realistic equations of state

Chapter 8

Real Gases

Compression FactorsReal gases do not obey the perfect gas equation exactly. The measure of the deviation from ideality of the behavior of a real gas is expressed as the compression factor Z:

RTPVTPZ m),( (8.1)

Real Gases

Z

P

200 K500 K

1000 K

200 K

1000 K

0 300 600 900

3

2

1

0

CH4

Z

P

H2

0 100 200 300

1.2

1

0.8

0

0oC

N2

CH4

Physical Chemistry

Real Gas Equations of State

RTbVVaP mm

2

2mm Va

bVRTP

(8.2)

van der Waals equation

Ideal Gas Law/Perfect Gas Equation:

PV = nRT (1.18)*

RTPVm

Real Gases

Physical Chemistry

Van der Waals Equation of State

RTbVVaP mm

))(( 2

RTPVm

Real Gases

Physical Chemistry

: to correct the effect of intermolecular attractive forces on the gas pressure

2m/a V

b: the volume excluded by intermolecular repulsive forces

Virial Equation of State

32

)()()(1mmm

m VTD

VTC

VTBRTPV (8.4)

Redlich-Kwong Equation

2/1)( TbVVa

bVRTP

mmm

(8.3)

Real Gases

Physical Chemistry

Real Gas Equations of State

The limited accuracy of the data allows evaluation of only B(T) and sometimes C(T).

Virial Equation of State

])(')(')('1[ 32 PTDPTCPTBRTPVm (8.5)222 )''(,' TRCBCRTBB (8.6)

BP

RTVm (8.7)low P

mmmm

mm

RTVa

VbRTVa

bVVZ

RTPV

/1

1vdW gas

2mm Va

bVRTP

(8.2)RTVm

Real Gases

Physical Chemistry

32

)()()(1mmm

m VTD

VTC

VTBRTPV (8.4)

Power series in 1/Vm

Power series in P

Gas Mixtures

2211222

2/121211

21 )(2 bxbxbandaxaaxxaxa (8.10)

For a mixture of two gases, 1 and 2, use a two-parameter equation,

(8.11)tot

m nVV mean molar volume

Real Gases

Physical Chemistry

x1 and x2: the mole fractions of the componentsb: a weighted average of b1 and b2

a: related to intermolecular attractions(a1a2)1/2: intermolecular interaction between gases 1 and 2

Isotherms of H2O

P

Vm

400 oC

U

RJ N

Y

374 oC300 oC200 oC

H2O

L + VL V

L

G

H

T S

K

MW

CondensationT < 374 oCgas condenses to liquid when PT = 300 oC

R(vapor)S(saturated vapor), P, V

S(saturated vapor)W(saturated liquid), P, V W(saturated liquid)Y(liquid), P , V

Real Gases

Physical Chemistry

t/℃

A

DC

0.00611

0.01

solid

gas

liquid

O

P / 1

0 5 Pa

374.2

218 atm

H2O phase diagram: P — T

99.974

1 atm

0.0024

I

R

S

Y

Tf TbT3

Real Gases

Physical Chemistry

400 oC

CondensationT 374 oCNo amount of compression will cause the separation out of a liquid phase in equil. with the gas.

T = 374 oCCritical temperature TcCritical pressure PcCritical volume Vm,cCritical constants Isotherms of H2O

P

Vm

U

RJ N

Y

374 oC300 oC200 oC

H2O

L + VL V

L

G

H

T S

K

MW

Real Gases

Physical Chemistry

Fig. 8.3

Critical constantsCritical T (Tc), Tc(CO2)=304.2 K

Critical P (Pc), Pc(CO2)=7.38 MPaCritical molar V (Vm,c), Vm,c(CO2)=94×10-6 m3·mol-1

Isotherms of CO2{P}

{Vm,c}

T3

c

Tc

gb a

l

T1 T2

Real Gases

Physical Chemistry

Table 8.1 Critical Constants Species Tc / K Pc / atm Vm,c / cm3·mol-1

Ar 150.7 48.3 74.6

Ne 44.4 27.2 41.7

N2 126.2 33.5 89.5

H2O 647.1 217.8 56.0

D2O 643.9 213.9 56.2

H2S 373.2 88.2 98.5

CO2 304.2 72.88 94.0

HCl 324.6 82.0 81.0

CH3OH 512.5 80.8 117

Real Gases

Physical Chemistry

FluidThere is a continuity between the gaseous and the liquid states. In recognition of this continuity, the term fluid is used to mean either a liquid or a gas.

An ordinary liquid can be viewed as a very dense gas. Only when both phases are present in the system is there a clear-cut distinction between liquid and gaseous states.

