In physics, energy (from Ancient Greek:

ἐνέργεια In philosophy, potentiality and actuality are a pair of closely connected principles which Aristotle used to analyze motion, causality, ethics, and physiology in his ''Physics'', ''Metaphysics'', '' Nicomachean Ethics'', and ''De Anima''. Th ...

, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a

physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...

, recognizable in the performance of work and in the form of heat and light. Energy is a conserved quantity—the law of

conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...

states that energy can be

converted Conversion or convert may refer to: Arts, entertainment, and media * "Conversion" (''Doctor Who'' audio), an episode of the audio drama ''Cyberman'' * "Conversion" (''Stargate Atlantis''), an episode of the television series * "The Conversion" ...

in form, but not created or destroyed. The unit of

measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...

for energy in the

International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...

(SI) is the joule (J). Common forms of energy include the kinetic energy of a moving object, the

potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...

stored by an object (for instance due to its position in a field), the

elastic energy Elastic energy is the mechanical potential energy stored in the configuration of a material or physical system as it is subjected to elastic deformation by work performed upon it. Elastic energy occurs when objects are impermanently compressed, ...

stored in a solid object,

chemical energy Chemical energy is the energy of chemical substances that is released when they undergo a chemical reaction and transform into other substances. Some examples of storage media of chemical energy include batteries, Schmidt-Rohr, K. (2018). "How ...

associated with chemical reactions, the radiant energy carried by electromagnetic radiation, and the

internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...

contained within a thermodynamic system. All living organisms constantly take in and release energy. Due to

mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...

, any object that has mass when stationary (called rest mass) also has an equivalent amount of energy whose form is called rest energy, and any additional energy (of any form) acquired by the object above that rest energy will increase the object's total mass just as it increases its total energy. Human civilization requires energy to function, which it gets from energy resources such as

fossil fuel A fossil fuel is a hydrocarbon-containing material formed naturally in the Earth's crust from the remains of dead plants and animals that is extracted and burned as a fuel. The main fossil fuels are coal, oil, and natural gas. Fossil fuels m ...

nuclear fuel Nuclear fuel is material used in nuclear power stations to produce heat to power turbines. Heat is created when nuclear fuel undergoes nuclear fission. Most nuclear fuels contain heavy fissile actinide elements that are capable of undergoing ...

, or

renewable energy Renewable energy is energy that is collected from renewable resources that are naturally replenished on a human timescale. It includes sources such as sunlight, wind, the movement of water, and geothermal heat. Although most renewable energy ...

. The Earth's climate and ecosystems processes are driven by the energy the planet receives from the Sun (although a small amount is also contributed by

geothermal energy Geothermal energy is the thermal energy in the Earth's crust which originates from the formation of the planet and from radioactive decay of materials in currently uncertain but possibly roughly equal proportions. The high temperature and pres ...

Forms

The total energy of a

system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...

can be subdivided and classified into potential energy, kinetic energy, or combinations of the two in various ways. Kinetic energy is determined by the

movement Movement may refer to: Common uses * Movement (clockwork), the internal mechanism of a timepiece * Motion, commonly referred to as movement Arts, entertainment, and media Literature * "Movement" (short story), a short story by Nancy Fu ...

of an object – or the composite motion of the components of an object – and

reflects the potential of an object to have motion, and generally is a function of the position of an object within a field or may be stored in the field itself. While these two categories are sufficient to describe all forms of energy, it is often convenient to refer to particular combinations of potential and kinetic energy as its own form. For example, the sum of translational and rotational kinetic and potential energy within a system is referred to as mechanical energy, whereas nuclear energy refers to the combined potentials within an atomic nucleus from either the nuclear force or the weak force, among other examples.

History

The word ''energy'' derives from the grc, ἐνέργεια, energeia, activity, operation, which possibly appears for the first time in the work of Aristotle in the 4th century BC. In contrast to the modern definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness and pleasure. In the late 17th century, Gottfried Leibniz proposed the idea of the lat, vis viva, or living force, which defined as the product of the mass of an object and its velocity squared; he believed that total ''vis viva'' was conserved. To account for slowing due to friction, Leibniz theorized that thermal energy consisted of the motions of the constituent parts of matter, although it would be more than a century until this was generally accepted. The modern analog of this property, kinetic energy, differs from ''vis viva'' only by a factor of two. Writing in the early 18th century,

Émilie du Châtelet Gabrielle Émilie Le Tonnelier de Breteuil, Marquise du Châtelet (; 17 December 1706 – 10 September 1749) was a French natural philosopher and mathematician from the early 1730s until her death due to complications during childbirth in 1749. ...

proposed the concept of

in the marginalia of her French language translation of Newton's '' Principia Mathematica'', which represented the first formulation of a conserved measurable quantity that was distinct from

momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...

