In

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^{3} = 0.001 m^{3}); the gas within is the air consisting of molecular nitrogen and oxygen only (thus a diatomic gas with 5 degrees of freedom, and so ); the compression ratio of the engine is 10:1 (that is, the 1 L volume of uncompressed gas is reduced to 0.1 L by the piston); and the uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27 °C, or 300 K, and a pressure of 1 bar = 100 kPa, i.e. typical sea-level atmospheric pressure).
: $\backslash begin\; \&\; P\_1\; V\_1^\backslash gamma\; =\; \backslash mathrm\_1\; =\; 100\backslash ,000~\backslash text\; \backslash times\; (0.001~\backslash text^3)^\backslash frac75\; \backslash \backslash \; \&\; =\; 10^5\; \backslash times\; 6.31\; \backslash times\; 10^~\backslash text\backslash ,\backslash text^\; =\; 6.31~\backslash text\backslash ,\backslash text^,\; \backslash end$
so the adiabatic constant for this example is about 6.31 Pa m^{4.2}.
The gas is now compressed to a 0.1 L (0.0001 m^{3}) volume, which we assume happens quickly enough that no heat enters or leaves the gas through the walls. The adiabatic constant remains the same, but with the resulting pressure unknown
: $P\_2\; V\_2^\backslash gamma\; =\; \backslash mathrm\_1\; =\; 6.31~\backslash text\backslash ,\backslash text^\; =\; P\; \backslash times\; (0.0001~\backslash text^3)^\backslash frac75,$
We can now solve for the final pressure
: $P\_2\; =\; P\_1\backslash left\; (\backslash frac\backslash right)^\backslash gamma\; =\; 100\backslash ,000~\backslash text\; \backslash times\; \backslash text^\; =\; 2.51\; \backslash times\; 10^6~\backslash text$
or 25.1 bar. This pressure increase is more than a simple 10:1 compression ratio would indicate; this is because the gas is not only compressed, but the work done to compress the gas also increases its internal energy, which manifests itself by a rise in the gas temperature and an additional rise in pressure above what would result from a simplistic calculation of 10 times the original pressure.
We can solve for the temperature of the compressed gas in the engine cylinder as well, using the ideal gas law, ''PV'' = ''nRT'' (''n'' is amount of gas in moles and ''R'' the gas constant for that gas). Our initial conditions being 100 kPa of pressure, 1 L volume, and 300 K of temperature, our experimental constant (''nR'') is:
: $\backslash frac\; =\; \backslash mathrm\_2\; =\; \backslash frac\; =\; 0.333~\backslash text\backslash ,\backslash text^3\backslash text^.$
We know the compressed gas has = 0.1 L and = , so we can solve for temperature:
: $T\; =\; \backslash frac\; =\; \backslash frac\; =\; 753~\backslash text.$
That is a final temperature of 753 K, or 479 °C, or 896 °F, well above the ignition point of many fuels. This is why a high-compression engine requires fuels specially formulated to not self-ignite (which would cause

Miscellaneous Scientific Papers

', edited by W.J. Millar, 1881, Charles Griffin, London. and adopted by Maxwell in 1871 (explicitly attributing the term to Rankine). The etymological origin corresponds here to an impossibility of transfer of energy as heat and of transfer of matter across the wall. The Greek word ἀδιάβατος is formed from privative ἀ- ("not") and διαβατός, "passable", in turn deriving from διά ("through"), and βαῖνειν ("to walk, go, come").

Adiabatic Processes

. HyperPhysics. * Thorngren, Dr. Jane R..

. Daphne – A Palomar College Web Server., 21 July 1995..

{{DEFAULTSORT:Adiabatic Process Thermodynamic processes Atmospheric thermodynamics Entropy

thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of t ...

, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...

or mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...

between the thermodynamic system
A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that are ...

and its environment. Unlike an isothermal process, an adiabatic process transfers energy to the surroundings only as work.. A translation may be founhere

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thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of t ...

, the adiabatic process supports the theory that explains the first law of thermodynamics
The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amou ...

