Chapter III: American (8)

S 2. The _Method of Mixture_ consists in imparting the quantity of
heat to be measured to a known mass of water, or some other standard
substance, contained in a vessel or calorimeter of known thermal
capacity, and in observing the rise of temperature produced, from
which data the quantity of heat may be found as explained in all
elementary text-books. This method is the most generally convenient
and most readily applicable of calorimetric methods, but it is not
always the most accurate, for various reasons. Some heat is generally
lost in transferring the heated body to the calorimeter; this loss may
be minimized by performing the transference rapidly, but it cannot be
accurately calculated or eliminated. Some heat is lost when the
calorimeter is raised above the temperature of its enclosure, and
before the final temperature is reached. This can be roughly estimated
by observing the rate of change of temperature before and after the
experiment, and assuming that the loss of heat is directly
proportional to the duration of the experiment and to the average
excess of temperature. It can be minimized by making the mixing as
rapid as possible, and by using a large calorimeter, so that the
excess of temperature is always small. The latter method was generally
adopted by J.P. Joule, but the rise of temperature is then difficult
to measure with accuracy, since it is necessarily reduced in nearly
the same proportion as the correction. There is, however, the
advantage that the correction is rendered much less uncertain by this
procedure, since the assumption that the loss of heat is proportional
to the temperature-excess is only true for small differences of
temperature. Rumford proposed to eliminate this correction by starting
with the initial temperature of the calorimeter as much below that of
its enclosure as the final temperature was expected to be above the
same limit. This method has been very generally recommended, but it is
really bad, because, although it diminishes the absolute magnitude of
the correction, it greatly increases the uncertainty of it and
therefore the probable error of the result. The coefficient of heating
of a calorimeter when it is below the temperature of its surroundings
is seldom, if ever, the same as the coefficient of cooling at the
higher temperature, since the convection currents, which do most of
the heating or cooling, are rarely symmetrical in the two cases, and
moreover, the duration of the two stages is seldom the same. In any
case, it is desirable to diminish the loss of heat as much as possible
by polishing the exterior of the calorimeter to diminish radiation,
and by suspending it by non-conducting supports, inside a polished
case, to protect it from draughts. It is also very important to keep
the surrounding conditions as constant as possible throughout the
experiment. This may be secured by using a large water-bath to
surround the apparatus, but in experiments of long duration it is
necessary to use an accurate temperature regulator. The method of
lagging the calorimeter with cotton-wool or other non-conductors,
which is often recommended, diminishes the loss of heat considerably,
but renders it very uncertain and variable, and should never be used
in work of precision. The bad conductors take so long to reach a
steady state that the rate of loss of heat at any moment depends on
the past history more than on the temperature of the calorimeter at
the moment. A more serious objection to the use of lagging of this
kind is the danger of its absorbing moisture. The least trace of damp
in the lagging, or of moisture condensed on the surface of the
calorimeter, may produce serious loss of heat by evaporation. This is
another objection to Rumford's method of cooling the calorimeter below
the surrounding temperature before starting. Among minor difficulties
of the method may be mentioned the uncertainty of the thermal capacity
of the calorimeter and stirrer, and of the immersed portion of the
thermometer. This is generally calculated by assuming values for the
specific heats of the materials obtained by experiment between 100
deg. C. and 20 deg. C. Since the specific heats of most metals
increase rapidly with rise of temperature, the values so obtained are
generally too high. It is best to make this correction as small as
possible by using a large calorimeter, so that the mass of water is
large in proportion to that of metal. Analogous difficulties arise in
the application of other calorimetric methods. The accuracy of the
work in each case depends principally on the skill and ingenuity of
the experimentalist in devising methods of eliminating the various
sources of error. The form of apparatus usually adopted for the
method of mixtures is that of Regnault with slight modifications, and
figures and descriptions are given in all the text-books. Among
special methods which have been subsequently developed there are two
which deserve mention as differing in principle from the common type.
These are (1) the constant temperature method, (2) the continuous flow
method.

The _constant temperature method of mixtures_ was proposed by N.
Hesehus (_Jour. Phys._, 1888, vii. p. 489). Cold water at a known
temperature is added to the calorimeter, immediately after dropping in
the heated substance, at such a rate as to keep the temperature of the
calorimeter constant, thus eliminating the corrections for the water
equivalent of the calorimeter and the external loss of heat. The
calorimeter is surrounded by an air-jacket connected to a petroleum
gauge which indicates any small change of temperature in the
calorimeter, and enables the manipulator to adjust the supply of cold
water to compensate it. The apparatus as arranged by F.A. Waterman is
shown in fig. 1 (_Physical Review_, 1896, iv. p. 161). A is the
calorimetric tube, B the air-jacket and L the gauge. H is an electric
heater for raising the body to a suitable temperature, which can swing
into place directly over the calorimeter. W is a conical can
containing water cooled by ice I nearly to 0 deg., which is swung over
the calorimeter as soon as the hot body has been introduced and the
heater removed. The cold water flow is regulated by a tap S with a
long handle O, and its temperature is taken by a delicate thermometer
with its bulb at G. The method is interesting, but the manipulations
and observations involved are more troublesome than with the ordinary
type of calorimeter, and it may be doubted whether any advantage is
gained in accuracy.