For a single-phase liquid system it is customary to define as a liquid a fluid whose temperature is below Tc and whose molar volume is less than Vm,c.If these two conditions are not met, the liquid is called a gas. So a further distinction between gas and vapor can be made, but these two words are used interchangeably in this book.

Real Gases

Physical Chemistry

Supercritical fluid

A supercritical fluid is one whose T and P satisfy

A supercritical fiquid usually has liquidlike density but its viscosity is much lower than typical for a liquid and diffusion coefficients in it are much higher than in liquids.

T > Tc and P > Pc

Real Gases

Physical Chemistry

Supercritical fluid

-60 -40 -20 0 20 40 60 80 100B

5

10

15

20

25

30

35

-56.6℃ tc=31.06℃ t/℃

o

C

gas

liquid

solid

100

200300

400

500600

700800

900100011001200p/MPa

P c=7.38MPa

A

0.518MPa

Supercritical CO2 is used commercially as a solvent to decaffeinate coffee.

Real Gases

Physical Chemistry CO2

Critical data and equations of state

Differentiating the van der Waals equation (8.2)

32

2)( mmTm V

abV

RTVP

and

0

TmVP 02

2

TmVP

At the critical point:

(8.12)

2mm Va

bVRTP

432

2 6)(

2

mmTm Va

bVRT

VP

Application of the conditions (8.12) gives

3,

2,

2)( cmcm

c

Va

bVRT

and 4

,3

,

3)( cmcm

c

Va

bVRT

(8.13)

Real Gases

Physical Chemistry


Division of the first equation in (8.13) by the second yields

2,, cmcm

cc V

abV

RTP

From van der Waals equation:

(8.14)

4,

3,

3,

2,

3

2

)(

)(

cm

cm

cm

c

cm

c

Va

Va

bVRT

bVRT

Use of (8.15) in the first equation in (8.13) gives

32 ,

,cm

cm

VbV

and32 272

4 ba

bRTc (8.16)

bV cm 3, (8.15)

RbaTc 27

8

Real Gases

Physical Chemistry


2,, cmcm

cc V

abV

RTP

Substitution of (8.15) and (8.16) into (8.14)

(8.14)

gives

22 279227/8

ba

ba

bbaPc

(8.16)

bV cm 3, (8.15)

RbaTc 27

8

(8.17)

Real Gases

Physical Chemistry

Critical data and equations of stateSubstitution of (8.15) and (8.16) into (8.14)

Three equations for two parameters, a and b

227baPc

(8.16)

bV cm 3, (8.15)

RbaTc 27

8

(8.17)

c

c

PTRa

6427 22

(8.18)c

c

PRTb8

vdW gas

Real Gases

Physical Chemistry

Critical data and equations of stateCombination of (8.15) to (8.17)

227baPc

(8.16)

bV cm 3, (8.15)

RbaTc 27

8

(8.17)

375.083,

c

cmcc RT

VPZ (8.19)

Real Gases

Physical Chemistry

Predicts the compressibility factor at the critical point

Van der waals equation


375.083,

c

cmcc RT

VPZ (8.19)

1, c

cmc

RTVP

ideal gas

c

c

c

c

PRT

PRTb 08664.0

3)12( 3/1

(8.20)c

c

c

c

PTR

PTRa

2/52

3/1

2/52

42748.0)12(9

(8.21)

333.031,

c

cmcc RT

VPZ (8.22)

R-K equation

Real Gases

Physical Chemistry

Van der waals equation

Selected equations of state

Equation Critical constants

Perfect gas

van der Waals

Berthelot

mVRTP

2mm Va

bVRTP

2mm TV

abV

RTP

227ba

2/1

332

121

baR

bRa

278

2/1

32

32

bRa

b3

b3

cV cTcP

Real Gases

Physical Chemistry

Selected equations of stateEquation Critical constants

Perfect gas

R-K

virial

mVRTP

2/1)( TbVVa

bVRTP

mmm

2

)()(1mmm VTC

VTB

VRTP

cV cTcP

Real Gases

Physical Chemistry

The law of corresponding statesThe critical constants are characteristic properties of gasesThe reduced variables of a gas by dividing the actual variable by the corresponding constant.

The observation that the real gases at the same reduced volume and reduced temperature exert the same reduced pressure is called the law (principle) of corresponding states.

,c

r PPP ,

,cm

mr V

VV ,c

r TTT (8.27)

reduced pressure

reduced volume

reduced temperature

),( rrr TPfV (8.28)

Real Gases

Physical Chemistry

realistic equations of state

Documents