Adiabatic process

In thermodynamics, an adiabatic processis one that occurs without transfer ofheat or matter between a thermodynamicsystem and its surroundings. In anadiabatic process, energy is transferredto its surroundings only as work.[1][2] Theadiabatic process provides a rigorousconceptual basis for the theory used toexpound the first law of thermodynamics,and as such it is a key concept inthermodynamics.

Some chemical and physical processesoccur so rapidly that they may beconveniently described by the term"adiabatic approximation", meaning thatthere is not enough time for the transferof energy as heat to take place to or fromthe system.[3]

By way of example, the adiabatic flametemperature is an idealization that usesthe "adiabatic approximation" so as toprovide an upper limit calculation oftemperatures produced by combustion ofa fuel. The adiabatic flame temperatureis the temperature that would beachieved by a flame if the process of

combustion took place in the absence ofheat loss to the surroundings.

A process that does not involve thetransfer of heat or matter into or out of asystem, so that Q = 0, is called anadiabatic process, and such a system issaid to be adiabatically isolated.[4][5] Theassumption that a process is adiabatic isa frequently made simplifyingassumption. For example, thecompression of a gas within a cylinder ofan engine is assumed to occur so rapidlythat on the time scale of thecompression process, little of the

Description

system's energy can be transferred outas heat to the surroundings. Even thoughthe cylinders are not insulated and arequite conductive, that process isidealized to be adiabatic. The same canbe said to be true for the expansionprocess of such a system.

The assumption of adiabatic isolation ofa system is a useful one, and is oftencombined with others so as to make thecalculation of the system's behaviourpossible. Such assumptions areidealizations. The behaviour of actualmachines deviates from theseidealizations, but the assumption of such"perfect" behaviour provide a useful first

approximation of how the real worldworks. According to Laplace, whensound travels in a gas, there is no timefor heat conduction in the medium andso the propagation of sound is adiabatic.For such an adiabatic process, themodulus of elasticity (Young's modulus)can be expressed as E = γP, where γ isthe ratio of specific heats at constant

pressure and at constant volume (γ = CpCv

) and P is the pressure of the gas .

Various applications of theadiabatic assumption

For a closed system, one may write thefirst law of thermodynamics as :

ΔU = Q + W, where ΔU denotes thechange of the system's internal energy, Qthe quantity of energy added to it as heat,and W the work done on it by itssurroundings.

If the system has rigid walls such thatwork cannot be transferred in or out(W = 0), and the walls of the systemare not adiabatic and energy is addedin the form of heat (Q > 0), and there isno phase change, the temperature ofthe system will rise.

If the system has rigid walls such thatpressure–volume work cannot bedone, and the system walls areadiabatic (Q = 0), but energy is added

as isochoric work in the form offriction or the stirring of a viscous fluidwithin the system (W > 0), and there isno phase change, the temperature ofthe system will rise.

If the system walls are adiabatic(Q = 0), but not rigid (W ≠ 0), and, in afictive idealized process, energy isadded to the system in the form offrictionless, non-viscous pressure–volume work, and there is no phasechange, the temperature of the systemwill rise. Such a process is called anisentropic process and is said to be"reversible". Fictively, if the process isreversed, the energy added as workcan be recovered entirely as work done

by the system. If the system contains acompressible gas and is reduced involume, the uncertainty of the positionof the gas is reduced, and seeminglywould reduce the entropy of thesystem, but the temperature of thesystem will rise as the process isisentropic (ΔS = 0). Should the work beadded in such a way that friction orviscous forces are operating within thesystem, then the process is notisentropic, and if there is no phasechange, then the temperature of thesystem will rise, the process is said tobe "irreversible", and the work added tothe system is not entirely recoverablein the form of work.

If the walls of a system are notadiabatic, and energy is transferred inas heat, entropy is transferred into thesystem with the heat. Such a processis neither adiabatic nor isentropic,having Q > 0, and ΔS > 0 according tothe second law of thermodynamics.

Naturally occurring adiabatic processesare irreversible (entropy is produced).

