The total energy of asystem can be subdivided and classified intopotential energy,kinetic energy, or combinations of the two in various ways. Kinetic energy is determined by themovement of an object – or thecomposite motion of the object's components – whilepotential energy reflects the potential of an object to have motion, generally being based upon the object's position within afield or what is stored within the field itself.[7]
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 androtational kinetic and potential energy within a system is referred to asmechanical energy, whereas nuclear energy refers to the combined potentials within an atomic nucleus from either thenuclear force or theweak force, among other examples.[8]
Some forms of energy (that an object or system can have as a measurable property)[9][10]
Thomas Young, the first person to use the term "energy" in the modern sense
The wordenergy derives from theAncient Greek:ἐνέργεια,romanized: energeia,lit.'activity, operation',[11] which possibly appears for the first time in the work ofAristotle 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.[12]
In the late 17th century,Gottfried Leibniz proposed the idea of theLatin:vis viva, or living force, which defined as the product of the mass of an object and its velocity squared; he believed that totalvis 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 fromvis viva only by a factor of two.[13] Writing in the early 18th century,Émilie du Châtelet proposed the concept ofconservation of energy in the marginalia of her French language translation of Newton'sPrincipia Mathematica, which represented the first formulation of a conserved measurable quantity that was distinct frommomentum, and which would later be called "energy".[14]
Albert Einstein's 1905 theory ofspecial relativity showed thatrest mass corresponds to anequivalent amount ofrest energy. This means thatrest mass can be converted to or from equivalent amounts of (non-material) forms of energy, for example, kinetic energy, potential energy, and electromagneticradiant energy. When this happens, rest mass is not conserved, unlike thetotal mass ortotal energy. All forms of energy contribute to the total mass and total energy. Thus, conservation of energy (total, including material orrest energy) andconservation of mass (total, not justrest) are one (equivalent) law. In the 18th century, these had appeared as two seemingly-distinct laws.[26][27]
Joule's apparatus for measuring the mechanical equivalent of heat. A descending weight attached to a string causes a paddle immersed in water to rotate.
Indimensional analysis, thebase units of energy are given by:Work = Force × Distance = M L2 T−2, with the fundamental dimensions of Mass M, Length L, and time T.[5] In theInternational System of Units (SI), the unit of energy is thejoule. It is aderived unit that is equal to the energy expended, or work done, in applying a force of onenewton through a distance of one metre.[1]
The SI unit ofpower, defined as energy per unit of time, is thewatt, which is one joule per second.[3] Thus, akilowatt-hour (kWh), which can be realized as the energy delivered by one kilowatt of power for an hour, is equal to 3.6 million joules.[33] TheCGS energy unit is theerg and theimperial and US customary unit is thefoot-pound.[34]
Inclassical mechanics, energy is a conceptually and mathematically useful property, as it is aconserved quantity. Several formulations of mechanics have been developed using energy as a core concept.
Work, a function of energy, is force times distance.[36]
This says that the work () is equal to theline integral of theforceF along a pathC; for details see themechanical work article. Work and thus energy isframe dependent. For example, consider a ball being hit by a bat. In thecenter-of-massreference 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.[37]
The total energy of a system is sometimes called theHamiltonian, afterWilliam Rowan Hamilton. The classical equations of motion can be written in terms of the Hamiltonian, even for highly complex or abstract systems.[38] These classical equations have direct analogs in nonrelativistic quantum mechanics.[39]
Another energy-related concept is called theLagrangian, afterJoseph-Louis Lagrange. This formalism is as fundamental as the Hamiltonian, and both can be used to derive the equations of motion or be derived from them. It was invented in the context ofclassical mechanics, but is generally useful in modern physics. The Lagrangian is defined as the kinetic energyminus the potential energy. Usually, the Lagrange formalism is mathematically more convenient than the Hamiltonian for non-conservative systems (such as systems with friction).[40]
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;[41] for example, dissipative systems with continuous symmetries need not have a corresponding conservation law.
