Mechanical Equivalents

Joule made the crucial addition of mechanical effects to his system of equivalences
by following a time-honored route to scientific discovery: he made a fortunate mistake. In the fourth of his electrochemistry papers he reported electric
potential data (voltages, in modern units) measured on voltaic cells whose
electrode reactions produced oxidation of zinc and other metals. He believed,
mistakenly, that these reaction potentials could be used in much the same way
as reaction heats: that for a given reaction the potential had the same value no
matter how the reaction was carried out. This interpretation is not sanctioned by
modern thermodynamics unless cell potentials are measured carefully (reversibly).
Joule and his contemporaries were unaware of this limitation, however, and
the mistake led Joule to calculate electrical and thermal equivalents for the process
in which dissolved oxygen is given “its elastic condition,” the reaction
O2 (solution) O2 (gas).
Joule’s result was an order of magnitude too large. But mistaken as it was quantitatively,
the calculation advanced Joule’s conceptual understanding immensely,
because he believed he had obtained electrical and thermal equivalents for a
mechanical effect, the evolution of oxygen gas from solution. In Joule’s fertile
imagination, this was suggestive. In the fourth electrochemistry paper, he remarked
that he had already thought of ways to measure mechanical equivalents.
He hoped to confirm the conclusion that “the mechanical and heating powers of
a current are proportional to each other.”
In this serendipitous way, Joule began the determinations of the mechanical
equivalent of heat for which he is best known today. The first experiments in
this grand series were performed in 1843, when Joule was twenty-four. In these
initial experiments, he induced an electrical current in a coil of wire by rotating
it mechanically in a strong magnetic field. The coil was contained in a glass tube
filled with water and surrounded by insulation, so any heating in the coil could
be measured by inserting a thermometer in the tube before and after rotating it
in the magnetic field. The induced current in the coil was measured by connecting
the coil to an external circuit containing a galvanometer. Although its
origin was entirely different, the induced current behaved the same way as the
voltaic current Joule had studied earlier: in both cases the current caused heating
that followed the I 2R-law.
In the final experiments of this design, the wheel of the induction device was
driven by falling weights for which the mechanical effect, measured as a mechanical
workcalculation, could be made directly in foot-pounds (abbreviated
ft-lb): one unit was equivalent to the workrequired to raise one pound one foot.
Heat was measured by a unit that fit the temperature measurements: one unit
raised the temperature of one pound of water 1 Fahrenheit (F). We will use the
term later attached to this unit, “British thermal unit,” or Btu.
In one experiment, Joule dropped weights amounting to 4 lb 12 oz ( 4.75 lb)
517 feet (the weights were raised and dropped many times), causing a temperature
rise of 2.46 F. He converted the weight of the glass tube, wire coil, and water
in which the temperature rise occurred all into a thermally equivalent weight of
water, 1.114 lb. Thus the heating effect was 2.46 F in 1.114 lb of water. If this
same amount of heat had been generated in 1 lb of water, the heating effect would
have been . Joule concluded that in this case (517)(4.75)
(2.46)(1.114)
2.74 F
1
ft-lb was equivalent to 2.74 Btu. He usually determined the mechanical workequivalent to 1 Btu. That number, which Thomson later labeled J to honor Joule,
was
(4.75)(517)
J 896 ft-lb per Btu
(2.74)
for this experiment. This was one determination of the mechanical equivalent of
heat. Joule did thirteen experiments of this kind and obtained results ranging
from J 587 to 1040 ft-lb per Btu, for which he reported an average value of 838
ft-lb per Btu. The modern “correct” value, it should be noted, is J 778 ft-lb per
Btu.
If the 27% precision achieved by Joule in these experiments does not seem
impressive, one can sympathize with Joule’s critics, who could not believe his
claims concerning the mechanical equivalent of heat. But the measurements Joule
was attempting set new standards for experimental difficulty. According to Reynolds,
the 1843 paper reported experiments that were more demanding than any
previously attempted by a physicist.
In any case, Joule was soon able to do much better. In 1845, he reported another,
much different determination of the mechanical equivalent of heat, which
agreed surprisingly well with his earlier measurement. In this second series of
experiments, he measured temperature changes, and calculated the heat produced,
when air was compressed. From the known physical behavior of gases
he could calculate the corresponding mechanical effect as workdone on the air
during the compression.
In one experiment involving compression of air, Joule calculated the workat
11230 ft-lb and a heating effect of 13.628 Btu from a measured temperature rise
of 0.344 F. The corresponding mechanical equivalent of heat was
11230
J 824 ft-lb per Btu.
13.628
Another experiment done the same way, in which Joule measured the temperature
change 0.128 F, gave the result J 796 ft-lb per Btu. Joule’s average for the
two experiments was 810 ft-lb per Btu. This was in impressive, if somewhat
fortuitous, agreement with the result J 838 ft-lb per Btu reported in 1843.
Joule also allowed compressed air to expand and do workagainst atmospheric
pressure. Temperature measurements were again made, this time with a temperature
decrease being measured. In one of these expansion experiments, Joule
measured the temperature change 0.1738 F and reduced this to 4.085 Btu. The
corresponding workcalculation gave 3357 ft-lb, so
3357
J 822 ft-lb per Btu.
