# Fundamentals of Energy

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## Introduction Lightning is the electric breakdown of air by strong electric fields and is a flow of energy. The electric potential energy in the atmosphere changes into thermal energy, light, and sound, which are other forms of energy.

In physics, energy (from the Greek ἐνέργεια - energeia, "activity, operation", from ἐνεργός - energos, "active, working") is a quantity that can be assigned to every particle, object, and system of objects as a consequence of the state of that particle, object or system of objects. Different forms of energy include kinetic, potential, thermal, gravitational, sound, elastic, light, and electromagnetic energy. The forms of energy are often named after a related force. German physicist Hermann von Helmholtz established that all forms of energy are equivalent - energy in one form can disappear but the same amount of energy will appear in another form. Energy is subject to a conservation law. Energy is a scalar physical quantity. In the International System of Units (SI), energy is measured in joules, but in some fields other units such as kilowatt-hours and kilocalories are also used.

Any form of energy can be transformed into another form. When energy is in a form other than thermal energy, it may be transformed with good or even perfect efficiency, to any other type of energy. With thermal energy, however, there are often limits to the efficiency of the conversion to other forms of energy, due to the second law of thermodynamics. As an example, oil is reacted with oxygen, potential energy is released, since new chemical bonds are formed in the products which are more powerful than those in the oil and oxygen. The released energy resulting from this process may be converted directly to electricity (as in a fuel cell) with good efficiency. Alternately it may be converted into thermal energy, if the oil is simply burned in order to heat the combustion gases to a certain temperature. In the latter case, however, some of the thermal energy can no longer be used to perform work at that temperature, and is said to be "degraded." As such, it exists in a form unavailable for further transformation. The remainder of the thermal energy may be used to produce any other type of energy, such as electricity.

In all such energy transformation processes, the total energy remains the same. Energy may not be created nor destroyed. This principle, the conservation of energy, was first postulated in the early 19th century, and applies to any isolated system. 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.

Although the total energy of a system does not change with time, its value may depend on the frame of reference. For example, a seated passenger in a moving airplane has zero kinetic energy relative to the airplane, but non-zero kinetic energy (and higher total energy) relative to the Earth.

Energy is subject to a strict global conservation law; that is, 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. (1)

• The total energy of a system can be subdivided and classified in various ways. For example, it is sometimes convenient to distinguish potential energy (which is a function of coordinates only) from kinetic energy (which is a function of coordinate time derivatives only). It may also be convenient to distinguish gravitational energy, electric energy, thermal energy, and other forms. These classifications overlap; for instance, thermal energy usually consists partly of kinetic and partly of potential energy.
• The transfer of energy can take various forms; familiar examples include work, heat flow, and advection, as discussed below.
• The word "energy" is also used outside of physics in many ways, which can lead to ambiguity and inconsistency. The vernacular terminology is not consistent with technical terminology. For example, while energy is always conserved (in the sense that the total energy does not change despite energy transformations), energy can be converted into a form, e.g., thermal energy, that cannot be utilized to perform work. When one talks about "conserving energy by driving less," one talks about conserving fossil fuels and preventing useful energy from being lost as heat. This usage of "conserve" differs from that of the law of conservation of energy.

### Energy and thermodynamics

#### Internal Energy

Internal energy is the sum of all microscopic forms of energy of a system. It is related to the molecular structure and the degree of molecular activity and may be viewed as the sum of kinetic and potential energies of the molecules; it comprises the following types of energy:

Type Composition of internal energy (U)
Sensible energy the portion of the internal energy of a system associated with kinetic energies (molecular translation, rotation, and vibration; electron translation and spin; and nuclear spin) of the molecules.
Latent energy phase of a system.
Chemical energy the internal energy associated with the different kinds of aggregation of atoms in matter.
Nuclear energy the tremendous amount of energy associated with the strong bonds within the nucleus of the atom itself.
Energy interactions those types of energies that are not stored in the system (e.g. heat transfer, mass transfer, and work), but are recognized at the system boundary as they cross it, representing gains or losses by a system during a process.
Thermal energy the sum of sensible and latent forms of internal energy.