, and which would later be called "energy". In 1807, Thomas Young was possibly the first to use the term "energy" instead of ''vis viva'', in its modern sense. Gustave-Gaspard Coriolis described " kinetic energy" in 1829 in its modern sense, and in 1853, William Rankine coined the term "

". The law of

was also first postulated in the early 19th century, and applies to any

isolated system In physical science, an isolated system is either of the following: # a physical system so far removed from other systems that it does not interact with them. # a thermodynamic system enclosed by rigid immovable walls through which neither m ...

. It was argued for some years whether heat was a physical substance, dubbed the

caloric Caloric is a brand of kitchen appliances, which dates back to 1903. History Caloric Corporation began as the Klein Stove Company in Philadelphia in 1890. The Caloric brand was introduced in 1903. It was reorganized in 1946 as the Caloric Stove C ...

, or merely a physical quantity, such as

. In 1845 James Prescott Joule discovered the link between mechanical work and the generation of heat. These developments led to the theory of conservation of energy, formalized largely by William Thomson ( Lord Kelvin) as the field of thermodynamics. Thermodynamics aided the rapid development of explanations of chemical processes by Rudolf Clausius, Josiah Willard Gibbs, and

Walther Nernst Walther Hermann Nernst (; 25 June 1864 – 18 November 1941) was a German chemist known for his work in thermodynamics, physical chemistry, electrochemistry, and solid state physics. His formulation of the Nernst heat theorem helped pave the wa ...

. It also led to a mathematical formulation of the concept of entropy by Clausius and to the introduction of laws of radiant energy by Jožef Stefan. According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time. Thus, since 1918, theorists have understood that the law of

is the direct mathematical consequence of the translational symmetry of the quantity conjugate to energy, namely time.

Units of measure

In 1843, James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. The most famous of them used the "Joule apparatus": a descending weight, attached to a string, caused rotation of a paddle immersed in water, practically insulated from heat transfer. It showed that the gravitational

lost by the weight in descending was equal to the

gained by the water through friction with the paddle. In the

(SI), the unit of energy is the joule, named after Joule. It is a derived unit. It is equal to the energy expended (or work done) in applying a force of one newton through a distance of one metre. However energy is also expressed in many other units not part of the SI, such as ergs,

calorie The calorie is a unit of energy. For historical reasons, two main definitions of "calorie" are in wide use. The large calorie, food calorie, or kilogram calorie was originally defined as the amount of heat needed to raise the temperature of on ...

British thermal unit The British thermal unit (BTU or Btu) is a unit of heat; it is defined as the amount of heat required to raise the temperature of one pound of water by one degree Fahrenheit. It is also part of the United States customary units. The modern SI ...

kilowatt-hour A kilowatt-hour (unit symbol: kW⋅h or kW h; commonly written as kWh) is a unit of energy: one kilowatt of power for one hour. In terms of SI derived units with special names, it equals 3.6 megajoules (MJ). Kilowatt-hours are a common bil ...

s and kilocalories, which require a conversion factor when expressed in SI units. The SI unit of energy rate (energy per unit time) is the watt, which is a joule per second. Thus, one joule is one watt-second, and 3600 joules equal one watt-hour. The CGS energy unit is the erg and the imperial and US customary unit is the foot pound. Other energy units such as the electronvolt, food calorie or thermodynamic kcal (based on the temperature change of water in a heating process), and BTU are used in specific areas of science and commerce.

Scientific use

Classical mechanics

In classical mechanics, energy is a conceptually and mathematically useful property, as it is a conserved quantity. Several formulations of mechanics have been developed using energy as a core concept. Work, a function of energy, is force times distance. :

W = \int_C \mathbf \cdot \mathrm \mathbf

This says that the work (

W

) is equal to the line integral of the

force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...

F along a path ''C''; for details see the mechanical work article. Work and thus energy is frame dependent. For example, consider a ball being hit by a bat. In the center-of-mass reference frame, the bat does no work on the ball. But, in the reference frame of the person swinging the bat, considerable work is done on the ball. The total energy of a system is sometimes called the Hamiltonian, after William Rowan Hamilton. The classical equations of motion can be written in terms of the Hamiltonian, even for highly complex or abstract systems. These classical equations have remarkably direct analogs in nonrelativistic quantum mechanics. Another energy-related concept is called the

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

, after

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaclassical mechanics, but is generally useful in modern physics. The Lagrangian is defined as the kinetic energy ''minus'' the potential energy. Usually, the Lagrange formalism is mathematically more convenient than the Hamiltonian for non-conservative systems (such as systems with friction). Noether's theorem (1918) states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalisation of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law.

Chemistry

In the context of

chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...

