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|English physicist James Joule's 1843 paddle wheel experiment, the best-known experimental determination of the mechanical equivalent of heat, wherein the work W of the falling weight was converted into a proportional amount of heat Q in the water measured by increase in degrees on the Celsius thermometer. ||A 50AD Alexandrian Hero-type aeolipile rigged to a pulley contraption so to be able to do mechanical work: namely the raising of a weight (mg) by a certain distance in height (h), during which process a certain quantity of heat Q is converted into a certain quantity of work W (mgh), in a certain proportionate ratio (J) of work to heat. |
where W is an amount of work expended in the production of heat Q (e.g. Thomson's cannon boring experiment, 1798) and or where W is an amount of work produced or obtained from heat Q (e.g. Papin's steam engine, 1690). The numerical value of J is independent of the manner in which the heat is produced by work (or work produced by heat).
This constant was first enunciated as a formal law of nature by German physicist Robert Mayer in 1842 who stated that the heat produced is proportional to he work done in its production; and conversely, when work is obtained from heat, the heat which disappears is in the same constant ratio to the work obtained.
The exact value of this constant was determined in 1843 by English physicist James Joule who found, through a number of all varieties of experiments, that an amount work W expended in the release of 778-pounds downward in gravitational height by one 1-foot results in an amount of heat generation or work-converted-into-heat quantified by a 1° F temperature increase in 1-pound of water. In unit form, Joules measurement is:
With the invention of standard SI units, in the 20th century, the value of J became defined as 4.186 10E7 ergs per calorie, and assigned a value of 1, after which all forms of energy became expressed in units of "joules". With this assignment of J=1, e.g. as was done by English chemist James Partington (1924), or possibly by someone (?) shortly before him, the original 19th century mechanical equivalent of heat ratio:
where the J is "assumed" or rather concealed in the proportion between the two. Hence, in SI notation, connection of this unit J=1, to the original 19th century experiments that derived this values becomes a bit camouflaged.
In its early formulation, the mechanical equivalent of heat went by the symbol a or A, depending on usage. In 1845, German mathematician Carl Holtzmann began to assign the letter "a" to the mechanical equivalent of heat and calculated values using methods similar to those of Mayer.  In his paper, Holtzmann stated:
“I call the unit of heat the heat which by its entrance into a gas can do the mechanical work a—that is, to use definite units, which can lift a kilograms through one meter.”
German physicist Rudolf Clausius, in 1850 memoir "On the Motive Power of Heat", begin to adopt this symbol use and terminology, defined as follows:
In this sense, it seems that A is defined as the inverse of J as defined in modern terms:
This detail, however, needs to be fact checked.
In event, the calculation of A was determined by a number of individuals, including: Benjamin Thomson (1798), Marc Séguin (1839), Robert Mayer (1842), Ludwig Colding (1843), James Joule (1843 to 1849), Carl Holtzmann (1845), and Gustave-Adolphe Hirn (1856).  The best known calculation was that performed by Joule in 1843 wherein the falling weight was attached to wound rope to a wooden paddle wheel immersed in a tub of water. When the weight fell, the paddle wheel turned, causing agitation in the water and as a result a temperature increase. 
In circa 1635, Italian physicist Giovanni Baliani showed that by placing an iron pot filled with water on a spinning metal disk it was possible to make water boil. This experiment is said to have been one of the first references to an experimental determination of the equivalence between heat and work. 
The earliest experiments showing that mechanical work could produce heat in a fixed ratio were the 1798 "cannon boring experiments" done by American-born English physicist Benjamin Thomson which showed that by continuously boring a cannon barrel with a dull drill bit, a seemingly unlimited supply of heat could be produced. Rumford published his results in a paper titled "An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction" presented to the Royal Society in 1798. Thomson found that the work of one horse during two and a half hours is sufficient to raise through 180° Fahrenheit 26.58 pounds of water. From this he calculated that one pound heated one degree is equivalent to 940 British units of work. 
