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An Ideal Gas is a Gas that is Ideal enough to be an Ideal Gas. The state of being an ideal gas is transitive. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec eu lorem vehicula, congue enim vel, consequat dui. Donec condimentum ligula a eleifend pellentesque. Donec eget tellus a nisl sodales cursus. Vestibulum sit amet nisl placerat, viverra lorem sit amet, tincidunt dolor. Cras pretium orci at vulputate molestie. Duis vitae viverra dolor. Aenean eros neque, luctus a urna vitae, dapibus viverra risus. In id elementum nibh. Praesent eu quam imperdiet, ornare dolor et, tristique neque. Phasellus a gravida nunc. Aliquam iaculis ex convallis tellus tempus, sit amet malesuada mi convallis. Interdum et malesuada fames ac ante ipsum primis in faucibus. Mauris odio quam, condimentum sit amet sollicitudin eu, tempus ut erat. Donec finibus, metus eu vehicula dapibus, sem sem lacinia augue, non fringilla turpis purus at risus. Etiam rutrum ligula leo, ultricies pharetra risus sollicitudin a.
{{short description|Mathematical model which approximates the behavior of real gases}}
{{Thermodynamics|cTopic=[[Thermodynamic system|Systems]]}}


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An '''ideal gas''' is a theoretical [[gas]] composed of many randomly moving [[point particle]]s that are not subject to [[Intermolecular force|interparticle interactions]].<ref name=tuckerman>{{Cite book|title=Statistical Mechanics: Theory and Molecular Simulation|edition=1st|isbn=978-0-19-852526-4|last=Tuckerman|first=Mark E.|year=2010|page=87}}</ref> The ideal gas concept is useful because it obeys the [[ideal gas law]], a simplified [[equation of state]], and is amenable to analysis under [[statistical mechanics]]. The requirement of zero interaction can often be relaxed if, for example, the interaction is [[Elastic collision|perfectly elastic]] or regarded as point-like collisions.


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Under various conditions of temperature and pressure, many [[real gas]]es behave qualitatively like an ideal gas where the gas molecules (or atoms for [[monatomic gas]]) play the role of the ideal particles. Many gases such as [[nitrogen]], [[oxygen]], [[hydrogen]], [[noble gas]]es, some heavier gases like [[carbon dioxide]] and mixtures such as air, can be treated like ideal gases within reasonable tolerances<ref name="boles">{{Cite book|title=Thermodynamics: An Engineering Approach|edition=4th|isbn=0-07-238332-1|last1=Cengel|first1=Yunus A.|last2=Boles|first2=Michael A.|page=[https://archive.org/details/thermodynamicsen00ceng_0/page/89 89]|url-access=registration|url=https://archive.org/details/thermodynamicsen00ceng_0/page/89}}</ref> over a considerable parameter range around [[standard temperature and pressure]]. Generally, a gas behaves more like an ideal gas at higher [[temperature]] and lower [[pressure]],<ref name=boles/> as the [[potential energy]] due to intermolecular forces becomes less significant compared with the particles' [[kinetic energy]], and the size of the molecules becomes less significant compared to the empty space between them. One [[mole (unit)|mole]] of an ideal gas has a capacity of 22.710947(13)&nbsp;litres<ref name=mvol >{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?mvol |title=CODATA Value: molar volume of ideal gas (273.15 K, 100 kPa) |accessdate=2017-02-07}}</ref> at standard temperature and pressure (a [[temperature]] of 273.15&nbsp;[[Kelvin|K]] and an [[absolute pressure]] of exactly 10<sup>5</sup>&nbsp;[[Pascal (unit)|Pa]]) as defined by [[International Union of Pure and Applied Chemistry|IUPAC]] since 1982.{{#tag:ref|Until 1982, STP was defined as a temperature of 273.15&nbsp;[[Kelvin|K]] and an [[absolute pressure]] of exactly 1&nbsp;[[Atmosphere (unit)|atm]]. The volume of one mole of an ideal gas at this temperature and pressure is 22.413962(13) litres.<ref name=mvolstd >{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?mvolstd |title=CODATA Value: molar volume of ideal gas (273.15 K, 101.325 kPa) |accessdate=2017-02-07}}</ref> IUPAC recommends that the former use of this definition should be discontinued;<ref name=IUPAC>{{cite journal |last1=Calvert |first1=J. G. |year=1990 |title=Glossary of atmospheric chemistry terms (Recommendations 1990) |journal=Pure and Applied Chemistry |volume=62 |issue=11 |pages=2167–2219 |doi=10.1351/pac199062112167 |doi-access=free }}</ref> however, some textbooks still use these old values.|group=note}}


