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Thermometer, Gas

THERMOMETER, GAS 8. The deviations of the gas thermometer from the absolute scale are so small that this instrument is now universally regarded as the ultimate standard in thermometry. It had, in fact, already been adopted for this purpose by Regnault and others, on a priori considerations, before the absolute scale itself had been invented. Although the indications of a gas thermometer are not absolutely independent of the changes of volume of the envelope or bulb in which the gas is contained, the effect of any uncertainty in this respect is minimized by the relatively large expansibility of the gas. The capricious changes of volume of the bulb, which are so great a difficulty in mercurial thermometry, are twenty times less important in the case of the gas thermometer. As additional reasons for the choice we have the great simplicity of the laws of gases, and the approximate equality of expansion and close agreement of the thermometric scales of all gases, provided that they are above their critical temperatures. Subject to this condition, at moderate pressures and provided that they are not dissociated or decomposed, all gases satisfy approximately the laws of Boyle and Charles. These two laws are combined in the characteristic equation of the gaseous state, viz., pv=RT, in which p is the pressure and v the volume of unit mass of the gas in question, and R is a constant which varies inversely as the molecular weight of the gas, and is approximately equal to the difference of the specific heats.

9. Practical Conditions. In practice it is not convenient to deal with unit mass, but with an arbitrary mass M occupying a space V, so that the specific volume =V/M. It is also necessary to measure the pressure p in terms of mercury columns, and not in absolute units. The numerical value of the constant R is adjusted to suit these conditions, but is of no consequence in thermometry, as we are concerned with ratios and differences only. The equation may be written in the form T=/>V/RM, but in order to satisfy the essential condition that T shall be a definite function of the temperature in the case of a gas which does not satisfy Boyle's law exactly, it is necessary to limit the application of the equation to special cases which lead to definite, but not necessarily identical, thermometric scales. There are three special cases of practical importance, corresponding to three essentially distinct experimental methods.

(i.) Volumetric Method (constant-pressure). In this method V is variable and p and M are constant. This method was employed by Gay-Lussac, and is typified in the ideal thermometer with reservoir of variable capacity designed by Lord Kelvin (Ency. Brit., ed. ix., vol. xi. p. 575, fig. 10). It corresponds to the method ordinarily employed in the common liquid-in-glass thermometer, but is not satisfactory in practice, owing to the difficulty of making a bulb of variable and measurable volume the whole of which can be exposed to the temperature to be measured.

(ii.) Manometric Method (constant-volume or density). In this method p is variable and V and M are constant. Variations of temperature are observed and measured by observing the corresponding variations of pressure with a mercury manometer, keeping a constant mass, M, of gas enclosed in a volume, V, which is constant except for the unavoidable but small expansion of the material of which the bulb is made.

(iii.) Gravimetric Method (constant-pressure). In this method M is variable and p and V are constant. This method is generally confounded with (i.) under the name of the constantpressure method, but it really corresponds to the method of the weight thermometer, or the " overflow " method, and is quite distinct from an experimental standpoint, although it leads to the same thermometric scale. In applying this method, the weight M of the vapour itself may be measured, as in Regnault's mercury-vapour thermometer, or in Deville and Troost's iodine-vapour thermometer. The best method of measuring the overflow is that of weighing mercury displaced by the gas. The mass of the overflow may also be estimated by observing its volume in a graduated tube, but this method is much less accurate.

In addition to the above, there are mixed methods in which both p and V or M are variable, such as those employed by Rudberg or Becquerel; but these are unsatisfactory for precision, as not leading to a sufficiently definite thermometric scale. There is also a variation of the constant-volume method (ii.), in which the pressure is measured by the volumetric compression of an equal mass of gas kept at a constant temperature, instead of by a manometer. This method is experimentally similar to (iii.), 'and gives the same equations, but a different thermometric scale from either (ii.) or (iii.). It will be considered with method (iii.), as the apparatus required is the same, and it is useful for testing the theory of the instrument. We shall consider in detail methods (ii.) and (iii.) only, as they are the most important for accurate work.

10. Construction of Apparatus. The manometric or constantvolume method was selected by Regnault as the standard, and has been most generally adopted since his time. His apparatus has not been modified except in points of detail. A description of his instrument will be found in most text-books on heat.