### Forms of energy

Classical mechanics distinguishes between potential energy, which is a function of the position of an object, and kinetic energy, which is a function of its movement. Both position and movement are relative to a frame of reference, which must be specified: this is often (and originally) an arbitrary fixed point on the surface of the Earth, the terrestrial frame of reference. It has been attempted to categorize all forms of energy as either kinetic or potential: this is not incorrect, but neither is it clear that it is a real simplification, as Feynman points out:

These notions of potential and kinetic energy depend on a notion of length scale. For example, one can speak of macroscopic potential and kinetic energy, which do not include thermal potential and kinetic energy. Also what is called chemical potential energy (below) is a macroscopic notion, and closer examination shows that it is really the sum of the potential and kinetic energy on the atomic and subatomic scale. Similar remarks apply to nuclear "potential" energy and most other forms of energy. This dependence on length scale is non-problematic if the various length scales are decoupled, as is often the case ... but confusion can arise when different length scales are coupled, for instance when friction converts macroscopic work into microscopic thermal energy.

#### Mechanical energy

Examples of the interconversion of energy
Mechanical energy is converted
into by
Mechanical energy Lever
Thermal energy Brakes
Electric energy Dynamo
Chemical energy Matches
Nuclear energy Particle accelerator

Mechanical energy manifest in many forms,but can be broadly classified into elastic potential energy and kinetic energy. The term potential energy is a very general term, because it exists in all force fields, such as gravitation, electrostatic and magnetic fields. Potential energy refers to the energy any object gets due to its position in a force field.

Potential energy, symbols Ep, V or Φ, is defined as the work done against a given force (= work of given force with minus sign) in changing the position of an object with respect to a reference position (often taken to be infinite separation). If F is the force and s is the displacement, $E_{\rm p} = -\int \mathbf{F}\cdot{\rm d}\mathbf{s}$

with the dot representing the scalar product of the two vectors.

The name "potential" energy originally signified the idea that the energy could readily be transferred as work—at least in an idealized system (reversible process, see below). This is not completely true for any real system, but is often a reasonable first approximation in classical mechanics.

The general equation above can be simplified in a number of common cases, notably when dealing with gravity or with elastic forces.

#### Elastic potential energy As a ball falls freely under the influence of gravity, it accelerates downward, its initial potential energy converting into kinetic energy. On impact with a hard surface the ball deforms, converting the kinetic energy into elastic potential energy. As the ball springs back, the energy converts back firstly to kinetic energy and then as the ball re-gains height into potential energy. Energy conversion to heat due to inelastic deformation and air resistance cause each successive bounce to be lower than the last.

Elastic potential energy is defined as a work needed to compress (or expand) a spring. The force, F, in a spring or any other system which obeys Hooke's law is proportional to the extension or compression, x,

F = − kx

where k is the force constant of the particular spring (or system). In this case, the calculated work becomes $E_{\rm p,e} = {1\over 2}kx^2$

only when k is constant. Hooke's law is a good approximation for behaviour of chemical bonds under normal conditions, i.e. when they are not being broken or formed.

#### Kinetic energy

Kinetic energy, symbols Ek, T or K, is the work required to accelerate an object to a given speed. Indeed, calculating this work one easily obtains the following: $E_{\rm k} = \int \mathbf{F} \cdot d \mathbf{x} = \int \mathbf{v} \cdot d \mathbf{p}= {1\over 2}mv^2$

At speeds approaching the speed of light, c, this work must be calculated using Lorentz transformations, which results in the following: $E_{\rm k} = m c^2\left(\frac{1}{\sqrt{1 - (v/c)^2}} - 1\right)$

Here the two terms on the right hand side are identified with the total energy and the rest energy of the object, respectively. This equation reduces to the one above it, at small (compared to c) speed. The kinetic energy is zero at v=0, so that at rest, the total energy is the rest energy. Thus, a resting mass has the amount of total energy equal to:

Erest = mc2

This energy is thus called rest mass energy.

