Adiabatic process: Difference between revisions

{{About|adiabatic processes in thermodynamics|the adiabatic theorem in quantum mechanics|Adiabatic theorem}}

{{Thermodynamics|cTopic=systems}}

An '''adiabatic process''' (''adiabatic'' {{etymology|grc|''{{Wikt-lang|grc|ἀδιάβατος}}'' ({{grc-transl|ἀδιάβατος}})|impassable}}) is a type of [[thermodynamic process]] whereby a transfer of [[energy]] between the [[thermodynamic system]] and its [[Environment (systems)|environment]] is accompanied neither by a transfer of [[entropy]] nor of amounts of constituents. Unlike an [[isothermal process]], an adiabatic process transfers energy to the surroundings only as [[work (thermodynamics)|work]] and/or mass flow.<ref name=Carathéodory>{{cite journal |last=Carathéodory |first=C. |author-link=Constantin Carathéodory |date=1909 |title=Untersuchungen über die Grundlagen der Thermodynamik |lang=de |trans-title=Investigation into the foundations of thermodynamics |journal=[[Mathematische Annalen]] |volume=67 |issue=3 |pages=355–386 |doi=10.1007/BF01450409 |s2cid=118230148 |url=https://zenodo.org/record/1428268 }} <br/>{{cite web |title=A translation may be found here |website=neo-classical-physics.info |url=http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf |archive-url=https://web.archive.org/web/20191012152205/http://neo-classical-physics.info/uploads/3/0/6/5/3065888/caratheodory_-_thermodynamics.pdf |archive-date=2019-10-12}} <br/>Also, a mostly reliable translation is to be found in {{cite book |last=Kestin |first=J. |year=1976 |section=Investigation into the foundations ... |title=The Second Law of Thermodynamics |publisher=Dowden, Hutchinson & Ross |location=Stroudsburg, Pennsylvania |lang=en-us |section-url=https://books.google.com/books?id=xwBRAAAAMAAJ&q=Investigation+into+the+foundations }}</ref><ref name=Bailyn>{{cite book |last=Bailyn |first=M. |year=1994 |title=A Survey of Thermodynamics |publisher=American Institute of Physics Press |isbn=0-88318-797-3 |location=New York, NY |page=21 |lang=en-us }} </ref>{{rp|style=ama|p=21}}<ref>{{cite journal |last1=Zanchini |first1=E. |last2=Beretta |first2=G.P. |year=2014 |title=Recent progress in the definition of thermodynamic entropy |journal=[[Entropy (journal)|Entropy]] |volume=16 |issue=3 |pages=1547–1570 |doi=10.3390/e16031547 |doi-access=free |arxiv=1403.5772 |bibcode=2014Entrp..16.1547Z }}</ref> As a key concept in [[thermodynamics]], the adiabatic process supports the theory that explains the [[first law of thermodynamics]]. The opposite term to "adiabatic" is ''diabatic''.

Some chemical and physical processes occur too rapidly for energy to enter or leave the system as heat, allowing a convenient "adiabatic approximation".<ref name=Bailyn/>{{rp|style=ama|pp=52–53}} For example, the [[adiabatic flame temperature]] uses this approximation to calculate the upper limit of [[fire|flame]] temperature by assuming combustion loses no heat to its surroundings.

A process without transfer of [[thermal energy]] (heat) to or from a system, so that {{math|1=''Q'' = 0}}, is called adiabatic, and such a system is said to be adiabatically isolated.<ref>{{cite book |last=Tisza |first=L. |title=Generalized Thermodynamics |date=1966 |publisher=MIT Press |location=Cambridge, Massachusetts |page=48 |language=en-us |quote=(adiabatic partitions inhibit the transfer of heat and mass) |author-link=László Tisza}}</ref><ref>Münster, A. (1970), p. 48: "mass is an adiabatically inhibited variable."</ref> 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 [[Pierre-Simon Laplace|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, the [[Elastic modulus|modulus of elasticity]] ([[Young's modulus]]) can be expressed as {{math| ''E'' {{=}} ''γP''}}, where {{mvar|γ}} is the [[Heat capacity ratio|ratio of specific heats]] at constant pressure and at [[constant volume]] ({{math| ''γ'' {{=}} {{sfrac|''C_p''|''C_v''}}}}) and {{math|''P''}} is the pressure of the gas.