A simple and convenient form of the instrument for general use is Jolly's (described in Poggendorif's Jubelband, p. 82, 1874), and represented in fig. 3. The two vertical tubes of the manometer are connected by an india-rubber tube properly strengthened by a cotton covering, and they can be made to slide vertically up and down a wooden pillar which supports them; they are provided with clamps for fixing them in any position and a tangent screw for fine adjustment. The connexion between the bulb and the manometer is made by means of a three-way tap. The scale of the instrument is engraved on the back of a strip of plane mirror before silvering, and the divisions are carried sufficiently far across the scale for the reflections of the two surfaces of the mercury to be visible behind the scale. Parallax can thus be avoided and an accurate reading obtained without the necessity of using a cathetometer. In order to allow for the expansion of the glass of the reservoir a weight-thermometer bulb is supplied with the instrument, made from another specimen of the same kind of glass, and the relative expansion of the mercury and the glass can thus be determined by the observer himself. The volume of the air-bulb and that of the capillary tube and the small portion of the manometer tube above the small beak of glass, the point of which serves as the fiducial mark, are determined by the instru- ment-makers. The improvements introduced by Chappuis, of the International Bureau at Sevres, in the construction of the constant-volume hydrogen thermometer selected by the committee for the determination of the normal scale, are described in the text-books (e.g. Watson's Physics). The most important is the combination of the manometer and th& barometer into a single instrument with a single scale, thus reducing the number of readings required. The level of the mercury in the branch of the manometer communicating with the bulb of the gas thermometer is adjusted in the usual manner up to a fixed contact-point, so as to reduce the contained gas to a constant volume. Simultaneously the barometer branch of the manometer is adjusted so that the surface of the mercury makes contact with another point fixed in the upper end of the barometer tube. The distance between the two contact-points, giving the pressure of the gas in the thermometer, is deduced from the reading of a vernier fixed relatively to the upper contactpoint. This method of reading the pressure is probably more accurate than the method of the cathetometer which is usually employed, but has the disadvantage of requiring a double adjustment.

11. Pressure Correction. In the practical application of the manometric method there are certain corrections peculiar to the method, of which account must be taken in work of precision. The volume of the bulb 4 is not accurately constant, but varies with change of pressure and tempera- FIG. 3.

ture. The thermal expansion of the bulb is common to all methods, and will be considered in detail later. The pressure correction is small, and is determined in the same manner as for a mercury thermometer. The value so determined, however, does not apply strictly except at the temperature to which it refers. If the pressure-coefficient were constant at all temperatures and equal to e, the pressure correction, dt, at any point / of the scale would be obtainable from the simple formula dt-epot^-iooi/Ta .... (10)

where p a is the initial pressure at the temperature TO. But as the coefficient probably varies in an unknown manner, the correction is somewhat uncertain, especially at high temperatures. Another very necessary but somewhat troublesome correction is the reduction of the manometer readings to allow for the varying temperatures of the mercury and scale. Since it is generally impracticable to immerse the manometer in a liquid bath to secure certainty and uniformity of temperature, the temperature must be estimated from the readings of mercury thermometers suspended in mercury tubes or in the air near the manometer. It is therefore necessary to work in a room specially designed to secure great constancy of temperature, and to screen the manometer with the utmost care from the source of heat in measurements of high temperature. Regnault considered that the limit of accuracy of correction was one-tenth of a millimetre of mercury, but it is probably possible to measure to one-hundredth as a mean of several readings under the best conditions, at ordinary temperatures.

1 2. Stem- Exposure. In all gas thermometers it is necessary in practice that the part of the gas in contact with the mercury or other liquid in the manometer should not be heated, but kept at a nearly constant temperature. The space above the mercury, together with the exposed portion of the capillary tube connecting the manometer with the thermometric bulb, may be called the " dead space." If the volume of the dead space is kept as nearly as possible constant by adjusting the mercury always up to a fixed mark, the quantity of air in this space varies nearly in direct proportion to the pressure, i.e. in proportion to the temperature of the thermometric bulb at constant volume. This necessitates the application of a stem exposure correction, the value of which is approximately given by the formula d<=r/(/-ioo)/T 2 , . . . . (n where r is the ratio of the volume of the dead space to the volume of the thermometric bulb, and T 2 is the mean temperature of the dead space, which is supposed to be constant. The magnitude of the correction is proportional to the ratio r, anc increases very rapidly at high temperatures. If the dead space is i per cent, of the bulb, the correction will amount to only one-tenth of a degree at 50 C., but reaches 5 at 445 C., anc 30 at 1000 C. It is for this reason important in high- temperature work to keep the dead space as small as possible anc to know its volume accurately. With a mercury manometer the volume is liable to a slight uncertainty on account of changes of shape in the meniscus, as it is necessary to use a wide tube in order to secure accurate measurements of pressure.

