Page 219 - Characterization and Properties of Petroleum Fractions - M.R. Riazi

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T and P three phases of a pure component can coexist all the
property. All molar properties are intensive properties and are
time. This temperature and pressure are known as triple point
related to total property as T1: IML 17:42 5. PVT RELATIONS AND EQUATIONS OF STATE 199
temperature and triple point pressure and are characteristics
θ t
(5.1) θ = of any pure compound and their values are given for many
n compounds [2, 3]. For example, for water the triple point tem-
t
where n is the number of moles, θ is a total property such as perature and pressure are 0.01 C and 0.6117 kPa (∼0.006 bar),
◦
t
volume, V , and θ is a molar property such as molar volume, respectively [3]. The most recent tabulation and formulation
V. The number of moles is related to the mass of the system, of properties of water recommended by International Asso-
m, through molecular weight by Eq. (1.6) as ciation for the Properties of Water and Steam (IAPWS) are
m given by Wagner and Pruss [4].
(5.2) n = A thermodynamic property that is defined to formulate the
M
first law of thermodynamics is called internal energy shown
If total property is divided by mass of the system (m), instead by U and has the unit of energy per mass or energy per mole
of n, then θ is called specific property. Both molar and spe- (i.e., J/mol). Internal energy represents both kinetic and po-
cific properties are intensive properties and they are related tential energies that are associated with the molecules and for
to each other through molecular weight.
any pure substance it depends on two properties such as T and
(5.3) Molar Property = Specific Property × M V. When T increases the kinetic energy increases and when
V increases the potential energy of molecules also increases
Generally thermodynamic relations are developed among and as a result U increases. Another useful thermodynamic
molar properties or intensive properties. However, once a mo- property that includes PV energy in addition to the internal
lar property is calculated, the total property can be calculated energy is enthalpy and is defined as
from Eq. (5.1).
The phase rule gives the minimum number of independent (5.5) H = U + PV
variables that must be specified in order to determine ther-
modynamic state of a system and various thermodynamic where H is the molar enthalpy and has the same unit as U.
properties. This number is called degrees of freedom and is Further definition of thermodynamic properties and basic re-
shown by F. The phase rule was stated and formulated by the lations are presented in Chapter 6.
American physicist J. Willard Gibbs in 1875 in the following
form [1]:
5.2 PVT RELATIONS
(5.4) F = 2 + N −
where is the number of phases and N is the number of For a pure component system after temperature and pressure,
components in the system. For example for a pure compo- a property that can be easily determined is the volume or
nent (N = 1) and a single phase ( = 1) system the degrees molar volume. According to the phase rule for single phase
of freedom is calculated as 2. This means when two intensive and pure component systems V can be determined from T
properties are fixed, the state of the systems is fixed and its and P:
properties can be determined from the two known parame-
ters. Equation (5.4) is valid for nonreactive systems. If there (5.6) V = f 1 (T, P)
are some reactions among the components of the systems, de-
grees of freedom is reduced by the number of reactions within where V is the molar volume and f 1 represents functional
the system. If we consider a pure gas such as methane, at least relation between V, T, and P for a given system. This equation
two intensive properties are needed to determine its thermo- can be rearranged to find P as
dynamic properties. The most easily measurable properties (5.7) P = f 2 (T, V)
are temperature (T) and pressure (P). Now consider a mix-
ture of two gases such as methane and ethane with mole frac- where the forms of functions f 1 and f 2 in the above two rela-
tions x 1 and x 2 (x 2 = 1 − x 1 ). According to the phase rule three tions are different. Equation (5.6) for a mixture of N compo-
properties must be known to fix the state of the system. In ad- nents with known composition is written as
dition to T and P, the third variable could be mole fraction of
one of the components (x 1 or x 2 ). Similarly, for a mixture with (5.8) P = f 3 (T, V, x 1 , x 2 , ... , x N−1 )
single phase and N components the number of properties that
must be known is N + 1 (i.e., T, P, x 1 , x 2 ,..., x N−1 ). When the where x i is the mole fraction of component i. Any mathemat-
number of phases is increased the degrees of freedom is de- ical relation between P, V, and T is called an equation of state
creased. For example, for a mixture of certain amount of ice (EOS). As will be seen in the next chapter, once the PVT rela-
and liquid water ( = 2, N = 1) from Eq. (5.4) we have F = 1. tion is known for a system all thermodynamic properties can
This means when only a single variable such as temperature be calculated. This indicates the importance of such relations.
is known the state of the system is fixed and its properties In general the PVT relations or any other thermodynamic re-
can be determined. Minimum value of F is zero. A system of lation may be expressed in three forms of (1) mathematical
pure component with three phases in equilibrium with each equations, (2) graphs, and (3) tables. The graphical approach
other, such as liquid water, solid ice, and vapor, has zero de- is tedious and requires sufficient data on each substance to
grees of freedom. This means the temperature and pressure construct the graph. Mathematical or analytical forms are
of the system are fixed and only under unique conditions of the most important and convenient relations as they can be

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