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The ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forces and molecular size becomes important. It also fails for most heavy gases, such as many [[refrigerant]]s,<ref name=boles/> and for gases with strong intermolecular forces, notably [[water vapor]]. At high pressures, the volume of a real gas is often considerably larger than that of an ideal gas. At low temperatures, the pressure of a real gas is often considerably less than that of an ideal gas. At some point of low temperature and high pressure, real gases undergo a [[phase transition]], such as to a [[liquid]] or a [[solid]]. The model of an ideal gas, however, does not describe or allow phase transitions. These must be modeled by more complex equations of state. The deviation from the ideal gas behavior can be described by a [[dimensionless quantity]], the [[compressibility factor]], {{mvar|Z}}.


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The ideal gas model has been explored in both the [[Newtonian dynamics]] (as in "[[kinetic theory of gases|kinetic theory]]") and in [[quantum mechanics]] (as a "[[gas in a box]]"). The ideal gas model has also been used to model the behavior of electrons in a metal (in the [[Drude model]] and the [[free electron model]]), and it is one of the most important models in statistical mechanics.

If the pressure of an ideal gas is reduced in a [[throttling process]] the temperature of the gas does not change. (If the pressure of a real gas is reduced in a throttling process, its temperature either falls or rises, depending on whether its [[Joule–Thomson (Kelvin) coefficient|Joule–Thomson coefficient]] is positive or negative.)

==Types of ideal gas==

There are three basic classes of ideal gas:{{citation needed|reason=to verify this whole section|date=June 2017}}
* the classical or [[Maxwell–Boltzmann statistics|Maxwell–Boltzmann]] ideal gas,
* the ideal quantum [[Bose gas]], composed of [[boson]]s, and
* the ideal quantum [[Fermi gas]], composed of [[fermion]]s.

The classical ideal gas can be separated into two types: The classical thermodynamic ideal gas and the ideal quantum Boltzmann gas. Both are essentially the same, except that the classical thermodynamic ideal gas is based on classical [[statistical mechanics]], and certain thermodynamic parameters such as the [[entropy]] are only specified to within an undetermined additive constant. The ideal quantum Boltzmann gas overcomes this limitation by taking the limit of the quantum Bose gas and quantum Fermi gas in the limit of high temperature to specify these additive constants. The behavior of a quantum Boltzmann gas is the same as that of a classical ideal gas except for the specification of these constants. The results of the quantum Boltzmann gas are used in a number of cases including the [[Sackur–Tetrode equation]] for the entropy of an ideal gas and the [[Saha ionization equation]] for a weakly ionized [[plasma (physics)|plasma]].

==Classical thermodynamic ideal gas==

The classical thermodynamic properties of an ideal gas can be described by two [[Equation of state|equations of state]]:<ref>{{cite book|last=Adkins|first=C. J.|date=1983|title=Equilibrium Thermodynamics|edition=3rd|publisher=Cambridge University Press|location=Cambridge, UK|isbn=0-521-25445-0|pages=116–120}}</ref><ref>{{cite book|last=Tschoegl|first=N. W.|date=2000|title=Fundamentals of Equilibrium and Steady-State Thermodynamics|publisher=Elsevier|location=Amsterdam|isbn=0-444-50426-5|page=88}}</ref>

===Ideal gas law===
{{ideal_gas_law_relationships.svg}}
{{main|Ideal gas law}}

The [[ideal gas law]] is the equation of state for an ideal gas, given by:
:<math display="block", qid=Q11432>PV = nRT\,</math>
where
* {{mvar|P}} is the [[pressure]]
* {{mvar|V}} is the [[Volume (thermodynamics)|volume]]
* {{mvar|n}} is the [[amount of substance]] of the gas (in [[Mole (unit)|moles]])
* {{mvar|R}} is the [[gas constant]] (0.08206&nbsp;[[litres|L]]·[[atmosphere|atm]]·[[kelvin|K]]<sup>−1</sup>·[[mole (unit)|mol]]<sup>−1</sup>)
* {{mvar|T}} is the [[absolute temperature]].

