(3) By observing the variations of the product pv with pres sure at constant temperature the deviations of different gases from Boyle's law are determined. Experiment shows that the rate of change of the product pv with increase of pressure, namely d(pv)/dp, is very nearly constant for moderate pressures such as those employed in gas thermometry. This implies that the char acteristic equation must be of the type v=F(0)/H-f(0) (15) in which F(0) and f(0) are functions of the temperature only to a first approximation at moderate pressures. The function F(0), representing the limiting value of pv at zero pressure, appears to be simply proportional to the absolute temperature for all gases. The function f(0), representing the defect of volume from the ideal volume, is the slope of the tangent at p = 0 to the isothermal of 0 on the pv, p diagram, and is sometimes called the "angular coefficient." It appears to be of the form b—c, in which b is a small constant quantity, the "co-volume," of the same order of magnitude as the volume of the liquid, and c depends on the co hesion or co-aggregation of the molecules, and diminishes for all gases continuously and indefinitely with rise of temperature. This method of investigation has been very widely adopted, especially at high pressures, but is open to the objection that the quantity b—c is a very small fraction of the ideal volume it the case of the permanent gases at moderate pressures, and its limiting value at p =0 is therefore difficult to determine accurately.
(4) By observing the cooling effect dO/dp, or the ratio of the fall of temperature to the fall of pressure under conditions of con stant total heat, when a gas flows steadily through a porous plug, it is possible to determine the variation of the total heat with pressure from the relation (See THERMODYNAMICS.) This method has the advantage of directly measuring the deviations from the ideal state, since Odv/d0 = v for an ideal gas, and the cooling effect vanishes. But the method is difficult to carry out, and has seldom been applied. Taken in conjunction with method (3), the observation of the cooling effect at different temperatures affords most valuable evi dence with regard to the variation of the defect of volume c—b from the ideal state. The formula assumed to represent the varia tions of c with temperature must be such as to satisfy both the observations on the compressibility and those on the cooling effect. It is possible, for instance, to choose the constants in van der Waals's formula to satisfy either (3) or (4) separately within the limits of experimental error, but they cannot be chosen so as to satisfy both. The simplest assumption to make with regard to c is that it varies inversely as some power n of the absolute tem perature, or that where is the value of c at the temperature O. In this case the expression Odv/d0— v takes the simple form (n+ 1)c—b. The values of n, c and b could be calcu lated from observations of the cooling effect SdO/dp alone over a sufficient range of temperature, but, owing to the margin of ex perimental error and the paucity of observations available, it is better to make use of the observations on the compressibility in addition to those on the cooling effect. It is preferable to calcu late the values of c and b directly from equation (16), in place of attempting to integrate the equation according to Kelvin's method because it is then easy to take account of the variation of the specific heat S, which is sometimes important.
Having found the most probable values of the quantities c, b and n from the experimental data, the calculation of the correc tion may be effected as follows : The temperature by gas ther mometer is defined by the relation T = pv/R, where the constant R is determined from the observations at o° and mo° C. The
characteristic equation in terms of absolute temperature 0 may be put in the form 0 =pv/R'+q, where q is a small quantity of the same dimensions as temperature, given by the relation The constant R' is determined, as before, by reference to the fundamental interval, which gives the relation R'/R = I + where qi, q0 are the values of q at too° and o° C respectively.
The correction to be added to the fundamental zero of the gas thermometer in order to deduce the value of the absolute zero (the absolute temperature corresponding to o° C) is given by the equation, The correction dt to be added to the centigrade temperature t by gas thermometer reckoned from o° C in order to deduce the corresponding value of the absolute temperature also reckoned from o° C is given by the relation, deduced from formula (I0), where q is the value at t° C of the deviation (c—b)p/R. The formulae may be further simplified if the index n is a simple integer such as f or 2. The values of the corrections for any given gas at different initial pressures are directly proportional to the pressure.