, energy is an attribute of a substance as a consequence of its atomic, molecular, or aggregate structure. Since a chemical transformation is accompanied by a change in one or more of these kinds of structure, it is usually accompanied by a decrease, and sometimes an increase, of the total energy of the substances involved. Some energy may be transferred between the surroundings and the reactants in the form of heat or light; thus the products of a reaction have sometimes more but usually less energy than the reactants. A reaction is said to be

exothermic In thermodynamics, an exothermic process () is a thermodynamic process or reaction that releases energy from the system to its surroundings, usually in the form of heat, but also in a form of light (e.g. a spark, flame, or flash), electricity (e ...

or exergonic if the final state is lower on the energy scale than the initial state; in the less common case of endothermic reactions the situation is the reverse. Chemical reactions are usually not possible unless the reactants surmount an energy barrier known as the activation energy. The ''speed'' of a chemical reaction (at a given temperature ''T'') is related to the activation energy ''E'' by the Boltzmann's population factor e^{−''E''/''kT''}; that is, the probability of a molecule to have energy greater than or equal to ''E'' at a given temperature ''T''. This exponential dependence of a reaction rate on temperature is known as the Arrhenius equation. The activation energy necessary for a chemical reaction can be provided in the form of thermal energy.

Biology

In biology, energy is an attribute of all biological systems, from the biosphere to the smallest living organism. Within an organism it is responsible for growth and development of a biological cell or

organelle In cell biology, an organelle is a specialized subunit, usually within a cell, that has a specific function. The name ''organelle'' comes from the idea that these structures are parts of cells, as organs are to the body, hence ''organelle,'' the ...

of a biological organism. Energy used in respiration is stored in substances such as carbohydrates (including sugars), lipids, and proteins stored by

cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...

. In human terms, the human equivalent (H-e) (Human energy conversion) indicates, for a given amount of energy expenditure, the relative quantity of energy needed for human metabolism, using as a standard an average human energy expenditure of 12,500 kJ per day and a basal metabolic rate of 80 watts. For example, if our bodies run (on average) at 80 watts, then a light bulb running at 100 watts is running at 1.25 human equivalents (100 ÷ 80) i.e. 1.25 H-e. For a difficult task of only a few seconds' duration, a person can put out thousands of watts, many times the 746 watts in one official horsepower. For tasks lasting a few minutes, a fit human can generate perhaps 1,000 watts. For an activity that must be sustained for an hour, output drops to around 300; for an activity kept up all day, 150 watts is about the maximum. The human equivalent assists understanding of energy flows in physical and biological systems by expressing energy units in human terms: it provides a "feel" for the use of a given amount of energy. Sunlight's radiant energy is also captured by plants as ''chemical potential energy'' in photosynthesis, when carbon dioxide and water (two low-energy compounds) are converted into carbohydrates, lipids, proteins and oxygen. Release of the energy stored during photosynthesis as heat or light may be triggered suddenly by a spark in a forest fire, or it may be made available more slowly for animal or human metabolism when organic molecules are ingested and

catabolism Catabolism () is the set of metabolic pathways that breaks down molecules into smaller units that are either oxidized to release energy or used in other anabolic reactions. Catabolism breaks down large molecules (such as polysaccharides, lipids, ...

is triggered by enzyme action. All living creatures rely on an external source of energy to be able to grow and reproduce – radiant energy from the Sun in the case of green plants and chemical energy (in some form) in the case of animals. The daily 1500–2000 Calories (6–8 MJ) recommended for a human adult are taken as food molecules, mostly carbohydrates and fats, of which glucose (C₆H₁₂O₆) and

stearin Stearin , or tristearin, or glyceryl tristearate is an odourless, white powder. It is a triglyceride derived from three units of stearic acid. Most triglycerides are derived from at least two and more commonly three different fatty acids. Like o ...

(C₅₇H₁₁₀O₆) are convenient examples. The food molecules are oxidized to carbon dioxide and water in the

mitochondria A mitochondrion (; ) is an organelle found in the Cell (biology), cells of most Eukaryotes, such as animals, plants and Fungus, fungi. Mitochondria have a double lipid bilayer, membrane structure and use aerobic respiration to generate adenosi ...

C6H12O6 + 6O2 -> 6CO2 + 6H2O C57H110O6 + (81 1/2) O2 -> 57CO2 + 55H2O and some of the energy is used to convert

ADP Adp or ADP may refer to: Aviation * Aéroports de Paris, airport authority for the Parisian region in France * Aeropuertos del Perú, airport operator for airports in northern Peru * SLAF Anuradhapura, an airport in Sri Lanka * Ampara Air ...

into

ATP ATP may refer to: Companies and organizations * Association of Tennis Professionals, men's professional tennis governing body * American Technical Publishers, employee-owned publishing company * ', a Danish pension * Armenia Tree Project, non ...

