1. MASS TRANSPORT
1.2 THERMODYNAMIC CONCEPTS
Thermodynamically speaking, the part of the universe we are studying is called the system
and surrounding is the remaining part of universe. The system typically consists of a
specified amount of chemical substance or substances, such as a given mass of a gas, liquid,
or solid. On the basis of its relations with the surrounding, the system can be classified as
isolated (neither matter nor energy can flow into or out of the system), closed (energy may be
exchanged with the surrounding, but not matter) and open (both energy and matter can enter
or leave the system). Additionally, the system can be defined homogeneous if it is
characterized by uniform properties or heterogeneous if it is constituted by two or more
phases (each one can be viewed as a homogeneous system). The thermodynamic energy U
associated to the system descends from the energy of its atoms and molecules. This, in turn,
depends on the electronic distribution within the material and on atomic or molecular motion,
namely translation (the movement of individual molecules in space), vibration (the
movement of atoms or groups of atoms with respect to each other within a molecule) and
rotation (the revolution of molecules about an axis). While translation energy represents the
kinetic part of U, vibrational, rotational and electronic energy represent the internal part of U.
Of course, U does not represent the whole system energy as kinetic energy (result of system
motion as a whole) and potential energy (result of system position) constitute system
mechanical energy and other forms of energy can be owned by the system as a whole (surface
energy and electromagnetic energy among others. In this sense, now, we are dealing with
simple systems). Accordingly, U can be seen as a measure of molecular and atomic
distributions and motions, as well as the electronic distribution within atoms and molecules.
It is also evident that U is an extensive system property as its value increases with system
mass (proportional to the number of system atoms) even if the mean atoms, molecules and
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electrons energy is the same. On the contrary, temperature T is an intensive system parameter
as it is a measure of the mean atoms, molecules and electrons energy and does not depend on
system mass. The condition of no translational, vibrational, rotational and electronic energy
corresponds to the absolute zero equal to -273.15 celsius degrees. Volume (V) and moles
number (ni) of each system component are other fundamental extensive system properties.
In order to describe a thermodynamic system, we need to define the walls separating it from
the surrounding and defining the boundary conditions or these external bonds. For example, a
rigid wall impedes V variations and this makes impossible work exchange between the
system and the surrounding. In addition, while a semi-impermeable membrane can be
restrictive with respect to the number of moles ni of the ith component, an adiabatic wall is
restrictive with respect to heat flux.
1.2.1 Thermodynamic Equilibrium
Upon boundary conditions modification, system intensive and extensive properties vary and
processes take place. When all system properties no longer change with time, the system is
said to be in thermodynamic equilibrium and the central problem of thermodynamic consists
in the determination of the new U, V and ni values. This problem is solved by means of the
second thermodynamic principle enunciated in an axiomatic form according to which for
each system it exists a function of the extensive parameters (U, V, ni), called entropy (S) and
defined for all system equilibrium conditions, characterized by the fact that a) the values
assumed by extensive parameters at equilibrium maximize S, b) S values corresponding to a
heterogeneous system is given by the sum of each phase entropy, c) S is, mathematically
speaking, a continue, differentiable and monotonic increasing function of U. This enunciation
allows writing down the following general equations:
m
S  Sj
j1
(4.1)
2

S j  S j U j ,Vj , nij
dS  0

j
and 1  i  r
d2S  0
(4.2)
(4.3)
where Sj, Uj and Vj represent, respectively, the entropy, the energy and the volume of the jth
system phase, m is phases number, r is system components number and nij denotes the moles
number relative to the ith component belonging to the jth phase. While eq.(4.1) expresses
entropy addittivity, eq.(4.3) imposes the S maximum condition at equilibrium. On the basis of
the above mentioned S properties, the following equations can be obtained:
U  U S ,V , ni 
i
r
 U 
 U 
 U 

dU  
dS  
dV   
dni


 S V ,n1 ,... nr
 V  S ,n1 ,... nr
i 1  ni  S ,V , n  n
j
i
(4.4)
(4.5)
Eq.(4.5) gives the opportunity of defining some important intensive parameters:
 U 
 T S , V , n1 ,....nr 


 S V ,n1 ,... nr
(4.6)
 U 

 PS , V , n1 ,....nr 

 V  S ,n1 ,... nr
(4.7)
 U 


 i S ,V , n1 ,....nr 
 ni  S ,V ,n j  ni
(4.8)
where T and P are, respectively, system temperature and pressure while i is the ith
component chemical potential. Accordingly, eq.(4.5) can be re-written as:
r
dU  TdS  PdV    i dni
i 1
(4.9)
In addition, on the basis of eqs.(4.6)-(4.8), it follows:
1
 S 



 U V ,n1 ,... nr T
(4.10)
1
P
 S 
 S 
 U 
 
*
  *  P  




T
T
 V U ,n1 ,... nr
 U V ,n1 ,... nr  V  S ,n1 ,... nr
(4.11)
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and, finally:
r
 S 
 S 
 S 

dS  
dU  
dV   
dni


 U V ,n1 ,... nr
 V U ,n1 ,... nr
i 1  ni U ,V ,n  n
j
i
dS 
r

dU P
 dV   i dni
T
T
i 1 T
(4.12)
(4.13)
It is now useful expressing eqs.(4.9) and (4.13) explicitly in terms of entropy (Sj), energy
(Uj), volume (Vj), temperature (Tj), pressure (Pj) and chemical potential (  ij ) referred to each
system phase j:
m
m
r
dU   dU j    TjdS j  PjdVj    ijdnij 
j 1
j 1
i 1

(4.14)

m
m  dU
r j
Pj
j
dS   dS j   
 dV j   i dnij 

j 1
j 1 T j
i 1 T j
Tj


(4.15)
Remembering the definition of isolated system, the following relations hold:
m
m
j 1
j 2
dU   dU j  0  dU 1    dU j
m
m
j 1
j 2
m
m
j 1
j 2
(4.16)
dV   dV j  0  dV1    dV j
dni   dnij  0  dni1    dnij
(4.17)
i
(4.18)
Consequently, system equilibrium conditions (eq.(4.3)) become:
 Pj P 
m  1
r  j
1  
1
dS      dU j    1 dV j    i  i dnij   0
T T 
j 1  T j
i 1  T j
T1 
T1  
1 
 j


(4.19)
As dS must be zero for any arbitrary set of dUj, dVj and d nij , we have the following
equilibrium conditions:
T1  Tj  T
P1 Pj

T1 Tj
j
 P1  Pj  P j
thermal equilibrium
(4.20)
mechanical equilibrium
(4.21)
4
1i   ij
 i, j
chemical equilibrium
(4.22)
Eqs.(4.20)-(4.22) express that an isolated, heterogeneous (simple) system made up by m
phases and r components is in thermodynamic equilibrium when each phase is characterized
by the same temperature and pressure and the chemical potential of each component is the
same in every phase system. On the basis of S and U properties, it can be demonstrated that
the S maximum condition required for the system equilibrium translates into a U minimum
condition and equilibrium conditions are always expressed by eqs.(4.20)-(4.22). Indeed, it is
easy to verify that the substitution of these equations into eq.(4.14) yields to dU = 0.
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