The _continuous flow method_ is specially applicable to the important
case of calorific value of gaseous fuel, where a large quantity of
heat is continuously generated at a nearly uniform rate by combustion.
Fig. 2 illustrates a recent type of gas calorimeter devised by C.V.
Boys (_Proc. R.S._, 1906, A. 77, p. 122). The heated products of
combustion from the burner B impinge on a metal box H, through which
water is circulating, and then pass downwards and outwards through a
spiral cooler which reduces them practically to the atmospheric
temperature. A steady stream of water enters the apparatus by the
inflow thermometer O, flows through the spiral coolers N and M, and
finally through the box H, where it is well mixed before passing the
outflow thermometer P. As soon as a steady state is reached, the
difference of temperature between the outflow and inflow thermometers,
multiplied by the current of water in grammes per minute gives the
heat per minute supplied by combustion. The gas current is
simultaneously observed by a suitable meter, which, with subsidiary
corrections for pressure, temperature, &c., gives the necessary data
for deducing calorific value.

A continuous flow calorimeter has been used by the writer for
measuring quantities of heat conveyed by conduction (see CONDUCTION OF
HEAT), and also for determining the variation of the specific heat of
water. In the latter case two steady currents of water at different
temperatures, say 0 deg. and 100 deg. are passed through an equalizer,
and the resulting temperature measured without mixing the currents,
which are then separately determined by weighing. This is a very good
method of comparing the mean specific heats over two ranges of
temperature such as 0-50, and 50-100, or 0-20 and 20-40, but it is not
so suitable as the electric method described below for obtaining the
actual specific heat at any point of the range.

S 3. _Method of Cooling._--A common example of this method is the determination of the specific heat of a liquid by filling a small calorimeter with the liquid, raising it to a convenient temperature, and then setting it to cool in an enclosure at a steady temperature, and observing the time taken to fall through a given range when the conditions have become fairly steady. The same calorimeter is afterwards filled with a known liquid, such as water, and the time of cooling is observed through the same range of temperature, in the same enclosure, under the same conditions. The ratio of the times of cooling is equal to the ratio of the thermal capacities of the calorimeter and its contents in the two cases. The advantage of the method is that there is no transference or mixture; the defect is that the whole measurement depends on the assumption that the rate of loss of heat is the same in the two cases, and that any variation in the conditions, or uncertainty in the rate of loss, produces its full effect in the result, whereas in the previous case it would only affect a small correction. Other sources of uncertainty are, that the rate of loss of heat generally depends to some extent on the rate of fall of temperature, and that it is difficult to take accurate observations on a rapidly falling thermometer. As the method is usually practised, the calorimeter is made very small, and the surface is highly polished to diminish radiation. It is better to use a fairly large calorimeter to diminish the rate of cooling and the uncertainty of the correction for the water equivalent. The surface of the calorimeter and the enclosure should be permanently blackened so as to increase the loss of heat by radiation as much as possible, as compared with the losses by convection and conduction, which are less regular. For accurate work it is essential that the liquid in the calorimeter should be continuously stirred, and also in the enclosure, the lid of which must be water-jacketed, and kept at the same steady temperature as the sides. When all these precautions are taken, the method loses most of the simplicity which is its chief advantage. It cannot be satisfactorily applied to the case of solids or powders, and is much less generally useful than the method of mixture.

S 4. _Method of Fusion._--The methods depending on change of state are theoretically the simplest, since they do not necessarily involve any reference to thermometry, and the corrections for external loss of heat and for the thermal capacity of the containing vessels can be completely eliminated. They nevertheless present peculiar difficulties and limitations, which render their practical application more troublesome and more uncertain than is usually supposed. They depend on the experimental fact that the quantity of heat required to produce a given change of state (e.g. to convert one gramme of ice at 0 deg. C. into water at 0 deg. C., or one gramme of water at 100 deg. C. into steam at 100 deg. C.) is always the same, and that there need be no change of temperature during the process. The difficulties arise in connexion with the determination of the quantities of ice melted or steam condensed, and in measuring the latent heat of fusion or vaporization in terms of other units for the comparison of observations. The earlier forms of ice-calorimeter, those of Black, and of Laplace and Lavoisier, were useless for work of precision, on account of the impossibility of accurately estimating the quantity of water left adhering to the ice in each case. This difficulty was overcome by the invention of the Bunsen calorimeter, in which the quantity of ice melted is measured by observing the diminution of volume, but the successful employment of this instrument requires considerable skill in manipulation. The sheath of ice surrounding the bulb must be sufficiently continuous to prevent escape of heat, but it must not be so solid as to produce risk of strain. The ideal condition is difficult to secure. In the practical use of the instrument it is not necessary to know both the latent heat of fusion of ice and the change of volume which occurs on melting; it is sufficient to determine the change of volume per calorie, or the quantity of mercury which is drawn into the bulb of the apparatus per unit of heat added. This can be determined by a direct calibration, by inserting a known quantity of water at a known temperature and observing the contraction, or weighing the mercury drawn into the apparatus. In order to be independent of the accuracy of the thermometer employed for observing the initial temperature of the water introduced, it has been usual to employ water at 100 deg. C., adopting as unit of heat the "mean calorie," which is one-hundredth part of the heat given up by one gramme of water in cooling from 100 deg. to 0 deg. C. The weight of mercury corresponding to the mean calorie has been determined with considerable care by a number of observers well skilled in the use of the instrument. The following are some of their results:--Bunsen, 15.41 mgm.; Velten, 15.47 mgm.; Zakrevski, 15.57 mgm.; Staub, 15.26 mgm. The explanation of these discrepancies in the fundamental constant is not at all clear, but they may be taken as an illustration of the difficulties of manipulation attending the use of this instrument, to which reference has already been made. It is not possible to deduce a more satisfactory value from the latent heat and the change of density, because these constants are very difficult to determine. The following are some of the values deduced by well-known experimentalists for the latent heat of fusion:--Regnault, 79.06 to 79.24 calories, corrected by Person to 79.43; Person, 79.99 calories; Hess, 80.34 calories; Bunsen, 80.025 calories. Regnault, Person and Hess employed the method of mixture which is probably the most accurate for the purpose. Person and Hess avoided the error of water sticking to the ice by using dry ice at various temperatures below 0 deg. C., and determining the specific heat of ice as well as the latent heat of fusion. These discrepancies might, no doubt, be partly explained by differences in the units employed, which are somewhat uncertain, as the specific heat of water changes rapidly in the neighbourhood of 0 deg. C; but making all due allowance for this, it remains evident that the method of ice-calorimetry, in spite of its theoretical simplicity, presents grave difficulties in its practical application.