#### Surface energy

If there is any kind of tension in a surface, such as a stretched sheet of rubber or material interfaces, it is possible to define surface energy. In particular, any meeting of dissimilar materials that don't mix will result in some kind of surface tension, if there is freedom for the surfaces to move then, as seen in capillary surfaces for example, the minimum energy will as usual be sought.

A minimal surface, for example, represents the smallest possible energy that a surface can have if its energy is proportional to the area of the surface. For this reason, (open) soap films of small size are minimal surfaces (small size reduces gravity effects, and openness prevents pressure from building up. Note that a bubble is a minimum energy surface but not a minimal surface by definition).

#### Sound energy

Sound is a form of mechanical vibration, which propagates through any mechanical medium.

#### Gravitational energy

The gravitational force near the Earth's surface varies very little with the height, h, and is equal to the mass, m, multiplied by the gravitational acceleration, g = 9.81 m/s². In these cases, the gravitational potential energy is given by

Ep,g = mgh

A more general expression for the potential energy due to Newtonian gravitation between two bodies of masses m1 and m2, useful in astronomy, is $E_{\rm p,g} = -G{{m_1m_2}\over{r}}$

where r is the separation between the two bodies and G is the gravitational constant, 6.6742(10)×10−11 m3kg−1s−2. In this case, the reference point is the infinite separation of the two bodies.

#### Thermal energy

Examples of the interconversion of energy
Thermal energy is converted
into by
Mechanical energy Steam turbine
Thermal energy Heat exchanger
Electric energy Thermocouple
Chemical energy Blast furnace
Nuclear energy Supernova

Thermal energy (of some media - gas, plasma, solid, etc.) is the energy associated with the microscopical random motion of particles constituting the media. For example, in case of monoatomic gas it is just a kinetic energy of motion of atoms of gas as measured in the reference frame of the center of mass of gas. In case of many-atomic gas rotational and vibrational energy is involved. In the case of liquids and solids there is also potential energy (of interaction of atoms) involved, and so on.

A heat is defined as a transfer (flow) of thermal energy across certain boundary (for example, from a hot body to cold via the area of their contact. A practical definition for small transfers of heat is $\Delta q = \int C_{\rm v}{\rm d}T$

where Cv is the heat capacity of the system. This definition will fail if the system undergoes a phase transition—e.g. if ice is melting to water—as in these cases the system can absorb heat without increasing its temperature. In more complex systems, it is preferable to use the concept of internal energy rather than that of thermal energy (see Chemical energy below).

Despite the theoretical problems, the above definition is useful in the experimental measurement of energy changes. In a wide variety of situations, it is possible to use the energy released by a system to raise the temperature of another object, e.g. a bath of water. It is also possible to measure the amount of electric energy required to raise the temperature of the object by the same amount. The calorie was originally defined as the amount of energy required to raise the temperature of one gram of water by 1 °C (approximately 4.1855 J, although the definition later changed), and the British thermal unit was defined as the energy required to heat one pound of water by 1 °F (later fixed as 1055.06 J).

#### Electrostatic energy

Examples of the interconversion of energy
Electric energy is converted
into by
Mechanical energy Electric motor
Thermal energy Resistor
Electric energy Transformer
Chemical energy Electrolysis
Nuclear energy Synchrotron

