The above formula shows that v+V-i- v"± ., the total number of molecules per unit volume, is determined when p, T and the constant R are given. Hence we have Avogadro's law : Different gases, at the same temperature and pressure, contain equal numbers of molecules per unit volume. According to the best determinations the number of molecules in a cubic centimetre of gas at a temperature of
C. and at a pressure of 76omm. of mercury (normal atmospheric pressure) is 2.705
Since the density of hydrogen under these con ditions is 0.00008987, it follows at once that the mass of the hydrogen molecule is 3.323X
grammes. The masses of other molecules are of course proportional to their molecular weights ; the molecule of oxygen, for instance, has a mass of 52 X z
grammes, and so on.
If v is the volume of a homo geneous mass of gas, and N the total number of its molecules, In this equation we have the combined laws of Boyle and Charles: When the temperature of a gas is kept constant the pressure varies inversely as the volume, and when the volume is kept con stant the pressure varies as the temperature. Since the volume at constant pressure is exactly proportional to the absolute tempera ture, it follows that the coefficients of expansion of all gases ought, to within the limits of error introduced by the assumptions on which we are working, to have the same value
Van der Waals's Equation.—The laws which have just been stated are obeyed very approximately, but not with perfect accuracy, by all gases of which the density is not too great or the temperature too low. Van der Waals, in a famous monograph, On the Continuity of the Liquid and Gaseous States (Leiden, 1873), has shown that the imperfections of the foregoing formula for the pressure may be traced to two causes :— (i) The calculation has not allowed for the finite size of the molecules, and their consequent interference with one another's motion, and (ii) The calculation has not allowed for the field of inter molecular force between the molecules, which, although small, is known to have a real existence. The presence of this field of force
results in the molecules, when they reach the boundary, being acted on by forces in addition to those originating in their impact with the boundary.
To allow for the first of these two factors, Van der Waals finds that v must be replaced by v —b , where b is four times the aggregate space occupied by all the molecules, while to allow for the second factor, p must be replaced by
Thus the pressure-volume-temperature relation is given by the equation
(v—b) =RNT which is known as Van der Waals's equation. This equation is found to be capable of representing the relation between p, v and T which is obtained experimentally over large ranges of values. (See CONDENSATION OF GASES.) A series of observations on any gas make it possible to determine the values of a and b for the gas in question. The value of a obtained in this way gives valuable information as to the field of force surrounding the mole cules ; the value of b gives still more valuable information, for it discloses the total volume occupied by all the molecules of the gas, thus providing information as to the sizes of the individual molecules. There are, however, other and better ways of deter mining the sizes of molecules, as we shall now see.