One of the chief difficulties in the practical use of the Bunsen
calorimeter is the continued and often irregular movement of the
mercury column due to slight differences of temperature, or pressure
between the ice in the calorimeter and the ice bath in which it is
immersed. C.V. Boys (_Phil. Mag._, 1887, vol. 24, p. 214) showed that
these effects could be very greatly reduced by surrounding the
calorimeter with an outer tube, so that the ice inside was separated
from the ice outside by an air space which greatly reduces the free
passage of heat. The present writer has found that very good results
may be obtained by enclosing the calorimeter in a vacuum jacket (as
illustrated in fig. 3), which practically eliminates conduction and
convection. If the vacuum jacket is silvered inside, radiation also is
reduced to such an extent that, if the vacuum is really good, the
external ice bath may be dispensed with for the majority of purposes.
If the inner bulb is filled with mercury instead of water and ice, the
same arrangement answers admirably as a Favre and Silbermann
calorimeter, for measuring small quantities of heat by the expansion
of the mercury.

The question has been raised by E.L. Nichols (_Phys. Rev._ vol. 8,
January 1899) whether there may not be different modifications of ice
with different densities, and different values of the latent heat of
fusion. He found for natural pond-ice a density 0.9179 and for
artificial ice 0.9161. J. Vincent (_Phil. Trans._ A. 198, p. 463) also
found a density .9160 for artificial ice, which is probably very
nearly correct. If such variations of density exist, they may
introduce some uncertainty in the absolute values of results obtained
with the ice calorimeter, and may account for some of the
discrepancies above enumerated.

S 5. The _Method of Condensation_ was first successfully applied by J. Joly in the construction of his steam calorimeter, a full description of which will be found in text-books. The body to be tested is placed in a special scale-pan, suspended by a fine wire from the arm of a balance inside an enclosure which can be filled with steam at atmospheric pressure. The temperature of the enclosure is carefully observed before admitting steam. The weight of steam condensed on the body gives a means of calculating the quantity of heat required to raise it from the atmospheric temperature up to 100 deg. C. in terms of the latent heat of vaporization of steam at 100 deg. C. There can be no appreciable gain or loss of heat by radiation, if the admission of the steam is sufficiently rapid, since the walls of the enclosure are maintained at 100 deg. C., very nearly. The thermal capacity of the scale-pan, &c., can be determined by a separate experiment, or, still better, eliminated by the differential method of counterpoising with an exactly similar arrangement on the other arm of the balance. The method requires very delicate weighing, as one calorie corresponds to less than two milligrammes of steam condensed; but the successful application of the method to the very difficult problem of measuring the specific heat of a gas at constant volume, shows that these and other difficulties have been very skilfully overcome. The application of the method appears to be practically limited to the measurements of specific heat between the atmospheric temperature and 100 deg. C. The results depend on the value assumed for the latent heat of steam, which Joly takes as 536.7 calories, following Regnault. Joly has himself determined the mean specific heat of water between 12 deg. and 100 deg. C. by this method, in terms of the latent heat of steam as above given, and finds the result .9952. Assuming that the mean specific heat of water between 12 deg. and 100 deg. is really 1.0011 in terms of the calorie at 20 deg. C. (see table, p. 66), the value of the latent heat of steam at 100 deg. C., as determined by Joly, would be 540.2 in terms of the same unit. The calorie employed by Regnault is to some extent uncertain, but the difference is hardly beyond the probable errors of experiment, since it appears from the results of recent experiments that Regnault made an error of the same order in his determination of the specific heat of water at 100 deg. C.

S 6. _Energy Methods._--The third general method of calorimetry, that based on the transformation of some other kind of energy into the form of heat, rests on the general principle of the conservation of energy, and on the experimental fact that all other forms of energy are readily and completely convertible into the form of heat. It is therefore often possible to measure quantities of heat indirectly, by measuring the energy in some other form and then converting it into heat. In addition to its great theoretical interest, this method possesses the advantage of being frequently the most accurate in practical application, since energy can be more accurately measured in other forms than in that of heat. The two most important varieties of the method are (_a_) mechanical, and (_b_) electrical. These methods have reached their highest development in connexion with the determination of the mechanical equivalent of heat, but they may be applied with great advantage in connexion with other problems, such as the measurement of the variation of specific heat, or of latent heats of fusion or vaporization.

S 7. _Mechanical Equivalent of Heat._--The phrase "mechanical equivalent of heat" is somewhat vague, but has been sanctioned by long usage. It is generally employed to denote the number of units of mechanical work or energy which, when completely converted into heat without loss, would be required to produce one heat unit. The numerical value of the mechanical equivalent necessarily depends on the particular units of heat and work employed in the comparison. The British engineer prefers to state results in terms of foot-pounds of work in any convenient latitude per pound-degree-Fahrenheit of heat. The continental engineer prefers kilogrammetres per kilogramme-degree-centigrade. For scientific use the C.G.S. system of expression in ergs per gramme-degree-centigrade, or "calorie," is the most appropriate, as being independent of the value of gravity. A more convenient unit of work or energy, in practice, on account of the smallness of the erg, is the _joule_, which is equal to 10.7 ergs, or one _watt-second_ of electrical energy. On account of its practical convenience, and its close relation to the international electrical units, the _joule_ has been recommended by the British Association for adoption as the absolute unit of heat. Other convenient practical units of the same kind would be the _watt-hour_, 3600 joules, which is of the same order of magnitude as the kilo-calorie, and the _kilowatt-hour_, which is the ordinary commercial unit of electrical energy.