The electric potential energy of given configuration of charges is defined as the work which must be done against the Coulomb force to rearrange charges from infinite separation to this configuration (or the work done by the Coulomb force separating the charges from this configuration to infinity). For two point-like charges Q1 and Q2 at a distance r this work, and hence electric potential energy is equal to: $E_{\rm p,e} = {1\over {4\pi\epsilon_0}}{{Q_1Q_2}\over{r}}$

where ε0 is the electric constant of a vacuum, 107/4πc0² or 8.854188…×10−12 F/m. If the charge is accumulated in a capacitor (of capacitance C), the reference configuration is usually selected not to be infinite separation of charges, but vice versa - charges at an extremely close proximity to each other (so there is zero net charge on each plate of a capacitor). The justification for this choice is purely practical - it is easier to measure both voltage difference and magnitude of charges on a capacitor plates not versus infinite separation of charges but rather versus discharged capacitor where charges return to close proximity to each other (electrons and ions recombine making the plates neutral). In this case the work and thus the electric potential energy becomes $E_{\rm p,e} = {{Q^2}\over{2C}}$

#### Electricity energy

If an electric current passes through a resistor, electric energy is converted to heat; if the current passes through an electric appliance, some of the electric energy will be converted into other forms of energy (although some will always be lost as heat). The amount of electric energy due to an electric current can be expressed in a number of different ways: $E = UQ = UIt = Pt = {{U^2}{t}\over{R}} = {I^2}Rt$

where U is the electric potential difference (in volts), Q is the charge (in coulombs), I is the current (in amperes), t is the time for which the current flows (in seconds), P is the power (in watts) and R is the electric resistance (in ohms). The last of these expressions is important in the practical measurement of energy, as potential difference, resistance and time can all be measured with considerable accuracy.

#### Magnetic energy

There is no fundamental difference between magnetic energy and electric energy: the two phenomena are related by Maxwell's equations. The potential energy of a magnet of magnetic moment m in a magnetic field B is defined as the work of magnetic force (actually of magnetic torque) on re-alignment of the vector of the magnetic dipole moment, and is equal: $E_{\rm p,m} = -m\cdot B$

while the energy stored in a inductor (of inductance L) when current I is passing via it is $E_{\rm p,m} = {1\over 2}LI^2$.

This second expression forms the basis for superconducting magnetic energy storage.

#### Electromagnetic Energy

Examples of the interconversion of energy
into by
Mechanical energy Solar sail
Thermal energy Solar collector
Electric energy Solar cell
Chemical energy Photosynthesis
Nuclear energy Mössbauer spectroscopy

Calculating work needed to create an electric or magnetic field in unit volume (say, in a capacitor or an inductor) results in the electric and magnetic fields energy densities: $u_e=\frac{\epsilon_0}{2} E^2$

and $u_m=\frac{1}{2\mu_0} B^2$,

in SI units.

Electromagnetic radiation, such as microwaves, visible light or gamma rays, represents a flow of electromagnetic energy. Applying the above expressions to magnetic and electric components of electromagnetic field both the volumetric density and the flow of energy in e/m field can be calculated. The resulting Poynting vector, which is expressed as $\mathbf{S} = \frac{1}{\mu} \mathbf{E} \times \mathbf{B},$

in SI units, gives the density of the flow of energy and its direction.

The energy of electromagnetic radiation is quantized (has discrete energy levels). The spacing between these levels is equal to

E = hν

where h is the Planck constant, 6.6260693(11)×10−34 Js, ν is the frequency of the radiation. This quantity of electromagnetic energy is usually called a photon. The photons which make up visible light have energies of 270–520 yJ, equivalent to 160–310 kJ/mol, the strength of weaker chemical bonds.

#### Chemical energy

Chemical energy is the energy due to associations of atoms in molecules and various other kinds of aggregates of matter. It may be defined as a work done by electric forces during re-arrangement of mutual positions of electric charges, electrons and protons, in the process of aggregation. So, basically it is electrostatic potential energy of electric charges. If the chemical energy of a system decreases during a chemical reaction, the difference is transferred to the surroundings in some form (often heat or light); on the other hand if the chemical energy of a system increases as a result of a chemical reaction - the difference then is supplied by the surroundings (usually again in form of heat or light). For example,

when two hydrogen atoms react to form a dihydrogen molecule, the chemical energy decreases by 724 zJ (the bond energy of the H–H bond);

when the electron is completely removed from a hydrogen atom, forming a hydrogen ion (in the gas phase), the chemical energy increases by 2.18 aJ (the ionization energy of hydrogen).

