Real Gases Part 2 - The Edelstein Center for the Analysis ... · 04/07/2010 · a gas may be liquefied Real Gases –Part 2 ... & 2 imaginary ones ... Similar to the Ideal Gas Equation

The Critical State (again): “Last Exit Before the Toll”

P

T

Phase Diagram of H2O

S

L

G Fluid

Tc

Explain

the 3

Physical Processes

critical point

triplepoint

Pc

Tc = the highest temperature at which

a gas may be liquefied

Real Gases – Part 2

1 © Prof. Zvi C. Koren 21.07.10

T<<Tc T=TcT<Tc

Tem

per

atu

re (

0C

)davg

davg = at+b

Gas Problems: 35. 2 © Prof. Zvi C. Koren 21.07.10

Experimental Isotherms

of CO2

At T = Tc,

substance is called

a “fluid” (זורם)

Note the behavior of

the gas at T > Tc !!!


L+V

L G

The Liquid-Vapor “Dome” & Boiling Points


L+V

L G

Consider 2 paths:B–i–f–A: 2-phase L-V path

B–D–C–E–A: 1-phase Fluid path

Principle ofContinuity of States

The liquid state can be

considered as a continuation

of the gaseous state:

That is, a liquid can be

considered as a highly

compressed (very dense) gas.

Thus, an equation of state for

a gas at high pressure may

also be applicable to the

liquid state.


Theoretical Isotherms of

CO2 According to the

van der Waals Equation

Experimental(from previous slide)

RT b-VV

aP 2


Equivalency of the

vdW Eqn. and

Experimental Results

Experimental

vdW

Equal Areas


Determination of the van der Waals Constants, a & b

Two Methods:

1. Algebraic –

Polynomial Properties

2. Differential –

Point of Inflection Properties

CO2

0 P

VP

VP

PbRTV

23

aba

3 roots. At:

T > Tc 1 real root& 2 imaginary ones

T < Tc 3 real roots (b,c,d)T = Tc 3 identical real roots (a)

1. Algebraic – Polynomial Properties:

vdW is a Cubic Eqn. in V:


vdW at (Tc,Vc):

STRATEGY:

• We already know that vdW is a cubic eqn. in V.

• So, create a general cubic eqn in V, and then compare coefficients:

Expand “V” about “Vc” for any order desired:

c

c

c

c

c

ccc

ccc

cc

ccc

P

abV

P

aV

P

bPRTV

P

ab

P

a

P

bPRT

VVV

, 3 , 3

0 V VV

0 V 3 V 3 V

32

23

3223

Comparing coefficients:

0 V 0 V V , TAt 3 cccc VVVT


cc PVa2

3

From before:

c

c

c

c

c

ccc

P

abV

P

aV

P

bPRTV , 3 , 3

32

a & b in terms of (Tc,Pc):

a & b in terms of (Vc,Pc):3cV

b

c

c

c

c

P

RTb

P

TRa

8 ,

64

2722

375.08

3

c

ccc

RT

VPZ

c

cc

T

VPR

3

8

according to vdW, for all gases:

van der Waals states that:

1. Zc is a constant, AND

2. Zc = 0.375

Experimentally:

Zc 0.3 (for many relatively non-polar gases)10 © Prof. Zvi C. Koren 21.07.10

2. Differential – Point of Inflection Properties:

CO2

P

V

0

(P/V)Tc=0

[(P/V)/V]Tc = (2P/V2)Tc = 0

Tc

0Vc

dP/dV 0

horizontal

point of inflection


(P/V)Tc=0

(2P/V2)Tc = 0

At the critical point (point of inflection):

van der Waals eqn.:

