From this equation, calculating the correlative values of p and v, we obtain isotherms which fully interpret the results obtained by Andrews. At temperatures above 82*5°, each value of p corresponds to only one value for v (the equation has only one real root) and the isotherms represent the gaseous state. At temperatures below 82*5° and for certain values of p the equation yields three real roots, that is, it corresponds to three different volumes. In fig. 7, the isotherm for 13-1° shows that under a pressure of about 50 atmospheres the same mass of carbon dioxide can occupy the volume v (in the gaseous state), the volume v l (after complete liquefaction) and the third volume v. This last corresponds to an unstable state of the material, for which a diminution in volume would correspond to a diminution of pressure. The volume v cannot be realised by homogeneous carbon dioxide.
At 82-5° and for a pressure of about 61 atmos., the equation again leads to three real roots, but now they coincide and correspond only to one volume — the critical state has been reached.
Calculation of the Characteristics oj the Critical State. — "When the general equation of the third degree (see above) gives three real and equal roots, we have the following relations :
8t> 2 = r
v 3 = s
Under the same conditions, that is, for a substance taken in its critical state, Van der Waals's equation gives us
P
3t> 2 = *, P
V* a ah , P
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