GAS THERMOMETRY
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= 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= pV/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• s ieeial'caaes which, lead'^to definite, but not necessarily identical, thermometric >sc es: There are three special cases of practical importance, corresponding to three essentially distinct experimental methods.
(i.) Volumetric Method (constantpressure).—In this method V is variable and p and M are constant. This method was employed by GayLussac, 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. 1o). It corresponds to the method ordinarily • employed in the common liquidinglass 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 (constantvolume 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 (constantpressure).—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 mercuryvapour thermometer, or in Deville and Troost's iodinevapour 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 constantvolume 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.
so. 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 textbooks on heat.
A simple and convenient form of the instrument for general use is Jolly's (described in Poggendortl's Jubelband, p. 82, 1874), and represented in fig. 3. The two vertical tubes of the manometer are connected by an indiarubber 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 threeway 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 weightthermometer 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 airbulb 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 instruxtleeufmakers., The. snptpveoagnts introduced by Chappuis„of the f,at pa~p }ai, Boxes%µ at Sevres, inf,,tlie cgastr1sption, pf thp,constantvb ti hydiogen thermometer selected' by the committee for the determination of the normal scale, are diekiibea in the textbooks (e.g. Watson's Physics). The most important is the combination of the manometer and the
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 contactpoint, 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 contactpoints, 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.
1I. 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 is not accurately constant, but varies with change of pressure and temperature. 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 pressurecoefficient were constant at all temperatures and equal
to e, the pressure correction, dt, at any point t of the scale would be obtainable from the simple formula
dt =epot(t– loo)/To . . (1o) where po is the initial pressure at the temperature T0. 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 onetenth of a millimetre of mercury, but it is probably possible to measure to onehundredth as a mean of several readings under the best conditions, at ordinary temperatures.
12. StemExposure.—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 stemexposure correction, the value of which is approximately given by the formula
dt=rt(t  too)/Tz, . . . (u)
where r is the ratio of the volume of the dead space to the volume of the thermometric bulb, and T2 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, and increases very rapidly at high temperatures. If the dead space is i per cent. of the bulb, the correction will amount to only onetenth of a degree at 5o° C., but reaches 5° at 445° C., and 3o° at x000° C. It is for this reason important in hightemperature work to keep the dead space as small as possible and 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 OilGauge.—It is possible to avoid this difficulty, and to make the dead space very small, by employing oil or sulphuric acid or other nonvolatile 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 FIG. 4.M ethod of adjusted to be the same in both tubes AB Compensation. 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 constantvolume thermometer are frequently quoted, and are generally accepted as entirely conclusive, but it is very easy to construct the constantpressure or gravimetric instrument in such a manner as to escape the objections which he urges against it. Briefly stated, his objections are as follows: (r) 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 isvery 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 constantpressure thermometer than in the constantvolume 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 o•o36° with the constantvolume thermometer at any point of the scale, but with the constantpressure 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 constantvolume 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 Differential 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 oilgauge 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 pressure, 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 di increases according to the formula
dt/t = (T/To) 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 temperatures is commonly urged as a serious objection to the method, but 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 is sensitive to onehundredth of a degree at food° C., which is far beyond the order of accuracy attainable in the application of the other corrections.
16. Method of Using the Instrument.—A form of gas thermo• meter constructed on the principles above laid down, with the
addition of a duplicate set of connecting tubes C for the elimination of the stemexposure correction by the method of automatic compensation already explained, is shown in fig. 5 (Prot. 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 w, 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+dW, 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,
ate=ti/To=(wi+dW,)/(W—w1), . (13)
Similarly if w is the overflow when the bulb is at any other temperature t, and the expansion of the bulb is dW, we have a precisely similar equation for determining t in terms of T0, but with t and w and dW substituted for t, and wt and dW,. In practice, if the pressures are not adjusted to exact equality, or if the volumes of
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 pressurecoefficient 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. I), as foundby 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 too° 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 1/To 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 t 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 t given by the general formula (t), already quoted in § 3. Let dT 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 (I) we have,
t +dt = loo (T —To ddT dTo)/(T' —To +dT1 —dTo),
from which, provided that the variations considered are small, we obtain the following general expression for the correction to t,
dt=(dT—dTo)—(dT,—dTo)t/zoo. . . (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 =pV/R(Mo—M2), . . (15)
where p is the observed pressure at any temperature t, 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 Mo contained in the apparatus the mass M2 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+dV, T =p(Vo+dV)/RM = (PVo/RM) (1+dV/Vo),
whence we obtain approximately
dT=TdV/Vo . . (16)
If the coefficient of expansion of the bulb is constant and equal to the fundamental coefficient f (the mean coefficient between o° and ioo° C.), we have simply dV/Vo=ft; and if we substitute this value in the general expression (14) for dt, we obtain
dt=(T—T,)ft=ft(t—foo) . (t7)
Provided that the correction can be expressed as a rational integral function of t, it is evident that it must contain the factors t and (t—too), since by hypothesis the scale must be correct at the fixed points o° and too° 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
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