.
Some chemical and physical processes occur too rapidly for energy to enter or leave the system as heat, allowing a convenient "adiabatic approximation".Bailyn, M. (1994), pp. 52–53. For example, the adiabatic flame temperature uses this approximation to calculate the upper limit of flame
A flame (from Latin '' flamma'') is the visible, gaseous part of a fire. It is caused by a highly exothermic chemical reaction taking place in a thin zone. When flames are hot enough to have ionized gaseous components of sufficient density they ...

temperature by assuming combustion loses no heat to its surroundings.
In meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...

and oceanography
Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynamics ...

, adiabatic cooling produces condensation of moisture or salinity, oversaturating the parcel. Therefore, the excess must be removed. There, the process becomes a ''pseudo-adiabatic process'' whereby the liquid water or salt that condenses is assumed to be removed upon formation by idealized instantaneous precipitation
In meteorology, precipitation is any product of the condensation of atmospheric water vapor that falls under gravitational pull from clouds. The main forms of precipitation include drizzle, rain, sleet, snow, ice pellets, graupel and hai ...

. The pseudoadiabatic process is only defined for expansion because a compressed parcel becomes warmer and remains undersaturated.
Description

A process without transfer of heat to or from a system, so that , is called adiabatic, and such a system is said to be adiabatically isolated. The simplifying assumption frequently made is that a process is adiabatic. For example, the compression of a gas within a cylinder of an engine is assumed to occur so rapidly that on the time scale of the compression process, little of the system's energy can be transferred out as heat to the surroundings. Even though the cylinders are not insulated and are quite conductive, that process is idealized to be adiabatic. The same can be said to be true for the expansion process of such a system. The assumption of adiabatic isolation is useful and often combined with other such idealizations to calculate a good first approximation of a system's behaviour. For example, according to Laplace, when sound travels in a gas, there is no time for heat conduction in the medium, and so the propagation of sound is adiabatic. For such an adiabatic process, themodulus of elasticity
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...

(Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...

) can be expressed as , where is the ratio of specific heats at constant pressure and at constant volume ( ) and is the pressure of the gas.
Various applications of the adiabatic assumption

For a closed system, one may write thefirst law of thermodynamics
The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amou ...

as : , where denotes the change of the system's internal energy, the quantity of energy added to it as heat, and the work done by the system on its surroundings.
*If the system has such rigid walls that work cannot be transferred in or out (), and the walls are not adiabatic and energy is added in the form of heat (), and there is no phase change, then the temperature of the system will rise.
*If the system has such rigid walls that pressure–volume work cannot be done, but the walls are adiabatic (), and energy is added as isochoric (constant volume) work in the form of friction or the stirring of a viscous fluid within the system (), and there is no phase change, then the temperature of the system will rise.
*If the system walls are adiabatic () but not rigid (), and, in a fictive idealized process, energy is added to the system in the form of frictionless, non-viscous pressure–volume work (), and there is no phase change, then the temperature of the system will rise. Such a process is called an isentropic process
In thermodynamics, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of heat or matter. Such an idealized proces ...

and is said to be "reversible". Ideally, if the process were reversed the energy could be recovered entirely as work done by the system. If the system contains a compressible gas and is reduced in volume, the uncertainty of the position of the gas is reduced, and seemingly would reduce the entropy of the system, but the temperature of the system will rise as the process is isentropic (). Should the work be added in such a way that friction or viscous forces are operating within the system, then the process is not isentropic, and if there is no phase change, then the temperature of the system will rise, the process is said to be "irreversible", and the work added to the system is not entirely recoverable in the form of work.
*If the walls of a system are not adiabatic, and energy is transferred in as heat, entropy is transferred into the system with the heat. Such a process is neither adiabatic nor isentropic, having , and according to the second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unless ...

