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Module 5: Volumetric properties of pure fluids
The Objective of this module is to explain the behavior of fluids with respect to changes in
thermodynamic properties. This module includes the description of
PVT behavior
Equations of state
Determination of equation of state parameters
Thermodynamic properties serve to define the state of a system completely. The
properties P, V and T are directly measurable, while the properties U, H, S etc.. are not directly
measurable. There exist relations between measurable and non-measurable properties as well as
among the measurable properties.
The PVT behavior of pure fluids can be presented graphically or mathematical relations between
properties.
Graphical representation of PVT behavior:
The P-V-T behavior of pure fluids can be represented on P vs V diagram and P-T diagram
P-V diagram:
Consider the thermodynamic state of a single component system, water as shown in the
following figure as a function of P and V. It shows the variation of molar volume (V) with
respect to Pressure at different constant temperature T1, T2, T3 and TC, where T3>TC>T2>T1.
Figure 1: PV diagram
Suppose that the initial state of the system is represented by point A. The change in the Volume
of water at a constant temperature T1 occurs along the isotherm ABDN.
AB – liquid region
BD – two phase region
DN- Vapor region
The slope of isotherm in water region is steep because water is incompressible and the change in
volume is negligible. When point B is reached, the liquid water begins to vaporize which results
in increase in volume from B to D.
At B=100% liquid
At D= 100% vapor
So the point B represents the saturated liquid state at which phase change from L to V, while
point D represents saturated vapor state at which phase change L to V ends. The locus of all such
points is the dome-shaped curve MCN.
MC = Saturated liquid curve
CN= Saturated vapor curve
It can be observed that the length of horizontal segment of the isotherm T1 in two phase region is
greater than the length of horizontal segment of isotherm T2, where T2 > T1. The length of the
horizontal segment of any given isotherm into two-phase region decreases when temperature of
the system is increased to TC, the horizontal segment reduces to a point (C). The point ‘C’ is
called the critical point and at this point the liquid and vapor phases can’t be distinguished from
each other. The critical point is called the critical isotherm.
Temperature at point ‘C’= Critical temperature.
Pressure at point ‘C’= Critical pressure.
TC = 647.3 K and PC = 220.5 bar for water, VC = 56x10-6 m3/mol
If T > TC, the substance is called gas.
Any extensive property can be expressed in terms of the quality of mixture.
As volume is an extensive property. It is additive
V = x VV + (1 - x) VL
x = quality of mixture = . .
x =
=
N – No .of moles
M – mass
V =
VV + (1 -
) VL
∴ =
P- T diagram /Phase diagram:- Lines 1 – 2 & 2 – C represent for a pure substance conditions of pressure and temperature at which solid and liquid phases exist in equilibrium with a vapor phase.
Line 1- 2: solid vapor equilibrium
2 – C: Liquid vapor equilibrium
2 – 3: Solid liquid equilibrium.
Line 1 -2 separates the solid and gas regions, the sublimation curve
Line 2-C separates liquid and gas vapor regions, the vaporization curve.
Line 2-3 separates solid and liquid regions, the fusion curve.
Point ‘C’ is called the critical point; its coordinates PC and TC are the highest pressure and
highest temperature at which a pure chemical species is observed to exist in vapor – liquid
equilibrium. The three phases coexist in equilibrium. According to the phase rule
At Triple point, F = 0, invariant
Two –phase lines, F = 1, Univariant
Single phase region, F= 2, divariant
Figure 2: Triple point of water
Liquid region
Solid region
Vapor region
Gas region
Fluid region
C
3
2(Triple point)
T C Temperature
Pressure
PC
1
The region existing at temperatures and pressures greater than TC and PC is marked off by
dashed lines and it is called fluid region and below which, it is called gas. The gas region is
divided into two parts as indicated by the dotted vertical line. A gas to the left of this line, which
can be condensed either by compression at constant temperature or by cooling at constant
pressure, is called vapor. A fluid existing at a temperature greater than TC is said to supercritical.
An example is atmospheric air.
Equations of state
Single phase region:-
The P – V – T behavior can be expressed mathematically
f (P, V,T) = 0 and such a relation is known as PVT equation of state. It relates pressure,
molar or specific volume, and temperature for a pure homogeneous fluid in equilibrium states.
An equation of state may be solved for any one of the 3 quantities P, V or T as a
function of the other two. For example, If V is considered a function of T and P
V =V (T, P)
dV = PT
V
dT +
TPV
dP
The partial derivates in this equation have definite physical meanings, commonly tabulated for
liquids.
Volume expansivity = V1
PTV
Isothermal compressibility K = - V1
TPV
The isotherms for the liquid phase on left side of figure 1 are very steep and closely spaced. Thus
both PT
V
and
TPV
and hence both and K are small.
2
For liquids is always positive (liquid water at 0 C and 4 C is an exception) and K is necessarily positive. At conditions not close to the critical point, and K are weak functions of temperature.