===Various applications of the adiabatic assumption===

For a closed system, one may write the [[first law of thermodynamics]] as {{math|Δ''U'' {{=}} ''Q'' − ''W''}}, where {{math|Δ''U''}} denotes the change of the system's internal energy, {{mvar|Q}} the quantity of energy added to it as heat, and {{mvar|W}} the work done by the system on its surroundings.

* If the system has such rigid walls that work cannot be transferred in or out ({{math|''W'' {{=}} 0}}), and the walls are not adiabatic and energy is added in the form of heat ({{math|''Q'' > 0}}), 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 ({{math|''Q'' {{=}} 0}}), and energy is added as [[Isochoric process|isochoric]] (constant volume) work in the form of friction or the stirring of a [[viscous]] fluid within the system ({{math|''W'' < 0}}), and there is no phase change, then the temperature of the system will rise.

* If the system walls are adiabatic ({{math|''Q'' {{=}} 0}}) but not rigid ({{math|''W'' ≠ 0}}), and, in a fictive idealized process, energy is added to the system in the form of frictionless, non-viscous pressure–volume work ({{math|''W'' < 0}}), and there is no phase change, then the temperature of the system will rise. Such a process is called an [[isentropic process]] 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 ({{math|Δ''S'' {{=}} 0}}). 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 {{math|''Q'' > 0}}, and {{math|Δ''S'' > 0}} according to the [[second law of thermodynamics]].

Adiabatic compression occurs when the pressure of a gas is increased by work done on it by its surroundings, e.g., a [[piston]] 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]] which rely on the lack of heat dissipation during the compression stroke to elevate the fuel vapor temperature sufficiently to ignite it.

Adiabatic expansion occurs when the pressure on an adiabatically isolated system is decreased, allowing it to expand in size, 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 expansion occurs in the Earth's atmosphere with [[orographic lifting]] and [[lee waves]], and this can form [[Pileus (meteorology)|pilei]] or [[lenticular cloud]]s.

Due in part to adiabatic expansion in mountainous areas, snowfall actually occurs in some parts of the [[Sahara desert]].<ref>{{cite web |last1=Knight |first1=Jasper |title=Snowfall in the Sahara desert: an unusual weather phenomenon |url=https://theconversation.com/snowfall-in-the-sahara-desert-an-unusual-weather-phenomenon-176037 |website=The Conversation |access-date=3 March 2022 |date=31 January 2022}}</ref>

In the Earth's convecting [[Earth's mantle|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.<ref>{{Cite book|title=Geodynamics|url=https://archive.org/details/geodynamics00dltu|url-access=limited|last=Turcotte and Schubert|publisher=Cambridge University Press|year=2002|isbn=0-521-66624-4|location=Cambridge|pages=[https://archive.org/details/geodynamics00dltu/page/n199 185]}}</ref>

The mathematical equation for an [[ideal gas]] undergoing a reversible (i.e., no entropy generation) adiabatic process can be represented by the [[polytropic process]] equation<ref name=Bailyn/>{{rp|style=ama|p=53}}

<math display="block"> P\ V^\gamma = \mathsf{constant}\ ,</math>

where {{mvar|P}} is pressure, {{mvar|V}} is volume, and {{mvar|γ}} is the [[adiabatic index]] or heat capacity ratio defined as

<math display="block"> \gamma = \frac{C_P}{C_V} = \frac{f + 2}{f} ~.</math>

Here {{mvar| C{{sub|P}} }} is the [[specific heat]] for constant pressure, {{mvar| C{{sub|V}} }} is the specific heat for constant volume, and {{mvar|f}} is the number of [[Degrees of freedom (physics and chemistry)|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, {{nobr|{{math|''γ'' {{=}} {{sfrac|5|3}}}},}} and for a diatomic gas (such as [[nitrogen]] and [[oxygen]], the main components of air), {{nobr|{{math|''γ'' {{=}} {{sfrac|7|5}}}}.}}<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html |title=Adiabatic Process |website=HyperPhysics |publisher=Georgia State University}}</ref> Note that the above formula is only applicable to classical ideal gases (that is, gases far above absolute zero temperature) and not [[Bose–Einstein condensate|Bose–Einstein]] or [[Fermionic condensate|Fermi gases]].

