﻿ Energy

# Energy

The energy ( Ancient Greek ἐν en "inside" and ἔργον ergon " work " ) is a fundamental physical quantity that plays a central role in all branches of physics as well as in engineering, chemistry, biology and business. Your SI unit is the joule. Energy is the one size that is retained because of the time invariance of the laws of nature, that is, the total energy of a closed system can neither be increased nor diminished ( conservation of energy ). Many introductory texts define energy in an illustrative, but not a general manner as the ability to do work.

Energy is needed to accelerate a body, or to move it against a force, for heating a substance, to compress a gas, to allow electrical current to flow and radiate electromagnetic waves. Plants, animals and humans need energy to live. Energy is also required for the operation of computer systems, telecommunications and for any economic production.

Energy can occur in various forms of energy. These include, for example, potential energy, kinetic energy, chemical energy or thermal energy. Energy can be converted from one form to another, except that the second law of thermodynamics for fundamental limits: thermal energy is limited convertible into other forms of energy and transferable between systems.

Due to the Hamiltonian equations of motion and the Schrödinger equation energy determines the time evolution of physical systems. According to the theory of relativity rest energy and mass are related by the formula.

• 4.1 converting thermal energy into mechanical work
• 4.2 Calculation of the maximum work ( exergy )
• 4.3 Energy and exergy flow diagram of power generation
• 4.4 Solar Energy
• 4.5 Combined Heat and Power (CHP )
• 9.1 Specific energy
• 9.2 Energy supply and consumption
• 9.3 Energy sources

## History of the term

The term first appeared in ancient Greece. Many thinkers dealt with the conversion of kinetic to potential energy in a pendulum (including Galileo Galilei, Christiaan Huygens, Evangelista Torricelli and Gottfried Wilhelm Leibniz ). Result was that kinetic and potential energy had to have an identical size. Leibniz - and later Immanuel Kant - formulated the principle of the conservation of energy. The modern term energy probably goes back to Thomas Young, who still used a purely mechanical connection in 1800 for energy. In connection with the steam engine, the idea that heat energy in many processes the cause of a moving energy, or mechanical work is responsible developed. The starting point was that the water is transferred by heat to the gaseous state, and the gas expansion is used to move a piston in a cylinder. Through the power of movement of the piston, the stored heat energy of the steam is reduced.

The physicist Nicolas Carnot realized that while performing mechanical work a change in volume of steam is required. He also found that the cooling of the hot water into the steam engine is done not only by heat conduction. These findings Carnot published in 1824 in a well-received book on the operating principle of the steam engine. Émile Clapeyron in 1834 brought Carnot's findings into a mathematical form and developed the graphical representation of the Carnot cycle process still used today.

Published in 1841, the German physician Julius Robert Mayer his idea that energy created nor destroyed neither, but can only be converted. He wrote to a friend: " My contention is ...: . Falling force, motion, heat, light, electricity and chemical difference in the Ponderabilien are one and the same object in different manifestations " The amount of heat that is lost in a steam engine, corresponded exactly to the mechanical work done by the machine. This is now known as the " energy conservation ", or " First law of thermodynamics ". The physicist Rudolf Clausius improved in 1854, the ideas about the energy conversion. He showed that only part of the heat energy can be converted into mechanical work. A body in which the temperature remains constant, can not afford mechanical work. Clausius developed the second law of thermodynamics and introduced the concept of entropy. According to the second law, it is impossible that heat from a colder to a hotter body goes.

Hermann von Helmholtz formulated in 1847 the principle of " on the conservation of force " and the impossibility of a perpetuum mobile ( perpetuus, lat forever; mobilis, Latin: mobile) 1 Art Many inventors then wanted to make machines more energy produced than was put into it. Helmholtz found his knowledge by operating with electric power from the galvanic elements, in particular a zinc / bromine cell. In later years he combined the entropy and the heat of a chemical conversion to the free energy.

Josiah Gibbs came in 1878 with similar findings in electrochemical cells. Chemical reactions take place only if the free energy is negative. By means of the free energy can be predicted and whether a chemical substance transformation is possible or how does the chemical equilibrium of a reaction at a temperature change.

