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platinum reservoir, for which f may be taken as 0•000025 nearly, the correction amounts to 0.0625° at 50° C., to 3.83° at 445° C., and to 22.5° at t000° C.
The value of the fundamental coefficient f can be determined with much greater accuracy than the coefficient over any other range of temperature. The most satisfactory method is to use the bulb itself as a mercury weight thermometer, and deduce the cubical expansion of the glass from the absolute expansion of mercury as determined by Regnault. Unfortunately the reductions of Regnault's observations by different calculators differ considerably even for the fundamental interval. The values of the fundamental coefficient range from .00018153 Regnault, and •00018210 Broch, to .00018253 Wullner. The extreme difference represents an uncertainty of about 4 per cent. (I in 25) in the expansion of the glass. This uncertainty is about too times as great as the probable error of the weight thermometer observations. But the expansion is even less certain beyond the limits of the fundamental interval. Another method of determining the expansion of the bulb is to observe the linear expansion of a tube or rod of the same material, and deduce the cubical expansion on the assumption that the expansion is isotropic. It is probable that the uncertainty involved in this assumption is greater in the case of glass or porcelain bulbs, on account of the difficulty of perfect annealing, than in the case of metallic bulbs.
Except for small ranges of temperature, the assumption of a constant coefficient of expansion is not sufficiently exact. It is therefore usual to assume that the coefficient is a linear function of the temperature, so that the whole expansion from 0° C. may be expressed in the form dV =t(a+bt)Vo, in which case the fundamental coefficient f=alIoob. Making this substitution in the formula already given, we obtain the whole correction
dt=(f+bT)t(tloo) . . (18)
It will be observed that the term involving b becomes of considerable importance at high temperatures. Unfortunately, it cannot be determined with the same accuracy as f, because the conditions of observation at the fixed points are much more perfect than at other temperatures. Provided that the range of the observations for the determination of the expansion is coextensive with the range of the temperature measurements for which the correction is required, the uncertainty of the correction will not greatly oxceed that of the expansion observed at any point of the range. It is not unusual, however, to deduce the values of b and f from observations confined to the range 0° to too° C., in which case an error of t per cent., in the observed expansion at 50° C., would mean an error of 6o per cent. at 445 or of 36o per cent. at I000° C. (Callendar, Phil. Mag. December 1899). Moreover, it by no means follows that the average value of b between 0° and too° C. should be the same as at higher or lower temperatures. The method of extrapolation would therefore probably lead to erroneous results in many cases, even if the value could be determined with absolute precision over the fundamental interval. It is probable that this expansioncorrection, which cannot be reduced or eliminated like many of the other corrections which have been mentioned, is the chief source of uncertainty in the realization of the absolute scale of temperature at the present time. The uncertainty is of the order of one part in five or ten thousand on the fundamental interval, but may reach 0.5° at 500° C., and 2° or 3° at I000° C.
18. Thermodynamical Correction.—Of greater theoretical interest, but of less practical importance on account of its smallness, is the reduction of the scale of the gas thermometer to the thermodynamical scale. The deviations of a gas from the ideal equation pv = R0 may be tested by a variety of different methods, which should be employed in combination to determine the form of the characteristic equation. The principal methods by which the problem has been attacked are the following:
(I) By the comparison of gas thermometers filled with different gases or with the same gas at different pressures (employing both Fravimetric and manometric methods) the differences in their indications are observed through as wide a range of temperature as possible. Regnault, employing this method, found that the differences in the scales of the permanent gases were so small as to be beyond the limits of accuracy of his observations. Applying greater refinements of measurement, Chappuis and others have succeeded in measuring small differences, which have an important bearing on the type of the characteristic equation. They show, for instance, that the equation of van der Waals, according to which all manometric gas thermometers should agree exactly in their indications, requires modification to enable it to represent the behaviour of gases even at moderate pressures.
(2) By measuring the pressure and expansion coefficients of different gases between o° and too° C. the values of the fundamental zero (the reciprocal of the coefficient of expansion or pressure) for each gas under different conditions may be observed and compared. The evidence goes to show that the values of the fundamental zero for all gases tend to the same limit, namely, the absolute zero, when the pressures are indefinitely reduced. The type of characteristic equation adopted must be capable of representing the variations of these coefficients.
(3) By observing the variations of the product pv with pressure 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 characteristic equation must be of the type
v=F(0)/p1 f(0) . . . . (19)
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= o to the isothermal of 0 on the pv, p diagram, and is sometimes called the " angular coefficient." It appears to be of the form bc, in which b is a small constant quantity, the " covolume," of the same order of magnitude as the volume of the liquid, and c depends on the cohesion or coaggregation 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 bc is a very small fraction of the ideal volume in the case of the permanent gases at moderate pressures, and its limiting value at p=o is therefore difficult to determine accurately.