In 1799, British chemist and physicist Humphry Davy conducted “ice-rubbing experiments”, where in a room colder than the freezing point of water, he generated heat or made ice melt by the mechanical rubbing of cubes together. By doing this, Davy demonstrated the conversion of work into heat and that indefinite amounts of heat could be generated from a body, this being contrary to caloric theory, which limits the amount. 
In 1841,German physician Robert Mayer sent the article “On the Quantitative and Qualitative Determination of Forces” to German physicist Johann Poggendorff, to be published in his Annalen der Physik and Chemie, in which he stated, among other things, that “motion is converted into heat”; but the paper was rejected. He followed this up with the 1842 paper “Remarks on the Forces of Inorganic Nature”, in which he gave the following figure for the mechanical equivalent of heat: 
“The warming of a given weight of water from 0˚C to 1˚C corresponds to the fall of an equal weight from the height of about 365 meters.”
In the early 1840s, English physicist James Joule repeated Davy’s ice-rubbing experiments and began to extrapolate this principle to various other work-producing experiments, such as chemical, mechanical, and electrical.
|A 2010 TutorVista.com video on James Joule's 1843 paddle wheel experimental calculation of the mechanical equivalent of heat.|
In 1843, Joule summarized his overall objective and theory by stating that:
“I shall lose no time in repeating and extending these experiments, being satisfied that the grand agents of nature are … indestructible; and that wherever mechanical force is expended, an exact equivalent of heat is always obtained.”
Joule's famous paddle wheel experiment was conducted in 1843.
In Joule's famous 1845 paper, entitled "On the Mechanical Equivalent of Heat" (or "On the Existence of an Equivalent Relation between Heat and the ordinary Forms of Mechanical Power"), he published the value A for the amount of work W required to produce a unit of heat Q. Joule contended that motion and heat were mutually interchangeable and that in every case, a given amount of work would generate the same amount of heat, regardless of the process. 
In 1843, Danish civil engineer and physicist Ludwig Colding stated that "the quantities of heat evolved are, in every case, proportional to the lost moving forces" though he did not calculate a mechanical equivalent of heat.  This postulate may have been based on his earlier 1839 experiments on the compressibility of water; later summarized with a review of other data on compression and friction of various materials. Into the late 1840s, in a series of quantitative experiments sponsored by the Royal Danish Academy of Sciences and Letters, Colding was able to obtain various values for the mechanical equivalent of heat. By 1850, Colding had obtained a value for the mechanical equivalent of heat, some 14% lower than the modern value (4.1860 J/cal) at a time when Joule had measured 4.159 J·cal-1.  A subsequent calculation by Colding in 1852 yielded a value only 3% below modern values.
In 1847, Joule measured the mechanical equivalent of heat on his honeymoon at Niagara Falls.
|2005 Oxford Dictionary of Science definition of the mechanical equivalent of heat. |
In 1856, French physicist Gustave-Adolphe Hirn’s conducted experiments in the determination of the mechanical equivalent of heat of a human being in working action. In particular, Hirn calculated a value for the mechanical equivalent of heat for a man doing work, i.e. running on a paddle-wheel like stair-climber treadmill, in a sealed chamber. To achieve this end, a man was placed in a hermetically closed chamber, and made to turn a wheel which could, at choice, revolve with or without doing work. The heat given out in the chamber was then ascertained by the ordinary calorimetric process. From these experiments, Hirn deduced a valuation of the mechanical equivalent of heat for animated motors; but the number which he obtained differed considerably from the standard obtained by Joule via physico-mechanical methods. 
In 1912, German physical chemist Otto Sackur, in his Thermochemistry and Thermodynamics (chapter three), gives a fairly good history and derivation of the mechanical equivalent of heat and its various calculations. 
Theorem of the equivalence of heat and work
In 1850, and over the next fifteen years, German physicist Rudolf Clausius began to use the mechanical equivalent of heat as a basis of his Mechanical Theory of Heat, which is considered the core book of modern thermodynamics. In his first article "On the Motive Power of Heat and on the Laws which can be Deduced from it for the Theory of Heat", Clausius begins by citing the mechanical equivalent of heat results of Holtzmann, Mayer, and Joule, but then then applies its logic to the heat-generating working action of the "working body" and the likely changes that result in the condition of the working body in a Carnot cycle. In this direction, by 1854 Clausius had enunciated what he called the theorem of the equivalence of heat and work as such: 
“Mechanical work may be transformed into heat, and conversely heat into work, the magnitude of the one being always proportional to that of the other.”