: The rest of the chemical energy of the carbohydrate or fat are converted into heat: the ATP is used as a sort of "energy currency", and some of the chemical energy it contains is used for other metabolism when ATP reacts with OH groups and eventually splits into ADP and phosphate (at each stage of a metabolic pathway, some chemical energy is converted into heat). Only a tiny fraction of the original chemical energy is used for work:These examples are solely for illustration, as it is not the energy available for work which limits the performance of the athlete but the power output (in case of a sprinter) and the

(in case of a weightlifter). :gain in kinetic energy of a sprinter during a 100 m race: 4 kJ :gain in gravitational potential energy of a 150 kg weight lifted through 2 metres: 3 kJ :Daily food intake of a normal adult: 6–8 MJ It would appear that living organisms are remarkably inefficient (in the physical sense) in their use of the energy they receive (chemical or radiant energy); most

machine A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to na ...

s manage higher efficiencies. In growing organisms the energy that is converted to heat serves a vital purpose, as it allows the organism tissue to be highly ordered with regard to the molecules it is built from. The second law of thermodynamics states that energy (and matter) tends to become more evenly spread out across the universe: to concentrate energy (or matter) in one specific place, it is necessary to spread out a greater amount of energy (as heat) across the remainder of the universe ("the surroundings"). Crystals are another example of highly ordered systems that exist in nature: in this case too, the order is associated with the transfer of a large amount of heat (known as the lattice energy) to the surroundings. Simpler organisms can achieve higher energy efficiencies than more complex ones, but the complex organisms can occupy ecological niches that are not available to their simpler brethren. The conversion of a portion of the chemical energy to heat at each step in a metabolic pathway is the physical reason behind the pyramid of biomass observed in ecology. As an example, to take just the first step in the food chain: of the estimated 124.7 Pg/a of carbon that is fixed by photosynthesis, 64.3 Pg/a (52%) are used for the metabolism of green plants, i.e. reconverted into carbon dioxide and heat.

Earth sciences

In geology, continental drift, mountain ranges, volcanoes, and earthquakes are phenomena that can be explained in terms of energy transformations in the Earth's interior, while

meteorological Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...

phenomena like wind, rain,

hail Hail is a form of solid precipitation. It is distinct from ice pellets (American English "sleet"), though the two are often confused. It consists of balls or irregular lumps of ice, each of which is called a hailstone. Ice pellets generally fal ...

, snow, lightning, tornadoes and hurricanes are all a result of energy transformations in our

atmosphere An atmosphere () is a layer of gas or layers of gases that envelop a planet, and is held in place by the gravity of the planetary body. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A s ...

brought about by

solar energy Solar energy is radiant light and heat from the Sun that is harnessed using a range of technologies such as solar power to generate electricity, solar thermal energy (including solar water heating), and solar architecture. It is an essenti ...

. Sunlight is the main input to Earth's energy budget which accounts for its temperature and climate stability. Sunlight may be stored as gravitational potential energy after it strikes the Earth, as (for example when) water evaporates from oceans and is deposited upon mountains (where, after being released at a hydroelectric dam, it can be used to drive turbines or generators to produce electricity). Sunlight also drives most weather phenomena, save a few exceptions, like those generated by volcanic events for example. An example of a solar-mediated weather event is a hurricane, which occurs when large unstable areas of warm ocean, heated over months, suddenly give up some of their thermal energy to power a few days of violent air movement. In a slower process,

radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...

of atoms in the core of the Earth releases heat. This thermal energy drives plate tectonics and may lift mountains, via orogenesis. This slow lifting represents a kind of gravitational potential

energy storage Energy storage is the capture of energy produced at one time for use at a later time to reduce imbalances between energy demand and energy production. A device that stores energy is generally called an accumulator or battery. Energy comes in ...

of the thermal energy, which may later be transformed into active kinetic energy during landslides, after a triggering event. Earthquakes also release stored elastic potential energy in rocks, a store that has been produced ultimately from the same radioactive heat sources. Thus, according to present understanding, familiar events such as landslides and earthquakes release energy that has been stored as potential energy in the Earth's gravitational field or elastic strain (mechanical potential energy) in rocks. Prior to this, they represent release of energy that has been stored in heavy atoms since the collapse of long-destroyed supernova stars (which created these atoms).