S 8. _Joule_.--The earlier work of Joule is now chiefly of historical
interest, but his later measurements in 1878, which were undertaken on
a larger scale, adopting G.A. Hirn's method of measuring the work
expended in terms of the torque and the number of revolutions, still
possess value as experimental evidence. In these experiments (see fig.
4) the paddles were revolved by hand at such a speed as to produce a
constant torque on the calorimeter _h_, which was supported on a float
_w_ in a vessel of water _v_, but was kept at rest by the couple due
to a pair of equal weights _k_ suspended from fine strings passing
round the circumference of a horizontal wheel attached to the
calorimeter. Each experiment lasted about forty minutes, and the rise
of temperature produced was nearly 3 deg. C. The calorimeter contained
about 5 kilogrammes of water, so that the rate of heat-supply was
about 6 calories per second. Joule's final result was 772.55
foot-pounds at Manchester per pound-degree-Fahrenheit at a temperature
of 62 deg. F., but individual experiments differed by as much as 1%.
This result in C.G.S. measure is equivalent to 4.177 joules per
calorie at 16.5 deg. C., on the scale of Joule's mercury thermometer.
His thermometers were subsequently corrected to the Paris scale by A.
Schuster in 1895, which had the effect of reducing the above figure to
4.173.

S 9. _Rowland_.--About the same time H.A. Rowland (_Proc. Amer. Acad._
xv. p. 75, 1880) repeated the experiment, employing the same method,
but using a larger calorimeter (about 8400 grammes) and a petroleum
motor, so as to obtain a greater rate of heating (about 84 calories
per second), and to reduce the importance of the uncertain correction
for external loss of heat. Rowland's apparatus is shown in fig. 5. The
calorimeter was suspended by a steel wire, the torsion of which made
the equilibrium stable. The torque was measured by weights O and P
suspended by silk ribbons passing over the pulleys n and round the
disk kl. The power was transmitted to the paddles by bevel wheels, f,
g, rotating a spindle passing through a stuffing box in the bottom of
the calorimeter. The number of revolutions and the rise of temperature
were recorded on a chronograph drum. He paid greater attention to the
important question of thermometry, and extended his researches over a
much wider range of temperature, namely 5 deg. to 35 deg. C. His
experiments revealed for the first time a diminution in the specific
heat of water with rise of temperature between 0 deg. and 30 deg. C.,
amounting to four parts in 10.000 per 1 deg. C. His thermometers were
compared with a mercury thermometer standardized in Paris, and with a
platinum thermometer standardized by Griffiths. The result was to
reduce the coefficient of diminution of specific heat at 15 deg. C. by
nearly one half, but the absolute value at 20 deg. C. is practically
unchanged. Thus corrected his values are as follows:--

Temperature 10 deg. 15 deg. 20 deg. 25 deg. 30 deg. 35 deg.
Joules per cal. 4.197 4.188 4.181 4.176 4.175 4.177

These are expressed in terms of the hydrogen scale, but the difference
from the nitrogen scale is so small as to be within the limits of
experimental error in this particular case. Rowland himself considered
his results to be probably correct to one part in 500, and supposed
that the greatest uncertainty lay in the comparison of the scale of
his mercury thermometer with the air thermometer. The subsequent
correction, though not carried out strictly under the conditions of
the experiment, showed that the order of accuracy of his work about
the middle of the range from 15 deg. to 25 deg. was at least 1 in
1000, and probably 1 in 2000. At 30 deg. he considered that, owing to
the increasing magnitude and uncertainty of the radiation correction,
there "might be a small error in the direction of making the
equivalent too great, and that the specific heat might go on
decreasing to even 40 deg. C." The results considered with reference
to the variation of the specific heat of water are shown in the curve
marked Rowland in Fig. 6.

S 10. _Osborne Reynolds and W.H. Moorby (Phil. Trans._, 1897, p. 381)
determined the mechanical equivalent of the mean thermal unit between
0 deg. and 100 deg. C., on a very large scale, with a Froude-Reynolds
hydraulic brake and a steam-engine of 100 h.p. This brake is
practically a Joule calorimeter, ingeniously designed to churn the
water in such a manner as to develop the greatest possible resistance.
The admission of water at 0 deg. C. to the brake was controlled by
hand in such a manner as to keep the outflow nearly at the
boiling-point, the quantity of water in the brake required to produce
a constant torque being regulated automatically, as the speed varied,
by a valve worked by the lifting of the weighted lever attached to the
brake.