This logic was then molded, via a number of arguments, into the (a) the conservation of energy and (b) the equivalence-value of all uncompensated transformations (entropy); or what are commonly known as the first and second laws of thermodynamics, respectively.
Theorem of the equivalence of transformations
In German physicist Rudolf Clausius' 1854 fourth memoir "On a Modified form of the Second Fundamental Theorem in the Mechanical Theory of Heat", he introduced what he called the theorem of the equivalence of transformations, in which the concept of irreversibility is rooted, and hence entropy increase.
Switch to J = 1
The original units of the mechanical equivalent of heat, calculated by James Joule, in a va In unit form, Joules measurement is:
With the invention of standard SI units, in the 20th century, the value of J became defined as 4.186 10E7 ergs per calorie, and assigned a value of 1, after which all forms of energy became expressed in units of "joules".
An earlier "absolute system of units" (c.1912) value for the mechanical equivalent of heat often used was:
where the unit of work 1 erg, in SI units, equals:
In sum, the original circa 1840s sense of the mechanical equivalent of heat means that the work energy released when a one pound weight falls through a height of 778 feet can affect a temperature increase of one degree in a pound of water, through a number of means of energy conversion.
The transition of original 1843 definition units of J to CGI units to the modern SI units of J = 1 has a bit of a blurry history to it. The switch seems to have begun to come about in circa 1910 when the inconsistencies in defining the unit of energy in terms of heat capacities of water became a less compelling as compared to the ability to measure quantities of energy more accurately by electrical methods; according to which the derivation and calculation for the new unit for J has something to do with Ohm's law (1827). In the midst of this, supposedly, in 1930 the decision was made to define the calorie arbitrarily by the relation: 
4.1850 absolute joules ≡ 1 calorie
In 1924, English chemical thermodynamicist James Partington defined J as:
where A was a commonly symbol for work, from the German word arbeit, meaning 'work', a notation employed by those such as German physical chemist Walther Nernst (1917). Partington then states that if the SGS system (work is ergs and heat in degree calories) and another unit system (work in gram-centimeters and heat in gram calories) with have different values of J. He then jumps to the conclusion that "with suitable units for A and Q, we can make J = 1, in which case A = Q."  The details of this jump need to be tracked down?
In sum, to review the mechanical equivalent of heat a constant factor relating the calorie (the c.g.s. unit of heat) to the joule (the unit of mechanical energy), equal to 4.1868 joules per calorie. It is redundant in the SI system of units, which measures heat and all forms of energy in joules. 
The unit of “calorie” was first defined by Nicolas Clément in 1824 as a unit of heat, entering French and English dictionaries between 1841 and 1867.
At some point along the way, the calorie was replaced by the SI unit of energy, the joule. Hence, sometime along the way, the units and definition of the mechanical equivalent of heat was reformulated, in standardized unit sense, in some way or another.
It seems, according to some, that a standardized value of 4.1860 J/cal was established in the early-to-mid 20th century, in the 1920s, supposedly, it was ultimately realized that the constant is simply the specific heat of water, a quantity that varies with temperature between the values of 4.17 and 4.22 J/g·°C. The change in unit was the result of the demise of the calorie as a unit in physics and chemistry.
Since early days the calorie or kilocalorie has been used as a unit of energy. In some circles, however, it is realised that this cannot be continued indefinitely and that in due course the joule is to be substituted for the calorie as the unit of energy in all nutritional work. Calories should then fall into disuse.
In the International System of Units (Système International d'Unités) called S.I., there are 6 basic units as adopted in 1954: the meter (M) for length, the kilogramme (Kg.) for mass, the second (s) for time, the ampere (A) for electric current, the kelvin (K) for thermodynamic temperature and the candela (cd) for luminous intensity (1).
