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XV. Students Simple Apparatus for Determining the Mechanical Equivalent of Heat. By Prof. W. E. AYRTON and H. C. HAYCRAFT*.
I. The Object to be Attained.
NOME time ago the authors considered the possibility of constructing an apparatus for the determination of the mechanical equivalent of heat which could be placed in the hands of junior students, and which would enable a sufficiently accurate result to be obtained without the introduction of troublesome corrections. For such a purpose the electrical method was naturally adopted; for now that the commercial values of the electrical units are known with considerable accuracy in the C.G.S. system, it is possible to measure energy in foot-pounds by means of a good commercial ammeter voltmeter and watch with greater ease and certainty than by any mechanical dynamometer.
Further, it has become easy to obtain as much electric power as is wanted for the experiment at a small cost, for the price of half a horse-power for ten minutes is only one-third of a penny, at 6d. per Board of Trade Unit. Hence there is not the practical objection to the electrical method that was so formidable when currents of 20 or 30 amperes could only be obtained by setting up a large battery of Grove or Bunsen cells. Indeed every properly organized physical laboratory is now provided with accumulators, from which a quarter or half a horse-power may be readily obtained for use in such experiments as those to be described; or, when accumulators are not available, power may be taken from the mains of one of the numerous electric-supply companies.
II. Design of the Apparatus.
The authors therefore set themselves to design an apparatus which, when used with a good commercial ammeter voltmeter, thermometer, and watch, would give the value of the mechanical equivalent of heat correct to one per cent. without any corrections having to be made even for the heat lost by radiation, convection, and conduction, and without any special manipulative skill being required on the part of the observer.
Broadly, the experiment consists in passing a known current through a resistance immersed in a known mass of water, and measuring the rise of temperature in a given time, and
*Communicated by the Physical Society: read November 23, 1894.
the average value of the P.D. between the terminals of the resistance.
It is evident that if the loss of heat during the experiment is to be small enough to be neglected in comparison with the quantity of heat generated, either the rise of temperature must be small, or must take place in a very short time, or the ratio of the cooling surface to the mass of the water heated must be small. As it is impracticable to reduce either the rise of temperature or the time to very small limits, and also to measure them with ease and accuracy, it is clear that the result is best obtained by using a large quantity of water, for the ratio of the surface area to the mass can then be made small enough to bring the error due to cooling within the required limits. But a large quantity of water necessarily involves the use of a large amount of electrical energy; and we thus arrive at the result that the accuracy attainable depends upon the amount of power at our disposal, and will be greater the greater the electric power that can be supplied.
In designing such an apparatus, then, the first thing to determine is the amount of electric power that can be used, and the details should then be arranged so as to get the least error in the result. In those cases in which an increased accuracy in one measurement involves a diminished accuracy in another, it is best to make the errors due to the two causes equal. For example, suppose that the time during which the electric energy is supplied is such that we can only measure it to one per cent., while the error due to cooling during the experiment is only per cent. We can clearly increase the accuracy of the result if we increase the time until the probable error in reading it is equal to the error due to cooling during that time, say 3 per cent. If we were still further to increase the time, the error due to cooling would increase and exceed per cent., and our result would therefore be less accurate. This equality of course does not apply to errors that are not interdependent, such as errors in reading volts and amperes each of these errors should independently be be made as small as possible.
The measurements to be made are as follows:
(a) The value of the constant current passed through the
(b) The average value of the P.D. between its terminals. (c) The mass of water heated, to which must be added the water-equivalent of the containing vessel, resistance-coil, and stirrer.
(d) The rise of temperature of the water.
(e) The time during which the current is passed. Phil. Mag. S. 5. Vol. 39. No. 237. Feb. 1895.
With the excellent electrical measuring-instruments now obtainable it is possible to measure either current or pressure with an accuracy much greater than one per cent.; indeed the Board of Trade undertake to measure them within one tenth part of one per cert. We can also measure the mass of water with considerable accuracy: any error, even a large one, made in determining the water-equivalent of the other bodies raised in temperature becomes of small consequence when the waterequivalent is added to the much larger and accurately measured mass. The measurements (a), (b), and (c), therefore, give us little trouble and do not affect the design of the apparatus.