The accompanying illustration (fig. 7) shows the brake lagged with
cotton-wool, and the 4-ft. lever to which the weights are suspended.
The power of the brake may be estimated by comparison with the size of
the rope pulley seen behind it on the same shaft. With 300 pounds on a
4-ft. lever at 300 revolutions per minute, the rate of generation of
heat was about 12 kilo-calories per second. In spite of the large
range of temperature, the correction for external loss of heat
amounted to only 5%, with the brake uncovered, and was reduced to less
than 2% by lagging. This is the special advantage of working on so
large a scale with so rapid a generation of heat. But, for the same
reason, the method necessarily presents peculiar difficulties, which
were not overcome without great pains and ingenuity. The principal
troubles arose from damp in the lagging which necessitated the
rejection of several trials, and from dissolved air in the water,
causing loss of heat by the formation of steam. Next to the radiation
loss, the most uncertain correction was that for conduction of heat
along the 4-in. shaft. These losses were as far as possible eliminated
by combining the trials in pairs, with different loads on the brake,
assuming that the heat-loss would be the same in the heavy and light
trials, provided that the external temperature and the gradient in the
shaft, as estimated from the temperature of the bearings, were the
same. The values deduced in this manner for the equivalent agreed as
closely as could be expected considering the impossibility of
regulating the external condition of temperature and moisture with any
certainty in an engine-room. The extreme variation of results in any
one series was only from 776.63 to 779.46 ft.-pounds, or less than
1/2%. This variation may have been due to the state of the lagging,
which Moorby distrusted in spite of the great reduction of the
heat-loss, or it may have been partly due to the difficulty of
regulating the speed of the engine and the water-supply to the brake
in such a manner as to maintain a constant temperature in the outflow,
and avoid variations in the heat capacity of the brake. Since hand
regulation is necessarily discontinuous, the speed and the temperature
were constantly varying, so that it was useless to take readings
nearer than the tenth of a degree. The largest variation recorded in
the two trials of which full details are given, was 4-9 deg. F. in two
minutes in the outflow temperature, and four or five revolutions per
minute on the speed. These variations, so far as they were of a purely
accidental nature, would be approximately eliminated on the mean of a
large number of trials, so that the accuracy of the final result would
be of a higher order than might be inferred from a comparison of
separate pairs of trials. Great pains were taken to discuss and
eliminate all the sources of constant error which could be foreseen.
The results of the light trials with 400 ft.-pounds on the brake
differ slightly from those with 600 ft.-pounds. This might be merely
accidental, or it might indicate some constant difference in the
conditions requiring further investigation. It would have been
desirable, if possible, to have tried the effect of a larger range of
variation in the experimental conditions of load and speed, with a
view to detect the existence of constant errors; but owing to the
limitations imposed by the use of a steam-engine, and the difficulty
of securing steady conditions of running, this proved to be
impossible. There can be no doubt, however, that the final result is
the most accurate direct determination of the value of the mean
calorie between 0 deg. and 100 deg. C. in mechanical units. Expressed
in joules per calorie the result is 4.1832, which agrees very closely
with the value found by Rowland as the mean over the range 15 deg. to
20 deg. C. The value 4.183 is independently confirmed in a remarkable
manner by the results of the electrical method described below, which
give 4.185 joules for the mean calorie, if Rowland's value is assumed
as the starting-point, and taken to be 4.180 joules at 20 deg. C.

S 11. _Electrical Methods._--The value of the international electrical units has by this time been so accurately determined in absolute measure that they afford a very good, though indirect, method of determining the mechanical equivalent of heat. But, quite apart from this, electrical methods possess the greatest value for calorimetry, on account of the facility and accuracy of regulating and measuring the quantity of heat supplied by an electric current. The frictional generation of heat in a metallic wire conveying a current can be measured in various ways, which correspond to slightly different methods. By Ohm's law, and by the definition of difference of electric pressure or potential, we obtain the following alternative expressions for the quantity of heat H in joules generated in a time T seconds by a current of C amperes flowing in a wire of resistance R ohms, the difference of potential between the ends of the wire being E = CR volts:--

H = ECT = C^2RT = E^2T/R (1).

The method corresponding to the expression C^2RT was adopted by Joule and by most of the early experimentalists. The defects of the earlier work from an electrical point of view lay chiefly in the difficulty of measuring the current with sufficient accuracy owing to the imperfect development of the science of electrical measurement. These difficulties have been removed by the great advances since 1880, and in particular by the introduction of accurate standard cells for measurements of electrical pressure.

S 12. _Griffiths_.--The method adopted by E.H. Griffiths (_Phil.
Trans._, 1893, p. 361), whose work threw a great deal of light on the
failure of previous observers to secure consistent results,
corresponded to the last expression E^2T/R, and consisted in
regulating the current by a special rheostat, so as to keep the
potential difference E on the terminals of the resistance R balanced
against a given number of standard Clark cells of the Board of Trade
pattern. The resistance R could be deduced from a knowledge of the
temperature of the calorimeter and the coefficient of the wire. But in
order to obtain trustworthy results by this method he found it
necessary to employ very rapid stirring (2000 revolutions per minute),
and to insulate the wire very carefully from the liquid to prevent
leakage of the current. He also made a special experiment to find how
much the temperature of the wire exceeded that of the liquid under the
conditions of the experiment. This correction had been neglected by
previous observers employing similar methods. The resistance R was
about 9 ohms, and the potential difference E was varied from three to
six Clark cells, giving a rate of heat-supply about 2 to 6 watts. The
water equivalent of the calorimeter was about 85 grammes, and was
determined by varying the quantity of water from 140 to 260 or 280
grammes, so that the final results depended on a difference in the
weight of water of 120 to 140 grammes. The range of temperature in
each experiment was 14 deg. to 26 deg. C. The rate of rise was
observed with a mercury thermometer standardized by comparison with a
platinum thermometer under the conditions of the experiment. The time
of passing each division was recorded on an electric chronograph. The
duration of an experiment varied from about 30 to 70 minutes. Special
observations were made to determine the corrections for the heat
supplied by stirring, and that lost by radiation, each of which
amounted to about 10% of the heat-supply. The calorimeter C, fig. 8,
was gilded, and completely surrounded by a nickel-plated steel
enclosure B, forming the bulb of a mercury thermo-regulator, immersed
in a large water-bath maintained at a constant temperature. In spite
of the large corrections the results were extremely consistent, and
the value of the temperature-coefficient of the diminution of the
specific heat of water, deduced from the observed variation in the
rate of rise at different points of the range 15 deg. to 25 deg.,
agreed with the value subsequently deduced from Rowland's experiments
over the same range, when his thermometers were reduced to the same
scale. Griffiths' final result for the average value of the calorie
over this range was 4.192 joules, taking the E.M.F. of the Clark cell
at 15 deg. C. to be 1.4342 volts. The difference from Rowland's value,
4.181, could be explained by supposing the E.M.F. of the Clark cells
to have in reality been 1.4323 volts, or about 2 millivolts less than
the value assumed. Griffiths subsequently applied the same method to
the measurement of the specific heat of aniline, and the latent heat
of vaporization of benzene and water.

