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Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)

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In This Chapter We Cover the Following Topics

Art. Content Page

9.1 Important Mathematical Relations 3

9.2 Gibbs and Helmholtz Functions 5

9.3 Maxwell's Equations 6

9.4 T.ds Equations 7

9.5 Difference in Heat Capacities 7

9.6 Ratio of Heat Capacities 9

9.7 Energy Equation 10

9.8 Joule-Kelvin Effect 13

9.9 Clausius-Clapeyron Equation 16

9.10 Evaluation of Thermodynamic Properties from an Equation of State 18

9.11 General Thermodynamic Considerations on an Equation of State 20

9.12 Chemical Potential 21

9.13 Fugacity 23

References:

1. M. J. Moran and H. N. Shapiro, Fundamentals of Engineering Thermodynamics, 6e,

John Wiley & Sons, Inc., New York, 2008.

2. G. J. Van Wylen, R. E. Sonntag, C. Borgnakke, Fundamentals of Thermodynamics,

John Wiley & Sons, Inc., New York, 1994.

3. J. P. Holman, Thermodynamics, 4e, McGraw-Hill, New York, 1988.

4. F. W. Sears, G. L. Salinger, Thermodynamics, Kinetic theory, and Statistical

Thermodynamics, 3e, Narosa Publishing House, New Delhi, 1998.

5. Y. A. Cengel and M. A. Boles, Thermodynamics: An Engineering Approach, 2e,

McGraw-Hill, New York, 1994.

6. E. Rathakrishnan, Fundamentals of Engineering Thermodynamics, 2e, PHI Learning

Private Limited, New Delhi, 2008.




2 Chapter 9: Thermodynamic Relations

7. P. K. Nag, Basic and Applied Thermodynamics, 1e, McGraw-Hill, New Delhi, 2010.

8. V Ganesan, Gas Turbine, 2e, Tata McGraw-Hill, New Delhi, 2003.

9. Y. V. C. Rao, An Introduction to Thermodynamics, 1e, New Age International (P)

Limited, Publishers, New Delhi, 1998.

10. Onkar Singh, Applied Thermodynamics, 2e, New Age International (P) Limited,

Publishers, New Delhi, 2006.

Please welcome for any correction or misprint in the entire manuscript and your

valuable suggestions kindly mail us [email protected].




3 Basic and Applied Thermodynamics By Brij Bhooshan

Objective of this section is to develop mathematical relations for estimation of various

thermodynamic properties such as u, h, s etc. for a compressible system.

Thermodynamic properties such as pressure, volume and temperature (P, V, T) etc. can

be directly measured experimentally while some other properties cannot be measured

directly and require thermodynamic relations for their determination. These

thermodynamic relations are the basis for getting useful thermodynamic properties.

In thermodynamics, an important task is to develop relationships which express the

nonmeasurable properties like U, H, A, G and S in terms of measurable properties P, V

and T. The number of partial derivatives of the type (∂S/∂p)V among the above eight

properties are 8C3 × 3! = 336.

9.1 IMPORTANT MATHEMATICAL RELATIONS

To define state of a simple compressible system of known mass and composition one

requires minimum two independent intensive properties. Thus, all intensive properties

can be determined through functions of the two independent intensive properties such

as,

Above are functions of two independent variables and can be given in general as, z = z(x,

y), where x, y are independent variables.

Exact Differential

In earlier discussions we have seen that the differential of any property should be exact.

Therefore, let us review calculus briefly.

If a relation exists among the variables x, y, and z, then z may be expressed as a function

of x and y, or exact differential of any function z shall be as given below for z being

continuous function of x and y.

or, dz = Mdx + Ndy

where, z, M and N are functions of x and y.

i.e. M is partial derivative of z with respect to x when variable y is held constant and N

is partial derivative of z with respect to y when variable x is held constant.

Here, since M and N have continuous first partial derivative therefore, order of

differentiation is immaterial for properties and second partial derivative can be given

as,

or,

This is the condition of exact (or perfect) differential. Thus, the test of exactness for any

property shall be, given by Eqn. (9.1).





Reciprocity Relation and Cyclic Relation

If a quantity f is a function of x, y, and z, and a relation exists among x, y, and z, then f is

a function of any two of x, y, and z. Similarly any one of x, y, and z may be regarded to be

a function of f and any one of x, y, and z.

