1 5.5.3 The van der Waals equation of state and the inter-molecular force (continued) (Hiroshi Matsuoka)

van der Waals’ bold application of his equation to high-density gases and liquids

Despite the above-mentioned caveat about the usefulness of the reduced van der Waals

equation of state near the critical point, van der Waals originally assumed that his equation of

state,

P =RT

v ! b!a

v2

,

to be good even at the critical point. He also defined the critical point associated with his

equation of state to be the point where the pressure function satisfy the following two conditions:

!

"P

"v

#

$ %

&

' ( T=T

c

v= vc

= 0 and

!

" 2P

"v 2#

$ %

&

' ( T=T

c

v= vc

= 0 .

These are actually the conditions that experimental data for real gases satisfy. Mathematically,

these conditions imply that near the critical point the pressure hardly changes as the molar

volume increases while the temperature is kept constant. Physically, they imply that the critical

point is where the molar volume can rapidly change in response to a slight change in the

pressure. In other words, the critical point is where the system is teetering on the edge between

the liquid and the gaseous phases. More on this later when we discuss phase transitions and

critical points.

By solving these two equations, van der Waals found the critical point to be parameterized by

the constants a and b:

Pc,v

c,T

c( ) =

a

27b2,3b,

8a

27Rb

!

" #

$

% & . (HW#5.5.5: show this)

so that his equation of state in terms of the reduced variables becomes

2

!

˜ P =8

3

˜ T

˜ v " 1 3( )"

9

8

1

˜ v 2

# $ %

& ' (

. (HW#5.5.6: show this)

We can easily check that this “reduced” equation of state satisfies

˜ P

˜ T =1, ˜ v = 1( ) = 1

and by comparing this “reduced” equation of state with the simplified universal equation of state

derived from the virial expansion,

!

ˆ P =1

Zc

ˆ T

ˆ v " ˆ b "

ˆ a

ˆ v 2

#

$ %

&

' ( ,

we can read off various parameters for the van der Waals equation of state:

Zc

vdW=3

8= 0.375 (cf. Z

c! 0.3)

ˆ a vdW

=9

8= 1.125 (cf. ˆ a ! 0.98)

ˆ b vdW

=1

3= 0.333 (cf. ˆ b ! 0.44 )

Clearly, the van der Waals equation of state cannot reproduce the experimentally observed

universal equation of state. However, it is much simpler so that it can provide us with a physical

understanding of the main features of the equation of state for classical simple gases. Therefore,

it should not be too surprising that van der Waals applied his equation of state even to the liquid

phase and received the Nobel prize in physics in 1910 for “his work on the equation of state for

gases and liquids.”

The physical meaning of Z (revisited) and the van der Waals equation of state

Z basically compares the molar volume v measured for a gas at

!

T,P( ) against the value

predicted by the ideal gas law or

!

vIdeal

" RT P :

3

!

Z =Pv

RT=v T,P( )RT

P

"

# $

%

& '

=v T,P( )

vIdeal

T,P( )

so that Z <1 implies v T, P( ) < vIdeal

T ,P( ) whereas Z >1 implies v T, P( ) > vIdeal

T ,P( ) . In the

figures below, we show Z and

!

ˆ v as functions of

!

ˆ P as predicted by the van der Waals equation of

state. ˆ T =1

ˆ T = 2

ˆ T =15

4 As shown in the left-hand side figures on the previous page, the van der Waals equation of state

at least qualitatively captures the main features of

!

Z ˆ P , ˆ T ( ). In the right-hand side figures, for

comparison, we also plot the curve for

!

ˆ v = 8 3( ) ˆ T ˆ P ( ) , which is the ideal gas law in terms of the

reduced variables for the van der Waals equation of state.

We now understand that at low pressures, the attractive forces among molecules bring Z below 1

or Z <1 while for high temperatures or high pressure the finite molecular volume or the

repulsive forces among molecules pushes Z above 1 or Z >1 .

Two parameters with dimensions of energy and volume and the reduced equation of state

As we have shown in the above, from the van der Waals equation of state,

!

P =RT

v " b"a

v2

,

we can derive the reduced van der Waals equation of state,

!

˜ P =8

3

˜ T

˜ v " 1 3( )"

9

8

1

˜ v 2

# $ %

& ' (

,

5 which is universal as it contains no parameter specific to a particular gas. We can also derive

universal reduced equations of state for various “empirical” equations of state that have been

used for engineering applications of gases. Examples of the empirical equations of state include

the Dieterici (1899) equation,

!

P =RT

v " bD

exp "aD

RTv

#

$ %

&

' ( ,

the Berthelot (1903) equation,

!

