Conservation of energy
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Conservation of energy (the first law of thermodynamics) is one of several conservation laws. It 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.
Although ancient philosophers as far back as Thales of Miletus may have had inklings of the First Law, it was first stated in its modern form by the German surgeon Julius Robert von Mayer (18141878) in his "Remarks On the Forces of Inorganic Nature" in Annalen der Chemie und Pharmacie, 43, 233 (1842). Mayer reached his conclusion on a voyage to the Dutch East Indies (now Indonesia), 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.
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Unfortunately for Mayer, his work was overlooked in favour of Joule's, and Mayer attempted to commit suicide. Later, Mayer's reputation was restored by a sympathetic account in John Tyndall's Heat: A Mode of Motion (1863).
A similar law was written in the privately published Die Erhaltung der Kraft (1847) by Hermann von Helmholtz.
With the discovery of special relativity by Albert Einstein, mass and energy were shown to be interchangeable by the famous equation E = mc^{2}. As a result, 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 timeinvariance of the laws of physics.
Within the realm of quantum mechanics, conservation of energy is violated for extremely small time scales due to the uncertainty principle.
Formulae
One formulation for the first law of thermodynamics is
 <math> Q = \Delta U + W \qquad \qquad \qquad (1) <math>
where Q is heat transferred into the system from the surroundings, W is work done by the system, and U is the internal energy of the system. This energy is mostly kinetic energy: the potential energy can be assumed to be negligible. Pressurevolume work (e.g. done by a gas on a piston) is defined to be
 <math> W = P \Delta V \qquad \qquad \qquad (2) <math>.
Equation (1) can be interpreted as follows: Q is heat energy being input into the system. The system then can use this incoming energy to do two things: (1) do work, or (2) increase its own internal energy. Here is an analogy: Q is income, which can then be spent to buy things (W), or it can be saved in a bank account (<math> \Delta U <math>).
If all the heat is used to do work (<math> Q = W <math> and <math> \Delta U = 0 <math>) then the system is undergoing an isothermal process, which means that its temperature remains constant. This is because the system's internal energy is proportional to its temperature.
If all the heat is used to increase internal energy, (<math> Q = \Delta U <math> and <math> W = 0 <math>) then the system is undergoing an isochoric process, also called isometric process. This is a process in which the system's volume is constant: <math> \Delta V = 0 <math> so that, according to equation (2), W = 0.
It is also possible for the heat energy to be used up partially by doing work and partially by increasing internal energy. Examples of such processes are the isobaric process and the adiabatic process.
Equation (1) is the one preferred by engineers. Another convention preferred by chemists is
 <math> \Delta U = Q + W \qquad \qquad \qquad (3) <math>
where W is work done on the system by the surroundings. In this case pressurevolume work is defined to be
 <math> W = P \Delta V \qquad \qquad \qquad (4) <math>.
Equation (3) can be interpreted to mean that heat Q and work W are energies being transferred into or out of the system. The system then responds by increasing its internal energy accordingly. In Equation (3) neither Q nor W are state functions. A state function does not depend on which particular thermodynamic process is chosen to connect the initial and final thermostatic states.
The law of conservation of energy excludes the possibility of perpetuum mobile of the first kind.
References
 Peter J. Nolan, Fundamentals of College Physics, 2nd ed., William C. Brown Publishers, 1995.
 Oxtoby & Nachtrieb, Principles of Modern Chemistry, 3rd edition, Saunders College Publishing, 1996.
 Papineau, D. (2002). Thinking about Consciousness. Oxford, UK, Oxford University Press.
See Also
Second law of thermodynamicscs:1. termodynamický zákon da:Termodynamikkens 1. lov de:Energieerhaltungssatz et:Energia jäävuse seadus es:Conservación de la energía id:Kekekalan energi it:Primo principio della termodinamica lt:Energijos tvermės dėsnis nl:Eerste wet van de thermodynamica ja:エネルギー保存の法則 pl:Zasada zachowania energii ru:Закон сохранения энергии sl:Prvi zakon termodinamike sv:Energiprincipen zh:热力学第一定律