Thus, if,

In differential form,

Substituting the expression of dy in the preceding equation

Again

Now let us consider three variables x, y, z such that any two of these are independent

variables. Thus, we can write

In differential form,

Combining above two relations we get

As x and z are independent variables so let us keep z constant and vary x, i.e. dz = 0 and

dx ≠ 0 which yields reciprocity relation as,

Equation (9.3) is termed as reciprocity relation.

Similarly, let us keep x constant and vary z i.e. dx = 0, dz ≠ 0 which shall be possible

only when;





Equation (9.4) is termed as cyclic relation.

Among the thermodynamic variables p, V, and T, the following relation holds good

9.2 GIBBS AND HELMHOLTZ FUNCTIONS

For a simple compressible system of fixed chemical composition thermodynamic

properties can be given from combination of first law and second law of thermodynamics

as,

Gibbs function (g) and Helmholtz function (f) are properties defined as below.

Gibbs function

Gibbs function is defined as G = H ‒ T.S

On unit mass basis i.e. specific Gibb’s function is given by

In differential form Gibbs function can be given as below for an infinitesimal reversible

process

for a reversible isothermal process,

or, also dG = Vdp – SdT for reversible isothermal process;

For a “reversible isobaric and isothermal process”, dp = 0, dT = 0 dG = 0

i.e. G = constant

‘Gibbs function’ is also termed as ‘Gibbs free energy’. For a reversible isobaric and

isothermal process Gibbs free energy remains constant or Gibbs function of the process

remains constant. Such reversible thermodynamic processes may occur in the processes

involving change of phase, such as sublimation, fusion, vaporization etc., in which Gibbs

free energy remains constant.

Helmholtz Function

Helmholtz function is defined as F = U – T·S

on unit mass basis i.e. specific Helmholtz function is given by





‘Helmholtz function’ is also called ‘Helmholtz free energy’. For any infinitesimal

reversible process, Helmholtz function can be given in differential form as,

or

For a reversible isothermal process df = – pdv

or,

or,

For a reversible isothermal and isochoric process, dT = 0, dV = 0, then df = 0 or, dF = 0

or, F = Constant

Above concludes that the Helmholtz free energy remains constant during a reversible

isothermal and isochoric process. Such processes may occur during chemical reactions

occurring isothermally and isochorically.

9.3 MAXWELL'S EQUATIONS

A pure substance existing in a single phase has only two independent variables. Of the

eight quantities p, V, T, S, U, H, F (Helmholtz function), and G (Gibbs function) any one

may be expressed as a function of any two others.

For a pure substance undergoing an infinitesimal reversible process

(a) dU = T.dS ‒ p.dV

(b) dH = dU + p.dV + V.dp = T.dS + V.dp

(c) dF = dU ‒ T.dS ‒ S.dT = ‒p.dV ‒ S.dT

(d) dG = dH ‒ T.dS ‒ S.dT = V.dp ‒ S.dT

Since U, H, F and G are thermodynamic properties and exact differentials of the type

Then

From above four equations for properties to be exact differentials, we can write

functions;

For differential of function ‘U’ to be exact;

For differential of function ‘H’ to be exact;

For differential of function ‘F’ to be exact;





For differential of function ‘G’ to be exact;

These four equations are known as Maxwell's equations. Maxwell relations have large

significance as these relations help in estimating the changes in entropy, internal

energy and enthalpy by knowing p, V and T.

9.4 T.dS EQUATIONS

Let entropy S be imagined as a function of T and V. Then S = S(T, V)

Since T(∂S/∂T)V = CV, heat capacity at constant volume, and Maxwell's third equation,

Then,

This is known as the first TdS equation.

Again let entropy S be imagined as a function of T and p. Then S = S(T, p)

Since T(∂S/∂T)p = Cp, heat capacity at constant pressure, and Maxwell's fourth equation,

Then,

This is known as the second TdS equation.

9.5 DIFFERENCE IN HEAT CAPACITIES

Equating the first and second TdS equations





Again let temperature T be imagined as a function of V and p. Then T = T(V, p)

Comparing above two equations, then we get

Both these equations give

From cyclic relation

This is a very important equation in thermodynamics. It indicates the following

important facts.

(a) Since

2

pT

V

is always positive, and

TV

p

for any substance is negative, ( pC ‒ vC )

is always positive. Therefore, Cp is always greater than vC .

(b) As T → 0 K, Cp → Cv or at absolute zero, Cp = Cv.

(c) When pT

V

= 0 (e.g., for water at 4°C, when density is maximum, or specific

volume minimum), Cp = Cv.