One can also use the ideal gas law to rewrite the above relationship between {{mvar|P}} and {{mvar|V}} as<ref name=Bailyn/>{{rp|style=ama|p=53}}

P^{1-\gamma}\ T^\gamma &= \mathsf{constant}\ ,\\

T\ V^{\gamma - 1} &= \mathsf{constant} ~.

where {{mvar|T}} is the absolute or [[thermodynamic temperature]].

The compression stroke in a [[gasoline engine]] can be used as an example of adiabatic compression. The model assumptions are: the uncompressed volume of the cylinder is {{nobr|one litre (1 L {{=}} 1000 cm{{sup|3}} {{=}} 0.001 m{{sup|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 {{nobr|{{math| ''γ'' {{=}} {{sfrac|7|5}}}} );}} the compression ratio of the engine is 10:1 (that is, the 1&nbsp;L volume of uncompressed gas is reduced to 0.1&nbsp;L by the piston); and the uncompressed gas is at approximately room temperature and pressure (a warm room temperature of ~27&nbsp;°C, or 300&nbsp;K, and a pressure of 1&nbsp;bar = 100&nbsp;kPa, i.e. typical sea-level atmospheric pressure).

P_1 V_1^\gamma &= \mathsf{constant}_1 \\

  & = 100\,000~\text{Pa} \times (0.001~\mathsf{m}^3)^\frac75 \\

  & = 10^5 \times 6.31 \times 10^{-5}~\mathsf{Pa}\,\mathsf{m}^{21/5} \\

  & = 6.31~\text{Pa}\,\mathsf{m}^{21/5}\ ,

so the adiabatic constant for this example is about {{nobr|6.31 Pa·m{{sup|4.2}} .}}

P_2 V_2^\gamma &= \mathsf{constant}_1 \\

&= 6.31~\mathsf{Pa}\,\mathsf{m}^{21/5} \\

&= P \times (0.0001~\mathsf{m}^3)^\frac75,

We can now solve for the final pressure<ref>{{cite book |last1=Atkins |first1=Peter |last2=de Paula |first2=Giulio |title=Atkins' Physical Chemistry |year=2006 |publisher=W.H. Freeman |isbn=0-7167-8759-8 |page=48 |edition=8th }} </ref>

P_2 &= P_1 \left( \frac{V_1}{V_2} \right)^\gamma \\

&= 100\ 000~\mathsf{Pa} \times {10}^{7/5} \\

&= 2.51 \times 10^6~\mathsf{Pa} \ ,

We can solve for the temperature of the compressed gas in the engine cylinder as well, using the ideal gas law, {{mvar|P V {{=}} n R T }} ({{mvar|n}} is amount of gas in moles and {{mvar|R}} the gas constant for that gas). Our initial conditions being 100&nbsp;kPa of pressure, 1&nbsp;L volume, and 300&nbsp;K of temperature, our experimental constant ({{mvar|n R}}{{hairsp}}) is:

\frac{P\ V}{T} &= \mathsf{constant}_2 \\

&= \frac{10^5~\mathsf{Pa} \times 10^{-3}~\mathsf{m}^3}{300~\mathsf{K}} \\

&= 0.333~\mathsf{Pa}\ \mathsf{m}^3\mathsf{K}^{-1} ~.