2V-V

RT P

a

b

0

V

2

-V

RT

V

P :)T,V,(PAt 3

c

2

c

c

T

ccc

c

a

b

0

V

6

-V

2RT

V

P :)T,V,(PAt 4

c

3

c

c

T

2

2

ccc

c

a

b

2ccc27

P , 27

8 T ,3 V

b

a

Rb

ab

c

c

c

c

P

RTb

P

TRa

8 ,

64

2722

8

3

c

ccc

RT

VPZ (as before)


Critical Constants and van der Waals Parameters

van der WaalsExperimentalb (L/mol)a (L2atm/mol2)ZcVc (L/mol)dc (g/cc)Pc (atm)tc (oC)Gas

0.235111.5132.4Ammonia

0.53148-122Argon

0.46073.030.98Carbon dioxide

0.31135-139Carbon monoxide

0.57376.1144.0Chlorine

0.2148.832.1Ethane

0.275563.1243.1Ethyl alcohol

0.2250.99.7Ethylene

0.06932.26-267.9Helium

0.031012.8-239.9Hydrogen

0.48425.9-228.7Neon

0.5265-94Nitric oxide

0.311033.5-147.1Nitrogen

0.43049.7-118.8Oxygen

0.22642.0196.81Propane

0.29241.6320.6Toluene

0.307219.5374.4Water


Principle of Corresponding States

Define dimensionless variables.

(An important “trick” in science and engineering.)

Define reduced variables: Pr=P/Pc, Vr=V/Vc, Tr=T/Tc,

Using, for example, van der Waals’ eqn., and recalling that:

c

cc

T

VPR

3

8cc PVa

23

3cV

b

RT -VV

P 2

b

a

Substitute these into vdW eqn.:

T3

8

3-V

V

3P 2

2

c

ccccc

T

VPVPV

(continued)14 © Prof. Zvi C. Koren 21.07.10

T3

8

3-V

V

3P 2

2

c

ccccc

T

VPVPV

PcVc:

cc

c

c T

T

V

V

V

V

P

P

3

8

3

132

2

rr

r

r TVV

P 8 133

2

van der WaalsReduced Equation

of State

Benefits of this eqn.:

1. No direct parameters specific for the gas molecules

2. Universally true

3. Similar to the Ideal Gas Equation of State.

Principle of Corresponding States:

Gases with the same reduced volume and the same reduced temperature

have the same reduced pressure; hence, they are in corresponding states.15 © Prof. Zvi C. Koren 21.07.10

Universal van der Waals

(Reduced Eqn. of State)

0.4

0.9

1.4

1.9

2.4

2.9

3.4

0.5 1 1.5 2 2.5Vr

Pr

0.9

0.95

1.0

1.05

1.1

1.15

1.4

ideal

log-log

0.1

1

10

100

0.1 1 10

log Vr

l o g

P r

PV=k, [T]

logP + logV=logk

logP = –logV+logk

Tr

Note the behavior of the gas at T > Tc !!!

1

slope = – 1

P vs. V


Universal van der Waals Equation

(Reduced vdW Equation of State)

3DAll Reduced Variables


How graph is obtained:

For N2, for example:

If Pr=3.0, P=3.0Pc=known

If Tr=1.0, T=1.0Tc=known

At this P & T, V is measured

Zexp = PV/RT=known

Experimental Z’sas a function of Pr

Tr =

Pr18 © Prof. Zvi C. Koren 21.07.10

Expansion and Compression Abilities

Coefficient of thermal expansionThermal expansivityVolume expansivity

Cubic expansion coefficient PT

V

V

1 βor α

• For an ideal gas, “expansivity” = 1/T

(Noteeach

factor)

Isothermal compressibility

TP

V

V

1 or

(“kappa”)

• For an ideal gas, = 1/P

(chem) (eng)

(eng) (chem)

Just Do it!

Just Do it!

Gas Problems: 36-42.


Documents

Real Gases Part 2 - The Edelstein Center for the Analysis ... · 04/07/2010 · a gas may be liquefied Real Gases –Part 2 ... & 2 imaginary ones ... Similar to the Ideal Gas Equation