2 Equation of State1

8/11/2019 2 Equation of State1

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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.


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Assumption:

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

motion;

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

3) no forces act on the molecules except during elastic

collisions of negligible duration.

PV = RT


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1. Van der Waals equation of state

2. RedlichKwong equation of state3. Soave modification of RedlichKwong

4. PengRobinson equation of state


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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 (Vb) instead of V.

We consider molecules attracting others within adistance of several molecules' radii affects pressure

we get dengan (P+ a/V2) instead of P.


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RTbVV

aP

2

where V is molar volume

The substance-specific constants a and b can be

calculated from the critical properties Pc, Tc, and Vcas

c

2

c

2

c

2

c

2

P

TR

64

27

P

TR

64

27a

c

c

c

c

P

TR

8

1

P

TR

8

1b


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Cubic form of vdW eos

2V

a

bV

RT

P

0

P

abV

P

aV

P

RTbV

23

0ABZAZ1BZ 23

2

r

r22

TP

6427

TRaPA

r

r

T

R

8

1

RT

bPB


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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

c

rT

TT

c

rP

PP

c

rV

VV


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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 stateprinciple demands that they behave alike.

In this case, the two conditions correspond to one

another, and we are to expect those gases to havethe same properties.


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Reduced form of vdW EOS:

rr2

r

r T81V3V

3

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 Vrregardless of the fluid.

As long as two gases are at corresponding states (samereduced 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.


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The compressibility factor at the critical point, which is

defined as

c

cc

cRT

VPZ

Zcis predicted to be a constant independent ofsubstance by many equations of state; the Van der Waals

equatione.g. predicts a value of 0.375
http://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equationhttp://en.wikipedia.org/wiki/Van_der_Waals_equation


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Substance Value

H2O 0.23

He 0.30

H2 0.30

Ne 0.29

N2 0.29Ar 0.29

Zcof various substances


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Standing-Katz

Compressibility

Factor Chart

Application of PCS


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The Redlich

Kwong equation is adequate for calculationof gas phase properties when:

bVVa

bVRTP

c

2

c

2

P

TR42748.0a

c

c

P

TR08662.0b

cc T2

T

P

P

21

rT


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Cubic form of RK eos

bVV

a

bV

RTP

0ABZBBAZZ 223

2

r

r

22

T

P42748.0

TR

PaA

r

r

T

P08662.0

RT

bPB


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bVV

a

bV

RTP

c

2c

2

P

TR42748.0ac

c

P

TR08662.0b

2

5.0r2 T115613.055171.148508.01

r2 T30288.0exp202.1:HorF


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Cubic form of SRK eos

bVV

a

bV

RTP

0ABZBBAZZ 223

2

r

r

22

T

P42748.0

TR

PaA

r

r

T

P08662.0

RT

bPB


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22bbV2V

abV

RTP

c

2

c

2

PTR45724.0a

c

c

P

TR07780.0b

25.0

r

2T12699.054226.137464.01


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Cubic form of PR eos

0BBABZB3B2AZB1Z 32223

2

r

r

22

T

P45724.0

TR

PaA

r

r

T

P07780.0

RT

bPB

22bbV2V

a

bV

RTP


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0cZcZcZ 01223

eos c2 c1 c0

vdW B1 A AB

RK 1 ABB2 AB

SRK 1 ABB2 AB

PR B1 A2B3B2 ABB2B3


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0cZcZcZ01

2

2

3

27

ccK

2

1

2

01232

c27cc9c2221L

4

L

27

KD

23

(determinant)

Calculate:

31

D2

LM

31

D2

LN


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

3

cNMZ 2

1

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

3

cNMZ 21

3

cNM

2

1ZZ 2

32


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

3

ck120cos

3

K2Z 2

i

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

k = 2 for i = 3

27K

4Lcos 3

2

1

The minus sign applies when B > 0,

The plus sign applies when B < 0.


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...

V

D

V

C

V

B1Z

32

2P'CP'B1

RT

PVZ

RT

B'B

2

2

RT

BC'C

3

3

RT

B2BC3D'D


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RTVaRTebVP

Rb4aT

c 22ceb4

aP b2Vc


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For mixtures, we apply the same equation,but weimpose 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 jijjim

ayya

i

iim byb

jiijij aak1a

2 Equation of State1

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