The ideal gas law is an extension of experimentally discovered [[gas laws]]. It can also be derived from microscopic considerations.

Real [[fluid]]s at low [[density]] and high [[temperature]] approximate the behavior of a classical ideal gas. However, at lower temperatures or a higher density, a real fluid deviates strongly from the behavior of an ideal gas, particularly as it [[Condensation|condenses]] from a gas into a liquid or as it [[deposition (phase transition)|deposits]] from a gas into a solid. This deviation is expressed as a [[compressibility factor]].

This equation is derived from
* [[Boyle's law]]: <math>V\propto\frac{1}{P}</math>;
* [[Charles's law]]: <math>V\propto T</math>;
* [[Avogadro's law]]: <math> V \propto n</math>.
After combining three laws we get
:<math>V \propto \frac{nT}{P}</math>
That is:
:<math>V = R\left(\frac{nT}{P}\right)</math>
:<math>PV = nRT</math>.

===Internal energy===
The other equation of state of an ideal gas must express [[Joule–Thomson effect#Joule's second law|Joule's second law]], that the internal energy of a fixed mass of ideal gas is a function only of its temperature. For the present purposes it is convenient to postulate an exemplary version of this law by writing:
:<math>U = \hat{c}_V nRT </math>

where
* {{mvar|U}} is the [[internal energy]]
* {{mvar|ĉ<sub>V</sub>}} is the dimensionless specific [[heat capacity]] at constant volume, approximately {{sfrac|3|2}} for a [[monatomic gas]], {{sfrac|5|2}} for [[diatomic]] gas, and 3 for non-linear molecules if we ignore quantum vibrational contribution. These formulas arise from application of the classical [[equipartition theorem]].<ref>{{Cite book|title=Non-equilibrium thermodynamics and statistical mechanics : foundations and applications|author=Attard, Phil|date=2012|publisher=Oxford University Press|isbn=9780191639760|oclc=810281588}}</ref>

That {{mvar|U}} for an ideal gas depends only on temperature is a consequence of the ideal gas law, although in the general case {{mvar|ĉ<sub>V</sub>}} depends on temperature and an integral is needed to compute {{mvar|U}}.

===Microscopic model===

In order to switch from macroscopic quantities (left hand side of the following equation) to microscopic ones (right hand side), we use
:<math>nR=N k_\mathrm{B}</math>
where
* <math>N</math> is the number of gas particles
* <math>k_\mathrm{B}</math> is the [[Boltzmann constant]] ({{val|1.381|e=−23|u=J·K<sup>−1</sup>}}).

The probability distribution of particles by velocity or energy is given by the [[Maxwell speed distribution]].

The ideal gas model depends on the following assumptions:
* The molecules of the gas are indistinguishable, small, hard spheres
* All collisions are elastic and all motion is frictionless (no energy loss in motion or collision)
* Newton's laws apply
* The average distance between molecules is much larger than the size of the molecules
* The molecules are constantly moving in random directions with a distribution of speeds
* There are no attractive or repulsive forces between the molecules apart from those that determine their point-like collisions
* The only forces between the gas molecules and the surroundings are those that determine the point-like collisions of the molecules with the walls
* In the simplest case, there are no long-range forces between the molecules of the gas and the surroundings.

The assumption of spherical particles is necessary so that there are no rotational modes allowed, unlike in a diatomic gas. The following three assumptions are very related: molecules are hard, collisions are elastic, and there are no inter-molecular forces. The assumption that the space between particles is much larger than the particles themselves is of paramount importance, and explains why the ideal gas approximation fails at high pressures.

