# Conservation of energy

Conservation of energy is possibly the most important, and certainly the most practically useful of several conservation laws in physics.

The law states that the total inflow of energy into a system must equal the total outflow of energy from the system, plus the change in the energy contained within the system. In other words, energy can be converted from one form to another, but it cannot be created or destroyed.

In thermodynamics, the first law of thermodynamics is a statement of the conservation of energy for thermodynamic systems.

The law of conservation of energy excludes the possibility of perpetuum mobile of the first kind.

## Historical development

Although ancient philosophers as far back as Thales of Miletus had inklings of the first law, it was the German Gottfried Wilhelm Leibniz during 1676-1689 who first attempted a mathematical formulation. Leibniz noticed that in many mechanical systems (of several masses, mi each with velocity vi) the quantity:

${\displaystyle \sum _{i}m_{i}v_{i}^{2}}$

was conserved. He called this quantity the vis viva or living force of the system. The principle represents an accurate statement of the approximate conservation of kinetic energy in many situations. However, many physicists were influenced by the prestige of Sir Isaac Newton in England and of René Descartes in France, both of whom had set great store by the conservation of momentum as a guiding principle. Thus the momentum:

${\displaystyle \,\!\sum _{i}m_{i}v_{i}}$

was held by the rival camp to be the conserved vis viva. It was largely engineers such as John Smeaton, Peter Ewart, Karl Hotzmann, Gustave-Adolphe Hirn and Marc Séguin who objected that conservation of momentum alone was not adequate for practical calculation and who made use of Leibniz's principle. The principle was also championed by some chemists such as William Hyde Wollaston.

Members of the academic establishment such as John Playfair were quick to point out that kinetic energy is clearly not conserved. This is obvious to a modern analysis based on the second law of thermodynamics but in the 18th and 19th centuries, the fate of the lost energy was still unknown. Gradually it came to be suspected that the heat inevitably generated by motion was another form of vis viva. In 1783, Antoine Lavoisier and Pierre-Simon Laplace reviewed the two competing theories of vis viva and caloric[1]. Count Rumford's 1798 observations of heat generation during the boring of cannons added more weight to the view that mechanical motion could be converted into heat. Vis viva now started to be known as energy, after the term was first used in that sense by Thomas Young in 1807.

The recalibration of vis visa to

${\displaystyle {\frac {1}{2}}\sum _{i}m_{i}v_{i}^{2}}$

was largely the result of the work of Gaspard-Gustave Coriolis and Jean-Victor Poncelet over the period 1819-1839. The former called the quantity quantité de travail and the latter, travail mécanique and both championed its use in engineering calculation.

In a paper Uber die Natur der Warme, published in the Zeitschrift für Physik in 1837, Karl Friedrich Mohr gave one of the earliest general statements of the doctrine of the conservation of energy in the words: "besides the 54 known chemical elements there is in the physical world one agent only, and this is called Kraft [energy]. It may appear, according to circumstances, as motion, chemical affinity, cohesion, electricity, light and magnetism; and from any one of these forms it can be transformed into any of the others."

A key stage in the development of the modern conservation principle was the demonstration of the mechanical equivalent of heat. The caloric theory maintained that heat could neither be created nor destroyed but conservation of energy entails the contrary principle that heat and mechanical work are interchangeable.

The mechanical equivalence principle was first stated in its modern form by the German surgeon Julius Robert von Mayer.[2] Mayer reached his conclusion on a voyage to the Dutch East Indies, where he found that his patients' blood was a deeper red because they were consuming less oxygen, and therefore less energy, to maintain their body temperature in the hotter climate. He had discovered that heat and mechanical work were both forms of energy, and later, after improving his knowledge of physics, he calculated a quantitative relationship between them.

File:Joule apparatus.png
Joule's apparatus for measuring the mechanical equivalent of heat

Meanwhile, in 1843 James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. In the most famous, now called the "Joule apparatus", a descending weight attached to a string caused a paddle immersed in water to rotate. He showed that the gravitational potential energy lost by the weight in descending was equal to the thermal energy (heat) gained by the water by friction with the paddle.

Over the period 1840-1843, similar work was carried out by engineer Ludwig A. Colding though it was little-known outside his native Denmark.

Both Joule's and Mayer's work suffered from resistance and neglect but it was Joule's that, perhaps unjustly, eventually drew the wider recognition.

For the dispute between Joule and Mayer over priority, see Mechanical equivalent of heat: Priority

Drawing on the earlier work of Joule, Sadi Carnot and Émile Clapeyron, in 1847, Hermann von Helmholtz postulated a relationship between mechanics, heat, light, electricity and magnetism by treating them all as manifestations of a single force (energy in modern terms). He published his theories in his book Über die Erhaltung der Kraft (On the Conservation of Force, 1847). The general modern acceptance of the principle stems from this publication.

