IDEAL AND REAL GASES 385

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\M-therm\Th8-1.pm5

vc^3 = ab

pc

,

and from these by a simple reduction, we have

vb

p a

b

T a

bR

c

c

c

=

=

=

U

V

|

|

|

W

|

|

|

3

27

8

27

2

.

...(8.27)

From these equations it follows that the critical volume, pressure, and temperature are all
completely determined by the constants of equation (8.19).
The equation (8.27) indicates the critical constants for a particular gas and leads to the
following results :
The values of a and b are also given by

a = 3pc vc^2 =^9

8

RTc vc =^27

64

. RT
p

c

c

2 2

...(i)

b = vc

3

= RT

p

c

(^8) c
...(ii)
and R =^8
3
pv
T
cc
c
...(iii)
Using the values of a, b and R in equation (8.23), and substituting in (8.26), we have for
carbon dioxide
pc = 61.2 atmospheres,
Tc = 305.3 K or 32.2°C.
It is frequently assumed that the approximate agreement between the calculated and ex-
perimental values of the critical temperature for carbon dioxide is a sufficient verification of Van
der Waals’ theory, but the constant b cannot be calculated with the required degree of accuracy
from Regnault’s experiments to make this an adequate test of the theory.
Also from equations (8.27), we have
pv
RT
cc
c

3
8
= 0.375
whereas experiment shows that about 0.27 as the average value of this ratio, varying considerably,
however, from gas to gas.
The Reduced Equation :
When the pressure, volume and temperature of the fluid are expressed as fractions of the
critical pressure, volume and temperature the reduced form of Van der Waals’ equation is ob-
tained. Thus, writing
p = epc =
ea
27 b^2 ,
v = nvc = 3nb,
T = mTc =
8
27
.
ma
bR
and substituting these values in eqn. (8.19), this reduces to