S 13. _Schuster and Gannon._--The method employed by A. Schuster and
W. Gannon for the determination of the specific heat of water in terms
of the international electric units (_Phil. Trans._ A, 1895, p. 415)
corresponded to the expression ECT, and differed in many essential
details from that of Griffiths. The current through a platinoid
resistance of about 31 ohms in a calorimeter containing 1500 grammes
of water was regulated so that the potential difference on its
terminals was equal to that of twenty Board of Trade Clark cells in
series. The duration of an experiment was about ten minutes, and the
product of the mean current and the time, namely CT, was measured by
the weight of silver deposited in a voltameter, which amounted to
about 0.56 gramme. The uncertainty due to the correction for the water
equivalent was minimized by making it small (about 27 grammes) in
comparison with the water weight. The correction for external loss was
reduced by employing a small rise of temperature (only 2.22 deg.), and
making the rate of heat-supply relatively rapid, nearly 24 watts. The
platinoid coil was insulated from the water by shellac varnish. The
wire had a length of 760 cms., and the potential difference on its
terminals was nearly 30 volts. The rate of stirring adopted was so
slow that the heat generated by it could be neglected. The result
found was 4.191 joules per calorie at 19 deg. C. This agrees very well
with Griffiths considering the difficulty of measuring so small a rise
of temperature at 2 deg. with a mercury thermometer. Admitting that the
electro-chemical equivalent of silver increases with the age of the
solution, a fact subsequently discovered, and that the E.M.F. of the
Clark cell is probably less than 1.4340 volts (the value assumed by
Schuster and Gannon), there is no difficulty in reconciling the result
with that of Rowland.

S 14. _H.L. Callendar and H.T. Barnes_ (_Brit. Assoc. Reports_, 1897
and 1899) adopted an entirely different method of calorimetry, as well
as a different method of electrical measurement. A steady current of
liquid, Q grammes per second, of specific heat, Js joules per degree,
flowing through a fine tube, A B, fig. 9, is heated by a steady
electric current during its passage through the tube, and the
difference of temperature d[theta] between the inflowing and the
outflowing liquid is measured by a single reading with a delicate pair
of differential platinum thermometers at A and B. The difference of
potential E between the ends of the tube, and the electric current C
through it, are measured on an accurately calibrated potentiometer, in
terms of a Clark cell and a standard resistance. If hd[theta] is the
radiation loss in watts we have the equation,

EC = JsQd[theta] + hd[theta] (2).

The advantage of this method is that all the conditions are steady, so
that the observations can be pushed to the limit of accuracy and
sensitiveness of the apparatus. The water equivalent of the
calorimeter is immaterial, since there is no appreciable change of
temperature. The heat-loss can be reduced to a minimum by enclosing
the flow-tube in a hermetically sealed glass vacuum jacket. Stirring
is effected by causing the water to circulate spirally round the bulbs
of the thermometers and the heating conductor as indicated in the
figure. The conditions can be very easily varied through a wide range.
The heat-loss hd[theta] is determined and eliminated by varying the
flow of liquid and the electric current simultaneously, in such a
manner as to secure approximately the same rise of temperature for two
or more widely different values of the flow of liquid. An example
taken from the _Electrician_, September 1897, of one of the earliest
experiments by this method on the specific heat of mercury will make
the method clearer. The flow-tube was about 1 metre long and 1 millim.
in diameter, coiled in a short spiral inside the vacuum jacket. The
outside of the vacuum jacket was immersed in a water jacket at a
steady temperature equal to that of the inflowing mercury.

SPECIFIC HEAT OF MERCURY BY CONTINUOUS ELECTRIC METHOD

+-----------+---------------+------+-------------+----------------+
|Flow of Hg.| Rise of Temp. |Watts.| Heat-loss. | Specific Heat. |
+-----------+---------------+------+-------------+----------------+
| gm./sec. | d[theta] | EC | hd[theta] | Per gm. deg. |
| 8.753 | 11.764 |14.862| 0.655 | \ .13780 joules|
| 4.594 | 12.301 | 7.912| 0.865 | / .03297 cals. |
+-----------+---------------+------+-------------+----------------+

It is assumed as a first approximation that the heat-loss is
proportional to the rise of temperature _d[theta]_, provided that
_d[theta]_ is nearly the same in both cases, and that the distribution
of temperature in the apparatus is the same for the same rise of
temperature whatever the flow of liquid. The result calculated on
these assumptions is given in the last column in joules, and also in
calories of 20 deg. C. The heat-loss in this example is large, nearly
4.5% of the total supply, owing to the small flow and the large rise
of temperature, but this correction was greatly reduced in subsequent
observations on the specific heat of water by the same method. In the
case of mercury the liquid itself can be utilized to conduct the
electric current. In the case of water or other liquids it is
necessary to employ a platinum wire stretched along the tube as
heating conductor. This introduces additional difficulties of
construction, but does not otherwise affect the method. The absolute
value of the specific heat deduced necessarily depends on the absolute
values of the electrical standards employed in the investigation. But
for the determination of relative values of specific heats in terms of
a standard liquid, or of the variations of specific heat of a liquid,
the method depends only on the constancy of the standards, which can
be readily and accurately tested. The absolute value of the E.M.F. of
the Clark cells employed was determined with a special form of
electrodynamometer (Callendar, _Phil. Trans._ A. 313, p. 81), and
found to be 1.4334 volts, assuming the ohm to be correct. Assuming
this value, the result found by this method for the specific heat of
water at 20 deg. C. agrees with that of Rowland within the probable
limits of error.