In the context ofchemistry,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 beexothermic orexergonic if the final state is lower on the energy scale than the initial state; in the less common case ofendothermic reactions the situation is the reverse.[42]
Chemical reactions are usually not possible unless the reactants surmount an energy barrier known as theactivation energy. Thespeed of a chemical reaction (at a given temperature T) is related to the activation energy E by the Boltzmann 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 theArrhenius equation. The activation energy necessary for a chemical reaction can be provided in the form of thermal energy.[43]
Inbiology, energy is an attribute of all biological systems, from the biosphere to the smallest living organism. It enables the growth, development, and functioning of a biologicalcell ororganelle in an organism. 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. Energy provided throughcellular respiration is stored in nutrients such ascarbohydrates (including sugars),lipids, andproteins bycells.[44]
Sunlight's radiant energy is captured by plants aschemical potential energy inphotosynthesis, when carbon dioxide and water (two low-energy compounds) are converted into carbohydrates, lipids, proteins, and oxygen.[45] 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 andcatabolism is triggered byenzyme action.[46]
Humans
Thebasal metabolism rate measures thefood energy expenditure per unit time byendothermic animals at rest.[47] In other words it is the energy required by body organs to perform normally. For humans,metabolic equivalent of task (MET) compares the energy expenditure per unit mass while performing a physical activity, relative to a baseline. By convention, this baseline is 3.5 mL of oxygen consumed per kg per minute, which is the energy consumed by a typical individual when sitting quietly.[48]
In human terms, thehuman equivalent (H-e) (Human energy conversion) indicates, for a given amount of energy expenditure, the relative quantity of energy needed for humanmetabolism, using as a standard an average human energy expenditure of 6,900 kJ per day and abasal metabolic rate of 80 watts.[citation needed] 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.[49] 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.[50]
The daily 1,600–3,000 calories (7–13 MJ) recommended for a human adult are taken as food molecules,[51] mostly carbohydrates and fats. Only a tiny fraction of the original chemical energy is used forwork:[note 1]
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
In growing organisms the energy that is converted to heat serves a vital purpose, as it allows the organism's tissue to be highly ordered with regard to the molecules it is built from. Thesecond 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").[note 2] Simpler organisms can achieve higher energy efficiencies than more complex ones, but the complex organisms can occupyecological 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 inecology. As an example, to take just the first step in thefood chain: of the estimated 124.7 Pg/a of carbon that isfixed byphotosynthesis, 64.3 Pg/a (52%) are used for the metabolism of green plants,[52] i.e. reconverted into carbon dioxide and heat.
Multicellular organisms such as humans have cell forms that are classified asEukaryote. These cells include anorganelle called themitochondria that generateschemical energy for the rest of the hosting cell. Ninety percent of the oxygen intake by humans is utilized by themitochondria, especially for nutrient processing.[53] The moleculeadenosine triphosphate (ATP) is the primary energy transporter in living cells, providing an energy source for cellular processes. It is continually being broken down and synthesized as a component of cellular respiration.[54]
Two examples of nutrients consumed by animals areglucose (C6H12O6) andstearin (C57H110O6). These food molecules are oxidized tocarbon dioxide andwater in the mitochondria:[55]and some of the energy is used to convertADP intoATP:[56][53]
ADP + HPO42− → ATP + H2O
The rest of the chemical energy of the nutrients 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 othermetabolism when ATP reacts with OH groups and eventually splits into ADP and phosphate (at each stage of ametabolic pathway, some chemical energy is converted into heat).
