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32 1 THE PROPERTIES OF GASES
15 20 25 30
1.0
1.5
2.0
2.5
3.0
273 K
373 K
V/dm3
p/
at
m
van der Waals
perfect
Figure 1.3
0.03 0.04 0.05 0.06 0.07
1.0
1.5
2.0
2.5
3.0
373 K
273 K
(1/V)/(dm−3)
p/
at
m
van der Waals
perfect
Figure 1.4
To �nd the turning points in Z as a function of Vm the derivative dZ/dVm is
set to zero. In computing the derivative, write the �rst term as Vm(Vm − b)−1
and recognise the the need to use the product rule
dZ
dVm
= −Vm
(Vm − b)2
+ 1
Vm − b
+ a
RTV 2m
= 0
hence V 2m(a − bRT) − 2abVm + ab2 = 0
To go to the second line the equation is multiplied through by RTV 2m(Vm−b)2.
�e resulting quadratic in Vm is solved in the usual way
Vm = 1
2(a − bRT)
[2ab ± (4a2b2 − 4ab2(a − bRT))1/2]
hence Vm = b
1 − bRT/a
⎡⎢⎢⎢⎣
1 ± (bRT
a
)
1/2⎤⎥⎥⎥⎦