The word energy was introduced in 1852 by the Scottish physicist William Rankine in its present sense in the physics corresponding to the above -mentioned ancient Greek meaning ( ἐν = in, inside and ἔργον = factory, knitting ). With this came a clear distinction between the concept of force. Based on considerations of Wilhelm Wien (1900 ), Max Abraham (1902 ), and Hendrik Lorentz (1904 ), Albert Einstein published in 1905 the realization that mass and energy are equivalent.

## Forms of energy and energy conversion

Energy may be included in a system in different ways. These options are called forms of energy. Examples of types of energy are kinetic energy, chemical energy, the electrical energy or potential energy. Various forms of energy can be converted into one another, wherein the sum of the amounts of energy of the different forms of energy before and after the energy conversion is always the same.

A transformation can only be carried out such that all other conservation parameters of the system before and after the conversion have the same value. For example, the conversion of kinetic energy through the conservation of momentum and the angular momentum of the system is limited. A gyro can only be slowed down and lose energy when it emits angular momentum simultaneously. Even at the molecular level, there are such restrictions. Many chemical reactions that would be energetically possible, do not run spontaneously, because they would violate the conservation of momentum. Other conserved quantities are the number of baryons and the number of leptons. To restrict the conversion of energy by nuclear reactions. The energy contained in the mass of matter can be fully converted into another form of energy with an equal amount of antimatter. Without antimatter conversion with the help of nuclear fission or nuclear fusion is possible only to a small extent.

The thermodynamics are with the second law of thermodynamics is another condition for conversion before: The entropy of a closed system can not decrease. Extraction of heat, without the parallel run other processes, means a slowdown. However, a lower temperature corresponds to a reduced entropy and is therefore contrary to the second law. In order to still convert heat into another form of energy, another part of the system must be heated in return for cooling. The conversion of thermal energy into other forms of energy therefore requires the constant temperature difference. Also can not be implemented in all of the stored heat quantity of the temperature difference. Heat engines are used to convert heat into mechanical energy. The ratio of the second law given by the maximum possible work to the consumed amount of heat is called the Carnot efficiency. It is all the greater the greater the temperature difference at which the heat engine operates.

Other conversions are not as strongly affected by the restrictions imposed by conservation laws and thermodynamics. This allows electrical energy with little technical effort almost completely into many other forms of energy. Electric motors convert, for example, into kinetic energy.

Most conversions are performed is not completely in a single form of energy, but it is a part of the energy is converted into heat. In mechanical applications, the heat is usually generated by friction. In electrical applications, the electric resistance or eddy currents are often the cause of the generation of heat. This heat is not normally used and referred to as a loss. In connection with an electric current may occur as an undesirable loss of the radiation of electromagnetic waves. The relationship between successfully converted energy output to energy is called efficiency.

In technical applications, a number of energy conversions is often coupled. In a coal-fired power plant the chemical energy of the coal is first converted by combustion into heat and transferred to water vapor. Turbines convert the heat of the steam into mechanical energy and in turn drive generators that convert mechanical energy into electrical energy.

## Energy in classical mechanics

In classical mechanics the energy of a system is its ability to work to afford. The work converts energy to switch between different forms of energy. The special shape of the Newtonian laws ensured that this does not change the sum of all energies. Friction and the problems associated with their energy losses are not considered in this analysis.

The Noether 's theorem allows a more general definition of energy that takes into account the aspect of energy conservation automatically. All natural laws of classical mechanics are invariant with respect to shifts in time. They are characterized by the fact that they are at all times unchanged in the same form. The Noether Theorem states that, there is a physical quantity about this symmetry with respect to shift in time, the value of which does not change with time. This size is the energy.

It follows from the principle of energy conservation and unavoidable energy losses due to friction, that it is impossible to build a mechanical machine that runs by itself as long as desired ( Perpetuum Mobile ). In addition, the conservation of energy allows, together with the momentum conservation statements about the result of collisions between objects, without the exact mechanism needs to be known in the collision.

### Energy and movement

The kinetic energy is the energy inherent in the state of motion of a body. It is proportional to the mass and the square of the velocity relative to the inertial frame in which to describe the body.

The amount of kinetic energy is therefore dependent on the position from which one describes the system. Often using an inertial system, which rests in relation to the ground.

An extensive body can perform not only a translational motion and rotational motion. The kinetic energy contained in the rotary movement is called rotational energy. This is proportional to the square of the angular velocity and the moment of inertia of the body.