Cosmology

In cosmology and astronomy the phenomena of

star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

nova A nova (plural novae or novas) is a transient astronomical event that causes the sudden appearance of a bright, apparently "new" star (hence the name "nova", which is Latin for "new") that slowly fades over weeks or months. Causes of the dramati ...

supernova A supernova is a powerful and luminous explosion of a star. It has the plural form supernovae or supernovas, and is abbreviated SN or SNe. This transient astronomical event occurs during the last evolutionary stages of a massive star or when ...

quasar A quasar is an extremely Luminosity, luminous active galactic nucleus (AGN). It is pronounced , and sometimes known as a quasi-stellar object, abbreviated QSO. This emission from a galaxy nucleus is powered by a supermassive black hole with a m ...

s and gamma-ray bursts are the universe's highest-output energy transformations of matter. All

stellar Stellar means anything related to one or more stars (''stella''). The term may also refer to: Arts, entertainment, and media * ''Stellar'' (magazine), an Irish lifestyle and fashion magazine * Stellar Loussier, a character from ''Mobile Suit Gun ...

phenomena (including solar activity) are driven by various kinds of energy transformations. Energy in such transformations is either from gravitational collapse of matter (usually molecular hydrogen) into various classes of astronomical objects (stars, black holes, etc.), or from nuclear fusion (of lighter elements, primarily hydrogen). The nuclear fusion of hydrogen in the Sun also releases another store of potential energy which was created at the time of the

Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...

. At that time, according to theory, space expanded and the universe cooled too rapidly for hydrogen to completely fuse into heavier elements. This meant that hydrogen represents a store of potential energy that can be released by fusion. Such a fusion process is triggered by heat and pressure generated from gravitational collapse of hydrogen clouds when they produce stars, and some of the fusion energy is then transformed into sunlight.

Quantum mechanics

In quantum mechanics, energy is defined in terms of the

energy operator In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry. Definition It is given by: \hat = i\hbar\frac It acts on the wave function (the ...

(Hamiltonian) as a time derivative of the wave function. The Schrödinger equation equates the energy operator to the full energy of a particle or a system. Its results can be considered as a definition of measurement of energy in quantum mechanics. The Schrödinger equation describes the space- and time-dependence of a slowly changing (non-relativistic) wave function of quantum systems. The solution of this equation for a bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of

quanta Quanta is the plural of quantum. Quanta may also refer to: Organisations * Quanta Computer, a Taiwan-based manufacturer of electronic and computer equipment * Quanta Display Inc., a Taiwanese TFT-LCD panel manufacturer acquired by AU Optronic ...

. In the solution of the Schrödinger equation for any oscillator (vibrator) and for electromagnetic waves in a vacuum, the resulting energy states are related to the frequency by Planck's relation:

E = h\nu

(where

h

is the Planck constant and

\nu

the frequency). In the case of an electromagnetic wave these energy states are called quanta of light or photons.

Relativity

When calculating kinetic energy ( work to accelerate a massive body from zero speed to some finite speed) relativistically – using Lorentz transformations instead of

Newtonian mechanics Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...

– Einstein discovered an unexpected by-product of these calculations to be an energy term which does not vanish at zero speed. He called it rest energy: energy which every massive body must possess even when being at rest. The amount of energy is directly proportional to the mass of the body:

E_0 = m_0 c^2 ,

where *''m''₀ is the rest mass of the body, *''c'' is the speed of light in vacuum, *

E_0

is the rest energy. For example, consider electron–

positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...

annihilation, in which the rest energy of these two individual particles (equivalent to their rest mass) is converted to the radiant energy of the photons produced in the process. In this system the matter and antimatter (electrons and positrons) are destroyed and changed to non-matter (the photons). However, the total mass and total energy do not change during this interaction. The photons each have no rest mass but nonetheless have radiant energy which exhibits the same inertia as did the two original particles. This is a reversible process – the inverse process is called pair creation – in which the rest mass of particles is created from the radiant energy of two (or more) annihilating photons. In general relativity, the

stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...

serves as the source term for the gravitational field, in rough analogy to the way mass serves as the source term in the non-relativistic Newtonian approximation. Energy and mass are manifestations of one and the same underlying physical property of a system. This property is responsible for the inertia and strength of gravitational interaction of the system ("mass manifestations"), and is also responsible for the potential ability of the system to perform work or heating ("energy manifestations"), subject to the limitations of other physical laws. In

classical physics Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...

, energy is a scalar quantity, the

canonical conjugate Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform dual (mathematics), duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an unc ...

to time. In special relativity energy is also a scalar (although not a

Lorentz scalar In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ve ...

but a time component of the energy–momentum 4-vector). In other words, energy is invariant with respect to rotations of space, but not invariant with respect to rotations of spacetime (= boosts).

Transformation

Energy may be transformed between different forms at various efficiencies. Items that transform between these forms are called transducers. Examples of transducers include a battery (from

to electric energy), a dam (from gravitational potential energy to kinetic energy of moving water (and the blades of a turbine) and ultimately to electric energy through an

electric generator In electricity generation, a generator is a device that converts motive power (mechanical energy) or fuel-based power (chemical energy) into electric power for use in an external circuit. Sources of mechanical energy include steam turbines, gas ...