We know the compressed gas has {{nobr|{{mvar|V}} {{=}} 0.1 L}} and {{nobr|{{mvar|P}} {{=}} {{val|2.51|e=6|u=Pa}},}} so we can solve for temperature:

T &= \frac{P\ V}{\mathsf{constant}_2} \\

&= \frac{2.51 \times 10^6~\mathsf{Pa} \times 10^{-4}~\text{m}^3}{0.333~\mathsf{Pa}\ \mathsf{m}^3\mathsf{K}^{-1}} \\

&= 753~\mathsf{K}.

The definition of an adiabatic process is that heat transfer to the system is zero, {{nobr|{{math|''δQ'' {{=}} 0}}.}} Then, according to the first law of thermodynamics,

{{NumBlk||<math display="block"> \mathrm{d}U + \delta W = \delta Q = 0\ , </math>|{{EquationRef|a1}}}}

where {{math|d''U''}} is the change in the internal energy of the system and {{math|''δW''}} is work done ''by'' the system. Any work {{nobr|({{math|''δW''}})}} done must be done at the expense of internal energy {{nobr|{{mvar|U}},}} since no heat {{mvar|δQ}} is being supplied from the surroundings. Pressure–volume work {{mvar|δW}} done ''by'' the system is defined as

{{NumBlk||<math display="block"> \delta W = P \ \mathrm{d}V ~.</math>|{{EquationRef|a2}}}}

However, {{mvar|P}} does not remain constant during an adiabatic process but instead changes along with {{mvar|V}}.

It is desired to know how the values of {{math|d''P''}} and {{math|d''V''}} relate to each other as the adiabatic process proceeds. For an ideal gas (recall ideal gas law {{nobr|{{math|''PV'' {{=}} ''nRT''}})}} the internal energy is given by

{{NumBlk||<math display="block"> U = \alpha\ n\ R\ T = \alpha\ P\ V\ , </math>|{{EquationRef|a3}}}}

where {{mvar|α}} is the number of degrees of freedom divided by 2, {{mvar|R}} is the [[universal gas constant]] and {{mvar|n}} is the number of moles in the system (a constant).

\mathrm{d} U &= \alpha\ n\ R\ \mathrm{d}T\\

                 & = \alpha\ \mathrm{d} (P\ V)\\

                 & = \alpha\ (P \ \mathrm{d}V + V \ \mathrm{d}P) ~.

Equation (a4) is often expressed as {{math|d''U'' {{=}} ''n C_V ''d''T''}} because {{nobr|{{math|''C_V'' {{=}} ''α R''}} .}}

<math display="block"> -P\ \mathrm{d}V = \alpha\ P\ \mathrm{d}V + \alpha\ V\ \mathrm{d}P\ ,</math>

factor {{math|−''P'' d''V''}}:

<math display="block"> -(\alpha + 1)\ P\ \mathrm{d}V = \alpha V\ \mathrm{d}P\ ,</math>

and divide both sides by {{mvar|P V }}:

<math display="block"> -(\alpha + 1)\ \frac{\mathrm{d}V}{V} = \alpha\ \frac{\mathrm{d}P}{P} ~.</math>

After integrating the left and right sides from {{math|''V''₀}} to {{mvar|V}} and from {{math|''P''₀}} to {{mvar|P}} and changing the sides respectively,

<math display="block"> \ln \left( \frac{P}{P_0} \right) = -\frac{\alpha + 1}{\alpha}\ \ln \left( \frac{V}{V_0} \right) ~.</math>

Exponentiate both sides, substitute {{math|{{sfrac|''α'' + 1|''α''}}}} with {{mvar|γ}}, the heat capacity ratio

<math display="block"> \left( \frac{P}{P_0} \right) = \left( \frac{V}{V_0} \right)^{-\gamma}\ ,</math>

<math display="block"> \left( \frac{P}{P_0} \right) = \left( \frac{V_0}{V} \right)^\gamma ~.</math>

<math display="block"> \left( \frac{P}{P_0} \right) \left( \frac{V}{V_0} \right)^\gamma = 1\ ,</math>

<math display="block"> P_0\ V_0^\gamma = P\ V^\gamma = \mathsf{constant} ~.</math>