### Energy and potential

Potential energy, also called potential energy of a body is due to its location to a force field, if it is a conservative force. This could be for example the Earth's gravity or the force field of a spring. The potential energy decreases in the direction of force to from and opposite to the force direction, perpendicular to the direction of the force is constant. The body moves from a point at which it has a high potential energy to a point where it is less, he does just as much physical work, as is its potential energy has decreased. This statement is true regardless of what the itinerary of the body passes from one to the other point.

The potential energy of a body with the mass in a homogeneous gravitational field with gravitational acceleration is proportional to the height above the origin of the coordinate system:

In free fall, this potential energy is converted into kinetic energy by the body is accelerated.

Since the coordinate origin can be chosen arbitrarily, the potential energy of the body is never absolute given and not measurable. Measurable are merely their changes.

For periodic motion is regularly turns potential into kinetic energy and back to potential energy. The pendulum is the potential energy, for example a maximum at the reversal points; the kinetic energy is zero here. If the thread just hangs vertically, the mass reaches its maximum speed and thus their maximum kinetic energy; the potential energy is at a minimum here. A planet has in his sonnenfernsten point, although the highest potential, but also the least kinetic energy. Up to the point nearest the Sun, his track speed just increased so much that the increase in kinetic energy compensates for the decrease in potential energy exactly.

Elastic energy, the potential energy of the displaced from its rest position, atoms or molecules in an elastically deformed body, for example a mechanical spring. Generally refers to the energy stored in the elastic or plastic deformation in the body ( or released ) as deformation energy.

## Energy in thermodynamics

Thermal energy is the energy that is stored in the disordered motion of the atoms or molecules of a substance. It is also colloquially referred to as " heat energy" or " heat content ". The conversion of thermal energy into other forms of energy is described by the thermodynamics. Here, between the energy contained in the system (internal energy, enthalpy ) and the heat, the system limit on the transported thermal energy, differentiated.

The sum of the thermal energy, vibrational energy in the body and binding energy is called internal energy. It is also distinguished as internal energy in many sources between the thermal internal energy, the chemical internal energy and nuclear energy, but leaves the framework of thermodynamics.

### Converting thermal energy into mechanical work

While all forms of energy under certain conditions can be completely converted into thermal energy (see # energy forms and conversions) (first law of thermodynamics ), this is not true in the reverse direction. The second law of thermodynamics here describes a very significant limitation ( Fig. 1). Depends on the temperature at which heat is available, only a more or less large portion of a cycle can be converted into mechanical work while the rest is discharged to the environment. In the technical thermodynamics, the convertible shares a form of energy are also called exergy. Exergy is not a state variable in the strict sense, because it depends not only on the state of the system but also on the state of the environment, which is given in a particular case, it must be assumed in general. Then can be illustrated track of exergy flow images of an energy conversion chain, where avoidable losses ( friction or other dissipative processes ) can be reported. In Figure 2 we see that in the conversion of chemical energy (100 % exergy ) into heat at an average temperature of 1000 ° C, the exergy content is only 80 %. If this energy is transferred as heat in a boiler to steam at 273 ° C, then only remain approximately 50% and at the transmission in a heated room at 20 ° C, only about 7%. In this case, an ambient temperature of 0 ° C was always assumed.

### Calculation of the maximum work ( exergy )

In calculating the exergetic proportion of thermal energy is necessary to consider whether the heat source has a constant temperature, as is the case in a boiling water reactor at about 270 ° C, or if the heat output from a struggling medium, flue gas takes place, . In the first case the exergetic content on the Carnot efficiency can be determined from the upper process temperature and the ambient temperature, otherwise you get the heat and the exergy of the surface integral, which from the TS diagram in Figure 3 and from the Ts- diagram can be seen in Figure 4. The formula is:

The relationship can also be read directly from the graphs. Where: T is the absolute temperature in K, S is the entropy in J / K, H is the enthalpy in J, Index 1: initial state index U: ambient condition.

The enthalpy difference is substantially (in this case ) is supplied from the fuel of the combustion air as a thermal energy. It appears as the area under the curve of the isobaric heat. The exergetic share is above the ambient temperature, the other non-usable component that is called " anergy " below this line. With the decrease of exergy in an energy conversion chain is also called a power devaluation.

