TITLE.PM5

IDEAL AND REAL GASES 387

dharm

\M-therm\Th8-1.pm5

b

a^

p a

v

F +

HG

I

(^2) KJ (v – b) =^1
2
F −
HG
I
KJ
b
v ,
which reduces to v = b
pv
a
2

• 2
  F
  HG
  I
  KJ
  Multiply each side of this equation by p, and put pv = y and p = x, and we obtain
  y = b
  y
  a
  x
  2


• 2
  F
  H
  G
  I
  K
  J or y(a – by) = 2abx
  The above expression gives the locus of minima and is
  a parabola with axis parallel to the x-axis as shown in
  Fig. 8.8.
  Consider the isothermal which goes through the point
  A. Here x = 0 and y =
  a
  b.
  Writing Van der Waals’ equation in terms of x and y,
  we have
  1 + 2
  F
  HG
  I
  KJ
  ax
  y (y – bx) = RT,
  and substituting the coordinates of the point A
  RT =
  a
  b
  or T =
  a
  bR ...(8.30)
  For temperatures above that given by equation (8.30) the minima lie in the region of nega-
  tive pressure, so an Amagat isothermal for a temperature equal to or greater than
  a
  bR will slope
  upwards along its whole length for increasing values of p, but for a temperature less than
  a
  bR the
  isothermals first dip to a minimum and then rise.
  Using the result from equation (8.27)
  Tc =
  8
  27
  a
  bR,
  we see that the limiting temperature for an isothermal to show a minimum is
  T =
  27

8. Tc
   The reason for Amagat finding no dip in the isothermals for hydrogen is now apparent. The
   critical temperature is 35 K, and therefore the limiting temperature above which minima do not
   occur is
   27
   8 × 35 = 118.1 K or – 155°C, and all Amagat’s experiments were conducted between 0°C
   and 100°C.
   The Cooling effect :
   The most gases show an inversion of the cooling effect at a certain temperature. The equa-
   tion of Van der Waals indicates at what temperature this occurs.
   Fig. 8.8