), and a heat engine (from heat to work). Examples of energy transformation include generating electric energy from heat energy via a steam turbine, or lifting an object against gravity using electrical energy driving a crane motor. Lifting against gravity performs mechanical work on the object and stores gravitational potential energy in the object. If the object falls to the ground, gravity does mechanical work on the object which transforms the potential energy in the gravitational field to the kinetic energy released as heat on impact with the ground. Our Sun transforms nuclear potential energy to other forms of energy; its total mass does not decrease due to that itself (since it still contains the same total energy even in different forms) but its mass does decrease when the energy escapes out to its surroundings, largely as radiant energy. There are strict limits to how efficiently heat can be converted into work in a cyclic process, e.g. in a heat engine, as described by

Carnot's theorem Carnot's theorem or Carnot's principle may refer to: In geometry: *Carnot's theorem (inradius, circumradius), describing a property of the incircle and the circumcircle of a triangle *Carnot's theorem (conics), describing a relation between triangl ...

and the second law of thermodynamics. However, some energy transformations can be quite efficient. The direction of transformations in energy (what kind of energy is transformed to what other kind) is often determined by entropy (equal energy spread among all available

degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

) considerations. In practice all energy transformations are permitted on a small scale, but certain larger transformations are not permitted because it is statistically unlikely that energy or matter will randomly move into more concentrated forms or smaller spaces. Energy transformations in the universe over time are characterized by various kinds of potential energy, that has been available since the

, being "released" (transformed to more active types of energy such as kinetic or radiant energy) when a triggering mechanism is available. Familiar examples of such processes include nucleosynthesis, a process ultimately using the gravitational potential energy released from the

gravitational collapse Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of gravity. Gravitational collapse is a fundamental mechanism for structure formatio ...

e to "store" energy in the creation of heavy isotopes (such as uranium and thorium), and nuclear decay, a process in which energy is released that was originally stored in these heavy elements, before they were incorporated into the solar system and the Earth. This energy is triggered and released in nuclear fission bombs or in civil nuclear power generation. Similarly, in the case of a chemical explosion, chemical potential energy is transformed to

kinetic Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to i ...

and thermal energy in a very short time. Yet another example is that of a pendulum. At its highest points the kinetic energy is zero and the gravitational potential energy is at its maximum. At its lowest point the kinetic energy is at its maximum and is equal to the decrease in

. If one (unrealistically) assumes that there is no friction or other losses, the conversion of energy between these processes would be perfect, and the pendulum would continue swinging forever. Energy is also transferred from potential energy (

E_p

) to kinetic energy (

E_k

) and then back to potential energy constantly. This is referred to as conservation of energy. In this

, energy cannot be created or destroyed; therefore, the initial energy and the final energy will be equal to each other. This can be demonstrated by the following: The equation can then be simplified further since

E_p = mgh

(mass times acceleration due to gravity times the height) and

E_k = \frac mv^2

(half mass times velocity squared). Then the total amount of energy can be found by adding

E_p + E_k = E_\text

Conservation of energy and mass in transformation

Energy gives rise to weight when it is trapped in a system with zero momentum, where it can be weighed. It is also equivalent to mass, and this mass is always associated with it. Mass is also equivalent to a certain amount of energy, and likewise always appears associated with it, as described in mass-energy equivalence. The formula ''E'' = ''mc''², derived by Albert Einstein (1905) quantifies the relationship between relativistic mass and energy within the concept of special relativity. In different theoretical frameworks, similar formulas were derived by J.J. Thomson (1881),

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

(1900), Friedrich Hasenöhrl (1904) and others (see Mass-energy equivalence#History for further information). Part of the rest energy (equivalent to rest mass) of matter may be converted to other forms of energy (still exhibiting mass), but neither energy nor mass can be destroyed; rather, both remain constant during any process. However, since

c^2

is extremely large relative to ordinary human scales, the conversion of an everyday amount of rest mass (for example, 1 kg) from rest energy to other forms of energy (such as kinetic energy, thermal energy, or the radiant energy carried by light and other radiation) can liberate tremendous amounts of energy (~

9\times 10^

joules = 21 megatons of TNT), as can be seen in nuclear reactors and nuclear weapons. Conversely, the mass equivalent of an everyday amount energy is minuscule, which is why a loss of energy (loss of mass) from most systems is difficult to measure on a weighing scale, unless the energy loss is very large. Examples of large transformations between rest energy (of matter) and other forms of energy (e.g., kinetic energy into particles with rest mass) are found in nuclear physics and particle physics. Often, however, the complete conversion of matter (such as atoms) to non-matter (such as photons) is forbidden by conservation laws.