P =RT

v " bB

"aB

RTv2

,

and the Redlich-Kwong (1949) equation,

!

P =RT

v " bRK

"aRK

RTv v + b( ).

Why can we derive a universal reduced equation of state for each of these equations of state?

By employing dimensional analysis, we can answer this question. Note first that all these

equations contain two parameters that carry physical dimensions. For example, the two

parameters, a and b, in the van der Waals equation of state carry the following physical

dimensions:

!

a[ ] = RT[ ] v[ ] = energy[ ] volume[ ] ,

and

!

b[ ] = v[ ] = volume[ ] .

We also find

!

bD[ ] = b

B[ ] = bRK[ ] = v[ ] = volume[ ] ,

!

aD[ ] = RT[ ] v[ ] = energy[ ] volume[ ],

6

!

aB[ ] = RT[ ]

2v[ ] = energy[ ]

2volume[ ] ,

and

!

aRK[ ] = RT[ ]

3 2v[ ] = energy[ ]

3 2volume[ ] .

Using the two parameters in each equation of state, we can then define a new parameter with the

dimension of energy by

!

" # a b ,

!

"D# a

DbD

,

!

"B# a

BbB

, and

!

"RK# a

RKbRK( )

2 3 .

In general, if an empirical equation of state contains two parameters,

!

" and b, with

dimensions of energy and volume just like

!

a b and b in the van der Waals equation of state, we

can show that this equation of state leads to a universal reduced equation of state as follows.

Since

!

P[ ] = energy[ ] volume[ ] = " b[ ] , we can express the pressure P as

!

P ="

bf ˆ T , ˆ v ( ),

where

!

f ˆ T , ˆ v ( ) is a dimensionless function of dimensionless reduced variables,

!

ˆ T and

!

ˆ v , which

are defined by

!

ˆ T " RT # and

!

ˆ v " v b. We then find the universal reduced equation of state:

!

ˆ P "Pb

#= f ˆ T , ˆ v ( ) .

As

!

RTc and

!

vc share physical dimensions with

!

" and b, respectively, using the following two

conditions,

!

"P

"v

#

$ %

&

' ( T=T

c

v= vc

= 0 and

!

" 2P

"v 2#

$ %

&

' ( T=T

c

v= vc

= 0 ,

we can also express

!

RTc,

!

vc, and

!

Pc in terms of

!

" and b:

!

RTc

= a numerical factor( )",

!

vc

= a numerical factor( )b , and

!

Pc

= a numerical factor( ) " b( ) . Consequently,

!

Zc

= PcvcRT

c( )

takes a particular numerical value for each empirical equation of state. As shown in HW#5.5.1,

the compressibility factor Z defined for each empirical equation of state by

!

Z " P T,v( )v RT( )

then becomes a function of

!

ˆ P and

!

ˆ T :

!

Z = Z ˆ P , ˆ T ( ) .

7 SUMMARY FOR SEC.5.5.3

1. For

!

6.6 > ˆ T >>1, approximating the universal reduced second virial coefficient

!

ˆ B ˆ T ( ) for the

classical simple gases as

!

ˆ B ˆ T ( ) = ˆ b "ˆ a

ˆ T ,

where

!

ˆ a = 0.98097 and

!

ˆ b = 0.44226, we derive the reduced van der Waals equation of state:

!

ˆ P =1

Zc

ˆ T

ˆ v " ˆ b "

ˆ a

ˆ v 2

#

$ %

&

' ( ,

which leads to the van der Waals equation of state:

!

P =RT

v " b"a

v2

,

where

!

a = RTcv

cˆ a and

!

b = vc

ˆ b .

2. In the van der Waals equation of state, the parameter b represents the effect of the finite

molecular size or the repulsive forces among molecules on the pressure whereas the

parameter a represents the effect of the attractive forces among molecules on the pressure.

3. Using the approximate form for

!

ˆ B ˆ T ( ) ,

!

ˆ B ˆ T ( ) = ˆ b "ˆ a

ˆ T ,

we can approximate the compressibility factor as

!

Z =1+ ˆ b "ˆ a

ˆ T

#

$ %

&

' (

1

ˆ v ,

8 with which we explain how Z depends on the inter-molecular force for

!

ˆ v >>1.

4. Modeling the inter-molecular potential energy using two parameters with dimensions of

energy and volume, we can construct a minimal model for the universality class of classical

simple gases by including the inter-molecular force due to an inter-molecular potential

energy and calculate the reduced second virial coefficient that agrees very well with the

experimentally obtained universal reduced second virial coefficient.

5. The van der Waals equation of state qualitatively captures the main features of

!

Z ˆ P , ˆ T ( ) even

beyond the low-density regime.