S 15. _Variation of Specific Heat of Water._--The question of the
variation of the specific heat of water has a peculiar interest and
importance in connexion with the choice of a thermal unit. Many of the
uncertainties in the reduction of older experiments, such as those of
Regnault, arise from uncertainty in regard to the unit in terms of
which they are expressed, which again depends on the scale of the
particular thermometer employed in the investigation. The first
experiments of any value were those of Regnault in 1847 on the
specific heat of water between 110 deg. C. and 192 deg. C. They were
conducted on a very large scale by the method of mixture, but showed
discrepancies of the order of 0.5%, and the calculated results in many
cases do not agree with the data. This may be due merely to deficient
explanation of details of tabulation. We may probably take the
tabulated values as showing correctly the rate of variation between
110 deg. and 190 deg. C., but the values in terms of any particular
thermal unit must remain uncertain to at least 0.5% owing to the
uncertainties of the thermometry. Regnault himself adopted the
formula,

s = 1 + 0.00004t + 0.0000009t^2 (Regnault), (3)

for the specific heat _s_ at any temperature _t_ C. in terms of the
specific heat at 0 deg. C. taken as the standard. This formula has
since been very generally applied over the whole range 0 deg. to 200
deg. C., but the experiments could not in reality give any information
with regard to the specific heat at temperatures below 100 deg. C. The
linear formula proposed by J. Bosscha from an independent reduction of
Regnault's experiments is probably within the limits of accuracy
between 100 deg. and 200 deg. C., so far as the mean rate of variation
is concerned, but the absolute values require reduction. It may be
written--

s = S_100 + .00023(t - 100) (Bosscha-Regnault) (4).

The work of L. Pfaundler and H. Platter, of G.A. Hirn, of J.C. Jamin
and Amaury, and of many other experimentalists who succeeded Regnault,
appeared to indicate much larger rates of increase than he had found,
but there can be little doubt that the discrepancies of their results,
which often exceeded 5%, were due to lack of appreciation of the
difficulties of calorimetric measurements. The work of Rowland by the
mechanical method was the first in which due attention was paid to the
thermometry and to the reduction of the results to the absolute scale
of temperature. The agreement of his corrected results with those of
Griffiths by a very different method, left very little doubt with
regard to the rate of diminution of the specific heat of water at 20
deg. C. The work of A. Bartoli and E. Stracciati by the method of
mixture between 0 deg. and 30 deg. C., though their curve is otherwise
similar to Rowland's, had appeared to indicate a minimum at 20 deg.
C., followed by a rapid rise. This lowering of the minimum was
probably due to some constant errors inherent in their method of
experiment. The more recent work of Ludin, 1895, under the direction
of Prof. J. Pernet, extended from 0 deg. to 100 deg. C., and appears
to have attained as high a degree of excellence as it is possible to
reach by the employment of mercury thermometers in conjunction with
the method of mixture. His results, exhibited in fig. 6, show a
minimum at 25 deg. C., and a maximum at 87 deg. C., the values being
.9935 and 1.0075 respectively in terms of the mean specific heat
between 0 deg. and 100 deg. C. He paid great attention to the
thermometry, and the discrepancies of individual measurements at any
one point nowhere exceed 0.3%, but he did not vary the conditions of
the experiments materially, and it does not appear that the well-known
constant errors of the method could have been completely eliminated by
the devices which he adopted. The rapid rise from 25 deg. to 75 deg.
may be due to radiation error from the hot water supply, and the
subsequent fall of the curve to the inevitable loss of heat by
evaporation of the boiling water on its way to the calorimeter. It
must be observed, however, that there is another grave difficulty in
the accurate determination of the specific heat of water near 100 deg.
C. by this method, namely, that the quantity actually observed is not
the specific heat _at_ the higher temperature _t_, but the _mean
specific heat_ over the range 18 deg. to _t_. The specific heat itself
can be deduced only by differentiating the curve of observation, which
greatly increases the uncertainty. The peculiar advantage of the
electric method of Callendar and Barnes, already referred to, is that
the specific heat itself is determined over a range of 8 deg. to 10
deg. at each point, by adding accurately measured quantities of heat
to the water at the desired temperature in an isothermal enclosure,
under perfectly steady conditions, without any possibility of
evaporation or loss of heat in transference. These experiments, which
have been extended by Barnes over the whole range 0 deg. to 100 deg.,
agree very well with Rowland and Griffiths in the rate of variation at
20 deg. C., but show a rather flat minimum of specific heat in the
neighbourhood of 38 deg. to 40 deg. C. At higher points the rate of
variation is very similar to that of Regnault's curve, but taking the
specific heat at 20 deg. as the standard of reference, the actual
values are nearly 0.56% less than Regnault's. It appears probable that
his values for higher temperatures may be adopted with this reduction,
which is further confirmed by the results of Reynolds and Moorby, and
by those of Ludin. According to the electric method, the whole range
of variation of the specific heat between 10 deg. and 80 deg. is only
0.5%. Comparatively simple formulae, therefore, suffice for its
expression to 1 in 10,000, which is beyond the limits of accuracy of
the observations. It is more convenient in practice to use a few
simple formulae, than to attempt to represent the whole range by a
single complicated expression:--

Below 20 deg. C. s = 0.9982 + 0.0000045(t - 40)^2 - 0.0000005(t - 20)^3.