Reversible and non-reversible transformations

Thermodynamics divides energy transformation into two kinds: reversible processes and irreversible processes. An irreversible process is one in which energy is dissipated (spread) into empty energy states available in a volume, from which it cannot be recovered into more concentrated forms (fewer quantum states), without degradation of even more energy. A reversible process is one in which this sort of dissipation does not happen. For example, conversion of energy from one type of potential field to another is reversible, as in the pendulum system described above. In processes where heat is generated, quantum states of lower energy, present as possible excitations in fields between atoms, act as a reservoir for part of the energy, from which it cannot be recovered, in order to be converted with 100% efficiency into other forms of energy. In this case, the energy must partly stay as thermal energy and cannot be completely recovered as usable energy, except at the price of an increase in some other kind of heat-like increase in disorder in quantum states, in the universe (such as an expansion of matter, or a randomization in a crystal). As the universe evolves with time, more and more of its energy becomes trapped in irreversible states (i.e., as heat or as other kinds of increases in disorder). This has led to the hypothesis of the inevitable thermodynamic heat death of the universe. In this heat death the energy of the universe does not change, but the fraction of energy which is available to do work through a heat engine, or be transformed to other usable forms of energy (through the use of generators attached to heat engines), continues to decrease.

Conservation of energy

The fact that energy can be neither created nor destroyed is called the law of

. In the form of the first law of thermodynamics, this states that a closed system's energy is constant unless energy is transferred in or out as work or heat, and that no energy is lost in transfer. 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. Whenever one measures (or calculates) the total energy of a system of particles whose interactions do not depend explicitly on time, it is found that the total energy of the system always remains constant. While heat can always be fully converted into work in a reversible isothermal expansion of an ideal gas, for cyclic processes of practical interest in heat engines the second law of thermodynamics states that the system doing work always loses some energy as waste heat. This creates a limit to the amount of heat energy that can do work in a cyclic process, a limit called the available energy. Mechanical and other forms of energy can be transformed in the other direction into thermal energy without such limitations. The total energy of a system can be calculated by adding up all forms of energy in the system. Richard Feynman said during a 1961 lecture: Most kinds of energy (with gravitational energy being a notable exception) are subject to strict local conservation laws as well. In this case, energy can only be exchanged between adjacent regions of space, and all observers agree as to the volumetric density of energy in any given space. There is also a global law of conservation of energy, stating that the total energy of the universe cannot change; this is a corollary of the local law, but not vice versa.''The Laws of Thermodynamics''
including careful definitions of energy, free energy, et cetera. This law is a fundamental principle of physics. As shown rigorously by Noether's theorem, the conservation of energy is a mathematical consequence of translational symmetry of time, a property of most phenomena below the cosmic scale that makes them independent of their locations on the time coordinate. Put differently, yesterday, today, and tomorrow are physically indistinguishable. This is because energy is the quantity which is

to time. This mathematical entanglement of energy and time also results in the uncertainty principle – it is impossible to define the exact amount of energy during any definite time interval (though this is practically significant only for very short time intervals). The uncertainty principle should not be confused with energy conservation – rather it provides mathematical limits to which energy can in principle be defined and measured. Each of the basic forces of nature is associated with a different type of potential energy, and all types of potential energy (like all other types of energy) appear as system mass, whenever present. For example, a compressed spring will be slightly more massive than before it was compressed. Likewise, whenever energy is transferred between systems by any mechanism, an associated mass is transferred with it. In quantum mechanics energy is expressed using the Hamiltonian operator. On any time scales, the uncertainty in the energy is by :

\Delta E \Delta t \ge \frac

which is similar in form to the

Heisenberg Uncertainty Principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...

(but not really mathematically equivalent thereto, since ''H'' and ''t'' are not dynamically conjugate variables, neither in classical nor in quantum mechanics). In particle physics, this inequality permits a qualitative understanding of virtual particles, which carry

. The exchange of virtual particles with real particles is responsible for the creation of all known fundamental forces (more accurately known as fundamental interactions). Virtual photons are also responsible for the electrostatic interaction between electric charges (which results in Coulomb's law), for

spontaneous Spontaneous may refer to: * Spontaneous abortion * Spontaneous bacterial peritonitis * Spontaneous combustion * Spontaneous declaration * Spontaneous emission * Spontaneous fission * Spontaneous generation * Spontaneous human combustion * Spontan ...

radiative decay of excited atomic and nuclear states, for the Casimir force, for the Van der Waals force and some other observable phenomena.