Sunlight is the main input toEarth's energy budget which accounts for its temperature and climate stability, after accounting for interaction with the atmosphere.[58] 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).[59] 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.[60]
In a slower process,radioactive decay of atoms in the core of the Earth releases heat, which supplies more than half of the planet'sinternal heat budget.[61] In the present day, thisradiogenic heat production was primarily driven by the decay ofUranium-235,Potassium-40, andThorium-232 some time in the past.[62] This thermal energy drivesplate tectonics and may lift mountains, viaorogenesis. This slow lifting represents a kind of gravitational potentialenergy storage 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.[63] 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).[64]
Early in a planet's history, theaccretion process provides impact energy that can partially or completely melt the body. This allows a planet to becomedifferentiated by chemical element. Chemical phase changes of minerals during formation provide additional internal heating. Over time the internal heat is brought to the surface then radiated away into space, cooling the body. Accretedradiogenic heat sources settle toward the core, providing thermal energy to the planet on ageologic time scale.[65] Ongoingsedimentation provides a persistent internal energy source forgas giant planets likeJupiter andSaturn.[66]
Cosmology
Incosmology and astronomy the phenomena ofstars,nova,supernova,quasars, andgamma-ray bursts are the universe's highest-output energy transformations of matter. Allstellar 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).[67]
Thenuclear fusion of hydrogen in the Sun also releases another store of potential energy which was created at the time of theBig Bang. 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.[68]
The accretion of matter onto acompact object is a very efficient means of generating energy fromgravitational potential. This behavior is responsible for some of the universe's brightest persistent energy sources.[69] ThePenrose process is a theoretical method by which energy could be extracted from a rotating black hole.[70]Hawking radiation is the emission ofblack-body radiation from a black hole, which results in a steady loss of mass and rotational energy. As the object evaporates, the temperature of this radiation is predicted to increase, speeding up the process.[71]
Inquantum mechanics, energy is defined in terms of theenergy operator(Hamiltonian) as a time derivative of thewave function. TheSchrödinger 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 anenergy level) which results in the concept ofquanta.[72]
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 thePlanck relation:, where is thePlanck constant and the frequency. In the case of an electromagnetic wave these energy states are called quanta of light orphotons. Formatter waves, the de Broglie relation yields, where is themomentum.[73]
Relativity
When calculating kinetic energy (work to accelerate amassive body from zerospeed to some finite speed) relativistically – usingLorentz transformations instead ofNewtonian mechanics – Einstein discovered an unexpected by-product of these calculations to be an energy term which does not vanish at zero speed. He called itrest 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:[74]
For example, considerelectron–positron 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 thematter andantimatter (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 calledpair creation – in which the rest mass of the particles is created from a sufficiently energetic photon near a nucleus.[75]
In general relativity, thestress–energy tensor 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.[76][page needed]
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"),[77] 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.
Examples of energy transformation include generatingelectric energy from heat energy via a steam turbine,[79] 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.[80] The Sun transformsnuclear 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 asradiant energy.[81]
There are strict limits to how efficiently heat can be converted intowork in a cyclic process, e.g. in a heat engine, as described byCarnot's theorem and thesecond law of thermodynamics.[82] However, some energy transformations can be quite efficient.[83] The direction of transformations in energy (what kind of energy is transformed to what other kind) is often determined byentropy (equal energy spread among all availabledegrees of freedom) considerations. In practice all energy transformations are permitted on a sufficiently small scale, but certain larger transformations are highly improbable because it is statistically unlikely that energy or matter will randomly move into more concentrated forms or smaller spaces.[84]
Energy transformations in the universe over time are characterized by various kinds of potential energy, that has been available since theBig Bang, being "released" (transformed to more active types of energy such as kinetic or radiant energy) when a triggering mechanism is available.[85] Familiar examples of such processes includenucleosynthesis, a process ultimately using the gravitational potential energy released from thegravitational collapse ofsupernovae to "store" energy in the creation of heavy isotopes (such asuranium andthorium), andnuclear 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.[86] This energy is triggered and released in nuclearfission bombs or in civil nuclear power generation. Similarly, in the case of achemical explosion,chemical potential energy is transformed tokinetic andthermal energy in a very short time.[87]
Yet another example of energy transformation is that of a simple gravitypendulum. At its highest points thekinetic energy is zero and thegravitational potential energy is at its maximum. At its lowest point thekinetic energy is at its maximum and is equal to the decrease inpotential energy. If one (unrealistically) assumes that there is nofriction or other losses, the conversion of energy between these processes would be perfect, and thependulum would continue swinging forever. Energy is transferred from potential energy () to kinetic energy () and then back to potential energy constantly. This is referred to as conservation of energy.