In the transfer of heat from the flue gas to the working medium, the water is vaporized and is superheated, exergy loss arises another. The maximum obtainable from the steam mass flow may for a mechanical power process with superheated steam of, for example 16 bar and 350 ° C must be calculated by the Carnot efficiency at this temperature. The result with an efficiency of 52% would be wrong. It would contradict the second law, since the mean temperature of the heat supply is lower in the water - steam cycle. If there is no internal heat transfer (regenerative feedwater heating ) from condensing steam in the feed water, such as in steam engines, in which the steam can be brought reversibly to water at ambient condition in the theoretical best case, we achieved at 15 ° C ambient temperature only a maximum efficiency of 34, 4%. The reversible guided Rankine process in Figure 4 with a steam pressure of 32 bar and condensation at 24 ° C, however, reached 37.2%. The real processes achieve only much lower efficiencies at these steam parameters.

### Energy and exergy flow diagram of power generation

Figure 5 shows a simplified energy flow diagram of power generation by a large steam power plant ( steam condition 260 545 ° C, feed water preheating to 276 ° C bar ) is compared with a corresponding Exergieflussbild with the distribution to the end user. It can be seen from the fact that a substantial part of the energy cancellation does not take place in the condenser or downstream of the cooling tower of the power plant, where the heat is removed, but in the conversion of chemical energy of the fuel into thermal energy (combustion ) and in heat transfer from the flue gas to water vapor. The numerical values ​​for the distribution of electricity are approximate and may differ slightly in each individual case.

### Solar energy

The solar energy that passes through radiation on the earth, learns on the way up to the surface a loss of exergy. While the internal energy of the sun is still practically consists of pure exergy at about 15 million K, the sun shines with a surface temperature of about 6000 K at the earth's surface whose temperature is to be set at about 300 K. So by concentrating the sun's rays in a collector you would - even in the high mountains, where the absorption by the Earth's atmosphere plays no role - on the temperature of the sun's surface not get out. This would result in about the Carnot factor, an efficiency of approximately 95%. Then, however, no energy would be transferred. The thermodynamic limit is less at a temperature of 2500 K absorber with an efficiency of about 85%. In practice, dissipative losses are added, from the absorption in the atmosphere, on the material properties of the crystalline cells to the ohmic resistance of the photovoltaic systems, so far only efficiencies of less than 20 % can be achieved. The highest currently achieved efficiency is 18.7%.

### Combined heat and power (CHP )

For heating, heat is usually required with only a small exergy. Therefore, heating is electric current through a resistance heater " energy waste ". Wherever mechanical energy or electricity is generated from heat and at the same time there is demand for heat, the use of waste heat for heating is more useful than the separate provision of heating. In a thermal power station, when it is operated with steam, the steam extracted from the turbine, the temperature of which is barely high enough to conduct the heat of condensation to the consumer via a district heating network. Alternatively (CHP) system uses the heat from stationary internal combustion engines in cogeneration plants. The heat pump is to be mentioned here. She turns to work to absorb heat (energy) from the environment and leave together with the drive work as heating at an appropriately high temperature. If ground water at 10 ° C is available as a heat source is available and there is a space to be heated at 20 ° C, a heat pump with Carnot cycle could be achieved by use of one kilowatt-hour drive work 29 kWh heat supply ( COP = 29). Reliable heat pumps, which are operated by alternately evaporating and condensing refrigerants at different pressures, achieve efficiency ratios of about 3 to 5

## Chemical Energy

Chemical energy as the energy form is referred to, which is stored in the form of a chemical compound in a carrier and can be released energy in chemical reactions. Thus, describing the energy associated with electrical forces in atoms and molecules, and can be divided into the one hand kinetic energy of the electrons in the atoms, and the other part of the electrical energy of the interaction of electrons and protons.

It is released in exothermic reactions and must be added for endothermic reactions.

## Energy in electrodynamics

, In an electric field, if there is no time-varying magnetic field is present, an electrical potential can be defined. A charge carrier then has a potential electrical (electrostatic ) energy that is proportional to the potential and its charge quantity. Since the zero point of the potential can be freely determined, the energy is not completely defined. For two points in the field potential but the difference in energy is independent of the choice of the potential of the zero point. In electrical engineering, usually the potential of the earth is chosen as the zero point of the potential scale.