The transfer of energy as work into anadiabatically isolated system can beimagined as being of two idealizedextreme kinds. In one such kind, there isno entropy produced within the system(no friction, viscous dissipation, etc.),and the work is only pressure-volume

work (denoted by P dV). In nature, thisideal kind occurs only approximately,because it demands an infinitely slowprocess and no sources of dissipation.

The other extreme kind of work isisochoric work (dV = 0), for which energyis added as work solely through frictionor viscous dissipation within the system.A stirrer that transfers energy to aviscous fluid of an adiabatically isolatedsystem with rigid walls, without phasechange, will cause a rise in temperatureof the fluid, but that work is notrecoverable. Isochoric work isirreversible.[6] The second law ofthermodynamics observes that a natural

process, of transfer of energy as work,always consists at least of isochoricwork and often both of these extremekinds of work. Every natural process,adiabatic or not, is irreversible, withΔS > 0, as friction or viscosity are alwayspresent to some extent.

The adiabatic compression of a gascauses a rise in temperature of the gas.Adiabatic expansion against pressure, ora spring, causes a drop in temperature. Incontrast, free expansion is an isothermalprocess for an ideal gas.

Adiabatic heating andcooling

Adiabatic heating occurs when thepressure of a gas is increased from workdone on it by its surroundings, e.g., apiston compressing a gas containedwithin a cylinder and raising thetemperature where in many practicalsituations heat conduction through wallscan be slow compared with thecompression time. This finds practicalapplication in diesel engines which relyon the lack of heat dissipation during thecompression stroke to elevate the fuelvapor temperature sufficiently to ignite it.

Adiabatic heating occurs in the Earth'satmosphere when an air mass descends,for example, in a katabatic wind, Foehn

wind, or chinook wind flowing downhillover a mountain range. When a parcel ofair descends, the pressure on the parcelincreases. Due to this increase inpressure, the parcel's volume decreasesand its temperature increases as work isdone on the parcel of air, thus increasingits internal energy, which manifests itselfby a rise in the temperature of that massof air. The parcel of air can only slowlydissipate the energy by conduction orradiation (heat), and to a firstapproximation it can be consideredadiabatically isolated and the process anadiabatic process.

Adiabatic cooling occurs when thepressure on an adiabatically isolatedsystem is decreased, allowing it toexpand, thus causing it to do work on itssurroundings. When the pressure appliedon a parcel of air is reduced, the air in theparcel is allowed to expand; as thevolume increases, the temperature fallsas its internal energy decreases.Adiabatic cooling occurs in the Earth'satmosphere with orographic lifting andlee waves, and this can form pileus orlenticular clouds.

Adiabatic cooling does not have toinvolve a fluid. One technique used toreach very low temperatures

(thousandths and even millionths of adegree above absolute zero) is viaadiabatic demagnetisation, where thechange in magnetic field on a magneticmaterial is used to provide adiabaticcooling. Also, the contents of anexpanding universe can be described (tofirst order) as an adiabatically coolingfluid. (See heat death of the universe.)

Rising magma also undergoes adiabaticcooling before eruption, particularlysignificant in the case of magmas thatrise quickly from great depths such askimberlites.[7]

Such temperature changes can bequantified using the ideal gas law, or the

hydrostatic equation for atmosphericprocesses.

In practice, no process is truly adiabatic.Many processes rely on a largedifference in time scales of the processof interest and the rate of heatdissipation across a system boundary,and thus are approximated by using anadiabatic assumption. There is alwayssome heat loss, as no perfect insulatorsexist.

Ideal gas (reversibleprocess)

The mathematical equation for an idealgas undergoing a reversible (i.e., noentropy generation) adiabatic processcan be represented by the polytropicprocess equation[3]

For a simple substance, during an adiabatic processin which the volume increases, the internal energy ofthe working substance must decrease

where P is pressure, V is volume, and forthis case n = γ, where

CP being the specific heat for constantpressure, CV being the specific heat forconstant volume, γ is the adiabatic index,and f is the number of degrees offreedom (3 for monatomic gas, 5 fordiatomic gas and collinear moleculese.g. carbon dioxide).

For a monatomic ideal gas, γ = 53 , and for

a diatomic gas (such as nitrogen andoxygen, the main components of air)γ = 7

5 .[8] Note that the above formula is

only applicable to classical ideal gasesand not Bose–Einstein or Fermi gases.