13. Compensation Method with Oil-Gauge. It is possible to avoid this difficulty, and to make the dead space very small by employing oil or sulphuric acid or other non-volatile liquid to confine the gas in place of mercury (Phil. Trans., A. 1887, p. 171) The employment of a liquid which wets the tube makes it possible to use a much smaller bore, and also greatly facilitates the reading of small changes of pressure. At the same time the instrument may be arranged so that the dead space correction is automatically eliminated with much greater accuracy than it can be calculated. This is effected as shown diagrammatically in fig. 4, by placing side by side with the tube AB, connecting the bulb B to the manometer A, an exact duplicate CD, closed at the end D, and containing liquid in the limb C, which is of the same size as the branch A of the manometer and in direct communication with it. The tube CD, which is called the compensating tube, contains a constant mass of gas under exactly similar conditions of volume and temperature to the tube AB. If therefore the level of the liquid is always r ; ti adjusted to be the same in both tubes AB *~uiiipLriSitLion. j .-.T-. . .. ....

and CD, the mass of gas contained in the dead space AB will also be constant, and is automatically eliminated from the equations, as they contain differences only.

14. Gravimetric Method. In the writer's opinion, the gravimetric or overflow method, although it has seldom been adopted, and is not generally regarded as the most accurate, is much to be preferred to the manometric method, especially for work at high temperatures. It is free from the uncertain corrections above enumerated as being peculiar to the manometric method. The apparatus is much simpler to manipulate and less costly to construct. If the pressure is kept constant and equal to the external atmospheric pressure, there is no strain of the bulb, which is particularly important at high temperatures. There is no dead space correction so long as the temperature of the dead space is kept constant. The troublesome operation of reading and adjusting the mercury columns of the manometer is replaced by the simpler and more accurate operation of weighing the mercury displaced, which can be performed at leisure. The uncertain correction for the temperature of the mercury in the manometer is entirely avoided.

The reasons which led Regnault to prefer the constant-volume thermometer are frequently quoted, and are generally accepted as entirely conclusive, but it is very easy to construct the constant-pressure or gravimetric instrument in such a manner as to escape the objections which he urges against it. Briefly stated, his objections are as follows: (i) Any error in the observation of the temperature of the gas in the overflow space produces a considerable error in the temperature deduced, when the volume of the overflow is large. This source of error is very simply avoided by keeping the whole of the overflow in melting ice, an expedient which also considerably simplifies the equations. It happened that Regnault's form of thermometer could not be treated in this manner, because he had to observe the level of the mercury in order to measure the pressure and the volume. It is much better, however, to use a separate gauge, containing oil or sulphuric acid, for observing small changes of pressure. The use of ice also eliminates the correction for the variation of density of the mercury by which the overflow is measured. (2) Regnault's second objection was that an error in the measurement of the pressure, or in reading the barometer, was more serious at high temperatures in the case of the constant-pressure thermometer than in the constant-volume method. Owing to the incessant variations in the pressure of the atmosphere, and in the temperature of the mercury columns, he did not feel able to rely on the pressure readings (depending on observations of four mercury surfaces with the cathetometer) to less than a tenth of a millimetre of mercury, which experience showed to be about the limit of accuracy of his observations. This would be equivalent to an error of 0-036 with the constant-volume thermometer at any point of the scale, but with the constant-pressure thermometer the error would be larger at higher temperatures, since the pressure does not increase in proportion to the temperature. This objection is really unsound, because the ideal condition to be aimed at is to keep the proportionate error dT/T constant. That the proportionate error diminishes with rise of temperature, in the case of the constant-volume thermometer, is really of no advantage, because we can never hope to be able to measure high temperatures with greater proportionate accuracy than ordinary temperatures. The great increase of pressure at high temperatures in the manometric method is really a serious disadvantage, because it becomes necessary to work with much lower initial pressures, which implies inferior accuracy at ordinary temperatures and in the determination of the initial pressure and the fundamental interval.