Arrangements for two electrical conductors, the electrostatic energy is proportional to the square of the difference of the electric potentials of the two conductors. Twice the constant of proportionality is called electrical capacitance. Capacitors are electrical devices, which have high capacity and, therefore, can save energy.

Equivalent to the view that the electrostatic energy is carried by charges, the interpretation that the energy is distributed over the empty space between the charges. The energy density, that is, the energy per volume element is proportional with this approach to the square of electric field strength. Located in the electric field, a dielectric, the energy is also proportional to the dielectric constant.

Moves a charge in vacuo to a place where there is a lower electrical potential, the kinetic energy of the charge increases just as much as the potential energy is lower. This happens, for example, with electrons in an electron tube in a X-ray tube or a cathode ray tube screen. The other hand, moving a charge along a potential gradient in a conductor, she gives her energy absorbed immediately in the form of heat to the head of medium. The power is proportional to the potential gradient and current.

Electrical energy can be transported by moving charge carriers without appreciable potential drop along conductors. This is the case with the aid flowing electrical energy from the power plant to the consumer, for example, in overhead power lines or power cables.

Magnetic energy contained in the magnetic fields, such as in the superconducting magnetic energy store.

In an ideal electrical resonant circuit stored energy is changing continuously between the electric and the magnetic form shape. At any given time is the sum of the partial energies equal to ( energy conservation ). Here, the pure electric magnetic, respectively, part of the energy has double the frequency of the electric oscillation.

## Energy in relativity theory

According to the special theory of relativity, the mass of a stationary object corresponds to a rest energy of

The rest energy is thus up to the factor ( speed of light squared ) of the mass equivalent. The rest energy can be converted in some events in other forms of energy, and vice versa. Thus, the reaction products of nuclear fission and nuclear fusion measurably lower mass than the starting materials. In elementary particle physics, the generation of particles and thus of rest energy is observed from other forms of energy vice versa.

In classical mechanics, the rest energy is not counted because it is irrelevant, as long as not convert particles into other particles.

The general theory of relativity generalizes further the concept of energy and contains a uniform representation of energies and impulses as sources for space curvatures of the energy -momentum tensor. For this can be described by the measurable contractions for an observer sizes win as energy density. For the study of the development of space-times the energy content is crucial. So you can predict the collapse of space-time to a singularity of energy conditions.

## Energy in quantum mechanics

In quantum mechanics, the Hamiltonian determines which energy can be measured on a physical system.

An electromagnetic wave can give only certain amounts of energy. This quantity is proportional to the frequency of the wave and for Planck's quantum of action:

Nuclear energy is the energy of binding of protons and neutrons in the nucleus. It is reacted in a nuclear reaction in the binding energy of the reaction products, ie, new nuclei, and in different types of radiation.

## Technical use of energy

Basically, a power generation is already not possible due to the energy conservation law. The term is used but still in economic life, to express the generation of a certain form of energy (eg electricity) from another form (for example, chemical energy in the form of coal). Similarly, there is, in the strict physical sense, no energy consumption, but economically meant is that the transition from a good usable primary energy (eg oil, gas, coal ) into a no longer reusable form of energy (eg waste heat into the environment). From saving energy is mentioned, if efficient processes are found that require less primary energy for the same purpose, or otherwise, for example by renunciation of consumption, primary energy use is reduced.

The physics describing the above casually introduced "energy consumption" with the exact concept of entropy. While in a closed system the energy is always maintained, the entropy increases with time or always remains at best constant. The higher the entropy, the worse is the energy available. Instead of entropy increase can clearly speak of energy inflation.

The law of increasing entropy in particular, prevents to convert heat energy directly into kinetic energy or electricity. Instead, a heat source and a heat sink ( = cooling) are always required. The maximum efficiency can be calculated from the temperature difference in accordance with Carnot.

The limiting case of an energy conversion without increase in entropy is called a reversible process. As an example, a nearly reversible energy conversion is called a satellite in an elliptical orbit around the Earth: At the highest point of the path he has high potential energy and low kinetic energy at the lowest point of the web, it is just the opposite. The conversion can be done thousands of times a year here without appreciable losses. In superconducting resonators, energy can be millions or even billions of times per second between radiant energy and electrical energy back and hergewandelt, also with losses of less than one per thousand per conversion.