The case of (d) and (e) is different. In order to measure a change of temperature by means of a thermometer with the accuracy required, either the change must be fairly large or the thermometer must be very sensitive; but as it is of little use to employ an exceedingly sensitive thermometer to measure the temperature of a liquid which is being locally and rapidly heated, even if the stirring is very efficient, it is necessary to use a fairly large rise of temperature. Similarly, if the circuit is to be closed and broken by hand, and the interval of time measured by an ordinary stop-watch such as would be found in a junior laboratory, there is a certain minimum time required to give the required accuracy of measurement. Also we must keep the ratio of heat lost to heat generated during the experiment equal to the probable error in the time or temperature-measurement; for as these three quantities are interdependent, the best condition is to make the percentage accuracy of the temperature measurement, the percentage accuracy of the time measurement, and the percentage heat lost of heat generated, equal.
Let W be the maximum number of watts at our disposal;
e the average emissivity of the cooling surface, or the
the minimum change of temperature that can be measured to 1 per cent. under the prescribed conditions ;
T the minimum time, in seconds, between closing and opening the switch that can be measured by the stop-watch to 1 per cent. ;
the rise of temperature of the water;
t the time during which the switch is closed;
* See Schedule to Final Report of Electrical Standards Committee, 1894.
and let per cent. be the maximum accuracy of result obtain
and the ratio of heat lost to heat received is 100
We also know, say from preliminary experiments, that the heat received in calories is about 0.24 time the energy in watt-seconds.
In equations (1) and (2) M and a are the only unknown quantities, for S is a function of M depending on the shape of the vessel containing the water. Thus for a cylindrical vessel of height equal to its diameter,
while for a spherical vessel,
We can therefore find both M and a from the equations, and since
It may be observed here that as S varies as M3, and a2
a is proportional to the 6th root of M or to the 6th root of W. Hence if the number of watts available is doubled the accuracy of the experiments is by no means doubled, but is only increased by about th.
In the particular case for which the apparatus was designed the number of watts available was about 300, the maximum current being 30 amperes. This determined the resistance of the coil or strip as of an ohm.
The average emissivity for small excess temperatures of a glass vessel standing on a felt base and containing water was obtained by taking cooling curves, the mean value being 0.000232 calorie per square centimetre of area per 1° C. excess temperature. The water was kept at a uniform temperature by means of a light wooden stirrer during these experiments.
, the minimum rise of temperature that can be measured to one per cent., was taken as 2.5 degrees, as it was not considered advisable to rely on the temperature measurements to more than of a degree. The thermometer used is read without the aid of a telescope, and is graduated in 20ths of a degree.
T, the minimum time that can be read to one per cent., was taken as a minute and a half, as it was thought that an error of nearly a second might be made in the measurement of the time between closing and opening the switch, and stopwatches are often a little doubtful as to their zero. From equation (2) we have at once
=2592 cubic centimetres.
Substituting in equation (1), we have
or the accuracy of measurement is 0.64 of one per cent. The rise of temperature adopted should therefore be 3.87 degrees, the time being 2 minutes 20 seconds. The numbers actually adopted in the experiments were 2000 cubic centim. of water and a time of 2 minutes; the smaller quantity of water being adopted because the resistance of the immersed strip when made proved to be rather less than ohm, the watts taken at
30 amperes being 260 instead of 300.
To ensure success in an experiment of this kind it is necessary that the water be as uniformly heated as possible, and that the stirring be very efficient; and after some consideration the authors decided that the best of all plans would be to use a movable conductor of considerable surface, and so shaped that it might itself be used to stir the liquid. By this means an exceedingly uniform rise of temperature may be produced, every particle of water receiving heat direct from the strip at practically the same rate. The following is a description of the apparatus as constructed by the authors.
A strip of manganin (chosen on account of its low tempera