For reversible adiabatic processes, it isalso true that[3]

 

where T is an absolute temperature. Thiscan also be written as[3]

 

Example of adiabaticcompression

The compression stroke in a gasolineengine can be used as an example ofadiabatic compression. The model

assumptions are: the uncompressedvolume of the cylinder is one litre (1 l =1000 cm3 = 0.001 m3); the gas within isthe air consisting of molecular nitrogenand oxygen only (thus a diatomic gaswith 5 degrees of freedom, and so γ = 7

5);

the compression ratio of the engine is10:1 (that is, the 1 l volume ofuncompressed gas is reduced to 0.1 l bythe piston); and the uncompressed gas isat approximately room temperature andpressure (a warm room temperature of~27 °C, or 300 K, and a pressure of 1 bar= 100 kPa, i.e. typical sea-levelatmospheric pressure).

 

so our adiabatic constant for thisexample is about 6.31 Pa m4.2.

The gas is now compressed to a 0.1 l(0.0001 m3) volume (we will assume thishappens quickly enough that no heat canenter or leave the gas through the walls).The adiabatic constant remains thesame, but with the resulting pressureunknown

so solving for P:

 

 

 

or 25.1 bar. Note that this pressureincrease is more than a simple 10:1compression ratio would indicate; this isbecause the gas is not only compressed,but the work done to compress the gasalso increases its internal energy, whichmanifests itself by a rise in the gastemperature and an additional rise inpressure above what would result from asimplistic calculation of 10 times theoriginal pressure.

We can solve for the temperature of thecompressed gas in the engine cylinder aswell, using the ideal gas law, PV = nRT (nis amount of gas in moles and R the gasconstant for that gas). Our initial

conditions being 100 kPa of pressure, 1 lvolume, and 300 K of temperature, ourexperimental constant (nR) is:

We know the compressed gas has V =0.1 l and P = 2.51 × 106 Pa, so we cansolve for temperature:

That is a final temperature of 753 K, or479 °C, or 896 °F, well above the ignitionpoint of many fuels. This is why a high-compression engine requires fuelsspecially formulated to not self-ignite

 

 

(which would cause engine knockingwhen operated under these conditions oftemperature and pressure), or that asupercharger with an intercooler toprovide a pressure boost but with a lowertemperature rise would be advantageous.A diesel engine operates under evenmore extreme conditions, withcompression ratios of 20:1 or more beingtypical, in order to provide a very high gastemperature, which ensures immediateignition of the injected fuel.

Adiabatic free expansion of agas

For an adiabatic free expansion of anideal gas, the gas is contained in aninsulated container and then allowed toexpand in a vacuum. Because there is noexternal pressure for the gas to expandagainst, the work done by or on thesystem is zero. Since this process doesnot involve any heat transfer or work, thefirst law of thermodynamics then impliesthat the net internal energy change of thesystem is zero. For an ideal gas, thetemperature remains constant becausethe internal energy only depends ontemperature in that case. Since atconstant temperature, the entropy isproportional to the volume, the entropy

increases in this case, therefore thisprocess is irreversible.

Derivation of P–V relation foradiabatic heating and cooling

The definition of an adiabatic process isthat heat transfer to the system is zero,δQ = 0. Then, according to the first law ofthermodynamics,

 

where dU is the change in the internalenergy of the system and δW is workdone by the system. Any work (δW) donemust be done at the expense of internalenergy U, since no heat δQ is beingsupplied from the surroundings.

Pressure–volume work δW done by thesystem is defined as

 

However, P does not remain constantduring an adiabatic process but insteadchanges along with V.

It is desired to know how the values ofdP and dV relate to each other as theadiabatic process proceeds. For an idealgas the internal energy is given by

 

where α is the number of degrees offreedom divided by two, R is theuniversal gas constant and n is the

number of moles in the system (aconstant).

Differentiating equation (3) and use ofthe ideal gas law, PV = nRT, yields

Equation (4) is often expressed asdU = nCV dT because CV = αR.

