EQUATION OF STATE

CHAPTER 2

• An equation of state is a relation between state variables

• It is a thermodynamic equation describing the state of matter under a given set of physical conditions.

• It is a constitutive equation which provides a mathematical relationship between two or more state functions associated with the matter, such as its temperature, pressure, volume, or internal energy.

• Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and even the interior of stars.

EQUATION OF STATE

Assumption:

1) the gas consists of a large number of molecules, which are in random motion and obey Newton's laws of motion;

2) the volume of the molecules is negligibly small compared to the volume occupied by the gas; and

3) no forces act on the molecules except during elastic collisions of negligible duration.

CLASSICAL IDEAL GAS LAW

PV = RT

CUBIC EQUATIONS OF STATE

1. Van der Waals equation of state

2. Redlich–Kwong equation of state

3. Soave modification of Redlich–Kwong

4. Peng–Robinson equation of state

VAN DER WAALS EQUATION OF STATE

The van der Waals equation may be considered as the ideal gas law, “improved” due to two independent reasons: Molecules are thought as particles with volume, not

material points. Thus V cannot be too little, less than some constant. So we get (V – b) instead of V.

We consider molecules attracting others within a distance of several molecules' radii affects pressure we get dengan (P + a/V2) instead of P.

RTbVVa

P 2

where V is molar volume

The substance-specific constants a and b can be calculated from the critical properties Pc, Tc, and Vc as

c

2c

2

c

2c

2

PTR

6427

PTR

6427

a

c

c

c

c

PTR

81

PTR

81

b

Cubic form of vdW eos

2Va

bVRT

P

0P

abV

Pa

VP

RTbV 23

0ABZAZ1BZ 23

2r

r22 T

P6427

TRaP

A

r

r

TR

81

RTbP

B

Principle of Corresponding States (PCS)

The principle of Corresponding States (PCS) was stated by van der Waals and reads: “Substances behave alike at the same reduced states. Substances at same reduced states are at corresponding states.”

Reduced properties provide a measure of the “departure” of the conditions of the substance from its own critical conditions and are defined as follows

cr T

TT

cr P

PP

cr V

VV

• The PCS says that all gases behave alike at the same reduced conditions.

• That is, if two gases have the same “relative departure” from criticality (i.e., they are at the same reduced conditions), the corresponding state principle demands that they behave alike.

• In this case, the two conditions “correspond” to one another, and we are to expect those gases to have the same properties.

Reduced form of vdW EOS:

rr2r

r T81V3V3

P

• This equation is “universal”.

• It does not care about which fluids we are talking about.

• Just give it the reduced conditions “Pr, Tr” and it will give you back Vr — regardless of the fluid.

• As long as two gases are at corresponding states (same reduced conditions), it does not matter what components you are talking about, or what is the nature of the substances you are talking about; they will behave alike.

The compressibility factor at the critical point, which is defined as

c

ccc RT

VPZ

Zc is predicted to be a constant independent of substance by many equations of state; the Van der Waals equation e.g. predicts a value of 0.375

Substance ValueH2O 0.23He 0.30H2 0.30Ne 0.29N2 0.29Ar 0.29

Zc of various substances

Standing-Katz Compressibility Factor Chart

Application of PCS

REDLICH-KWONG EOS

The Redlich–Kwong equation is adequate for calculation of gas phase properties when:

bVVa

bVRT

P

c

2c

2

PTR

42748.0a

c

c

PTR

08662.0b

cc T2T

PP

21rT

Cubic form of RK eos

bVVa

bVRT

P

0ABZBBAZZ 223

2r

r22 T

P42748.0

TRPa

A

r

r

TP

08662.0RTbP

B

SOAVE-REDLICH-KWONG EOS

bVVa

bVRT

P

c

2c

2

PTR

42748.0a c

c

PTR

08662.0b

25.0r

2 T115613.055171.148508.01

r2 T30288.0exp202.1:HorF

Cubic form of SRK eos

bVVa

bVRT

P

0ABZBBAZZ 223

2r

r22 T

P42748.0

TRPa

A

r

r

TP

08662.0RTbP

B

PENG-ROBINSON EOS

22 bbV2Va

bVRT

P

c

2c

2

PTR

45724.0a

c

c

PTR

07780.0b

25.0r

2 T12699.054226.137464.01

Cubic form of PR eos

0BBABZB3B2AZB1Z 32223

2r

r22 T

P45724.0

TRPa

A

r

r

TP

07780.0RTbP

B

22 bbV2Va

bVRT

P

SOLVING CUBIC EQUATION

0cZcZcZ 012

23

eos c2 c1 c0

vdW – B – 1 A – ABRK – 1 A – B – B2 – ABSRK – 1 A – B – B2 – ABPR B – 1 A – 2B – 3B2 AB – B2 – B3

0cZcZcZ 012

23

27c

cK21

2

01232 c27cc9c2

221

L

4L

27K

D23

(determinant)

Calculate:

31

D2L

M

31

D2L

N

Case 1: D > 01 real root and 2 imaginary roots

3c

NMZ 21

Case 2: D = 0 three real roots and at least two are equal

3c

NMZ 21

3c

NM21

ZZ 232

Case 3: D < 0three, distinct, real roots

3c

k120cos3K

2Z 2i

Where k = 0 for i = 1k = 1 for i = 2k = 2 for i = 3

27K4L

cos 3

21

The minus sign applies when B > 0, The plus sign applies when B < 0.

NON CUBIC EQUATIONS OF STATE

VIRIAL EOS

...VD

VC

VB

1Z 32

2P'CP'B1RTPV

Z

RTB

'B 2

2

RTBC

'C

3

3

RTB2BC3D

'D

DIETERICI EOS

RTVaRTebVP

Rb4a

Tc 22c eb4a

P b2Vc

MIXTURE

• For mixtures, we apply the same equation, but we impose certain mixing rules to obtain “a” and “b”, which are functions of the properties of the pure components.

• We create a new “pseudo” pure substance that has the average properties of the mixture.

i j

ijjim ayya

i

iim byb

jiijij aak1a