(4) By observing the cooling effect d0/dp, or the ratio of the fall of temperature to the fall of pressure under conditions of constant 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
Sde/dp = 0dv/d0  v . . . . (20)
(See THERMODYNAMICS, § to, equation 15.) This method has the advantage of directly measuring the deviations from the ideal state, since 0dv/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 evidence with regard to the variation of the defect of volume c b from the ideal state. The formula assumed to represent the variations 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 temperature, or that c = co(Oo/O)°, where co is the value of c at the temperature 0o. In this case the expression 0dv/d0v takes the simple form (n+ t)c b. The values of n, c and b could be calculated from observations of the cooling effect Sd0/dp alone over a sufficient range of temperature, but, owing to the margin of experimental 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 calculate the values of c and b directly from equation (20), in place of attempting to integrate the equation according to Kelvin's method (Ency. But. ed. ix. vol. xi. p. 573), because it is then easy to take account of the variation of the specific heat S, which is sometimes important.
Calculation of the Correction.—Having found the most probable values of the quantities c, b and n from the experimental data, the calculation of the correction may be very simply effected as follows: The temperature by gas thermometer is defined by the relation T=pv/R,where the constant R is determined from the observations at o° and loo° C. The characteristic equation in terms of absolute temperature 0 may be put in the forme = pv/R'+q, where q is a small quantity of the same dimensions as temperature, given by the relation
q=(cb)p/R . . . . (21)
The constant R' is determined, as before, by reference to the fundamental interval, which gives the relation R'/R = I +(q1 qo)/Ioo, where qi, qo are the values of q at Ioo° and 0° C. respectively.
The correction to be added to the fundamental zero To of the gas thermometer in order to deduce the value of the absolute zero 00 (the absolute temperature corresponding to 0° C.) is given by the equation,
00 To= go  (gi  go)Oo/too . . . (22)
The correction dt to be added to the centigrade temperature t by gas thermometer reckoned from 0° C. in order to deduce the corresponding value of the absolute temperature also reckoned from 0° C. is given by the relation, deduced from formula (14),
dt = (q  go)  (qi  go) t/Ioo, . . . (23)
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 t or 2. The values of the corrections for any given gas at different initial pressures are directly proportional to the pressure.
Values of the Corrections.—If we take for the gas hydrogen the values c =1.5 c.c. at o° C., b =8•o c.c., with the index n = I.5, which satisfy the observations of joule and Thomson on the cooling effect, and those of Regnault, Amagat and Chappuis on the compressibility, the values of the absolute zero Bo, calculated from Chappuis's values of the pressure and expansion coefficients at too ems. initial pressure, are found to be 273.10° and 273.05° respectively, the reciprocals of the coefficients themselves being 273.03 and 273.22. The corrections are small and of opposite signs. For nitrogen, taking co=1.58, b—1.14, n=1,5, we find similarly 273.10° and 273.23° for the absolute zero, the correction eo—To in this case amounting to nearly I°. The agreement is very good considering the difficulty of determining the small deviations c and b, and the possible errors of the expansion and pressurecoefficients. It appears certain that the value of the absolute zero is within a few hundredths of a degree of 273.10°. Other observations confirm this result within the limits of experimental error. The value of the index n has generally been taken as equal to 2 for diatomic gases, but this does not satisfy either the observations on the cooling effect or those on the compressibility so well as n= 1.5, although it makes comparatively little difference to the value of the absolute zero. The value deduced from Travers's observation of the pressurecoefficient of helium is 273.13°, taking n=4, which is the probable value of the index.for a monatomic gas. The application of the method to the condensible gas carbonic acid is interesting as a test of the method (although the gas itself is not suited for thermometry), because its deviations from the ideal state are so large and have been so carefully studied. The observations of joule and Thomson on the cooling effect give co =3.70 c.c., b =o.58 c.c., n=2, provided that allowance is made for the variation of the specific heat with temperature as determined by Regnault and Wiedemann. Chappuis's values of the pressure and expansion coefficients agree in giving 273.05° for the absolute zero, the values of the corrections Bo—To being 4.6° and 5.8° respectively.
The values of the scale correction dt deduced from these formulae agree with those experimentally determined by Chappuis in the case of carbonic acid within the limits of agreement of the observations themselves. The calculated values for nitrogen and hydrogen give rather smaller differences than those found experimentally, but the differences themselves are of the same order as the experimental errors. The deviations of hydrogen and helium from the absolute scale between o° and loo° C. are of the order of .00i° only, and beyond the limits of accuracy of experiment. Even at 25o° C. (near the boilingpoint of hydrogen) the corrections of the constant volume hydrogen and helium thermometers are only a tenth of a degree, but, as they are of opposite signs, the difference amounts to onefifth of a degree at this point, which agrees approximately with that observed by Travers. For a fuller discussion of the subject, together with tables of corrections, the reader may refer to papers by Callendar, Phil. Mag. v. p. 48 (1903), and D. Berthelot, Pratt et Mem. Bur. Int. Paris, xiii. (1903). Berthelot assumes a similar type of equation to that given above, but takes n=2 in all cases, following the socalled law of corresponding states. This assumption is of doubtful validity, and might give rise to relatively large errors in the case of monatomic gases.