(d) For an ideal gas, pV = mRT

Then Eq (9.11) gives

or,

In single phase region the specific volume can also be given as function of T & p and the

differential of function V shall be,

or,

The above differential form of specific volume indicates that it depends upon partial

derivatives of specific volume with respect to temperature and pressure. Partial

derivatives of V with respect to temperature can be related to “volume expansivity” or

“coefficient of volume expansion” as below,





Partial derivative of specific volume with respect to pressure can be related to

“isothermal compressibility”, as below.

Inverse of isothermal compressibility is called “isothermal bulk modulus”,

Thus, volume expansivity gives the change in volume that occurs when temperature

changes while pressure remains constant. Isothermal compressibility gives change in

volume when pressure changes while temperature remains constant. These volume

expansivity and isothermal compressibility are thermodynamic properties.

Similarly, the change in specific volume with change in pressure isentropically is also

called “isentropic compressibility” or “adiabatic compressibility”, Mathematically

Reciprocal of isentropic compressibility is called “isentropic bulk modulus,”

Now, using equation (9.11), (9.13) and (9.14), then we have

or,

Above difference in specific heat expression is termed as “Mayer relation” and helps in

getting significant conclusion such as,

The difference between specific heats is zero at absolute zero temperature i.e.

specific heats at constant pressure and constant volume shall be same at

absolute zero temperature (T = 0 K).

Specific heat at constant pressure shall be generally more than specific heat at

constant volume i.e., Cp ≥ Cv. It may be attributed to the fact that ‘α’ the

isothermal compressibility shall always be +ve and volume expansivity ‘β’ being

squared in (Cp – Cv) expression shall also be +ve. Therefore (Cp – Cv), shall be

either zero or positive value depending upon magnitudes of V, T, β and .

For incompressible substances having dV = 0, the difference (Cp – Cv) shall be

nearly zero. Hence, specific heats at constant pressure and at constant volume

are identical.

9.6 RATIO OF HEAT CAPACITIES

At constant S, the two TdS equations become





Since γ > 1,

Therefore, the slope of an isentropic is greater than that of an isotherm on p-v diagram

(Diagram 9.1). For reversible and adiabatic compression, the work done is

Diagram 9.1

For reversible and isothermal compression, the work done would be

For polytropic compression with 1 < n < γ, the work done will be between these two

values. So, isothermal compression requires minimum work.

The adiabatic compressibility ( ) is defined as

9.7 ENERGY EQUATION

For a system undergoing an infinitesimal reversible process between two equilibrium

states, the change of internal energy is

Substituting the first TdS equation

T = C

S = C

V

1

4

P 3

2T 2s





If U = U(T, V)

Comparing Eqns. (9.20) and (9.21), then we get

This is known as the energy equation.

Application 1 of the Equation (9.22)

For an ideal gas,

U does not change when V changes at T = C.

Since

U does not change either when p changes at T = C. So the internal energy of an ideal gas

is a function of temperature only, as shown earlier in Chapter 8.

Another important point to note is that in Eq. (9.20), for an ideal gas

Therefore

holds good for an ideal gas in any process (even when the volume changes). But for any

other substance

is true only when the volume is constant and dV = 0.

Similarly

and





As shown for internal energy, it can be similarly proved from Eq. (9.24) that the

enthalpy of an ideal gas is not a function of either volume or pressure

but a function of temperature alone.

Since for an ideal gas,

and

the relation dH = CpdT is true for any process (even when the pressure changes).

However, for any other substance the relation dH = CpdT holds good only when the

pressure remains constant or dp = 0.

Application 2 of the Equation (9.22)

Thermal radiation in equilibrium with the enclosing walls possesses an energy that

depends only on the volume and temperature. The energy density (u), defined as the

ratio of energy to volume, is a function of temperature only, or

The electromagnetic theory of radiation states that radiation is equivalent to a photon

gas and it exerts a pressure, and that the pressure exerted by the black-body radiation

in an enclosure is given by

Black-body radiation is thus specified by the pressure, volume, and temperature of the

radiation.

Since

By substituting in the energy Eq. (9.20)

or

where b is a constant. This is known as the Stefan-Boltzmann Law.

Since U = uV = VbT4





and

from the first TdS equation

For a reversible isothermal change of volume, the heat to be supplied reversibly to keep

temperature constant

For a reversible adiabatic change of volume

or

If the temperature is one-half the original temperature, the volume of black-body

radiation is to be increased adiabatically eight times its original volume so that the

radiation remains in equilibrium with matter at that temperature.