It is common to quote the changes in chemical energy for one mole of the substance in question: typical values for the change in molar chemical energy during a chemical reaction range from tens to hundreds of kilojoules per mole.

Since the industrial revolution, the burning of coal, oil, natural gas or products derived from them has been a socially significant transformation of chemical energy into other forms of energy. the energy "consumption" (one should really speak of "energy transformation") of a society or country is often quoted in reference to the average energy released by the combustion of these fossil fuels:

1 tonne of coal equivalent (TCE) = 29.3076 GJ = 8,141 kilowatt hour

1 tonne of oil equivalent (TOE) = 41.868 GJ = 11,630 kilowatt hour

On the same basis, a tank-full of gasoline (45 litres, 12 gallons) is equivalent to about 1.6 GJ of chemical energy. Another chemically based unit of measurement for energy is the "tonne of TNT", taken as 4.184 GJ. Hence, burning a tonne of oil releases about ten times as much energy as the explosion of one tonne of TNT: fortunately, the energy is usually released in a slower, more controlled manner.

Simple examples of storage of chemical energy are batteries and food. When food is digested and metabolized (often with oxygen), chemical energy is released, which can in turn be transformed into heat, or by muscles into kinetic energy.

#### Nuclear energy

Nuclear potential energy, along with electric potential energy, provides the energy released from nuclear fission and nuclear fusion processes. The result of both these processes are nuclei in which the more-optimal size of the nucleus allows the nuclear force (which is opposed by the electromagnetic force) to bind nuclear particles more tightly together than before the reaction.

The Weak nuclear force (different from the strong force) provides the potential energy for certain kinds of radioactive decay, such as beta decay.

The energy released in nuclear processes is so large that the relativistic change in mass (after the energy has been removed) can be as much as several parts per thousand.

Nuclear particles (nucleons) like protons and neutrons are not destroyed (law of conservation of baryon number) in fission and fusion processes. A few lighter particles may be created or destroyed (example: beta minus and beta plus decay, or electron capture decay), but these minor processes are not important to the immediate energy release in fission and fusion. Rather, fission and fusion release energy when collections of baryons become more tightly bound, and it is the energy associated with a fraction of the mass of the nucleons (but not the whole particles) which appears as the heat and electromagnetic radiation generated by nuclear reactions. This heat and radiation retains the "missing" mass, but the mass is missing only because it escapes in the form of heat and light, which retain the mass and conduct it out of the system where it is not measured.

The energy from the Sun, also called solar energy, is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million metric tons of solar matter per second into light, which is radiated into space, but during this process, the number of total protons and neutrons in the sun does not change. In this system, the light itself retains the inertial equivalent of this mass, and indeed the mass itself (as a system), which represents 4 million tons per second of electromagnetic radiation, moving into space. Each of the helium nuclei which are formed in the process are less massive than the four protons from they were formed, but (to a good approximation), no particles or atoms are destroyed in the process of turning the sun's nuclear potential energy into light.

### Transformations of energy

One form of energy can often be readily transformed into another with the help of a device- for instance, a battery, from chemical energy to electric energy; a dam: gravitational potential energy to kinetic energy of moving water (and the blades of a turbine) and ultimately to electric energy through an electric generator. Similarly, in the case of a chemical explosion, chemical potential energy is transformed to kinetic energy 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 maximum. At its lowest point the kinetic energy is at maximum and is equal to the decrease of potential energy. If one (unrealistically) assumes that there is no friction, the conversion of energy between these processes is perfect, and the pendulum will continue swinging forever.