.
Naturally occurring adiabatic processes are irreversible (entropy is produced).
The transfer of energy as work into an adiabatically isolated system can be imagined as being of two idealized extreme kinds. In one such kind, no entropy is produced within the system (no friction, viscous dissipation, etc.), and the work is only pressure-volume work (denoted by ). In nature, this ideal kind occurs only approximately because it demands an infinitely slow process and no sources of dissipation.
The other extreme kind of work is isochoric work (), for which energy is added as work solely through friction or viscous dissipation within the system. A stirrer that transfers energy to a viscous fluid of an adiabatically isolated system with rigid walls, without phase change, will cause a rise in temperature of the fluid, but that work is not recoverable. Isochoric work is irreversible. The second law of thermodynamics observes that a natural process, of transfer of energy as work, always consists at least of isochoric work and often both of these extreme kinds of work. Every natural process, adiabatic or not, is irreversible, with , as friction or viscosity are always present to some extent.
Adiabatic heating and cooling

The adiabatic compression of a gas causes a rise in temperature of the gas. Adiabatic expansion against pressure, or a spring, causes a drop in temperature. In contrast,free expansion
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...

is an isothermal
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and ...

process for an ideal gas.
Adiabatic heating occurs when the pressure of a gas is increased by work done on it by its surroundings, e.g., a piston
A piston is a component of reciprocating engines, reciprocating pumps, gas compressors, hydraulic cylinders and pneumatic cylinders, among other similar mechanisms. It is the moving component that is contained by a cylinder and is made gas-t ...

compressing a gas contained within a cylinder and raising the temperature where in many practical situations heat conduction through walls can be slow compared with the compression time. This finds practical application in diesel engines
The diesel engine, named after Rudolf Diesel, is an internal combustion engine in which ignition of the fuel is caused by the elevated temperature of the air in the cylinder due to mechanical compression; thus, the diesel engine is a so-cal ...

which rely on the lack of heat dissipation during the compression stroke to elevate the fuel vapor temperature sufficiently to ignite it.
Adiabatic heating occurs in the Earth's atmosphere
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing for ...

when an air mass
In meteorology, an air mass is a volume of air defined by its temperature and humidity. Air masses cover many hundreds or thousands of square miles, and adapt to the characteristics of the surface below them. They are classified according to la ...

descends, for example, in a katabatic wind
A katabatic wind (named from the Greek word κατάβασις '' katabasis'', meaning "descending") is a drainage wind, a wind that carries high-density air from a higher elevation down a slope under the force of gravity. Such winds are somet ...

, Foehn wind
A Foehn or Föhn (, , ), is a type of dry, relatively warm, downslope wind that occurs in the lee (downwind side) of a mountain range.
It is a rain shadow wind that results from the subsequent adiabatic warming of air that has dropped most of ...

, or chinook wind
Chinook winds, or simply Chinooks, are two types of prevailing warm, generally westerly winds in western North America: Coastal Chinooks and interior Chinooks. The coastal Chinooks are persistent seasonal, wet, southwesterly winds blowing in from ...

flowing downhill over a mountain range. When a parcel of air descends, the pressure on the parcel increases. Because of this increase in pressure, the parcel's volume decreases and its temperature increases as work is done on the parcel of air, thus increasing its internal energy, which manifests itself by a rise in the temperature of that mass of air. The parcel of air can only slowly dissipate the energy by conduction or radiation (heat), and to a first approximation it can be considered adiabatically isolated and the process an adiabatic process.
Adiabatic cooling occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand, thus causing it to do work on its surroundings. When the pressure applied on a parcel of gas is reduced, the gas in the parcel is allowed to expand; as the volume increases, the temperature falls as its internal energy decreases. Adiabatic cooling occurs in the Earth's atmosphere with orographic lifting and lee waves
In meteorology, lee waves are atmospheric stationary waves. The most common form is mountain waves, which are atmospheric internal gravity waves. These were discovered in 1933 by two German glider pilots, Hans Deutschmann and Wolf Hirth, above ...

