Chapter III: The Principle of Energy in Physics

We can know only from _experience_ that mechanical processes produce other physical transformations, or _vice versa_. The attention was first directed to the connexion of mechanical processes, especially the performance of work, with changes of thermal conditions by the invention of the steam-engine, and by its great technical importance. Technical interests and the need of scientific lucidity meeting in the mind of S. Carnot led to the remarkable development from which thermodynamics flowed. It is simply _an accident of history_ that the development in question was not connected with the practical applications of _electricity_.

In the determination of the maximum quantity of _work_ that, generally, a heat-machine, or, to take a special case, a steam-engine, can perform with the expenditure of a _given_ amount of heat of combustion, Carnot is guided by mechanical analogies. A body can do work on being heated, by expanding under pressure. But to do this the body must receive heat from a _hotter_ body. Heat, therefore, to do work, must pass from a hotter body to a colder body, just as water must fall from a higher level to a lower level to put a mill-wheel in motion. Differences of temperature, accordingly, represent forces able to do work exactly as do differences of height in heavy bodies. Carnot pictures to himself an ideal process in which no heat flows away unused, that is, without doing work. With a given expenditure of heat, accordingly, this process furnishes the maximum of work. An analogue of the process would be a mill-wheel which scooping its water out of a higher level would slowly carry it to a lower level without the loss of a drop. A peculiar property of the process is, that with the expenditure of the same work the water can be raised again exactly to its original level. This property of _reversibility_ is also shared by the process of Carnot. His process also can be reversed by the expenditure of the same amount of work, and the heat again brought back to its original temperature level.

Suppose, now, we had _two_ different reversible processes _A_, _B_, such that in _A_ a quantity of heat, _Q_, flowing off from the temperature _t₁_ to the lower temperature _t₂_ should perform the work _W_, but in _B_ under the same circumstances it should perform a greater quantity of work _W_ + _W'_; then, we could join _B_ in the sense assigned and _A_ in the reverse sense into a _single_ process. Here _A_ would reverse the transformation of heat produced by _B_ and would leave a surplus of work _W'_, produced, so to speak, from nothing. The combination would present a perpetual motion.

With the feeling, now, that it makes little difference whether the mechanical laws are broken directly or indirectly (by processes of heat), and convinced of the existence of a _universal_ law-ruled connexion of nature, Carnot here excludes for the first time from the province of _general_ physics the possibility of a perpetual motion. _But it follows, then, that the quantity of work W, produced by the passage of a quantity of heat Q from a temperature t₁ to a temperature t₂, is independent of the nature of the substances as also of the character of the process, so far as that is unaccompanied by loss, but is wholly dependent upon the temperature t₁, t₂.

This important principle has been fully confirmed by the special researches of Carnot himself (1824), of Clapeyron (1834), and of Sir William Thomson (1849), now Lord Kelvin. The principle was reached _without any assumption whatever_ concerning the nature of heat, simply by the exclusion of a perpetual motion. Carnot, it is true, was an adherent of the theory of Black, according to which the sum-total of the quantity of heat in the world is constant, but so far as his investigations have been hitherto considered the decision on this point is of no consequence. Carnot's principle led to the most remarkable results. W. Thomson (1848) founded upon it the ingenious idea of an "absolute" scale of temperature. James Thomson (1849) conceived a Carnot process to take place with water freezing under pressure and, therefore, performing work. He discovered, thus, that the freezing point is lowered 0·0075° Celsius by every additional atmosphere of pressure. This is mentioned merely as an example.

About twenty years after the publication of Carnot's book a further advance was made by J. R. Mayer and J. P. Joule. Mayer, while engaged as a physician in the service of the Dutch, observed, during a process of bleeding in Java, an unusual redness of the venous blood. In agreement with Liebig's theory of animal heat he connected this fact with the diminished loss of heat in warmer climates, and with the diminished expenditure of organic combustibles. The total expenditure of heat of a man at rest must be equal to the total heat of combustion. But since _all_ organic actions, even the mechanical actions, must be set down to the credit of the heat of combustion, some connexion must exist between mechanical work and expenditure of heat.

Joule started from quite similar convictions concerning the galvanic battery. A heat of association equivalent to the consumption of the zinc can be made to appear in the galvanic cell. If a current is set up, a part of this heat appears in the conductor of the current. The interposition of an apparatus for the decomposition of water causes a part of this heat to disappear, which on the burning of the explosive gas formed, is reproduced. If the current runs an electromotor, a portion of the heat again disappears, which, on the consumption of the work by friction, again makes its appearance. Accordingly, both the heat produced and the work produced, appeared to Joule also as connected with the consumption of material. The thought was therefore present, both to Mayer and to Joule, of regarding heat and work as equivalent quantities, so connected with each other that what is lost in one form universally appears in another. The result of this was a _substantial_ conception of heat and of work, and _ultimately a substantial conception of energy_. Here every physical change of condition is regarded as energy, the destruction of which generates work or equivalent heat. An electric charge, for example, is energy.

In 1842 Mayer had calculated from the physical constants then universally accepted that by the disappearance of one kilogramme-calorie 365 kilogramme-metres of work could be performed, and _vice versa_. Joule, on the other hand, by a long series of delicate and varied experiments beginning in 1843 ultimately determined the mechanical equivalent of the kilogramme-calorie, more exactly, as 425 kilogramme-metres.

If we estimate every change of physical condition by the _mechanical work_ which can be performed upon the _disappearance_ of that condition, and call this measure _energy_, then we can measure all physical changes of condition, no matter how different they may be, with the same common measure, and say: _the sum-total of all energy remains constant_. This is the form that the principle of excluded perpetual motion received at the hands of Mayer, Joule, Helmholtz, and W. Thomson in its extension to the whole domain of physics.

After it had been proved that heat must _disappear_ if mechanical work was to be done at its expense, Carnot's principle could no longer be regarded as a complete expression of the facts. Its improved form was first given, in 1850, by Clausius, whom Thomson followed in 1851. It runs thus: "If a quantity of heat _Q'_ is transformed into work in a reversible process, _another_ quantity of heat _Q_ of the absolute[55] temperature _T₁_ is lowered to the absolute temperature _T₂_." Here _Q'_ is dependent only on _Q_, _T₁_, _T₂_, but is independent of the substances used and of the character of the process, so far as that is unaccompanied by loss. Owing to this last fact, it is sufficient to find the relation which obtains for some one well-known physical substance, say a gas, and some definite simple process. The relation found will be the one that holds generally. We get, thus,

_Q'/(Q' + Q) = (T₁-T₂)/T₁_ (1)

that is, the quotient of the available heat _Q'_ transformed into work divided by the sum of the transformed and transferred heats (the total sum used), the so-called _economical coefficient_ of the process, is,

_(T₁-T₂)/T₁_.