Now substitute equations (2) and (4) intoequation (1) to obtain

 

factorize −P dV:

 

and divide both sides by PV:

 

 

After integrating the left and right sidesfrom V0 to V and from P0 to P andchanging the sides respectively,

 

Exponentiate both sides, substitute α + 1α

with γ, the heat capacity ratio

 

and eliminate the negative sign to obtain

 

Therefore,

 

and

 

Derivation of P–T relation foradiabatic heating and cooling

Substituting the ideal gas law into theabove, we obtain

 

which simplifies to

 

Derivation of discrete formula

The change in internal energy of asystem, measured from state 1 to state2, is equal to

At the same time, the work done by thepressure–volume changes as a resultfrom this process, is equal to

 

Since we require the process to beadiabatic, the following equation needsto be true

 

 

By the previous derivation,

Rearranging (4) gives

 

Substituting this into (2) gives

 

Integrating,

 

Substituting γ = α + 1α ,

 

α 

Rearranging,

 

Using the ideal gas law and assuming aconstant molar quantity (as oftenhappens in practical cases),

By the continuous formula,

 

 

or

 

Substituting into the previous expressionfor W,

Substituting this expression and (1) in (3)gives

Simplifying,

 

 

 

 

An adiabat is a curve of constant entropyin a diagram. Some properties ofadiabats on a P–V diagram are indicated.These properties may be read from theclassical behaviour of ideal gases, exceptin the region where PV becomes small

 

Graphing adiabats

(low temperature), where quantumeffects become important.

1. Every adiabat asymptoticallyapproaches both the V axis and the Paxis (just like isotherms).

2. Each adiabat intersects each isothermexactly once.

3. An adiabat looks similar to anisotherm, except that during anexpansion, an adiabat loses morepressure than an isotherm, so it has asteeper inclination (more vertical).

4. If isotherms are concave towards thenorth-east direction (45°), then adiabatsare concave towards the east north-east(31°).

5. If adiabats and isotherms are graphedat regular intervals of entropy andtemperature, respectively (like altitude ona contour map), then as the eye movestowards the axes (towards the south-west), it sees the density of isothermsstay constant, but it sees the density ofadiabats grow. The exception is very nearabsolute zero, where the density ofadiabats drops sharply and they becomerare (see Nernst's theorem).

The following diagram is a P–V diagramwith a superposition of adiabats andisotherms:

 

The isotherms are the red curves and theadiabats are the black curves.

The adiabats are isentropic.

Volume is the horizontal axis andpressure is the vertical axis.

Etymology

Look up adiabatic in Wiktionary, thefree dictionary.

The term adiabatic (/ˌædiəˈbætɪk/) is ananglicization of the Greek termἀδιάβατος "impassable" (used byXenophon of rivers). It is used in thethermodynamic sense by Rankine(1866),[9][10] and adopted by Maxwell in1871 (explicitly attributing the term toRankine).[11] The etymological origincorresponds here to an impossibility oftransfer of energy as heat and of transferof matter across the wall.

The Greek word ἀδιάβατος is formedfrom privative ἀ- ("not") and διαβατός,"passable", in turn deriving from διά

("through"), and βαῖνειν ("to walk, go,come").[12]

The adiabatic process has beenimportant for thermodynamics since itsearly days. It was important in the workof Joule, because it provided a way ofnearly directly relating quantities of heatand work.

For a thermodynamic system that isenclosed by walls that do not allow masstransfer, energy can pass in and out onlyas heat or work. Thus a quantity of workcan be related almost directly to an

Conceptual significance inthermodynamic theory

equivalent quantity of heat in a cycle oftwo limbs. The first is an isochoricadiabatic work process that adds to thesystem's internal energy. Then anisochoric and workless heat transferreturns the system to its original state.The first limb adds a definite amount ofenergy and the second removes it.Accordingly, Rankine measured quantityof heat in units of work, rather than as acalorimetric quantity .[13] In 1854,Rankine used a quantity that he called"the thermodynamic function" that laterwas called entropy, and at that time hewrote also of the "curve of notransmission of heat",[14] which he latercalled an adiabatic curve.[9] Besides it

two isothermal limbs, Carnot's cycle hastwo adiabatic limbs.