Energy gives rise to weight and is equivalent to matter and vice versa. The formula E = mc², derived by Albert Einstein (1905) quantifies the relationship between mass and rest energy within the concept of special relativity. In different theoretical frameworks, similar formulas were derived by J. J. Thomson (1881), Henri Poincaré (1900), Friedrich Hasenöhrl (1904) and others (see Mass-energy equivalence#History for further information). Since c2 is extremely large relative to ordinary human scales, the conversion of ordinary amount of mass (say, 1 kg) to other forms of energy can liberate tremendous amounts of energy (~9x1016 joules), as can be seen in nuclear reactors and nuclear weapons. Conversely, the mass equivalent of a unit of energy is minuscule, which is why a loss of energy from most systems is difficult to measure by weight, unless the energy loss is very large. Examples of energy transformation into matter (particles) are found in high energy nuclear physics.

In nature, transformations of energy can be fundamentally classed into two kinds: those that are thermodynamically reversible, and those that are thermodynamically irreversible. A reversible process in thermodynamics is one in which no 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 exitations 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 heat, 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 in time, more and more of its energy becomes trapped in irreversible states (i.e., as heat or other kinds of increases in disorder). This has been referred to as 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 produce work through a heat engine, or be transformed to other usable forms of energy (through the use of generators attached to heat engines), grows less and less.

### Law of conservation of energy

Energy is subject to the law of conservation of energy. According to this law, energy can neither be created (produced) nor destroyed by itself. It can only be transformed.

Most kinds of energy (with gravitational energy being a notable exception)(2) are also 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.(3) Conservation of energy is the mathematical consequence of translational symmetry of time (that is, the indistinguishability of time intervals taken at different time) - see Noether's theorem. (4)

According to energy conservation law 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.

This law is a fundamental principle of physics. It follows from the 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 canonical 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. 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.

### Energy and life

Any living organism relies on an external source of energy—radiation from the Sun in the case of green plants; chemical energy in some form in the case of animals—to be able to grow and reproduce. The daily 1500–2000 Calories (6–8 MJ) recommended for a human adult are taken as a combination of oxygen and food molecules, the latter mostly carbohydrates and fats, of which glucose (C6H12O6) and stearin (C57H110O6) are convenient examples. The food molecules are oxidised to carbon dioxide and water in the mitochondria

C6H12O6 + 6O2 → 6CO2 + 6H2O
C57H110O6 + 81.5O2 → 57CO2 + 55H2O

and some of the energy is used to convert ADP into ATP

ADP + HPO42− → ATP + H2O

The rest of the chemical energy in the carbohydrate or fat is converted into heat: the ATP is used as a sort of "energy currency", and some of the chemical energy it contains when split and reacted with water, is used for other metabolism (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: (a).

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: 3kJ

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 energy or radiation), and it is true that most real machines 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")(b). 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: 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. (5)

### Units

Throughout the history of science, energy has been expressed in several different units such as ergs and calories. At present, the accepted unit of measurement for energy is the SI unit of energy, the joule. In addition to the joule, other units of energy include the kilowatt hour (kWh) and the British thermal unit (Btu). These are both larger units of energy. One kWh is equivalent to exactly 3.6 million joules, and one Btu is equivalent to about 1055 joules.(6)

## References

(1) Berkeley Physics Course Volume 1. Charles Kittel, Walter D Knight and Malvin A Ruderman

(2) The Laws of Thermodynamics including careful definitions of energy, free energy, et cetera.

(5) Ito, Akihito; Oikawa, Takehisa (2004). "Global Mapping of Terrestrial Primary Productivity and Light-Use Efficiency with a Process-Based Model." in Shiyomi, M. et al. (Eds.) Global Environmental Change in the Ocean and on Land. pp. 343–58.

(6) Ristinen, Robert A., and Kraushaar, Jack J. Energy and the Environment. New York: John Wiley & Sons, Inc., 2006.

(7) http://en.wikipedia.org/wiki/Energy - wikipedia.com

## Notes

(a) 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 of the sprinter and the force of the weightlifter. A worker stacking shelves in a supermarket does more work (in the physical sense) than either of the athletes, but does it more slowly.

(b) 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.