, and this can form pilei or lenticular clouds.
Due in part to adiabatic cooling in mountainous areas, snowfall infrequently occurs in some parts of the Sahara desert.
Adiabatic cooling does not have to involve a fluid. One technique used to reach very low temperatures (thousandths and even millionths of a degree above absolute zero) is via adiabatic demagnetisation, where the change in magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...

on a magnetic material is used to provide adiabatic cooling. Also, the contents of an expanding universe can be described (to first order) as an adiabatically cooling fluid. (See heat death of the universe
The heat death of the universe (also known as the Big Chill or Big Freeze) is a hypothesis on the ultimate fate of the universe, which suggests the universe will evolve to a state of no thermodynamic free energy, and will therefore be unab ...

.)
Rising magma also undergoes adiabatic cooling before eruption, particularly significant in the case of magmas that rise quickly from great depths such as kimberlite
Kimberlite is an igneous rock and a rare variant of peridotite. It is most commonly known to be the main host matrix for diamonds. It is named after the town of Kimberley in South Africa, where the discovery of an diamond called the Star of ...

s.
In the Earth's convecting mantle (the asthenosphere) beneath the lithosphere, the mantle temperature is approximately an adiabat. The slight decrease in temperature with shallowing depth is due to the decrease in pressure the shallower the material is in the Earth.
Such temperature changes can be quantified using the ideal gas law, or the hydrostatic equation for atmospheric processes.
In practice, no process is truly adiabatic. Many processes rely on a large difference in time scales of the process of interest and the rate of heat dissipation across a system boundary, and thus are approximated by using an adiabatic assumption. There is always some heat loss, as no perfect insulators exist.
Ideal gas (reversible process)

The mathematical equation for anideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...

undergoing a reversible (i.e., no entropy generation) adiabatic process can be represented by the polytropic process equation
:$P\; V^\backslash gamma\; =\; \backslash text,$
where is pressure, is volume, and is the adiabatic index or heat capacity ratio defined as
: $\backslash gamma\; =\; \backslash frac\; =\; \backslash frac.$
Here is the specific heat for constant pressure, is the specific heat for constant volume, and is the number of degrees of freedom (3 for a monatomic gas, 5 for a diatomic gas or a gas of linear molecules such as carbon dioxide).
For a monatomic ideal gas, , and for a diatomic gas (such as nitrogen
Nitrogen is the chemical element with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at seven ...

and oxygen
Oxygen is the chemical element with the chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen Group (periodic table), group in the periodic table, a highly Chemical reaction, reactive nonmetal, and an oxidizing a ...

, the main components of air), . Note that the above formula is only applicable to classical ideal gases (that is, gases far above absolute zero temperature) and not Bose–Einstein or Fermi gases.
One can also use the ideal gas law to rewrite the above relationship between and as
: $P^\; T^\backslash gamma\; =\; \backslash text,$
: $TV^\; =\; \backslash text.$
where ''T'' is the absolute or thermodynamic temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wo ...

.
Example of adiabatic compression

The compression stroke in agasoline engine
A petrol engine (gasoline engine in American English) is an internal combustion engine designed to run on petrol (gasoline). Petrol engines can often be adapted to also run on fuels such as liquefied petroleum gas and ethanol blends (such as ' ...

can be used as an example of adiabatic compression. The model assumptions are: the uncompressed volume of the cylinder is one litre (1 L = 1000 cmengine knocking
In spark ignition internal combustion engines, knocking (also knock, detonation, spark knock, pinging or pinking) occurs when combustion of some of the air/fuel mixture in the cylinder does not result from propagation of the flame front ignite ...

when operated under these conditions of temperature and pressure), or that a supercharger
In an internal combustion engine, a supercharger compresses the intake gas, forcing more air into the engine in order to produce more power for a given displacement.
The current categorisation is that a supercharger is a form of forced induc ...

with an intercooler
An intercooler is a heat exchanger used to cool a gas after compression. Often found in turbocharged engines, intercoolers are also used in air compressors, air conditioners, refrigeration and gas turbines.
Internal combustion engines
M ...