In many processes, which were still connected in the past with high losses ergo significant increase in entropy, technological progress is increasingly lower losses. Thus, an energy saving bulb or LED turns electricity into light more efficiently than an incandescent lamp. A heat pump formed by use of heat from the environment at a given electric power many times more heat than a conventional electric heater at the same power. In other areas of the prior art, but for quite some time close to the theoretical maximum, so that only small improvements are possible here. To turn good electric motors over 90 percent of the injected current into usable mechanical energy, and only a small part in useless heat.

Saving energy means in the physical sense, to minimize the energy devaluation and entropy increase in energy conversion or energy use.

### Specific energy

Specific means in the natural sciences " to a particular base relation " ( related size). The specific energy relative to a certain characteristic of a system which can be described by a physical quantity.

According to DIN 5485, the specific energy is specifically referenced to ground, and the volumetric energy density of the dimensional -related name.

• Energy per volume in J / m³ (Dimension): enthalpy (thermodynamics ), specific latent heat: heat of fusion, heat of vaporization, heat of crystallization or the corresponding enthalpies ( Material Science ), calorific value and net calorific value ( energy technology ), specific compaction energy ( material science ), specific energy of explosive
• Energy per mass in J / kg (Dimension): specific work, specific latent heat (thermodynamics ), calorific value and net calorific value of solid fuels, specific energy of the energy storage (energy technology), Electric capacity and energy density of the plate capacitor (electrical engineering), specific energy of the point mass ( mechanics)

Referred to as non -specific, but as molar thermodynamics and chemical substance-related energy values ​​:

• Energy per amount of substance in J / mol (Dimension): molar latent heat (thermodynamics )

### Energy supply and consumption

With energy supply and consumption, the use of different energies in good usable for human forms is called. The forms of energy most frequently used by people are thermal energy and electrical energy. The human needs are directed primarily to the areas of heating, food preparation and operation of equipment and machinery for food relief. Here, the theme of locomotion and the consumption of fossil energy sources, for example, in vehicles is significant.

The various energy sources over signal lines reach the consumer, such as typically electric power, natural gas, district heating and district heating, or they are largely stored and any transportable, such as coal and lignite, heating oils, fuels (petrol, diesel fuels ), industrial gases, nuclear fuels ( uranium), biomass (wood).

The energy requirement is the world very differently and in industrialized countries is many times higher than for example in the Third World (see list of countries with the highest energy consumption). In industrially developed countries, companies have dealt with the production and supply of energy for general consumption since the 19th century. This is the central generation of electricity and the transmission to the individual consumer to the fore. Furthermore, the procurement, transport and transformation of fuel for heating purposes important industries.

Approximately 40 percent of global energy demand will be met by electrical energy. Leaders within this portion are electric with around 20 percent drives. After illumination with 19 percent, the air conditioning at 16 percent and the information technology is involved with 14 percent of global electricity demand.

## Units

In addition to the SI unit joule were and are depending on the application, other units of energy in use. Watt second (Ws) and volt-ampere second ( VAs ) are identical to the Joule. Also identical with the Joule is the newton meter (Nm). However, since the Newton meter is the SI unit for torque, it is rarely used for specifying energies.

The electron volt (eV ) is used in atomic physics, nuclear physics and elementary particle physics for specifying particle energies and energy levels. Rare comes in atomic physics before the Rydberg. The cgs unit erg is often used in theoretical physics.

The calorie was usual in the calorimetry and is still used today to indicate the physiological caloric value of food. In kilowatt-hours ( kWh) of energy utilities measure the amount of energy delivered to the customer. The coal-fired unit and the oil unit are used to indicate the energy content of primary energy sources. With the TNT equivalent to measure the explosive power of explosives.

The following conversion table respectively the left- specified unit is equal to the number of times the above unit:

### Orders of magnitude

Energy is a quantity that can assume a different many orders of magnitude value in everyday life. Examples are:

## Formulas

• Potential energy of a stretched spring (hence also called clamping energy ):
• Electrical energy in a circuit:
• Energy of a charged plate capacitor:
• Magnetic field energy of a current flowing through an ideal coil:
• Relativistic energy of a free particle of mass with velocity:
• Energy of light quanta (photons)
• Energy of an earthquake:
• Work ( energy change ) is the integral of the force along the distance covered:
• The work done in a system work in the time interval can also be defined on the performance of:
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