15. Compensated Diferential Gas Thermometer. The chief advantage of the gravimetric method, which Regnault and others appear to have missed, is that it is possible to make the measurements altogether independent of the atmospheric pressure and of the observation of mercury columns. This is accomplished by using, as a standard of constant pressure, a bulb S, fig. 5, containing a constant mass of gas in melting ice, side by side with the bulb M, in which the volume of the overflow is measured. The pressure in the thermometric bulb T is adjusted to equality with the standard by means of a delicate oil-gauge G of small bore, in which the difference of pressure is observed by means of a- cathetometer microscope. This kind of gauge permits the rapid observation of small changes of sressure, and is far more accurate and delicate than the mercury manometer. The fundamental measurement of the volume of the overflow in terms of the weight of mercury displaced at o C. involves a single weighing made at leisure, and requires no temperature correction. The accuracy obtainable at ordinary temperatures in this measurement is about ten times as great as that attainable under the best conditions with the mercury manometer. At higher temperatures the relative accuracy diminishes in proportion to the absolute temperature, or the error <ft increases according to the formula dt/t=-(T/T<,)dw/w, .... (12)

where w is the weight of the overflow and dw the error. This diminution of the sensitiveness of the method at high temperaures is commonly urged as a serious objection to the method, >ut the objection is really without weight in practice, as the possible accuracy of measurement is limited by other conditions. So far as the weighing alone is concerned, the method s sensitive to one-hundredth of a degree at 1000 C., which is ar beyond the order of accuracy attainable in the application if the other corrections.

1 6. Method of Using the Instrument. A form of gas thermometer constructed on the principles above laid down, with the addition of a duplicate set of connecting tubes C for the elimination of the stem-exposure correction by the method of automatic compensation already explained, is shown in fig. 5 (Proc. R. S. vol. 50, p. 243; Preston's Heat, p. 133).

In setting up the instrument, after cleaning, and drying and calibrating the bulbs and connecting tubes, the masses of gas on the two sides are adjusted as nearly as possible to equality, in order that any changes of temperature in the two sets of connecting tubes may compensate each other. This is effected with all the bulbs in melting ice, by adjusting the quantities of mercury in the bulbs M and S and equalizing the pressures. The bulb T is then heated in steam to determine the fundamental interval. A weight Wi of mercury is removed from the overflow bulb M in order to equalize the_ pressures again. If W is the weight of the mercury at o C which would be required to fill the bulb T at o C., and if W+ <fWi is the weight of mercury at o which would be required to fill a volume equal to that of the bulb in steam at h, we have the following equation for determining the coefficient of expansion a, or the fundamental zero To, o/i=/i/T = (/ 1 +<iWi)/(W-Wi) I . . . (13)

Similarly if tv is the overflow when the bulb is at any other temperature t, and the expansion of the bulb is <AV, we have a precisely similar equation for determining t in terms of To, but with t and w and <AV substituted for ti and Wi and dWj. In practice, if the pressures are not adjusted to exact equality, or if the volumes of FIG. 5. Compensated Differential Gas Thermometer.

the connecting tubes do not exactly compensate, it is only necessary to include in w a small correction dw, equivalent to the observed difference, which need never exceed one part in ten thousand.

"It is possible to employ the same apparatus at constant volume as well as at constant pressure, but the manipulation is not quite so simple, in consequence of the change of pressure. Instead of removing mercury from the overflow bulb M in connexion with the thermometric bulb, mercury is introduced from a higher level into the standard bulb S so as to raise its pressure to equality with that of T at constant volume. The equations of this method are precisely the same as those already given, except that w now signifies the " inflow " weight introduced into the bulb S, instead of the overflow weight from M. It is necessary, however, to take account of the pressure-coefficient of the bulb T, and it is much more important to have the masses of gas on the two sides of the apparatus equal than in the other case. The thermometric scale obtained in this method differs slightly from the scale of the manometric method, on account of the deviation of the gas compressed at o C. from Boyle's law, but it is easy to take account of this with certainty.

