








(22)
that is, at constant
, , ,Tp n
j
i

,the chemical potential of component i in a mixture equals its
partial molar Gibbs energy.
By cross-differentiation in (17) the temperature- and pressure-dependence of µ
i
can be
derived as

, ,
, , ,
,
i
i
pn
i
Tp n
is
ji
S

is
TH
TT







(25)
The pressure-dependence of mi is also obtained from (17):

, ,
, ,
,
i
i
i
Tp n
Tn
ji
is
V
v
pn





ln
ii i
RT p


(28)
where
0
i

is an integration constant that is independent of p
i
;
0
(1)
iii
p


 , its value
depending on the units in which p
i
is expressed. Similarly, without giving the derivation
here, it is mentioned that for the chemical potential of component i in an ideal solution

0
ln
ii i
RT c



are defined per unit X
i
, c
i
,
and p
i
, respectively, and their values are therefore independent of the configurations of i in
the mixture. They do depend on the interactions between i and the other components and
therefore on the types of substances in the mixture. Because X
i
, c
i
, and p
i
are expressed in
different units, the values for µ
i
and
0
i

differ (Keller J.U., 2005)
The RTln X
i
term in Eq. (30) or, for that matter, the RTlnc
i
and RTlnp
i

ln
ii i
ssRX
(32)

The partial molar entropy of i is composed of a part
0
i
s ,which is independent of the
configurations of i in the mixture but dependent on the interactions of i with the other
components, and a part Rln X
i
, which takes into account the possible configurations of i. It
follows that the RTlnX
i
(or RTlnc
i
or RTlnp
i
) term in the expressions for µ
i
stems from the
configurationally possibilities as well.

Thermodynamics – Interaction Studies – Solids, Liquids and Gases

208
3. Basic thermodynamics of interfaces
For an open system of variable surface area, the Gibbs free energy must depend on
composition, temperature, T, pressure, p, and the total surface area, A:



 
 
 


 
 
 



(34)
The first two partial differentials refer to constant composition, so we my use the general
definitions:
GHTSUPVTS

  (35)
To obtain

,
p
n
j
G
S
T






   

(38)
where the chemical potential µ
i
is defined as:

,,
i
i
T
p
n
j
G
n







(39)
and the surface energy γ as:

,,Tpn

W., 2003). This is especially clear when it is understood that m
i is a simple function of
concentration, that is:

0
ln
ii i
kT C

 (41)
for dilute mixtures, where m
i o is the standard chemical potential of component ‘i’, usually 1
M for solutes and 1 atm for gas mixtures. This equation is based on the entropy associated
with a component in a mixture and is at the heart of why we generally plot measurable
changes in any particular solution property against the log of the solute concentration,
rather than using a linear scale. Generally, only substantial changes in concentration or
pressure produce significant changes in the properties of the mixture. (For example,
consider the use of the pH scale.)
(Koopal L.K., and et al. 1994).
3.1 Thermodynamics for closed systems
The First Law of Thermodynamics is the law of conservation of energy; it simply requires
that the total quantity of energy be the same both before and after the conversion. In other
words, the total energy of any system and its surroundings is conserved. It does not place
any restriction on the conversion of energy from one form to another. The interchange of
heat and work is also considered in this first law. In principle, the internal energy of any
system can be changed, by heating or doing work on the system. The First Law of
Thermodynamics requires that for a closed (but not isolated) system, the energy changes of
the system be exactly compensated by energy changes in the surroundings. Energy can be
exchanged between such a system and its surroundings in two forms: heat and work. Heat
and work have the same units (joule, J) and they are ways of transferring energy from one

are many different ways that energy can be stored in a body by doing work on it:
volumetrically by compressing it; elastically by straining it; electrostatically by charging it;
by polarizing it in an electric field E; by magnetizing it in a magnetic field H; and
chemically, by changing its composition to increase its chemical potential. In interface
science, the formation of a new surface area is also another form of doing work. Each
example is a different type of work – they all have the form that the (differential) work
performed is the change in some extensive variable of the system multiplied by an intensive
variable. In thermodynamics, the most studied work type is pressure–volume work,
W
PV
, on
gases performed by compressing or expanding the gas confined in a cylinder under a piston.
All other work types can be categorized by a single term,
non-pressure–volume work, W
non-PV
.
Then,
W is expressed as the sum of the pressure–volume work, W
PV
, and the non pressure–
volume work,
W
non-PV
, when many types of work are operative in a process (Miladinovic N.,
Weatherley L.R. 2008).
Equation (11) states that the internal energy, Δ
U depends only on the initial and final states
and in no way on the path followed between them. In this form, heat can be defined as
the
work-free transfer of internal energy from one system to another


expression.
3.2 Derivation of the gibbs adsorption isotherm
Let us consider the interface between two phases, say between a liquid and a vapor, where a
solute (i) is dissolved in the liquid phase. The real concentration gradient of solute near the
interface may look like Figure 10.1. When the solute increases in concentration near the
surface (e.g. a surfactant) there must be a surface excess of solute
i
n

, compared with the
bulk value continued right up to the interface. We can define a surface excess concentration
(in units of moles per unit area) as:

Thermodynamics of Interfaces

211

i
i
n
A


(44)

Fig. 1. Diagram of the variation in solute concentration at an interface between two phases. Fig. 2. Diagrammatic illustration of the change in surface energy caused by the addition of a


ii i
dd



  (46)
The change in mi is caused by the change in bulk solute concentration. This is the Gibbs
surface tension equation. Basically, these equations describe the fact that increasing the
chemical potential of the adsorbing species reduces the energy required to produce new
surface (i.e. γ). This, of course, is the principal action of surfactants, which will be discussed
in more detail in a later section. Using this result let us now consider a solution of two
components

11 22
ddd



   (47)
and hence the adsorption excess for one of the components is given by

1
1
,
2
T




gives

11
111
ln
TT
cRT
RT
ccc

  


  

  
(50)
Then substitution in (49) leads to the result:

1
1
11
1
ln
TT
c
RT c RT c


 


1
1
1
2ln
TT
RT c







(52)
because the bulk chemical potentials of both ions change with concentration of the
surfactant.
4. Fundamentals of pure component adsorption equilibrium
Adsorption equilibria information is the most important piece of information in
understanding an adsorption process. No matter how many components are present in the
system, the adsorption equilibria of pure components are the essential ingredient for the
understanding of how many those components can be accommodated by a solid adsorbent.
With this information, it can be used in the study of adsorption kinetics of a single
component, adsorption equilibria of multicomponent systems, and then adsorption kinetics
of multicomponent systems. In this section, we present the fundamentals of pure
component equilibria. Various fundamental equations are shown, and to start with the
proceeding we will present the most basic theory in adsorption: the Langmuir theory (1918).
This theory allows us to understand the monolayer surface adsorption on an ideal surface.
By an ideal surface here, we mean that the energy fluctuation on this surface is periodic and
the magnitude of this fluctuation is larger than the thermal energy of a molecule (kT), and