Energy transfer

Closed systems

Energy transfer can be considered for the special case of systems which are

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

to transfers of matter. The portion of the energy which is transferred by conservative forces over a distance is measured as the work the source system does on the receiving system. The portion of the energy which does not do work during the transfer is called heat.Although heat is "wasted" energy for a specific energy transfer (see: waste heat), it can often be harnessed to do useful work in subsequent interactions. However, the maximum energy that can be "recycled" from such recovery processes is limited by the second law of thermodynamics. Energy can be transferred between systems in a variety of ways. Examples include the transmission of electromagnetic energy via photons, physical collisions which transfer kinetic energy,The mechanism for most macroscopic physical collisions is actually electromagnetic, but it is very common to simplify the interaction by ignoring the mechanism of collision and just calculate the beginning and end result. tidal interactions, and the conductive transfer of thermal energy. Energy is strictly conserved and is also locally conserved wherever it can be defined. In thermodynamics, for closed systems, the process of energy transfer is described by the

first law "First Law" is a science fiction short story by American writer Isaac Asimov, first published in the October 1956 issue of ''Fantastic Universe'' magazine and later collected in ''The Rest of the Robots'' (1964) and ''The Complete Robot'' (1982). ...

:There are several sign conventions for this equation. Here, the signs in this equation follow the IUPAC convention. where

E

is the amount of energy transferred,

W

represents the work done on or by the system, and

Q

represents the heat flow into or out of the system. As a simplification, the heat term,

Q

, can sometimes be ignored, especially for fast processes involving gases, which are poor conductors of heat, or when the thermal efficiency of the transfer is high. For such adiabatic processes, This simplified equation is the one used to define the joule, for example.

Open systems

Beyond the constraints of closed systems, open systems can gain or lose energy in association with matter transfer (this process is illustrated by injection of an air-fuel mixture into a car engine, a system which gains in energy thereby, without addition of either work or heat). Denoting this energy by

E_\text

, one may write

Thermodynamics

Internal energy

Internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...

is the sum of all microscopic forms of energy of a system. It is the energy needed to create the system. It is related to the potential energy, e.g., molecular structure, crystal structure, and other geometric aspects, as well as the motion of the particles, in form of kinetic energy. Thermodynamics is chiefly concerned with changes in internal energy and not its absolute value, which is impossible to determine with thermodynamics alone.I. Klotz, R. Rosenberg, ''Chemical Thermodynamics – Basic Concepts and Methods'', 7th ed., Wiley (2008), p. 39

First law of thermodynamics

The first law of thermodynamics asserts that the total energy of a system and its surroundings (but not necessarily thermodynamic free energy) is always conserved and that heat flow is a form of energy transfer. For homogeneous systems, with a well-defined temperature and pressure, a commonly used corollary of the first law is that, for a system subject only to pressure forces and heat transfer (e.g., a cylinder-full of gas) without chemical changes, the differential change in the internal energy of the system (with a ''gain'' in energy signified by a positive quantity) is given as :

\mathrmE = T\mathrmS - P\mathrmV\,

, where the first term on the right is the heat transferred into the system, expressed in terms of temperature ''T'' and entropy ''S'' (in which entropy increases and its change d''S'' is positive when heat is added to the system), and the last term on the right hand side is identified as work done on the system, where pressure is ''P'' and volume ''V'' (the negative sign results since compression of the system requires work to be done on it and so the volume change, d''V'', is negative when work is done on the system). This equation is highly specific, ignoring all chemical, electrical, nuclear, and gravitational forces, effects such as

advection In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is al ...

of any form of energy other than heat and ''PV''-work. The general formulation of the first law (i.e., conservation of energy) is valid even in situations in which the system is not homogeneous. For these cases the change in internal energy of a ''closed'' system is expressed in a general form by :

\mathrmE=\delta Q+\delta W

where

\delta Q

is the heat supplied to the system and

\delta W

is the work applied to the system.

Equipartition of energy

The energy of a mechanical

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...

(a mass on a spring) is alternately

and

. At two points in the oscillation

cycle Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in soc ...

it is entirely kinetic, and at two points it is entirely potential. Over a whole cycle, or over many cycles, average energy is equally split between kinetic and potential. This is an example of the equipartition principle: the total energy of a system with many degrees of freedom is equally split among all available degrees of freedom, on average. This principle is vitally important to understanding the behavior of a quantity closely related to energy, called entropy. Entropy is a measure of evenness of a

distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...

of energy between parts of a system. When an isolated system is given more degrees of freedom (i.e., given new available

energy state A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The te ...

s that are the same as existing states), then total energy spreads over all available degrees equally without distinction between "new" and "old" degrees. This mathematical result is part of the second law of thermodynamics. The second law of thermodynamics is simple only for systems which are near or in a physical equilibrium state. For non-equilibrium systems, the laws governing the systems' behavior are still debatable. One of the guiding principles for these systems is the principle of maximum entropy production. It states that nonequilibrium systems behave in such a way as to maximize their entropy production.

Notes

References

External links

*
Differences between Heat and Thermal energy
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