For the foundations of thermodynamics,the conceptual importance of this wasemphasized by Bryan,[15] byCarathéodory,[1] and by Born.[16] Thereason is that calorimetry presupposes atype of temperature as already definedbefore the statement of the first law ofthermodynamics, such as one based onempirical scales. Such a presuppositioninvolves making the distinction betweenempirical temperature and absolutetemperature. Rather, the definition ofabsolute thermodynamic temperature is

best left till the second law is available asa conceptual basis.[17]

In the eighteenth century, the law ofconservation of energy was yet to be fullyformulated or established, and the natureof heat was debated. One approach tothese problems was to regard heat,measured by calorimetry, as a primarysubstance that is conserved in quantity.By the middle of the nineteenth century, itwas recognized as a form of energy, andthe law of conservation of energy wasthereby also recognized. The view thateventually established itself, and iscurrently regarded as right, is that the lawof conservation of energy is a primary

axiom, and that heat is to be analyzed asconsequential. In this light, heat cannotbe a component of the total energy of asingle body because it is not a statevariable, but, rather, is a variable thatdescribes a process of transfer betweentwo bodies. The adiabatic process isimportant because it is a logicalingredient of this current view.[17]

This present article is written from theviewpoint of macroscopicthermodynamics, and the word adiabaticis used in this article in the traditional

Divergent usages of theword adiabatic

way of thermodynamics, introduced byRankine. It is pointed out in the presentarticle that, for example, if a compressionof a gas is rapid, then there is little timefor heat transfer to occur, even when thegas is not adiabatically isolated by adefinite wall. In this sense, a rapidcompression of a gas is sometimesapproximately or loosely said to beadiabatic, though often far fromisentropic, even when the gas is notadiabatically isolated by a definite wall.

Quantum mechanics and quantumstatistical mechanics, however, use theword adiabatic in a very different sense,one that can at times seem almost

opposite to the classical thermodynamicsense. In quantum theory, the wordadiabatic can mean something perhapsnear isentropic, or perhaps near quasi-static, but the usage of the word is verydifferent between the two disciplines.

On the one hand, in quantum theory, if aperturbative element of compressivework is done almost infinitely slowly (thatis to say quasi-statically), it is said tohave been done adiabatically. The idea isthat the shapes of the eigenfunctionschange slowly and continuously, so thatno quantum jump is triggered, and thechange is virtually reversible. While theoccupation numbers are unchanged,

nevertheless there is change in theenergy levels of one-to-onecorresponding, pre- and post-compression, eigenstates. Thus aperturbative element of work has beendone without heat transfer and withoutintroduction of random change within thesystem. For example, Max Born writes"Actually, it is usually the 'adiabatic' casewith which we have to do: i.e. the limitingcase where the external force (or thereaction of the parts of the system oneach other) acts very slowly. In this case,to a very high approximation

 

that is, there is no probability for atransition, and the system is in the initialstate after cessation of the perturbation.Such a slow perturbation is thereforereversible, as it is classically."[18]

On the other hand, in quantum theory, if aperturbative element of compressivework is done rapidly, it randomly changesthe occupation numbers of theeigenstates, as well as changing theirshapes. In that theory, such a rapidchange is said not to be adiabatic, andthe contrary word diabatic is applied to it.One might guess that perhaps Clausius,if he were confronted with this, in thenow-obsolete language he used in his

day, would have said that "internal work"was done and that 'heat was generatedthough not transferred'.

In classical thermodynamics, such arapid change would still be calledadiabatic because the system isadiabatically isolated, and there is notransfer of energy as heat. The strongirreversibility of the change, due toviscosity or other entropy production,does not impinge on this classical usage.

Thus for a mass of gas, in macroscopicthermodynamics, words are so used thata compression is sometimes loosely orapproximately said to be adiabatic if it israpid enough to avoid heat transfer, even

if the system is not adiabatically isolated.But in quantum statistical theory, acompression is not called adiabatic if itis rapid, even if the system isadiabatically isolated in the classicalthermodynamic sense of the term. Thewords are used differently in the twodisciplines, as stated just above.