In thisisolated system, 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:
4
The equation can then be simplified further since (mass times acceleration due to gravity times the height) and (half mass times velocity squared). Then the total amount of energy can be found by adding.[88]
Part of the rest energy (equivalent to rest mass) ofmatter 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 is extremely large relative to ordinary human scales, the conversion of an everyday amount of rest mass 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, as can be seen innuclear reactors and nuclear weapons.[89] For example, 1 kg of rest mass equals9×1016 joules, equivalent to 21.5 megatonnes of TNT.[90]
Conversely, the mass equivalent of an everyday amount energy is minuscule. Examples of large-scale transformations between the rest energy of matter and other forms of energy are found innuclear physics andparticle physics. The complete conversion of matter, such as atoms, to non-matter, such as photons, occurs during interaction withantimatter.[91]
Reversible and non-reversible transformations
Thermodynamics divides energy transformation into two kinds:reversible processes andirreversible 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.[92]
At the atomic scale, thermal energy is present in the form of motion and vibrations of individual atoms and molecules. When heat is generated, radiation excites lower energy states of these atoms and their surrounding fields. This heating process acts as a reservoir for part of the applied energy, from which it cannot be converted with 100% efficiency into other forms of energy.[93] According to the second law of thermodynamics, this heat can only be completely recovered as usable energy at the price of an increase in some other kind of heat-like disorder in quantum states.
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 thermodynamicheat 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 aheat engine, or be transformed to other usable forms of energy (through the use of generators attached to heat engines), continues to decrease.[94]
The fact that energy can be neither created nor destroyed is called the law ofconservation of energy. In the form of thefirst law of thermodynamics, this states that aclosed system's energy is constant unless energy is transferred in or out aswork orheat, 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.[95]
While heat can always be fully converted into work in a reversible isothermal expansion of an ideal gas, for cyclic processes of practical interest inheat engines thesecond law of thermodynamics states that the system doing work always loses some energy aswaste heat. This creates a limit to the amount of heat energy that can do work in a cyclic process, a limit called theavailable energy. Mechanical and other forms of energy can be transformed in the other direction intothermal energy without such limitations.[96] The total energy of a system can be calculated by adding up all forms of energy in the system.
There is a fact, or if you wish, alaw, governing all natural phenomena that are known to date. There is no known exception to this law – it is exact so far as we know. The law is called theconservation of energy. It states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.
Most kinds of energy (with gravitational energy being a notable exception)[98] 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.[96][97]
This law is a fundamental principle of physics. As shown rigorously byNoether's theorem, the conservation of energy is a mathematical consequence oftranslational symmetry of time,[99] 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 iscanonical conjugate 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 withenergy 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 systemmass, 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.[100]
Energy transfer can be considered for the special case of systems which areclosed to transfers of matter. The portion of the energy which is transferred byconservative forces over a distance is measured as thework the source system does on the receiving system. The portion of the energy which does not do work during the transfer is calledheat.[note 3] Energy can be transferred between systems in a variety of ways. Examples include the transmission ofelectromagnetic energy via photons, physical collisions which transferkinetic energy,[note 4]tidal interactions,[107] and the conductive transfer ofthermal energy.[108]
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 thefirst law:[note 5][108]
1
where is the amount of energy transferred, represents the work done on or by the system, and represents the heat flow into or out of the system. As a simplification, the heat term,, can sometimes be ignored, especially for fast processes involving gases, which are poor conductors of heat, or when thethermal efficiency of the transfer is high. For suchadiabatic processes,
2
This simplified equation is the one used to define thejoule, 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, one may write:[109]
Internal energy 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.[110]
First law of thermodynamics
Thefirst law of thermodynamics asserts that the total energy of a system and its surroundings (but not necessarilythermodynamic free energy) is always conserved[111] 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 topressure 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 again in energy signified by a positive quantity) is given as:[112]
where the first term on the right is the heat transferred into the system, expressed in terms oftemperatureT andentropyS (in which entropy increases and its change dS 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 isP and volumeV (the negative sign results since compression of the system requires work to be done on it and so the volume change, dV, is negative when work is done on the system).