4.085
Joule did two more experiments of this kind and measured the temperature
changes 0.081 F and 0.0855 F, giving J 814 and J 760 ft-lb per Btu.
When Joule’s colleagues looked at these results, the first thing they noticed
was the accuracy claimed for measurements of very small temperature changes.
In Joule’s time, accurate measurement of one-degree temperature changes was difficult enough. Joule reported temperature changes of tenths of a degree with
three or four significant digits, and based his conclusions on such tiny changes.
As William Thomson remarked, “Joule had nothing but hundredths of a degree
to prove his case by.” Yet, most of Joule’s claims were justified. He made temperature
measurements with mercury thermometers of unprecedented sensitivity
and accuracy. He told the story of the thermometers in an autobiographical note:
“It was needful in these experiments to use thermometers of greater exactness
and delicacy than any that could be purchased at that time. I therefore determined
to get some calibrated on purpose after the manner they had been by
Regnault. In this I was ably seconded by Mr. Dancer [J. B. Dancer, a well-known
Manchester instrument maker], at whose workshop I attended every morning for
some time until we completed the first accurate thermometers which were ever
made in England.”
Joule demonstrated the heat-mechanical-workequivalence with a third gas
expansion experiment that incorporated one of his most ingenious experimental
designs. In this experiment, two constant-volume copper vessels, one evacuated
and the other pressurized with air, were connected with a valve. The connected
vessels were placed in a calorimeter, the valve opened, and the usual temperature
measurements made. In this case, Joule could detect no net temperature change.
Air expanding from the pressurized vessel was cooled slightly, and air flowing
into the evacuated vessel was slightly heated, but no net temperature change was
observed.
This was what Joule expected. Because the combined system consisting of the
two connected vessels was closed and had a fixed volume, all of the workwas
done internally, in tandem between the two vessels. Workdone by the gas in one
vessel was balanced by workdone on the gas in the other; no net workwas done.
Heat equivalent to zero workwas also zero, so Joule’s concept of heatmechanical-
workequivalence demanded that the experiment produce no net
thermal effect, as he observed.
The next stage in Joule’s relentless pursuit of an accurate value for the mechanical
equivalent of heat, which he had begun in 1847, was several series of
experiments in which he measured heat generated by various frictional processes.
The frictional effects were produced in a water-, mercury-, or oil-filled
calorimeter by stirring with a paddle-wheel device, the latter being driven by
falling weights, as in the 1843 experiments. The workdone by the weights was
converted directly by the paddle-wheel stirrer into heat, which could be measured
on a thermometer in the calorimeter.
Of all Joule’s inventions, this experimental design, which has become the bestknown
monument to his genius, made the simplest and most direct demonstration
of the heat-mechanical-workequivalence. This was the Joule technique reduced
to its essentials. No complicated induction apparatus was needed, no
calculational approximations, just falling weights and one of Joule’s amazingly
accurate thermometers.
With the paddle-wheel device and water as the calorimeter liquid, Joule obtained
J 773.64 ft-lb per Btu from a temperature rise of 0.563 F. Using mercury
in the calorimeter, he obtained J 773.762 and 776.303 ft-lb per Btu. In two
further series of experiments, Joule arranged his apparatus so the falling weights
caused two cast-iron rings to rub against each other in a mercury-filled calorimeter;
the results J 776.997 and 774.880 ft-lb per Btu were obtained.
Joule described his paddle-wheel experiments in 1847 at an Oxford meeting of the British Association for the Advancement of Science. Because his previous
papers had aroused little interest, he was asked to make his presentation as brief
as possible. “This I endeavored to do,” Joule recalled later, “and a discussion not
being invited the communication would have passed without comment if a
young man had not risen in the section, and by his intelligent observations created
a lively interest in the new theory.”
The silence was finally broken. The young man was William Thomson, recently
installed as professor of natural philosophy at Glasgow University. Thomson
had reservations about Joule’s work, but he also recognized that it could not
be ignored. “Joule is, I am sure, wrong in many of his ideas,” Thomson wrote to
his father, “but he seems to have discovered some facts of extreme importance,
as for instance, that heat is developed by the friction of fluids.” Thomson recalled
in 1882 that “Joule’s paper at the Oxford meeting made a great sensation. Faraday
was there, and was much struckby it, but did not enter fully into the new views.
. . . It was not long after when Stokes told me he was inclined to be a Joulite.”
George Stokes was another rising young physicist and mathematician, in 1847 a
fellow at Pembroke College, Cambridge, and in two years to be appointed Lucasian
Professor of Mathematics, the chair once occupied by Newton.
During the three years following the Oxford meeting, Joule rose from obscurity
to a prominent position in the British scientific establishment. Recognition came
first from Europe: a major French journal, Comptes Rendu, published a short
account of the paddle-wheel experiments in 1847, and in 1848 Joule was elected
a corresponding member of the Royal Academy of Sciences at Turin. Only two
other British scientists, Faraday and William Herschel, had been honored by the
Turin Academy. In 1850, when he was thirty-one, Joule received the badge of
British scientific acceptance: election as a fellow of the Royal Society.
After these eventful years, Joule’s main research effort was a lengthy collaboration
with Thomson, focusing on the behavior of expanding gases. This was one
of the first collaborative efforts in history in which the talents of a theorist and
those of an experimentalist were successfully and happily united.