to provide a pressure boost but with a lower temperature rise would be advantageous. A diesel engine operates under even more extreme conditions, with compression ratios of 16:1 or more being typical, in order to provide a very high gas temperature, which ensures immediate ignition of the injected fuel.
Adiabatic free expansion of a gas

For an adiabatic free expansion of an ideal gas, the gas is contained in an insulated container and then allowed to expand in a vacuum. Because there is no external pressure for the gas to expand against, the work done by or on the system is zero. Since this process does not involve any heat transfer or work, the first law of thermodynamics then implies that the net internal energy change of the system is zero. For an ideal gas, the temperature remains constant because the internal energy only depends on temperature in that case. Since at constant temperature, the entropy is proportional to the volume, the entropy increases in this case, therefore this process is irreversible.Derivation of ''P''–''V'' relation for adiabatic heating and cooling

The definition of an adiabatic process is that heat transfer to the system is zero, . Then, according to the first law of thermodynamics, where is the change in the internal energy of the system and is work done ''by'' the system. Any work () done must be done at the expense of internal energy , since no heat is being supplied from the surroundings. Pressure–volume work done ''by'' the system is defined as However, does not remain constant during an adiabatic process but instead changes along with . It is desired to know how the values of and relate to each other as the adiabatic process proceeds. For an ideal gas (recall ideal gas law ) the internal energy is given by where is the number of degrees of freedom divided by 2, is theuniversal gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment pe ...

and is the number of moles in the system (a constant).
Differentiating equation (a3) yields
Equation (a4) is often expressed as because .
Now substitute equations (a2) and (a4) into equation (a1) to obtain
: $-P\; \backslash ,\; dV\; =\; \backslash alpha\; P\; \backslash ,\; dV\; +\; \backslash alpha\; V\; \backslash ,\; dP,$
factorize :
: $-(\backslash alpha\; +\; 1)\; P\; \backslash ,\; dV\; =\; \backslash alpha\; V\; \backslash ,\; dP,$
and divide both sides by :
: $-(\backslash alpha\; +\; 1)\; \backslash frac\; =\; \backslash alpha\; \backslash frac.$
After integrating the left and right sides from to and from to and changing the sides respectively,
: $\backslash ln\; \backslash left(\; \backslash frac\; \backslash right)\; =\; -\backslash frac\; \backslash ln\; \backslash left(\; \backslash frac\; \backslash right).$
Exponentiate both sides, substitute with , the heat capacity ratio
: $\backslash left(\; \backslash frac\; \backslash right)\; =\; \backslash left(\; \backslash frac\; \backslash right)^,$
and eliminate the negative sign to obtain
: $\backslash left(\; \backslash frac\; \backslash right)\; =\; \backslash left(\; \backslash frac\; \backslash right)^\backslash gamma.$
Therefore,
: $\backslash left(\; \backslash frac\; \backslash right)\; \backslash left(\; \backslash frac\; \backslash right)^\backslash gamma\; =\; 1,$
and
: $P\_0\; V\_0^\backslash gamma\; =\; P\; V^\backslash gamma\; =\; \backslash mathrm.$
At the same time, the work done by the pressure–volume changes as a result from this process, is equal to
Since we require the process to be adiabatic, the following equation needs to be true
By the previous derivation,
Rearranging (b4) gives
:$P\; =\; P\_1\; \backslash left(\backslash frac\; \backslash right)^\backslash gamma.$
Substituting this into (b2) gives
:$W\; =\; \backslash int\_^\; P\_1\; \backslash left(\backslash frac\; \backslash right)^\backslash gamma\; \backslash ,dV.$
Integrating we obtain the expression for work,
:$W\; =\; P\_1\; V\_1^\backslash gamma\; \backslash frac\; =\; \backslash frac.$
Substituting in second term,
:$W\; =\; -\backslash alpha\; P\_1\; V\_1^\backslash gamma\; \backslash left(\; V\_2^\; -\; V\_1^\; \backslash right).$
Rearranging,
:$W\; =\; -\backslash alpha\; P\_1\; V\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right).$
Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases),
:$W\; =\; -\backslash alpha\; n\; R\; T\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right).$
By the continuous formula,
:$\backslash frac\; =\; \backslash left(\backslash frac\backslash right)^,$
or
:$\backslash left(\backslash frac\backslash right)^\; =\; \backslash frac.$
Substituting into the previous expression for ,
:$W\; =\; -\backslash alpha\; n\; R\; T\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right).$
Substituting this expression and (b1) in (b3) gives
:$\backslash alpha\; n\; R\; (T\_2\; -\; T\_1)\; =\; \backslash alpha\; n\; R\; T\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right).$
Simplifying,
:$T\_2\; -\; T\_1\; =\; T\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right),$
:$\backslash frac\; -\; 1\; =\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1,$
:$T\_2\; =\; T\_1\; \backslash left(\; \backslash frac\; \backslash right)^.$
Derivation of discrete formula and work expression