6. An empirical equation of state such as that of van der Waals with two parameters with

dimensions of energy and volume leads to a universal reduced equation of state.

Answers for the homework questions in Sec.5.5.3

HW#5.5.3

!

ˆ P =ˆ T

Zcˆ v

1+

ˆ B ˆ T ( )ˆ v

"

#

$ $

%

&

' '

=ˆ T

Zcˆ v

1+1

ˆ v

ˆ b (ˆ a

ˆ T

)

* +

,

- .

"

# $

%

& ' =

1

Zc

ˆ T

ˆ v 1+

ˆ b

ˆ v

)

* +

,

- . (

ˆ a

ˆ v 2

"

# $ $

%

& ' '

=1

Zc

ˆ T

ˆ v

1

1( ˆ b ˆ v

)

* +

,

- . (

ˆ a

ˆ v 2

"

# $ $

%

& ' '

=1

Zc

ˆ T

ˆ v ( ˆ b (

ˆ a

ˆ v 2

)

* +

,

- .

HW#5.5.4

!

P = Pc

ˆ P =P

c

Zc

ˆ T

ˆ v " ˆ b "

ˆ a

ˆ v 2

#

$ %

&

' ( =

Pc

Pcv

c

RTc

#

$ %

&

' (

ˆ T

ˆ v " ˆ b "

ˆ a

ˆ v 2

#

$ %

&

' (

=RT

c

ˆ T

vcˆ v " v

c

ˆ b "

RTcˆ a

vcˆ v

2=

RT

v " vc

ˆ b "

RTcv

cˆ a

v2

Comparing this equation with

!

P =RT

v " b"a

v2

,

we find a = RT

cv

cˆ a and b = v

c

ˆ b .

9 HW#5.5.5

The critical point is defined by

!

"P

"v

#

$ %

&

' ( T=T

c

v= vc

= 0 and

!

" 2P

"v 2#

$ %

&

' ( T=T

c

v= vc

= 0 .

By substituting the van der Waals equation into these equations, we get

!

0 ="P

"v

#

$ %

&

' ( T=T

c

v= vc

= )RT

c

vc) b( )

2+ 2

a

vc

3 (1)

and

!

0 =" 2P

"v 2#

$ %

&

' ( T=T

c

v= vc

= 2RT

c

vc) b( )

3) 6

a

vc

4= 2

RTc

vc) b( )

3) 3

a

vc

4

* + ,

- ,

. / ,

0 ,

or

!

RTc

vc" b( )

3= 3

a

vc

4 (2)

From Eq.(1), we find

!

Tc

= 2a v

c" b( )

2

Rvc

3. (3)

By substituting Eq.(3) into Eq.(2), we get

!

R

vc" b( )

32a v

c" b( )

2

Rvc

3

# $ %

& %

' ( %

) % = 3

a

vc

4,

from which we find

!

2vc

= 3 vc" b( ) so that

!

vc

= 3b. (4) By substituting Eq.(4) into Eq.(3), we then get

!

Tc

= 2a v

c" b( )

2

Rvc

3= 2

a 3b " b( )2

R 3b( )3

=8a

27Rb. (5)

Finally, by substituting Eq.(4) and Eq.(5) into the van der Waals equation of state, we can find the critical pressure:

!

Pc

=RT

c

vc" b

"a

vc

2=

8a

27b

3b " b"

a

3b( )2

=4

27"1

9

#

$ %

&

' ( a

b2

=a

27b2

10 Therefore, the critical point is located at

!

Pc,v

c,T

c( ) =a

27b2,3b,

8a

27Rb

"

# $

%

& '

HW#5.5.6

By substituting

!

P = Pc

ˆ P ,

!

T = Tc

ˆ T , and

!

v = vcˆ v into the original van der Waals equation of

state, we find

!

Pc

ˆ P =R T

c

ˆ T ( )v

cˆ v " b

"a

vcˆ v ( )

2

so that

!

ˆ P =RT

c

Pcv

c

"

# $

%

& '

ˆ T

ˆ v ( b vc

(a

RTcv

c

"

# $

%

& '

1

ˆ v 2

) * +

, - .

=1

Zc

ˆ T

ˆ v ( b 3b( )(

a

R8a

27Rb

"

# $

%

& ' 3b( )

"

#

$ $ $ $

%

&

' ' ' '

1

ˆ v 2

)

*

/ /

+

/ /

,

-

/ /

.

/ /

=8

3

ˆ T

ˆ v (1 3(

9

8

"

# $ %

& '

1

ˆ v 2

) * +

, - .

=8 ˆ T

3ˆ v (1(

3

ˆ v 2

or

!

ˆ P =8 ˆ T

3ˆ v "1"

3

ˆ v 2