VIRIAL equations of state:
Isotherms for gases and vapors, lying above and to the right of CN in fig.1 , are relatively
simple curves for which V decreases as P increases. Here, the product PV for a given T should
be much more nearly constant and hence represented analytically as a function of P.
This suggests expressing PV for an isotherm by a power series in P.
PV =a + bP +cP2+ ----------------
If b = a B1, c = a C1 etc ----------
PV = a (1 + B1 P+ C1 P2 + D1P3 + ---------)
Where a, B1, C1 are constants for a given temperature and a given chemical species.
In principle, equation 3 is an infinite series. However, in practice, a finite number of
terms are used. In fact, PVT data show that at low pressure truncation after two terms often
provides satisfactory results.
Two forms of the Virial equation:
A usual auxiliary thermodynamic property is defined by the equation
=
= Z
This dimensionless ratio is called the compressibility factor. With this definition and a= RT
Equation becomes Z=1 + B1 P+ C1 P2 + D1 P3 + --------
An alternative expression for Z is also in common use
3
3
Z=1+VB + 2V
C + 3VD +------------------
Both of these equations are known as Virial expansions, and the Parameters B, C, D & B, C, D
are called Virial coefficients.
B & B Second Virial coefficient
C & C1 Third Virial coefficient.
For a given gas the Virial coefficients are functions of temperature only.
Many other equations of state have been proposed for gases, but the Virial equations are
the only ones firmly based on statistical mechanics, which provides physical significance to the
Virial coefficients. Thus, for the expansion in V1 the term
VB arises on account of interactions
between pairs of molecules; The 2VC term, on account of three – body interactions.etc…..
because two – body interactions are many times more common than three – body interactions
are more numerous than four – body interaction etc….The contributions to Z of the successively
higher – ordered terms decreases rapidly.
For an ideal gas PV = RT and
Application of the Virial equations:-
A more common form of Virial equation
Z = 1 + B P
Z = = 1 + (B1 = )
This equation expresses a direct proportionality between Z and P, and is often applied to vapors
at subcritical temperatures up to their saturation pressures.
Equation as well may be truncated to two terms for application at low pressures.
Z = = 1 +
4
Z =1
4
6
5
However, equation is more convenient in application. The second Virial coefficient B is substance dependent and a function of temperature.
For pressures above the range of applicability of equation , but below the critical pressure,
the Virial equation truncated to three terms often provides excellent result.
Z = = 1 + + 2VC
Values of C, like those of B depend on the gas and on temperature. However much less is known
about third coefficients than second Virial coefficients.
Correlations for Compressibility factor:
Theorem of corresponding states and Accentric factor:
Experimental observation shows that compressibility factors Z for different fluids exhibit
similar behavior at reduced temperature Tr and reduced pressure Pr.
Tr = CT
T
Pr =
CPP
These dimensionless thermodynamic coordinates provide the basis for the simplest form of the
theorem of corresponding states.
“All fluids when compared at the same reduced temperature and reduced pressure, have
approximately the same compressibility factor, and all deviate from ideal – gas behavior to
about the same degree”
Corresponding – states correlations of Z based on this theorem are called two – parameter
correlations, because they require use of the two reducing parameters Tr and Pr. Although these
correlations are very nearly exact for the simple fluids (Argon, krypton and xenon) systematic
deviations are observed for more complex fluids. Appreciable improvement results from
introduction of a third corresponding – state parameter, a characteristic of molecular structure;
the most popular such parameter is the “accentric factor” introduced by Pitzer.
5
5
The accentric factor for a pure chemical species is defined with reference to its
vapor pressure.
r
satr
Td
Pd1
log = S
Where sat
rP - is the reduced vapor pressure
Tr - reduced saturation temperature
S- slope of plot of log Prsat vs
rT1
Note that” log” denotes a logarithm to the base 10. If the two-parameter theorem of
corresponding states were generally valid, the slope S would be the same for all fluids. This is
observed not to be true; each fluid has its own characteristic valve of S, which could in principle
serve as a third corresponding state parameter. However, Pitzer noted that all vapor pressure data
for the simple fluids lie on the same line when plotted. Data for other fluids define other lines
whose locations can be fixed in relation to the line for the simple fluids (SF) by the difference.
log satrp (SF) - logp sat
r
The acentric factor is defined at this difference evaluated at T r =0.7
= -1.0 - log(p satr )Tr=0.7
Therefore can be determined for any fluid from Tc, Pc and a single vapor pressure
measurement made at Tr = 0.7
Figure3: Variation of slope for complex fluids
The definition of makes its value zero for Ar, Kr and Xe.
"All fluids having the same value of , when compared at the same Tr and Pr, have
about the same value of Z, and all deviate from ideal gas behavior to about the same
degree”
Generalized correlations:
Equations of state have the advantage that they provide analytical expressions for PVT relations.
As we will see more and more, these analytical expressions can be used to obtain various
thermodynamic properties. However, correlations have also been developed, which can be used
to get PVT graphically, (or numerically, through tables).