9.8 JOULE-KELVIN EFFECT

A gas is made to undergo continuous throttling process by a valve, as shown in Diagram

9.2. The pressures and temperatures of the gas in the insulated pipe upstream and

downstream of the valve are measured with suitable manometers and thermometers.

Diagram 9.2 Joule-Thomson expansion

Diagram 9.3 Isenthalpic states of a gas

T

h = C

States before throttling

States after throttling

Valve

Insulation





Let Pi and Ti, be the arbitrarily chosen pressure and temperature before throttling and let

them be kept constant. By operating the valve manually, the gas is throttled successively to

different pressures and temperatures pf1, Tf1; pf2, Tf2; pf3, Tf3 and so on. These are then

plotted on the T-p coordinates as shown in Diagram 9.3. All the points represent equilibrium

states of some constant mass of gas, say, 1 kg, at which the gas has the same enthalpy.

The curve passing through all these points is an isenthalpic curve or an isenthalpe. It is

not the graph of a throttling process, but the graph through points of equal enthalpy.

The initial temperature and pressure of the gas (before throttling) are then set to new

values, and by throttling to different states, a family of isenthalpes is obtained for the

gas, as shown in Diagrams 9.4 and 9.5. The curve passing through the maxima of these

isenthalpes is termed as the inversion curve.

Diagram 9.4 Isenthalpic curve and the inversion curve

Diagram 9.5 Inversion and saturated curve on T-s graph

This graphical representation of isenthalpic curve gives the Joule-Thomson coefficient

by its slope at any point. Slope may be positive, negative or zero at different points on

the curve. The points at which slope has zero value or Joule-Thomson coefficient is zero

are termed as “inversion points” or “inversion states”. Temperature at these inversion

states is referred as “inversion temperature”. Locii of these inversion states is named as

“inversion line”. Temperature at the intersection of inversion line with zero pressure line

is termed as “maximum inversion temperature”.

The numerical value of the slope of an isenthalpe on a T-p diagram at any point is

named as the Joule-Kelvin coefficient or Joule-Thomson coefficient (μj). Thus the locus of

Inversion curve ( ) T

Critical point

Saturation curve

Liquid vapour region

Constant enthalpy curve

Liq.

T

Maximum inversion

temperature

Heating

region

Inversion curve ( )

Critical point

Vap.

Cooling

region

Constant enthalpy curve (isenthalpes)





all points at which μj is zero is the inversion curve. The region inside the inversion curve

where μj > 0 is named as the cooling region and the region outside where μj < 0 is named

as the heating region. So,

Joule-Thomson coefficient is defined as the rate of change of temperature with pressure

during an isenthalpic process or throttling process.

Mathematically evaluating the consequence of μj we see,

for μj > 0, temperature decreases during the process.

for μj = 0, temperature remains constant during the process.

for μj < 0, temperature increases during the process.

The difference in enthalpy between two neighbouring equilibrium states is

and the second TdS equation (per unit mass)

The second term in the above equation stands only for a real gas, because for an ideal

gas, dh = cp dT.

For an ideal gas pv = RT

There is no change in temperature when an idea' gas is made to undergo a Joule-Kelvin

expansion (i.e. throttling).

For achieving the effect of cooling by Joule-Kelvin expansion, the initial temperature of

the gas must be below the point where the inversion curve intersects the temperature

axis, i.e. below the maximum inversion temperature. For nearly all substances, the

maximum inversion temperature is above the normal ambient temperature, and hence

cooling can be obtained by the Joule-Kelvin effect. In the case of hydrogen and helium,

however, the gas is to be precooled in heat exchangers below the maximum inversion

temperature before it is throttled. For liquefaction the gas has to be cooled below the

critical temperature.

Let the initial state of gas before throttling be at A (Diagram 9.6). The change in

temperature may be positive, zero, or negative, depending upon the final pressure after

throttling. If the final pressure lies between A and B, there will be a rise in temperature

or heating effect. If it is at C, there will be no change in temperature. If the final

pressure is below pc, there will be a cooling effect, and if the final pressure is pD, the

temperature drop will be (TA ‒ TD). Maximum temperature drop will occur if the initial

state lies on the inversion curve. In Diagram 9.6, it is (TB ‒ TD).