This equation is highly specific, ignoring all chemical, electrical, nuclear, and gravitational forces, effects such asadvection of any form of energy other than heat andPV-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 aclosed system is expressed in a general form by:[108]
where is the heat supplied to the system and is the work applied to the system.
Equipartition of energy
The energy of a mechanicalharmonic oscillator (a mass on a spring) is alternatelykinetic andpotential energy. At two points in the oscillationcycle it is entirely kinetic, and at two points it is entirely potential.[88] Over a whole cycle, or over many cycles, average energy is equally split between kinetic and potential. This is an example of theequipartition principle: the total energy of a system with many degrees of freedom is equally split among all available degrees of freedom, on average.[113]
This principle is vitally important to understanding the behavior of a quantity closely related to energy, calledentropy. Entropy is a measure of evenness of adistribution of energy between parts of a system. When an isolated system is given more degrees of freedom (i.e., given new availableenergy states 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 thesecond law of thermodynamics. The second law of thermodynamics is simple only for systems which are near or in a physicalequilibrium 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 ofmaximum entropy production.[114][115] It states that nonequilibrium systems behave in such a way as to maximize their entropy production.[116]
^These examples are solely for illustration, as it is not the energy available for work which limits the performance of the athlete but thepower output (in case of a sprinter) and theforce (in case of a weightlifter).
^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 thelattice energy) to the surroundings.
^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 thesecond law of thermodynamics.
^The mechanism for most macroscopic physical collisions is actuallyelectromagnetic, but it is very common to simplify the interaction by ignoring the mechanism of collision and just calculate the beginning and end result.
^abFuller, Ą. J. Baden (2014). Hammon, P. (ed.).Engineering Field Theory. The Commonwealth and international library. Applied electricity and electronics division. Elsevier.ISBN978-1-4831-8700-6.
^Harper, Douglas."Energy".Online Etymology Dictionary.Archived from the original on October 11, 2007. RetrievedMay 1, 2007.
^Chen, Chung-Hwan (September 1956). "Different meanings of the term Energeia in the philosophy of Aristotle".Philosophy and Phenomenological Research.17 (1):56–65.doi:10.2307/2104687.JSTOR2104687.
^McDonough, Jeffrey K. (2021)."Leibniz's Philosophy of Physics". In Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy (Fall 2021 ed.). Metaphysics Research Lab, Stanford University. Retrieved2025-07-25.
^Smith, Crosbie (1998).The Science of Energy – a Cultural History of Energy Physics in Victorian Britain. The University of Chicago Press.ISBN978-0-226-76420-7.
^Ruedenberg, Klaus; Schwarz, W. H. Eugen (February 13, 2013). "Three Millennia of Atoms and Molecules".Pioneers of Quantum Chemistry. ACS Symposium Series. Vol. 1122. American Chemical Society. pp. 1–45.doi:10.1021/bk-2013-1122.ch001.ISBN978-0-8412-2716-3.
^Cohen, Barbara Janson; Hull, Kerry L. (2020).Memmler's Structure & Function of the Human Body, Enhanced Edition (12th ed.). Jones & Bartlett Learning. p. 375.ISBN978-1-284-59160-6.
^Lehninger, Albert L. (1960). "The Enzymic and Morphological Organiation [sic?] of the Mitochondria".Pediatrics.26 (3):466–475.doi:10.1542/peds.26.3.466.
^Ahmadi, Pouria; Dincer, Ibrahim (2018)."Energy Optimization". In Dincer, Ibrahim (ed.).Comprehensive Energy Systems. Vol. 1. Elsevier. pp. 1139–1140.ISBN978-0-12-814925-6.
^Hogan, Craig J. (December 6, 2012)."Energy flow in the universe". In Crittenden, Robert G.; Turok, Neil G. (eds.).Structure Formation in the Universe. NASA Science Series. Kluwer Academic Publishers. pp. 283–292.ISBN978-94-010-0540-1.
^Schmidt, G. R.; et al. (September 2000). "Antimatter Requirements and Energy Costs for Near-Term Propulsion Applications".Journal of Propulsion and Power.16 (5): 923.doi:10.2514/2.5661.hdl:2060/19990110316.
.png)