{{NumBlk||<math display="block"> \Delta U = \alpha\ R\ n\ T_2 - \alpha\ R\ n\ T_1 = \alpha\ R\ n\ \Delta T ~.</math>|{{EquationRef|b1}}}}

{{NumBlk||<math display="block"> W = \int_{V_1}^{V_2}\ P\ \mathrm{d}V ~.</math>|{{EquationRef|b2}}}}

{{NumBlk||<math display="block"> \Delta U + W = 0 ~.</math>|{{EquationRef|b3}}}}

{{NumBlk||<math display="block"> P\ V^\gamma = \mathsf{constant} = P_1\ V_1^\gamma ~.</math>|{{EquationRef|b4}}}}

<math display="block"> P = P_1\ \left(\frac{V_1}{V} \right)^\gamma ~.</math>

<math display="block"> W = \int_{V_1}^{V_2} P_1\ \left(\frac{V_1}{V} \right)^\gamma\ \mathrm{d}V ~.</math>

<math display="block">

\begin{align}

W &= P_1\ V_1^\gamma\ \frac{V_2^{1-\gamma} - V_1^{1-\gamma}}{1 - \gamma} \\[1ex]

&= \frac{P_2\ V_2 - P_1\ V_1}{1 - \gamma}.

\end{align}

</math>

Substituting {{math|''γ'' {{=}} {{sfrac|''α'' + 1|''α''}}}} in the second term,

<math display="block"> W = -\alpha\ P_1\ V_1^\gamma\ \left( V_2^{1-\gamma} - V_1^{1-\gamma} \right) ~.</math>

<math display="block"> W = -\alpha\ P_1\ V_1\ \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right) ~.</math>

<math display="block"> W = -\alpha\ n\ R\ T_1 \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right) ~.</math>

<math display="block"> \frac{P_2}{P_1} = \left(\frac{V_2}{V_1}\right)^{-\gamma}\ ,</math>

<math display="block"> \left(\frac{P_2}{P_1}\right)^{-\frac{1}{\gamma}} = \frac{V_2}{V_1} ~.</math>

Substituting into the previous expression for {{mvar|W}},

<math display="block"> W = -\alpha\ n\ R\ T_1 \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right) ~.</math>

<math display="block"> \alpha\ n\ R\ (T_2 - T_1) = \alpha\ n\ R\ T_1\ \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right) ~.</math>

T_2 - T_1 &=  T_1\ \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right)\ , \\

\frac{T_2}{T_1} - 1 &=  \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1\ , \\

T_2 &= T_1\ \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} ~.

{{NumBlk||<math display="block"> W = \int_{V_1}^{V_2}\ P\ \mathrm{d}V ~.</math>|{{EquationRef|c2}}}}

{{NumBlk||<math display="block"> \Delta U + W = 0 ~.</math>|{{EquationRef|c3}}}}

{{NumBlk||<math display="block"> P\ V^\gamma = \text{constant} = P_1\ V_1^\gamma ~.</math>|{{EquationRef|c4}}}}

<math display="block"> P = P_1\ \left(\frac{V_1}{V} \right)^\gamma ~.</math>

<math display="block"> W = \int_{V_1}^{V_2} P_1\ \left(\frac{V_1}{V} \right)^\gamma\ \mathrm{d}V ~.</math>

<math display="block"> W = P_1\ V_1^\gamma \frac{V_2^{1-\gamma} - V_1^{1-\gamma}}{1 - \gamma} = \frac{P_2\ V_2 - P_1\ V_1}{1 - \gamma} ~.</math>

<math display="block"> W = -\alpha\ P_1 V_1^\gamma\ \left( V_2^{1-\gamma} - V_1^{1-\gamma} \right) ~.</math>

<math display="block"> W = -\alpha\ P_1\ V_1\ \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right) ~.</math>

<math display="block"> W = -\alpha\ n\ R\ T_1\ \left( \left( \frac{V_2}{V_1} \right)^{1-\gamma} - 1 \right) ~.</math>

<math display="block"> \frac{P_2}{P_1} = \left(\frac{V_2}{V_1}\right)^{-\gamma}\ ,</math>