The change in internal energy of a system, measured from state 1 to state 2, is equal to : At the same time, the work done by the pressure–volume changes as a result from this process, is equal to Since we require the process to be adiabatic, the following equation needs to be true By the previous derivation, Rearranging (c4) gives :$P\; =\; P\_1\; \backslash left(\backslash frac\; \backslash right)^\backslash gamma.$ Substituting this into (c2) gives :$W\; =\; \backslash int\_^\; P\_1\; \backslash left(\backslash frac\; \backslash right)^\backslash gamma\; \backslash ,dV.$ Integrating we obtain the expression for work, :$W\; =\; P\_1\; V\_1^\backslash gamma\; \backslash frac\; =\; \backslash frac.$ Substituting in second term, :$W\; =\; -\backslash alpha\; P\_1\; V\_1^\backslash gamma\; \backslash left(\; V\_2^\; -\; V\_1^\; \backslash right).$ Rearranging, :$W\; =\; -\backslash alpha\; P\_1\; V\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right).$ Using the ideal gas law and assuming a constant molar quantity (as often happens in practical cases), :$W\; =\; -\backslash alpha\; n\; R\; T\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right).$ By the continuous formula, :$\backslash frac\; =\; \backslash left(\backslash frac\backslash right)^,$ or :$\backslash left(\backslash frac\backslash right)^\; =\; \backslash frac.$ Substituting into the previous expression for , :$W\; =\; -\backslash alpha\; n\; R\; T\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right).$ Substituting this expression and (c1) in (c3) gives :$\backslash alpha\; n\; R\; (T\_2\; -\; T\_1)\; =\; \backslash alpha\; n\; R\; T\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right).$ Simplifying, :$T\_2\; -\; T\_1\; =\; T\_1\; \backslash left(\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1\; \backslash right),$ :$\backslash frac\; -\; 1\; =\; \backslash left(\; \backslash frac\; \backslash right)^\; -\; 1,$ :$T\_2\; =\; T\_1\; \backslash left(\; \backslash frac\; \backslash right)^.$Graphing adiabats

An adiabat is a curve of constantentropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...

in a diagram. Some properties of adiabats on a ''P''–''V'' diagram are indicated. These properties may be read from the classical behaviour of ideal gases, except in the region where ''PV'' becomes small (low temperature), where quantum effects become important.
# Every adiabat asymptotically
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

approaches both the ''V'' axis and the ''P'' axis (just like isotherms).
# Each adiabat intersects each isotherm exactly once.
# An adiabat looks similar to an isotherm, except that during an expansion, an adiabat loses more pressure than an isotherm, so it has a steeper inclination (more vertical).
# If isotherms are concave towards the north-east direction (45°), then adiabats are concave towards the east north-east (31°).
# If adiabats and isotherms are graphed at regular intervals of entropy and temperature, respectively (like altitude on a contour map), then as the eye moves towards the axes (towards the south-west), it sees the density of isotherms stay constant, but it sees the density of adiabats grow. The exception is very near absolute zero, where the density of adiabats drops sharply and they become rare (see Nernst's theorem).
The right diagram is a ''P''–''V'' diagram with a superposition of adiabats and isotherms:
The isotherms are the red curves and the adiabats are the black curves.
The adiabats are isentropic.
Volume is the horizontal axis and pressure is the vertical axis.
Etymology

