��VAN DER WAALS'S EQUATION 5|&
If in the general equation for a gas we introduce these corrected values for volume and pressure, we obtain Van der Waals's equation :
��(p+$) <•-»)-«
��Or arranging the equation in descending values of v we get:
\ p / p p
that is, it becomes an equation of the third degree, like
v 3 — q v 2 + r v — 8 = 0,
and so, according to the values of the variables (p and t), the solution of the equation leads to three real roots for v or one real root (and two imaginary).
It must be observed that in using Van der Waals's equation we do not express, as usual, the pressures in grams per square centimetre of surface and volumes in cubic centimetres ; p and a are expressed in atmospheres, v and b are fractions of the volume occupied by the gas under normal conditions of temperature and pressure. Hence the constant b has a correlative value. It is hardly
volume of the surface layer, n times fewer molecules, each of which will be attracted by n times fewer neighbours. The total attraction will therefore be n* times smaller, that is, inversely proportional to the square of the change of volume. This reasoning may seem to be imperfect because it omits entirely the variation of the attraction due to change in the intermolecular distances. One might also say
that the expression ~ was justified a posteriori, by the consequences
of Yam, der Waals's equation, rather than a priori, by good scientific reasoning.
The term — may attain very high values, exceeding even the
external pressure p. We shall presently see that for carbon dioxide a = 0*00874. Now at 21-5° carbon dioxide, liquefied by a pressure of 60 atmospheres, is reduced to about 0003 of its normal
volume. Hence the value of % in this case is !?' J^ = 971 atmo-
spheres,
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