<math display="block"> \left(\frac{P_2}{P_1}\right)^{-\frac{1}{\gamma}} = \frac{V_2}{V_1} ~.</math>

<math display="block"> W = -\alpha\ n\ R\ T_1\ \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right) ~.</math>

<math display="block"> \alpha\ n\ R\ (T_2 - T_1) = \alpha\ n\ R\ T_1\ \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right) ~.</math>

T_2 - T_1 &=  T_1\ \left( \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1 \right)\ , \\

\frac{T_2}{T_1} - 1 &=  \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} - 1\ , \\

T_2 &= T_1\ \left( \frac{P_2}{P_1} \right)^{\frac{\gamma-1}{\gamma}} ~.

An adiabat is a curve of constant [[entropy]] in a diagram. Some properties of adiabats on a {{math|''P'' – ''V''}} diagram are indicated. These properties may be read from the classical behaviour of ideal gases, except in the region where {{mvar|P V}} becomes small (low temperature), where quantum effects become important.

# Every adiabat [[asymptotically]] approaches both the {{mvar|V}}&nbsp;axis and the {{mvar|P}}&nbsp;axis (just like [[isotherms]]).

# If isotherms are concave towards the north-east direction (45° from {{mvar|V}}&nbsp;axis), then adiabats are concave towards the east north-east (31° from {{mvar|V}}&nbsp;axisaxis).

The term ''adiabatic'' ({{IPAc-en|ˌ|æ|d|i|ə|ˈ|b|æ|t|ɪ|k}}) is an anglicization of the [[Ancient Greek|Greek]] term {{math|ἀδιάβατος}} "impassable" (used by [[Xenophon]] of rivers). It is used in the thermodynamic sense by [[William John Macquorn Rankine|Rankine]] (1866),<ref name="Rankine 1866">{{cite journal |author-link=William John Macquorn Rankine |last=Rankine |first=W.J.McQ. |date=27 July 1866 |title=On the theory of explosive gas engines |journal=The Engineer |page=467}} Page number from the reprint in {{cite book |author-link=William John Macquorn Rankine |last=Rankine |first=W.J.McQ. |year=1881 |title=[[iarchive:miscellaneoussci00rank|Miscellaneous Scientific Papers]] |editor-first=W.J. |editor-last=Millar |publisher=Charles Griffin |place=London, UK }} </ref><ref name= "Partington 122">{{Citation

| first = J.R.

The Greek word {{math|ἀδιάβατος}} is formed from [[privative a|privative {{math|ἀ-}}]] ("not") and {{math|διαβατός}}, "passable", in turn deriving from {{math|διά}} ("through"), and {{math|βαῖνειν}} ("to walk, go, come").<ref>{{cite book |author1-link=Henry Liddell |last1=Liddell |first1=H.G. |author2-link=Robert Scott (philologist) |last2=Scott |first2=R. |year=1940 |title=[[A Greek-English Lexicon]] |publisher=Clarendon Press |place=Oxford, UK }} </ref>

Furthermore, in [[atmospheric thermodynamics]], a diabatic process is one in which heat is exchanged.<ref>{{cite web |title=diabatic process |publisher=[[American Meteorological Society]] |url=https://glossary.ametsoc.org/wiki/Diabatic_process |access-date=24 November 2020 }}</ref> An adiabatic process is the opposite – a process in which no heat is exchanged.

<math display="block">c_1^2 = 1, ~ c_2^2=0, ~ c_3^2=0,\ ... ~.</math>

{{reflist|25em}}

* Thorngren, Dr. Jane R. "[http://daphne.palomar.edu/jthorngren/adiabatic_processes.htm Adiabatic Processes]". Daphne – A Palomar College Web Server, 21 July 1995. {{Webarchive|url=https://web.archive.org/web/20110509121743/http://daphne.palomar.edu/jthorngren/adiabatic_processes.htm |date=2011-05-09 }}

== External links ==

* [http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html#c1: Article in HyperPhysics Encyclopaedia]

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