The term ''adiabatic'' () is an anglicization of the Greek term ἀδιάβατος "impassable" (used byXenophon
Xenophon of Athens (; grc, Ξενοφῶν ; – probably 355 or 354 BC) was a Greek military leader, philosopher, and historian, born in Athens. At the age of 30, Xenophon was elected commander of one of the biggest Greek mercenary armies o ...

of rivers).
It is used in the thermodynamic sense by Rankine (1866), Rankine, W.J.McQ. (1866). On the theory of explosive gas engines, ''The Engineer'', July 27, 1866; at page 467 of the reprint in Miscellaneous Scientific Papers

', edited by W.J. Millar, 1881, Charles Griffin, London. and adopted by Maxwell in 1871 (explicitly attributing the term to Rankine). The etymological origin corresponds here to an impossibility of transfer of energy as heat and of transfer of matter across the wall. The Greek word ἀδιάβατος is formed from privative ἀ- ("not") and διαβατός, "passable", in turn deriving from διά ("through"), and βαῖνειν ("to walk, go, come").

Conceptual significance in thermodynamic theory

The adiabatic process has been important for thermodynamics since its early days. It was important in the work of Joule because it provided a way of nearly directly relating quantities of heat and work. Energy can enter or leave a thermodynamic system enclosed by walls that preventmass transfer
Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtratio ...

only as heat or work. Therefore, a quantity of work in such a system can be related almost directly to an equivalent quantity of heat in a cycle of two limbs. The first limb is an isochoric adiabatic work process increasing the system's internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...

; the second, an isochoric and workless heat transfer returning the system to its original state. Accordingly, Rankine measured quantity of heat in units of work, rather than as a calorimetric quantity . In 1854, Rankine used a quantity that he called "the thermodynamic function" that later was called entropy, and at that time he wrote also of the "curve of no transmission of heat", which he later called an adiabatic curve. Besides its two isothermal limbs, Carnot's cycle has two adiabatic limbs.
For the foundations of thermodynamics, the conceptual importance of this was emphasized by Bryan, by Carathéodory, and by Born. The reason is that calorimetry presupposes a type of temperature as already defined before the statement of the first law of thermodynamics, such as one based on empirical scales. Such a presupposition involves making the distinction between empirical temperature and absolute temperature. Rather, the definition of absolute thermodynamic temperature is best left till the second law is available as a conceptual basis.
In the eighteenth century, the law of conservation of energy was not yet fully formulated or established, and the nature of heat was debated. One approach to these problems was to regard heat, measured by calorimetry, as a primary substance that is conserved in quantity. By the middle of the nineteenth century, it was recognized as a form of energy, and the law of conservation of energy was thereby also recognized. The view that eventually established itself, and is currently regarded as right, is that the law of conservation of energy is a primary axiom, and that heat is to be analyzed as consequential. In this light, heat cannot be a component of the total energy of a single body because it is not a state variable but, rather, a variable that describes a transfer between two bodies. The adiabatic process is important because it is a logical ingredient of this current view.
Divergent usages of the word ''adiabatic''

This present article is written from the viewpoint of macroscopic thermodynamics, and the word ''adiabatic'' is used in this article in the traditional way of thermodynamics, introduced by Rankine. It is pointed out in the present article that, for example, if a compression of a gas is rapid, then there is little time for heat transfer to occur, even when the gas is not adiabatically isolated by a definite wall. In this sense, a rapid compression of a gas is sometimes approximately or loosely said to be ''adiabatic'', though often far from isentropic, even when the gas is not adiabatically isolated by a definite wall.Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