19. Limitations.—In the application of the gas thermometer to the measurement of high temperatures certain difficulties are encountered which materially limit the range of measurement and the degree of accuracy attainable. These may be roughly classified under the heads—(1) changes in the volume of the bulb; (2) leakage, occlusion and porosity; (3) chemical change and dissociation. The difficulties arise partly from defects in the materials available for the bulb, and partly from the small mass of gas enclosed. The troubles due to irregular changes of volume of glass bulbs, which affect the mercury thermometer at ordinary temperatures, become so exaggerated at higher points of the scale as to be a serious source of trouble in gas thermometry in spite of the twentyfold larger expansion. For instance, the volume of a glass bulb will be diminished by from onequarter to onehalf of 1 per cent. the first time it is heated to the temperature of boiling sulphur (445° C.). This would not matter so much if the volume then remained constant. Unfortunately, the volume continues to change, especially in the case of hard glass, each time it is heated, by amounts which cannot be predicted, and which are too large to neglect. The most accurate method of taking account of these variations in a series of observations, without recalibrating and refilling and cleaning the bulb, is to assume the known constant value of the coefficient of expansion of the gas, and to calculate the volume of the bulb at any time by taking observations in ice and steam (Phil. Trans. A. 1891, vol. 132, p. 124). Similar changes take place with porcelain at higher temperatures.
Metallic bulbs are far more perfect than glass bulbs in this respect. It is probable that silica bulbs would be the most perfect. The writer suggested the use of this material (in the Journ. Iron and Steel Inst. for 1892), but failed to construct bulbs of sufficient size. W. A. Shenstone, however, subsequently succeeded, and there seems to be a good prospect that this difficulty will soon be minimized. The difficulties of leakage and porosity occur chiefly with porcelain bulbs, especially if they are not perfectly glazed inside. A similar difficulty occurs with metallic bulbs of platinum or platinumiridium, in the case of hydrogen, which passes freely through the metal by occlusion at high temperatures. The difficulty can be avoided by substituting either nitrogen or preferably argon or helium as the thermometric material at high temperatures. With many kinds of glass and porcelain the chemical action of hydrogen begins to be appreciable at temperatures as low as Zoo° or 300° C. In any case, if metallic bulbs are used, it is absolutely necessary to protect them from furnace gases which may contain hydrogen. This can be effected either by enclosing the bulb in a tube of porcelain, or by using some method of electric heating which cannot give rise to the presence of hydrogen. At very high temperatures it is probable that the dissociation of diatomic gases like nitrogen might begin to be appreciable before the limit of resistance of the bulb itself was reached. It would probably be better, for this reason, to use the monatomic and extremely inert gases argon or helium.
20. Other Methods.—Many attempts have been made to overcome the difficulties of gas pyrometry by adopting other methods of measurement. Among the most interesting may be mentioned: (i.) The variation in the wavelength of sound. The objection to this method is the difficulty of accurately observing the wavelength, and of correcting for the expansion of the material of the tubes in which it is measured. There is the further objection that the velocity varies as the square root of the absolute temperature. (ii.) A similar method, but more promising, is the variation of the refractivity of a gas, which can be measured with great accuracy by an interference method. Here again there is difficulty in determining the exact length of the heated column of gas, and in maintaining the temperature uniform throughout a long column at high temperatures. These difficulties have been ingeniously met by D. Berthelot (Comptes Rendus, 1895, 120, p. 831). But the method is not easy to apply, and the degree of accuracy attainable is probably inferior to the bulb methods. (iii.) Methods depending on the effusion and transpiration of gases through fine orifices and tubes have been put in practice by Barus and by the writer. The method of transpiration, when the resistance of the tube through which the current of gas is passed is measured on the Wheatstone bridge principle (Nature, 23rd March 1899), is extremely delicate, and the apparatus may be made very small and sensitive, but the method cannot be used for extrapolation at high temperatures until the law of increase of resistance has been determined with certainty. This may be successfully accomplished in the near future, but the law is apparently not so simple as is usually supposed.
On account of these and similar difficulties, the limit of gas thermometry at the present time must be placed at 15oo° C., or even lower, and the accuracy with which temperatures near 1.000° C. are known does not probably exceed 2° C. Although measurements can be effected with greater consistency than this by means of electrical pyrometers, the absolute values corresponding to those temperatures must remain uncertain to this extent, inasmuch as they depend on observations made with the gas thermometer.
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