Diagram 9.6 Maximum cooling by Joule- Kelvin expansion

The volume expansivity is

So the Joule-Kelvin coefficient μj is given by, from Eq. (9.26)

or,

For an ideal gas,

There are two inversion temperatures for each pressure, e.g. T1 and T2 at pressure p

(Diagram 9.4).

9.9 CLAUSIUS-CLAPEYRON EQUATION

Let us look upon phase change at fixed temperature and pressure and estimate changes

in specific entropy, internal energy and enthalpy during phase change. If x is the

fraction of initial phase i which has been transformed into final phase f, then

where s and v are linear functions of x.

For reversible phase transition, the heat transferred per mole (or per kg) is the latent

heat, given by

which indicates the change in entropy.

A phase change of the first order is known as any phase change that satisfies the

following requirements:

(a) There are changes of entropy and volume.

(b) The first-order derivatives of Gibbs function change discontinuously.

Let us start with one of Maxwell relations;

A

p

T

D

C

Heating

region

Cooling region

Inversion curve

B

isenthalpe





For pure substances we have seen that during phase transformation at some

temperature the pressure is saturation pressure. Thus pressure is also independent of

specific volume and can be determined by temperature alone. Hence,

Here (∂p/∂T)sat is the slope of saturation curve on pressure-temperature (p–T) diagram

at some point determined by fixed constant temperature during phase transformation

and is independent of specific volume.

Diagram 9.7

Substituting in the Maxwell relation.

Thus, during vaporization i.e. phase transformation from liquid to vapour state, above

relation can be given as,

From differential form of specific enthalpy,

for phase change occurring at constant pressure and temperature,

for saturated liquid to dry vapour transformation,

Substituting hfg/T in place of entropy in (∂p/∂T)sat, then, we get

P

Vapour

Solid

T

T

P

Slope

Liquid





Above equation is termed as Clapeyron equation. It can be used for determination of

change in enthalpy during phase change i.e. hfg from the p, v and T values which can be

easily measured. Thus, Clapeyron equation can also be used for “sublimation process” or

“melting occurring at constant temperature and pressure” just by knowing slope of

saturation curve on p-T diagram, temperature and change in specific volume.

Hence, for initial state ‘i’ getting transformed into final state ‘f’ due to phase

transformation at constant pressure and temperature, general form of Clapeyron

equation:

At low pressure during liquid-vapour transformation it is seen that specific volume of

saturated liquid state is very small as compared to dry vapour state, i.e. vf <<< vg. Also

at low pressure the substance in vapour phase may be treated as perfect gas. Therefore,

Clapeyron equation can be modified in the light of two approximations of “vf being

negligible compared to vg at low pressures” and ideal gas equation of state during

vapour phase at low pressure, vg = RT/p.

Clapeyron equation thus becomes, Clausius-Clapeyron equation as given here,

Above equation is termed as Clausius-Clapeyron equation.

Now integrating between initial state ‘i’ to final state ‘f’, then we have

Clausius-Clapeyron equation is thus a modified form of Clapeyron equation based upon

certain approximations and is valid for low pressure phase transformations of liquid-

vapour or solid-vapour type.

9.10 EVALUATION OF THERMODYNAMIC PROPERTIES FROM AN

EQUATION OF STATE

Apart from calculating pressure, volume, or temperature, an equation of state can also

be used to evaluate other thermodynamic properties such as internal energy, enthalpy

and entropy. The property relations to be used are:





Diagram 9.8

Integrations of the differential relations of the properties p, v and T in the above

equations are carried out with the help of an equation of state. The changes in

properties are independent of the path and depend only on the end states. Let us

consider that the change in enthalpy per unit mass of a gas from a reference state 0 to

, T0 having enthalpy, h0 to some other state B at p, T with enthalpy h is to be

calculated (Diagram 9.8). The reversible path 0B may be replaced for convenience by

either path 0-a-B or path 0-b-B, both also being reversible.

Path 0-a-B:

From Eqn. 9.31(b),

On addition.