All the other units are derived from these 6 basic units such as the unit for force as the newton (N) (Kgm/s²), the unit of energy in any form is the joule (J) (Nm) and the unit of power as the watt (W) (J/s). The joule was adopted as the unit for electrical work, heat, mechanical work and energy in 1948 at the 9th General Conference on Weights and Measures, avoiding the calorie as far as possible. This unit was also formally approved in 1960 in the International System of Units (SI). Also the International Standards Organization (ISO) has recommended its adoption as the preferred unit for energy, as well as the USA National Bureau of Standards, the British Standards Institution and the Royal Society. 
1. Sackur, Otto. (1917). A Textbook of Thermo-chemistry and Thermodynamics (pg. 74). MacMillan.
2. McCulloch, Richard S. (1876). Treatise on the Mechanical Theory of Heat - and its Applications to the Steam-Engine, etc. New York: D. Van Nostrand Publishers.
3. Joule, James P. (1845). "On the Mechanical Equivalent of Heat", Brit. Assoc. Rep., trans. Chemical Sect, p.31, read before the British Association at Cambridge, June.
4. Milestones in Thermodynamics – Thermal Physics, University of Notre Dame.
5. (a) Mayer, J. Robert. (1842). “Remarks on the Forces of Inorganic Nature”. Annalender Chemie und Pharmacie, Justin von Liebigs’ journal.
(b) Caneva, Kenneth L. (1993). Robert Mayer and the Conservation of Energy, (index: “heat: mechanical equivalence of, pgs. 25, 27-28, 37-38, 194, 233-34, 252, 261-62). Princeton, New Jersey: Princeton University Press.
6. Colding, Ludwig A. (1843). "Nogle saetninger om kraefterne" ("Theses concerning force"), read at the Royal Danish Academy of Sciences and Letters, published as Colding (1856).
7. (a) Dahl, P.F. (1981). "Colding, Ludwig August" in Gillespie, C.C. (ed.) (1981). Dictionary of Scientific Biography, Supplement I, New York: Charles Screibner's Sons, 84-87.
(b) Joule, James P. (1850). Philosophical Transactions of the Royal Society of London 140(1): 61-82.
8. Holtzmann, Carl von. (1845). “Ueber die Warme und Elaslicitat der gase und Dampfe” (“On the Heat and elasticity of Gases and Vapours”), Mannheim: Taylor’s Scientific Memoirs, iv. 189; also Pogg. Ann., vol. 72a.
9. Marey, Étienne-Jules. (1973). La Machine Animale (Animal Mechanism: A Treatise on Terrestrial and Aërial Locomotion), (pg. 13-18). D. Appleton and Co.
10. Daintith, John. (2005). Oxford Dictionary of Science (pg. 510). New York: Oxford University Press.
11. (a) Clausius, Rudolf. (1850). "On the Motive Power of Heat, and on the Laws which may be deduced from it for the Theory of Heat", Poggendorff's Annalen der Physick, LXXIX, 368, 500.
(b) Clausius, Rudolf. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies. London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.
12. Giovanni Battista Baliani (Italian → English) – Wikipedia. 13. Aeolipile – Wikipedia.
14. Engraving of Joule's apparatus for measuring the mechanical equivalent of heat from the August 1869 issue of Harper's New Monthly Magazine, No. 231.
15. The Adoption of Joules as Units of Energy – FAO.org.
16. Mechanical equivalent of heat – Hutchinson’s Encyclopedia.
17. Partington, James R. (1924). Chemical Thermodynamics: An Introduction to General Thermodynamics and its Applications to Chemistry. D. Van Nostrand.
18. Glasstone, Samuel B. (1946). Thermodynamics for Chemists (J=1 etymology, pg. 440). D. Van Nostrand Co.
● Venkataraman, G. (1994). A Hot Story (pg. 8). Orient Blackswan.
● Mechanical equivalent of heat – Wikipedia.
● On the Mechanical Equivalent of Heat (historical) – Encyclopedia of Earth.