Another use to which the same apparatus may be put is the accurate comparison of the scales of two different gases at constant volume by a differential method. It is usual to effect this comparison indirectly, by comparing the gas thermometers separately with a mercury thermometer, or other secondary standard. But by using a pair of bulbs like M and S simultaneously in the same bath, and measuring the small difference of pressure with an oilgauge, a higher order of accuracy may be attained in the measurement of the small differences than by the method of indirect comparison. For instance, in the curves representing the difference between the nitrogen and hydrogen scales (fig. l), as found by Chappuis by comparison of the nitrogen and hydrogen thermometers with the mercury thermometer, it is probable that the contrary flexure of the curve between 70 and 100 C. is due to a minute error of observation, which is quite as likely to be caused by the increasing aberrations of the mercury thermometer at these temperatures as by the difficulties of the manometric method. It may be taken as an axiom in all such cases that it is better to measure the small difference itself directly than to deduce it from the much more laborious observations of the separate magnitudes concerned.

17. Expansion Correction. In the use of the mercury thermometer we are content to overlook the modification of the scale due to the expansion of the envelope, which is known as Poggendorff's correction, or rather to include it in the scale correction. In the case of the gas thermometer it is necessary to determine the expansion correction separately, as our object is to arrive at the closest approximation possible to the absolute scale. It is a common mistake to imagine that if the rate of expansion of the bulb were uniform, the scale of the apparent expansion of the gas would be the same as the scale of the real expansion in other words, that the correction for the expansion of the bulb would affect the value of the coefficient of expansion i/7"o only, and would be without effect on the value of the temperature t deduced. A result of this kind would be produced by a constant error in the initial pressure on the manometric method, or by a constant error in the initial volume on the volumetric method, or by a constant error in the fundamental interval on any method, but not by a constant error in the coefficient of expansion of the bulb, which would produce a modification of the scale exactly analogous to Poggendorff's correction. The correction to be applied to the value of t in any case to allow for any systematic error or variation in the data is easily found by differentiating the formula for / with respect to the variable considered. Another method, which is in some respects more instructive, is the following :

Let T be the function of the temperature which is taken as the basis of the scale considered, then we have the value of /given by the general formula (l), already quoted in 3. Let <ff be the correction to be added to the observed value of T to allow for any systematic change or error in the measurement of any of the data on which the value of T depends, and let dt be the corresponding correction produced in the value of t, then substituting in formula (l) we have, t+dt = ioo(T-To+<fT -<ZT )/(Ti -To+<TTi -<fT), from which, provided that the variations considered are small, we obtain the following general expression for the correction to t, d/=(dT-dTo)-(<2T,-dTo)</ioo. . . (14)

It is frequently simpler to estimate the correction in this manner, rather than by differentiating the general formula.

In the special case of the gas thermometer the value of T is given by the formula T=pV/RM=/.V/R(M -M 2 ), . . . (15)

where p is the observed pressure at any temperature /, V the volume of the thermometric bulb, and M the mass of gas remaining in the bulb. The quantity M cannot be directly observed, but is deduced by subtracting from the whole mass of gas M contained in the apparatus the mass M 2 which is contained in the dead space and overflow bulb. In applying these formulae to deduce the effect of the expansion of the bulb, we observe that if dV is the expansion from o C., and Vo the volume at o C., we may write V = Vo-hfV, T =p(V +<AO/RM = (Vo/RM) (i +</V/Vo), whence we obtain approximately <fT = TdV/Vo .... (16)

If the coefficient of expansion of the bulb is constant and equal to the fundamental coefficient / (the mean coefficient between o and 100 C.), we have simply <ZV/Vo=/<; and if we substitute this value in the general expression (14) for dt, we obtain dt = (T-T,)/< =ft(t- 100) . (17)

Provided that the correction can be expressed as a rational integral function of /, it is evident that it must contain the factors t and (/ 100), since by hypothesis the scale must be correct at the fixed points o and 100 C., and the correction must vanish at these points. It is clear from the above that the scale of the gas thermometer is not independent of the expansion of the bulb even in the simple case where the coefficient is constant. The correction is by no means unimportant. In the case of an average glass or platinum reservoir, for which / may be taken as 0-000025 nearly, the correction amounts to -0-0625* at 5O C., to 3-83 at 445 C., and to 22-5 at 1000 C.

