Now experimental evidence had been accumulating – from that furnished by the work of Benjamin Thomson (Count Rumford) and Sir Humphrey Davy just before 1800 to that of Robert Mayer and James Prescott Joule in the 1840's -- that heat and mechanical work are simply two different manifestations of the same entity, that both are forms of energy; and that when one form is transformed into the other the ratio of the amounts so converted is in all cases the same. This statement is an expression of what is known as the First Law of Thermodynamics. When heat energy is supplied to a body from without, it in general expands (doing work) and grows hotter (increasing its "internal", or "intrinsic" energy).
Thus if these three forms of energy are expressed in the same unit and increases in each are reckoned as positive, by the first law for any transformation of energy in either direction the resulting change in intrinsic energy is the difference between the heat energy supplied the body and the work it performs. Where no heat energy is supplied the work is performed at the expense of the intrinsic energy and the stock of mechanical energy external and internal is unchanged or conserved. When a body is put through a cyclic process which returns it periodically to its initial state, there can be no change in its intrinsic energy and hence the net heat energy supplied must equal the work performed; that is, its whole energy in both forms is unchanged or conserved. Hence the first law may be regarded as an expression of the general principle of the "conservation of energy," valid for transformations between the two forms, heat and mechanical energy.
With the final triumph of the first law in the late 1840's, it be came necessary
to correct Clapeyron's mistaken derivation of the Carnot function. Instead
of the relation assumed from the hydraulic analogy that the amounts of heat
energy received and rejected by working substance are equal, the first law
requires that their difference be the thermal equivalent of the external work
done, or
W= H – h
On the basis of this relation Thomson showed (in 1848) that by choosing a series of heat reservoirs of uniformly descending temperatures and by supposing that reversible engines each doing the same amount of external work are operated between them in such a manner that the heat rejected by one becomes the heat received by the next in the series, we would ultimately arrive at a point where there would no longer remain any heat to be rejected. The temperature of this last reservoir would then be the lowest conceivable -- the absolute zero of temperature.
A scale of temperatures thus based on the amounts of work performed by such a series of reversible engines then leads to the reciprocal of the temperature of the hottest reservoir as the value of Carnot's function, and yields for the efficiency of a reversible engine the expression
Work done/Heat received = (Heat rejected-Heat received)/Heat received
Work done/Heat received = Trej-Trec/Trec;
where Trec - Temp. of reception
Trej - Temp of rejection
The scale of temperature thus specified is variously called the "work," the "absolute," the "thermodynamic," or the "Kelvin" scale.
In what follows a temperature is always to be understood as one measured on this scale whose "zero point" is some 273 degrees below that of the centigrade or about 460 degrees below that of the Fahrenheit scale. Thus it was twenty-four years after the enunciation of Carnot's principle that it received its exact quantitative expression, and another two years before it can be said to have been placed upon its final firm foundation by Clausius' formulation of a precise statement of the second law.
The further development of the fundamental theory consists in the interpretation of the equation just given and in the combination of the expressions for the two laws into a single equation. It was from the alternative form of the above expression for the efficiency of a reversible engine,
Heat rejected/Temp. of rejection = Heat received/Temp.of reception
that Clausius derived the concept of "entropy." This quantity is one of the characteristics specifying the state of a substance, as do its temperature or pressure, its volume or intrinsic energy. It may be defined as the ratio of the heat energy taken in or given out in reversible changes of the condition of a body to the absolute temperature at which the change takes place. Another definition is possibly more instructive.
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