and quantum statistical mechanics, however, use the word ''adiabatic'' in a very different sense, one that can at times seem almost opposite to the classical thermodynamic sense. In quantum theory, the word ''adiabatic'' can mean something perhaps near isentropic, or perhaps near quasi-static, but the usage of the word is very different between the two disciplines.
On the one hand, in quantum theory, if a perturbative element of compressive work is done almost infinitely slowly (that is to say quasi-statically), it is said to have been done ''adiabatically''. The idea is that the shapes of the eigenfunctions change slowly and continuously, so that no quantum jump is triggered, and the change is virtually reversible. While the occupation numbers are unchanged, nevertheless there is change in the energy levels of one-to-one corresponding, pre- and post-compression, eigenstates. Thus a perturbative element of work has been done without heat transfer and without introduction of random change within the system. For example, Max Born writes "Actually, it is usually the 'adiabatic' case with which we have to do: i.e. the limiting case where the external force (or the reaction of the parts of the system on each other) acts very slowly. In this case, to a very high approximation
:$c\_1^2=1,\backslash ,\backslash ,c\_2^2=0,\backslash ,\backslash ,c\_3^2=0,\backslash ,...\backslash ,,$
that is, there is no probability for a transition, and the system is in the initial state after cessation of the perturbation. Such a slow perturbation is therefore reversible, as it is classically."
On the other hand, in quantum theory, if a perturbative element of compressive work is done rapidly, it randomly changes the occupation numbers of the eigenstates, as well as changing their shapes. In that theory, such a rapid change is said not to be ''adiabatic'', and the contrary word ''diabatic'' is applied to it. One might guess that perhaps Clausius, if he were confronted with this, in the now-obsolete language he used in his day, would have said that "internal work" was done and that 'heat was generated though not transferred'.
Furthermore, in atmospheric thermodynamics, a diabatic process is one in which heat is exchanged.
In classical thermodynamics, such a rapid change would still be called adiabatic because the system is adiabatically isolated, and there is no transfer of energy as heat. The strong irreversibility of the change, due to viscosity or other entropy production, does not impinge on this classical usage.
Thus for a mass of gas, in macroscopic thermodynamics, words are so used that a compression is sometimes loosely or approximately said to be adiabatic if it is rapid enough to avoid heat transfer, even if the system is not adiabatically isolated. But in quantum statistical theory, a compression is not called adiabatic if it is rapid, even if the system is adiabatically isolated in the classical thermodynamic sense of the term. The words are used differently in the two disciplines, as stated just above.
See also

* Fire piston *Heat burst
In meteorology, a heat burst is a rare atmospheric phenomenon characterized by a sudden, localized increase in air temperature near the Earth's surface. Heat bursts typically occur during night-time and are associated with decaying thunderstor ...

; Related physics topics
* First law of thermodynamics
The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amou ...

* Entropy (classical thermodynamics)
* Adiabatic conductivity
* Adiabatic lapse rate
The lapse rate is the rate at which an atmospheric variable, normally temperature in Earth's atmosphere, falls with altitude. ''Lapse rate'' arises from the word ''lapse'', in the sense of a gradual fall. In dry air, the adiabatic lapse rate is ...

* Total air temperature
* Magnetic refrigeration
* Berry phase
; Related thermodynamic processes
* Cyclic process
* Isobaric process
* Isenthalpic process
An isenthalpic process or isoenthalpic process is a process that proceeds without any change in enthalpy, ''H''; or specific enthalpy, ''h''.
Overview
If a steady-state, steady-flow process is analysed using a control volume, everything outside t ...

* Isentropic process
In thermodynamics, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of heat or matter. Such an idealized proces ...

* Isochoric process
In thermodynamics, an isochoric process, also called a constant-volume process, an isovolumetric process, or an isometric process, is a thermodynamic process during which the volume of the closed system undergoing such a process remains constant. ...

* Isothermal process
* Polytropic process
* Quasistatic process
References

;General * * Nave, Carl Rod.Adiabatic Processes

. HyperPhysics. * Thorngren, Dr. Jane R..

. Daphne – A Palomar College Web Server., 21 July 1995..

External links

{{DEFAULTSORT:Adiabatic Process Thermodynamic processes Atmospheric thermodynamics Entropy