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⎛ n ⎞ ⎛ n ⎞
Correction term ∝ ⎜ ⎟ ⎜ ⎟
⎝ V ⎠ ⎝ V ⎠
2
n
or Correction term = a
...( 3)
2
V
Where a is the proportionality constant and is a
measure of the forces of attraction between the
molecules. Thus
2
n
p i = p + a
...(4)
2
V
The unit of the term an 2 /V 2 will be the same as that of
the pressure. Thus, the SI unit of a will be Pa m 6 mol –2 .
It may be conveniently expressed in kPa dm 6 mol –2 .
When the expressions as given by Eqs (1) and (4) are
substituted in the ideal gas equation p i V i = nRT, we
get
⎛
2
⎞
⎜
n a
p + ⎟ (V – nb) = nRT ...(5)
2
⎝ V ⎠
This equation is applicable to real gases and is known
as the van der Waals equation.
Values of van der Waals Constants :
The constants a and b in van der Waals equation are
called van der Waals constants and their values
depend upon the nature of the gas. They
Van Der Waals Constants
a
Gas 6 2
kPa dm mol –
H 2
He
N 2
O 2
Cl 2
NO
NO 2
H 2 O
CH 4
C 2 H 6
C 3 H 8
C 4 H 10 (n)
C 4 H 10 (iso)
C 5 H 12 (n)
CO
CO 2
21.764
3.457
140.842
137.802
657.903
135.776
535.401
553.639
228.285
556.173
877.880
1466.173
1304.053
1926.188
150.468
363.959
b
dm
3 mol –1
0.026 61
0.023 70
0.039 13
0.031 83
0.056 22
0.027 89
0.044 24
0.030 49
0.042 78
0.063 80
0.084 45
0.122 6
0.114 2
0.146 0
0.039 85
0.042 67
are characteristics of the gas. The values of these
constants are determined by the critical constants of
the gas. Actually, the so-called constant vary to some
extent with temperature and this shows that the van
der Waals equation is not a complete solution of the
behaviour of real gases.
Applicability of the Van Der Waals Equation :
Since the van der Waals equation is applicable to real
gases, it is worth considering how far this equation
can explain the experimental behaviours of real
gases. The van der Waals equation for 1 mole of a
gas is
⎛ ⎞
⎜
a
p + ⎟ (V 2
m – b) = RT ..(i)
⎝ V m ⎠
At low pressure When pressure is low, the volume is
sufficiently large and b can be ignored in comparison
to V m in Eq. (i). Thus, we have
⎛ ⎞
⎜
a
⎟
a
p + V 2
m = RT or pV m + =RT
⎝ V m ⎠
V m
a
or Z = 1 –
...(ii)
VmRT
From the above equation it is clear that in the low
pressure region, Z is less than 1. On increasing the
pressure in this region, the value of the term
(a/V m RT) increase as V is inversely proportional to p.
Consequently, Z decreases with increase of p.
At high pressure When p is large , V m will be small
and one cannot ignore b in comparison to V m .
2
However, the term a / V m may be considered
negligible in comparison to p in Eq. (i) Thus,
pb
p(V m – b) = RT or Z = 1 + ...(iiii)
RT
Here Z is greater than 1 and increases linearly with
pressure. This explains the nature of the graph in the
high pressure region.
A high temperature and low pressure If
temperature is high, V m will also be sufficiently large
2
and thus the term a / V m will be negligibly small. At
this stage, b may also be negligible in comparison to
V m . Under these conditions, Eq. (i) reduces to an
ideal gas equation of state:
pV m = RT
Hydrogen and helium The value of a is extremely
small for these gases as they are difficult to liquefy.
Thus, we have the equation of state as p(V m – b) = RT,
obtained from the van der Waals equation by
2
ignoring the term a / V m . Hence, Z is always greater
than 1 and it increases with increase of p.
The van dar Waals equation is a distinct
improvement over the ideal gas law in that it gives
qualitative reasons for the deviations from ideal
behaviour. However, the generality of the equation is
lost as it contains two constants, the values of which
depend upon the nature of the gas.
XtraEdge for IIT-JEE 35 MAY 2010