Example:
Pitzer, Lee and Kestler Correlations:
Correlations for compressibility factors
1
Log P satr
2
0 1.2 1.4 1.6 2.0
Slope=-2.3 (Ar,Kr,Xe)
Slope=-3.2
1/Tr
Zo is evaluated for simple fluids (inert gases) from experimental data. Z' is evaluated from non-
simple fluids data. [3 parameter corresponding states and the correlations are valid only for non-
polar and slightly polar substances]
Pitzer correlation for second virial coefficient
CUBIC EQUATION OF STATE:
If an equation of state is to represent the PVT behavior of both liquids and vapors, it must
encompass a wide range of temperatures and pressures. Yet it must not be so complex as to
present excessive analytical difficulties in application. Polynomial equations that are cubic in
molar volume offer a comprise between generality and simplicity that is suitable to many
purposes. Cubic equations are in part the simplest equations capable of representing both liquid
and vapor behavior.
The Vander Waals equation of state:
The first practical cubic equation of state was proposed by J.D Vander Waals
2Va
bVRTP
_______ (8)
a – The intermolecular force of attraction between the molecules
b-- Excluded volume or the volume unavailable for molecular motion
For a=0 & b=0, the above equation reduces to the ideal gas equation.
Given values of a and b for a particular fluid, one can calculate P as a function of V for
various valves of T. We can discuss with the help of PV diagram.
Figure 4: PV behavior predicted by Vander Walls
For the isotherm T2 > TC : Pressure is a monotonically decreasing function with
increasing molar volume
For T = TC: Pressure remains constant at C (C critical point with VC)
For T1<TC: Pressure decreases rapidly in the sub cooled region with increasing V. After
crossing the saturated line, it goes through minimum, rises to a maximum, and then
decreases. Crossing the saturated vapor line and continuing downward into the super
heated vapor region.
Cubic equations of state have three volume roots, of which two may be complex.
Physically meaningful values of V are always real, positive and greater than constant b.
For an isotherm T > TC – solution for V at any positive value of p yields only one real
positive root
For isotherms T = TC (critical isotherm) - Solution for V will give one such root except at
critical pressure, where there are three roots equal to VC.
For isotherms at T<TC – The equation may exhibit one or three real roots, depending on
the pressure.
For saturation pressure Psat; Vsat(liq) and Vsat
(vap) are the roots
For other pressure (For the lines above and below dashed horizontal line) – The smallest root is
liquid like volume, and the largest is a vapor- like volume. The third root, lying between the
other values, is of no significance.
Determination of equation of state parameter:-
The parameters of a cubic equation can be found from the value s of TC and PC, because
the critical isotherm exhibits a horizontal inflection at critical point.
crTVP
= 0 & crTV
P
2
2
=0
By solving the Vander Walls equation, we will get
VC=83
PcRTc , a=
6427
PcTcR 22
, b=81
PcRTc
For compressibility factor at critical conditions
83
C
CCC RT
VPZ
Redlich – Kwong equation (RK equation):-
It is a two parameter equation of state
P= bV
RT
- )(2
1
bVVT
a
a =cP
TcR 224278.0
and b=
PcRTc0866.0
It is the best two parameter equation of state used for estimation of PVT behavior of real gases.
Soave – Redlich- Kwong (SRK) equation of state:
Soave proposed a modification to RK equation to improve its accuracy by introducing a third parameter, α
bVVa
bVRTP
Where, C
C
PTR
a224278.0
C
C
PRT
b08664.0
2
21
2 1176.057.148.01
rT
Where is accentric factor
This equation is widely used to predict the properties of hydrocarbons
Peng-Robinson (PR) Equation of state:
bVbbVVa
bVRTP
C
C
PTR
a224572.0
and C
C
PRT
b0778.0
2
21
2 1267.0542.13746.01
rT
This equation is used to predict the properties of hydrocarbons and inorganic gases such as N2, O2 etc…
Exercise: 1. Calculate the pressure developed by 1 k mol, of NH3 gas contained in a vessel of 0.6 m3
volume at a constant temperature of 473 K by using a. The ideal gas equation b. The Vander Walls equation, a = 0.4233 Nm4/mol2, b = 3.73*10-5 m3/mol
2. Calculate the molar volume of saturated liquid and saturated vapor of n-octane at
427.85K is 0.215 MPa. Assume that it follows Vander Walls equation of state
Data: a = 3.789 Pa m6 / mol2 and b = 2.37*10-4 m3/mol
3. Generally, volume expansivity β and isothermal compressibility depends on T and P. Prove that
PT TP
4. Based on definition of compressibility factor, for which values of Z there will be
intermolecular attraction and repulsions? Explain?
5. The saturation pressure of water at 180 0C is 1.0027 MPa. The critical constants of water are TC = 647.3 K, PC = 221.2 bar, calculate the accentric factor of water.
6. Derive the relationship between Virial coefficients BI, CI, DI are related to B, C, D