Similarly, for

Path 0-b-B:

Equation (11.27) is preferred to Eq. (11.28) since cp at lower pressure can be

conveniently measured. Now,

Again,

T

P

0( , )

b(p, )

a( , T)

= C

p = C

= C

B(p, T)





Substituting in Eq. (9.32),

To find the entropy change, Eq. (9.31c) is integrated to yield:

9.11 GENERAL THERMODYNAMIC CONSIDERATIONS ON AN EQUATION OF

STATE

Certain general characteristics are common to all gases. These must be clearly observed

in the developing and testing of an equation of state. It is edifying to discuss briefly

some of the more important ones:

1. Any equation of state must reduce to the ideal gas equation as pressure

approaches zero at any temperature. This is clearly seen in a generalized

compressibility factor chart in which all isotherms converge to the point z = 1 at

zero pressure. Therefore,

Also, as seen from effect of clearance on volumetric efficiency graph, the reduced

isotherms approach the line z = 1 as the temperature approaches infinity, or:

2. The critical isotherm of an equation of state should have a point of inflection at

the critical point on p-v coordinates, or

3. The isochores of an equation of state on a p-T diagram should be essentially

straight, or:

An equation of state can predict the slope of the critical isochore of a fluid. This

slope is identical with the slope of the vaporization curve at the critical point.





From the Clapegron equation, dp/dT =Δs/Δv, the slope of the vaporization curve

at the critical point becomes:

Therefore, the vapour-pressure slope at the critical point, dp/dT, is equal to the

slope of the critical isochore [Diagram 9.9].

Diagram 9.9 P-T graph with isochoric lines

4. The slopes of the isotherms of an equation of state on a Z-p compressibility factor

chart as p approaches zero should be negative at lower temperatures and positive

at higher temperatures. At the Boyle temperature, the slope is zero as p

approaches zero, or

An equation of state should predict the Boyle temperature which is about 2.54 TC

for many gases.

An isotherm of maximum slope on the Z-p plot as p approaches zero, called the

foldback isotherm, which is about 5TC for many gases, should be predicted by an

equation of state, for which:

where TF is the foldback temperature. As temperature increases beyond TF the

slope of the isotherm decreases, but always remains positive.

5. An equation of state should predict the Joule-Thomson coefficient, which is

For the inversion curve, = 0,

9.12 CHEMICAL POTENTIAL

In case of multicomponent systems such as non-reacting gas mixtures the partial molal

properties are used for describing the behaviour of mixtures and solutions. Partial molal

properties are intensive properties of the mixture and can be defined as,

T

P

Critical point





where X is extensive property for multi component system in single phase.

X = X(T, p, n) i.e. function of temperature, pressure and no. of moles of each component

nk refers to the all n values with varying k values and are kept constant except for ni.

In multicomponent systems the partial molal Gibbs function for different constituents

are termed as “chemical potential” for particular constituent. Chemical potential, μi can

be defined for ith component as,

where G, ni, nk, T and P have usual meanings. Chemical potential being a partial molal

property is intensive property.

Also, it can be given as,

Thus, for non reacting gas mixture the expression for internal energy, enthalpy,

Helmholtz function can be given using G defined as above,

Internal energy,

Enthalpy,

Helmholtz function,

Writing differential of G considering it as function of (T, p, n1, n2, ... nj)

From definition of Gibbs function dG = Vdp – SdT, for T = constant,

for pressure as constant,

Therefore,

Also from Eq (9.38) we can write differential as,





From two differential of function G we get,

Above equation is also called Gibbs-Duhem equation.

9.13 FUGACITY

From earlier discussions for a single component system one can write,

or, μ = G/n ⇒Chemical potential for pure substance = Gibbs function per mole.

For Gibbs function written on unit mole basis,

For constant temperature

If single component system is perfect gas then,

Here chemical potential may have any value depending upon the value of pressure.

Above mathematical formulation is valid only for perfect gas behaviour being exhibited

by the system. For a real gas above mathematical equation may be valid if pressure is

replaced by some other property is named as ‘fugacity’.

Fugacity was first used by Lewis.

Fugacity denoted by ‘ℱ’ can be substituted for pressure in above equation,

ℱ

For constant pressure using

and above equation, we get

ℱ

Thus, for a limiting case when ideal gas behaviour is approached the fugacity of a pure

component shall equal the pressure in limit of zero pressure.

ℱ

For an ideal gas ℱ = p.

For real gas, equation of state can be given using compressibility factor as,

Substituting the fugacity function,

ℱ





ℱ

or,

ℱ

Here as p →0 the Z →1

Also we have seen

ℱ

or,

ℱ

or,

ℱ

Integrating between very low pressure p* and high pressure p.

ℱ

ℱ

ℱ

ℱ ℱ

Here for very low pressure, ℱ* = p*

or,

ℱ

When low pressure is 1 atm then the ratio (ℱ/ ℱ*) is termed as “activity”.

bbaassiicc aanndd aapppplliieedd ... data/thermodynamics... · thermodynamic properties such as...

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