Cyclic process

First law of thermodynamics

Heat burst

Isobaric process

Isenthalpic process

Isentropic process

See also

Isochoric process

Isothermal process

Polytropic process

Entropy (classical thermodynamics)

Quasistatic process

Total air temperature

Magnetic refrigeration

1. Carathéodory, C. (1909)."Untersuchungen über die Grundlagen derThermodynamik". MathematischeAnnalen. 67: 355–386.doi:10.1007/BF01450409 .. A translationmay be found here . Also a mostly reliabletranslation is to be found in Kestin, J.

References

(1976). The Second Law ofThermodynamics. Stroudsburg, PA:Dowden, Hutchinson & Ross.

2. Bailyn, M. (1994). A Survey ofThermodynamics. New York, NY:American Institute of Physics Press.p. 21. ISBN 0-88318-797-3.

3. Bailyn, M. (1994), pp. 52–53.

4. Tisza, L. (1966). GeneralizedThermodynamics. Cambridge, MA: MITPress. p. 48. “(adiabatic partitions inhibitthe transfer of heat and mass)”

5. Münster, A. (1970), p. 48: "mass is anadiabatically inhibited variable."

6. Münster, A. (1970). ClassicalThermodynamics. Translated by

Halberstadt, E. S. London: Wiley–Interscience. p. 45. ISBN 0-471-62430-6.

7. Kavanagh, J. L.; Sparks, R. S. J. (2009)."Temperature changes in ascendingkimberlite magmas" . Earth and PlanetaryScience Letters. Elsevier. 286 (3–4): 404–413. Bibcode:2009E&PSL.286..404K .doi:10.1016/j.epsl.2009.07.011 .Retrieved 18 February 2012.

8. Adiabatic Processes .

9. Rankine, W.J.McQ. (1866). On thetheory of explosive gas engines, TheEngineer, July 27, 1866; at page 467 ofthe reprint in Miscellaneous ScientificPapers , edited by W.J. Millar, 1881,Charles Griffin, London.

10. Partington, J. R. (1949), An AdvancedTreatise on Physical Chemistry., volume 1,Fundamental Principles. The Properties ofGases, London: Longmans, Green and Co.,p. 122

11. Maxwell, J. C. (1871), Theory of Heat(first ed.), London: Longmans, Green andCo., p. 129

12. Liddell, H.G., Scott, R. (1940). A Greek-English Lexicon, Clarendon Press, OxfordUK.

13. Rankine, W. J. M. (1854). "On thegeometrical representation of theexpansive action of heat, and theory ofthermodynamic engines". Proc. Roy. Soc.

144: 115–175. Miscellaneous ScientificPapers p. 339

14. Rankine, W. J. M. (1854). "On thegeometrical representation of theexpansive action of heat, and theory ofthermodynamic engines". Proc. Roy. Soc.144: 115–175. Miscellaneous ScientificPapers p. 341 .

15. Bryan, G. H. (1907). Thermodynamics.An Introductory Treatise dealing mainlywith First Principles and their DirectApplications . Leipzig: B. G. Teubner.

16. Born, M. (1949). "Natural Philosophyof Cause and Chance" . London: OxfordUniversity Press.

GeneralSilbey, Robert J.; et al. (2004). Physicalchemistry. Hoboken: Wiley. p. 55.ISBN 978-0-471-21504-2.

Broholm, Collin. "Adiabatic freeexpansion". Physics & Astronomy @

17. Bailyn, M. (1994). "Chapter 3". ASurvey of Thermodynamics. New York,NY: American Institute of Physics. ISBN 0-88318-797-3.

18. Born, M. (1927). "Physical aspects ofquantum mechanics". Nature. 119: 354–357. Bibcode:1927Natur.119..354B .doi:10.1038/119354a0 . (Translation byRobert Oppenheimer.)

Johns Hopkins University. N.p., 26 Nov.1997. Web. 14 Apr.

Nave, Carl Rod. "Adiabatic Processes".HyperPhysics. N.p., n.d. Web. 14 Apr.2011. [1] .

Thorngren, Dr. Jane R.. "AdiabaticProcesses". Daphne – A PalomarCollege Web Server. N.p., 21 July 1995.Web. 14 Apr. 2011. [2] .

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