From 20 deg. to 60 deg., s = 0.9982 + 0.0000045(t - 40)^2 (5).

/ s = 0.9944 + .00004t + 0.0000009t^2
Above 60 deg. to 200 deg. < (Regnault corrd.)
\ s = 1.000 + 0.00022(t - 60), (Bosscha corrd.)

The addition of the cubic term below 20 deg. is intended to represent
the somewhat more rapid change near the freezing-point. This effect is
probably due, as suggested by Rowland, to the presence of a certain
proportion of ice molecules in the liquid, which is also no doubt the
cause of the anomalous expansion. Above 60 deg. C. Regnault's formula
is adopted, the absolute values being simply diminished by a constant
quantity 0.0056 to allow for the probable errors of his thermometry.
Above 100 deg. C., and for approximate work generally, the simpler
formula of Bosscha, similarly corrected, is probably adequate.

The following table of values, calculated from these formulae, is
taken from the _Brit. Assoc. Report_, 1899, with a slight modification
to allow for the increase in the specific heat below 20 deg. C. This
was estimated in 1899 as being equivalent to the addition of the
constant quantity 0.20 to the values of the total heat h of the liquid
as reckoned by the parabolic formula (5). This quantity is now, as the
result of further experiments, added to the values of h, and also
represented in the formula for the specific heat itself by the cubic
term.

SPECIFIC HEAT OF WATER IN TERMS OF UNIT AT 20 deg. C. 4.180 JOULES

+-----------+-------+--------+---------+----------+
| t deg. C. |Joules.| s. | h | Rowland. |
+-----------+-------+--------+---------+----------+
| 0 deg. | 4.208 | 1.0094 | 0 | 0 |
| 5 deg. | 4.202 | 1.0054 | 5.037 | 5.037 |
| 10 deg. | 4.191 | 1.0027 | 10.056 | 10.058 |
| 15 deg. | 4.184 | 1.0011 | 15.065 | 15.068 |
| 20 deg. | 4.180 | 1.0000 | 20.068 | 20.071 |
| 25 deg. | 4.177 | 0.9992 | 25.065 | 25.067 |
| 30 deg. | 4.175 | 0.9987 | 30.060 | 30.057 |
| 35 deg. | 4.173 | 0.9983 | 35.052 | 35.053 |
| 40 deg. | 4.173 | 0.9982 | 40.044 | |
| 50 deg. | 4.175 | 0.9987 | 50.028 | |
| 60 deg. | 4.180 | 1.0000 | 60.020 | |
| 70 deg. | 4.187 | 1.0016 | 70.028 | |
| 80 deg. | 4.194 | 1.0033 | 80.052 | |
| 90 deg. | 4.202 | 1.0053 | 90.095 | Shaw |
| 100 deg. | 4.211 | 1.0074 | 100.158 | Regnault |
| 120 deg. | 4.231 | 1.0121 | 120.35 | 120.73 |
| 140 deg. | 4.254 | 1.0176 | 140.65 | 140.88 |
| 160 deg. | 4.280 | 1.0238 | 161.07 | 161.20 |
| 180 deg. | 4.309 | 1.0308 | 181.62 | 182.14 |
| 200 deg. | 4.341 | 1.0384 | 202.33 | |
| 220 deg. | 4.376 | 1.0467 | 223.20 | |
+-----------+-------+--------+---------+----------+

The unit of comparison in the following table is taken as the specific
heat of water at 20 deg. C. for the reasons given below. This unit is
taken as being 4.180 joules per gramme-degree-centigrade on the scale
of the platinum thermometer, corrected to the absolute scale as
explained in the article THERMOMETRY, which has been shown to be
practically equivalent to the hydrogen scale. The value 4.180 joules
at 20 deg. C. is the mean between Rowland's corrected result 4.181 and
the value 4.179, deduced from the experiments of Reynolds and Moorby
on the assumption that the ratio of the mean specific heat 0 deg. to
100 deg. to that at 20 deg. is 1.0016, as given by the formulae
representing the results of Callendar and Barnes. This would indicate
that Rowland's corrected values should, if anything, be lowered. In
any case the value of the mechanical equivalent is uncertain to at
least 1 in 2000.

The mean specific heat, over any range of temperature, may be obtained
by integrating the formulae between the limits required, or by taking
the difference of the corresponding values of the total heat h, and
dividing by the range of temperature. The quantity actually observed
by Rowland was the total heat. It may be remarked that starting from
the same value at 5 deg., for the sake of comparison, Rowland's values
of the total heat agree to 1 in 5000 with those calculated from the
formulae. The values of the total heat observed by Regnault, as
reduced by Shaw, also show a very fair agreement, considering the
uncertainty of the units. It must be admitted that it is desirable to
redetermine the variation of the specific heat above 100 deg. C. This
is very difficult on account of the steam-pressure, and could not
easily be accomplished by the electrical method. Callendar has,
however, devised a continuous method of mixture, which appears to be
peculiarly adapted to the purpose, and promises to give more certain
results. In any case it may be remarked that formulae such as those of
Jamin, Henrichsen, Baumgartner, Winkelmann or Dieterici, which give
far more rapid rates of increase than that of Regnault, cannot
possibly be reconciled with his observations, or with those of
Reynolds and Moorby, or Callendar and Barnes, and are certainly
inapplicable above 100 deg. C.