The value of the fundamental coefficient / 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 Wiillner. The extreme difference represents an uncertainty of about ^ per cent, (i in 25) in the expansion of the glass. This uncertainty is about 100 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 o C. may be expressed in the form dV =t(a-\-bt)Vo, in which case the fundamental coefficient /=o + ioo6. Making this substitution in the formula already given, we obtain the whole correction dt = (f+bT)t(t 100) . . . (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 /, 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 co-extensive with the range of the temperature measurements for which the correction is required, the uncertainty of the correction will not greatly exceed that of the expansion observed at any point of the range. It is not unusual, however, to deduce the values of b and / from observations confined to the range o to 100 C.,in which case an error of I per cent., in the observed expansion at 50 C., would mean an error of 60 per cent, at 445, or of 360 per cent, at 1000 C. ( Callendar, Phil. Mag. December 1899). Moreover, it by no means follows that the average value of b between o and 100 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 expansion correction, 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 1000 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 = RO 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 :

(1) By the comparison of gas thermometers filled with different gases or with the same gas at different pressures (employing both gravimetric 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 100 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)jdp, 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 .... (19)

in which F(0) and f(ff) 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(6), representing the defect of volume from the ideal volume, is the slope of the tangent at p=o to the isothermal of 8 on the pv, p diagram, and is sometimes called the " angular coefficient." It appears to be of the form b c, in which 6 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 cohesion 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 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 d8/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 Sd6/dp=edvld6-v (20)

(See THERMODYNAMICS, 10, equation 15.) This method has the advantage of directly measuring the deviations from the ideal state, since Odv/d8v 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(0 /0) n , where Ct, is the value of c at the temperature 0o- In this case the expression 0dv/d9 v takes the simple form (n+i)c-b. The values of n, c and b could be calculated from observations of the cooling effect SdS/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. Brit. 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 p and 100 C. The characteristic equation in terms of absolute' temperature 8 may be put in the iormO = pv/R'+q, where q is a small quantity of the same dimensions as temperature, given by the relation q = (c-b)p/R . . . . (21)

The constant R' is determined, as before, by reference to the fundamental interval, which gives the relation R'/R = I + (<?i ~ 3o)/ioo, where 31, q are the values of q at 100 and o 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 ft> (the absolute temperature corresponding to o C.) is given by the equation, 0o-T =3o- (31-30)60/100 . . . (22)

The correction At 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 C. is given by the relation, deduced from formula (14), <W = (3-3o)-(3i-3o)//lOO, . _ _ (23)

where 3 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 I 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., 6 = 8-0 c.c., with the index n = 1-5, 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 flo, calculated from Chappuis's values of the pressure and expansion coefficients at 100 cms. initial pressure, are found to be 273-10 and 273-05 respectively, the reciprocals of the coefficients themselves being 2 73'3 an d 273-22. The corrections arc small and of opposite signs. For nitrogen, taking 0=1-58, 6=1-14, = i'5i we find similarly 273-10 and 273-13 for the absolute zero, the correction 0o To in this case amounting to nearly 1. 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 = i-5, although it makes comparatively little difference to the value of the absolute zero. The value deduced from Travers's observation of the pressure-coefficient of helium is 273-13, taking n = J, 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 = 3-76 c.c., 6=0-58 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 0o-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 100 C. are of the order of -001 only, and beyond the limits of accuracy of experiment. Even at -250 C. (near the boiling-point of hydrogen) the corrections of the constant volume hydrogen and helium thermometers are only a tenth of a degree, but, as they are 01 opposite signs, the difference amounts to one-fifth 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, Trav. el 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 so-called 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 (i) 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 one-quarter to one-half of i 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. 182, 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 platinum-indium, 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 200 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 wave-length of sound. The objection to this method is the difficulty of accurately observing the wave-length, 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 (Complex 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 1500 C., or even lower, and the accuracy with which temperatures near 1000 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.

Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)

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