In 1877, Peter Guthrie Tait claimed that the principle originated with Sir Isaac Newton, based on a creative reading of propositions 40 and 41 of the Philosophiae Naturalis Principia Mathematica. This is now generally regarded as nothing more than an example of Whig history.

## Modern physics

With the discovery of special relativity by Albert Einstein, it was found that the energy is one component of an energy-momentum 4-vector. The energy component is not conserved, nor is the 3-vector momentum conserved, but rather the energy-momentum 4-vector is conserved. Only for situations in which velocities are much smaller than the speed of light are the energy and momentum 3-vector separately conserved. In addition, the relativistic energy of a massive particle contains a rest mass in addition to its kinetic energy of motion. This rest mass can sometimes be converted into kinetic energy via the famous equation ${\displaystyle E=mc^{2}}$ so that the rule of conservation of energy was shown to be a special case of a more general rule, the conservation of mass and energy, which is now usually just referred to as conservation of energy.

Conservation of energy can be shown through Noether's theorem to be the result of the time-invariance of the laws of physics (=time has no effect on any physical process).

Within the realm of quantum mechanics, conservation of energy is not applicable when energy can not be defined (say, for time scales shorter than the uncertainty principle defines).

## The first law of thermodynamics

Main article: First law of thermodynamics

For a thermodynamic system with a fixed number of particles, the first law of thermodynamics may be stated as:

${\displaystyle \delta Q=dU+\delta W\,}$

where ${\displaystyle \delta Q}$ is the amount of energy added to the system by a heating process, ${\displaystyle \delta W}$ is the amount of energy lost by the system due to work done by the system on its surroundings and ${\displaystyle dU}$ is the increase in the internal energy of the system.

The δ's before the heat and work terms are used to indicate that they describe an increment of energy which is to be interpreted somewhat differently than the dU increment of internal energy. Work and heat are processes which add or subtract energy, while the internal energy U is a particular form of energy associated with the system. Thus the term "heat energy" for ${\displaystyle \delta Q}$ means "that amount of energy added as the result of heating" rather than referring to a particular form of energy. Likewise, the term "work energy" for ${\displaystyle \delta w}$ means "that amount of energy lost as the result of work". The most significant result of this distinction is the fact that one can clearly state the amount of internal energy posessed by a thermodynamic system, but one cannot tell how much energy has flowed into or out of the system as a result of its being heated or cooled, nor as the result of work being performed on or by the system.

The first law can be written exclusively in terms of system variables. The work done by the system may be written

${\displaystyle \delta w=P\,dV}$

where P is the pressure and dV is a small change in the volume of the system, each of which are system variables. The heat energy may be written

${\displaystyle \delta q=T\,dS}$

where T is the temperature and dS is a small change in the entropy of the system. Temperature and entropy are also system variables.

## Notes

1. ^  Lavoisier, A.L. & Laplace, P.S. (1780) "Memoir on Heat", Académie Royal des Sciences pp4-355
2. ^  von Mayer, J.R. (1842) "Remarks on the forces of inorganic nature" in Annalen der Chemie und Pharmacie, 43, 233

## References

### Modern accounts

• Kroemer, Herbert; Kittle, Charles (1980). Thermal Physics (2nd ed.), W. H. Freeman Company. ISBN 0716710889.
• Nolan, Peter J. (1996). Fundamentals of College Physics, 2nd ed., William C. Brown Publishers.
• Oxtoby & Nachtrieb (1996). Principles of Modern Chemistry, 3rd ed., Saunders College Publishing.
• Papineau, D. (2002). Thinking about Consciousness, Oxford University Press: Oxford.
• Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed.), Brooks/Cole. ISBN 0534408427.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.), W. H. Freeman. ISBN 0716708094.

### History of ideas

• Cardwell, D.S.L. (1971). From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age, Heinemann: London. ISBN 0435541501.
• Guillen, M. (1999). Five Equations That Changed the World. ISBN 0349110646.
• Hiebert, E.N. (1981). Historical Roots of the Principle of Conservation of Energy, Ayer Co Pub. ISBN 0405138806.
• Kuhn, T.S. (1957) “Energy conservation as an example of simultaneous discovery”, in M. Clagett (ed.) Critical Problems in the History of Science pp.321–56
• Smith, C. (1998). The Science of Energy: Cultural History of Energy Physics in Victorian Britain, Heinemann: London. ISBN 0485114313.

### Classic accounts

• Mach, E. (1872). History and Root of the Principles of the Conservation of Energy, Open Court Pub. Co., IL.
• Poincaré, H. (1905). Science and Hypothesis, Walter Scott Publishing Co. Ltd; Dover reprint, 1952. ISBN 0486602214., Chapter 8, "Energy and Thermo-dynamics"