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Joule, and others, continued in other experiments, noting the pressure changes caused by electrically-heated gases, and achieved similar results. This in turn led to a number of other scientists to research in this area, among them the German physicists Rudolf Clausius and Hermann von Helmholtz in the 1840's. This led to the general acceptance of [[Conservation of Energy]] as a clear and precise principle. Clausius stated in 1850:
Joule, and others, continued in other experiments, noting the pressure changes caused by electrically-heated gases, and achieved similar results. This in turn led to a number of other scientists to research in this area, among them the German physicists Rudolf Clausius and Hermann von Helmholtz in the 1840's. This led to the general acceptance of [[Conservation of Energy]] as a clear and precise principle. Clausius stated in 1850:
:''"In any process, energy can be changed from one form to another, but it is never created or destroyed."''
:''"In any process, energy can be changed from one form to another, but it is never created or destroyed."''
This can be expressed mathematically as
<math>
\,dU = \delta Q + \delta W
</math>
where <math>\,dU</math> is the infinitesimal change in internal energy, <math>\delta Q</math> is the heat that flows into the system and <math>\delta W</math> is the work done on the system.
This principle notably included heat as a form of energy, which was the first law of thermodynamics. Hot objects contain potential energy in the form of their heat, and all the usual rules of transformation between potential and kinetic energy apply. When an electric current is passed through a resistor, the electrical energy is converted to heat energy, and the resistor gets hotter. Everything seemed to work out accurately.
This principle notably included heat as a form of energy, which was the first law of thermodynamics. Hot objects contain potential energy in the form of their heat, and all the usual rules of transformation between potential and kinetic energy apply. When an electric current is passed through a resistor, the electrical energy is converted to heat energy, and the resistor gets hotter. Everything seemed to work out accurately.
Of course, in all experiments that attempt to tally energy (or other properties) accurately, one must be careful to avoid outside interference. Newton's formula <math>F = ma\,</math> requires that no unaccounted-for forces are acting on the object. Similarly, in thermodynamics, one must take into account any possible sources of heat into or out of the entity under test. This is often described by saying that the law of thermodynamics apply only to "isolated", or "closed", systems. If a system can interact with some external entity, that entity's properties must be taken into account.
Of course, in all experiments that attempt to tally energy (or other properties) accurately, one must be careful to avoid outside interference. Newton's formula <math>F = ma\,</math> requires that no unaccounted-for forces are acting on the object. Similarly, in thermodynamics, one must take into account any possible sources of heat into or out of the entity under test. This is often described by saying that the law of thermodynamics apply only to "isolated", or "closed", systems. If a system can interact with some external entity, that entity's properties must be taken into account.
Scientists had mostly figured out a linear relationship between temperature rise and change in heat energy. This is called the ''specific heat'', in joules per degree of temperature rise per gram of substance. For example, the specific heat of water is 4.2 joules per gram per degree of temperature rise.
Scientists had mostly figured out a linear relationship between temperature rise and change in heat energy. This is called the ''specific heat'', in joules per degree of temperature rise per gram of substance. For example, the specific heat of water is 4.2 joules per gram per degree Celsius of temperature rise.
Heat engines (that is, steam engines) were being used on an industrial scale by then, but scientists still didn't know what temperature really meant. And they didn't know why it only flows "downhill". Also, while other forms of energy (as in running electricity through a resistor) could be converted to heat with essentially perfect efficiency, converting the other way (as in a steam engine) was very inefficient. No one knew why.
Heat engines (that is, steam engines) were being used on an industrial scale by then, but scientists still didn't know what temperature really meant. And they didn't know why it only flows "downhill". Also, while other forms of energy (as in running electricity through a resistor) could be converted to heat with essentially perfect efficiency, converting the other way (as in a steam engine) was very inefficient. No one knew why.
The clues that unraveled this mystery came from the study of gases, which had been going on for some time before. [[Boyle's Law]], formulated in the 1660's, stated that, for a given sample of gas at a fixed temperature, the pressure was inversely proportional to the volume. That is,
The clues that unravelled this mystery came from the study of gases, which had been going on for some time before. [[Boyle's Law]], formulated in the 1660's, stated that, for a given sample of gas at a fixed temperature, the pressure was inversely proportional to the volume. That is,
:<math>PV = constant</math> where the constant depends on the amount and type of the gas sample
:<math>PV = constant</math>
where the constant depends on the amount and type of the gas sample
[[Charles' Law]], formulated in the 1780's, stated that, for a given sample of gas at a fixed pressure, the volume was directly proportional the the "absolute" temperature. That is,
[[Charles' Law]], formulated in the 1780's, stated that, for a given sample of gas at a fixed pressure, the volume was directly proportional the the "absolute" temperature. That is,
:<math>V/T = constant</math> where the constant depends on the amount and type of the gas sample
:<math>V/T = constant</math>
where the constant depends on the amount and type of the gas sample
This required that the temperature scale be modified. The necessary scale was known as absolute temperature, now known as [[International System of Units#Kelvin|Kelvin scale]]. All thermodynamic measurements are in Kelvins.
This required that the temperature scale be modified. The necessary scale was known as absolute temperature, now known as [[International System of Units#Kelvin|Kelvin scale]]. All thermodynamic measurements are in Kelvins.