Mod 5

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