==Heat capacity==
The dimensionless [[Heat capacity#Heat capacity of compressible bodies|heat capacity]] at constant volume is generally defined by

:<math>\hat{c}_V=\frac{1}{n R}T\left(\frac{\partial S}{\partial T}\right)_V=\frac{1}{n R}\left(\frac{\partial U}{\partial T}\right)_V </math>

where {{mvar|S}} is the [[entropy]]. This quantity is generally a function of temperature due to intermolecular and intramolecular forces, but for moderate temperatures it is approximately constant. Specifically, the [[Equipartition Theorem]] predicts that the constant for a monatomic gas is {{mvar|''ĉ<sub>V</sub>''}}&nbsp;=&nbsp;{{sfrac|3|2}} while for a diatomic gas it is {{mvar|''ĉ<sub>V</sub>''}}&nbsp;=&nbsp;{{sfrac|5|2}} if vibrations are neglected (which is often an excellent approximation). Since the heat capacity depends on the atomic or molecular nature of the gas, macroscopic measurements on heat capacity provide useful information on the microscopic structure of the molecules.

The dimensionless heat capacity at constant pressure of an ideal gas is:

:<math>\hat{c}_P =\frac{1}{n R}T\left(\frac{\partial S}{\partial T}\right)_P= \frac{1}{n R}\left(\frac{\partial H}{\partial T}\right)_P = \hat{c}_V+1</math>

where {{math|''H'' {{=}} ''U'' + ''PV''}} is the [[enthalpy]] of the gas.

Sometimes, a distinction is made between an ideal gas, where {{mvar|''ĉ<sub>V</sub>''}} and {{mvar|''ĉ<sub>P</sub>''}} could vary with temperature, and a [[perfect gas]], for which this is not the case.

The ratio of the constant volume and constant pressure heat capacity is the [[adiabatic index]]

:<math>\gamma = \frac{c_P}{c_V} </math>

For air, which is a mixture of gases, this ratio is 1.4.

==Entropy==
Using the results of [[thermodynamics]] only, we can go a long way in determining the expression for the [[entropy]] of an ideal gas. This is an important step since, according to the theory of [[thermodynamic potential]]s, if we can express the entropy as a function of {{mvar|U}} ({{mvar|U}} is a thermodynamic potential), volume {{mvar|V}} and the number of particles {{mvar|N}}, then we will have a complete statement of the thermodynamic behavior of the ideal gas. We will be able to derive both the ideal gas law and the expression for internal energy from it.

Since the entropy is an [[exact differential]], using the [[chain rule]], the change in entropy when going from a reference state 0 to some other state with entropy {{mvar|S}} may be written as {{math|Δ''S''}} where:

:<math>\Delta S = \int_{S_0}^{S}dS
=\int_{T_0}^{T} \left(\frac{\partial S}{\partial T}\right)_V\!dT
+\int_{V_0}^{V} \left(\frac{\partial S}{\partial V}\right)_T\!dV
</math>

where the reference variables may be functions of the [[number of particles]] {{mvar|N}}. Using the definition of the [[heat capacity]] at constant volume for the first differential and the appropriate [[Maxwell relation]] for the second we have:

:<math>\Delta S
=\int_{T_0}^{T} \frac{C_V}{T}\,dT+\int_{V_0}^{V}\left(\frac{\partial P}{\partial T}\right)_VdV.
</math>

Expressing {{mvar|C<sub>V</sub>}} in terms of {{mvar|''ĉ<sub>V</sub>''}} as developed in the above section, differentiating the ideal gas equation of state, and integrating yields:

:<math>\Delta S
= \hat{c}_VNk\ln\left(\frac{T}{T_0}\right)+Nk\ln\left(\frac{V}{V_0}\right)
</math>

which implies that the entropy may be expressed as:

:<math>S= Nk\ln\left(\frac{VT^{\hat{c}_V}}{f(N)}\right)
</math>

where all constants have been incorporated into the logarithm as {{math|''f''(''N'')}} which is some function of the particle number {{mvar|N}} having the same dimensions as {{mvar|VT<sup>ĉ<sub>V</sub></sup>}} in order that the argument of the logarithm be dimensionless. We now impose the constraint that the entropy be extensive. This will mean that when the extensive parameters ({{mvar|V}} and {{mvar|N}}) are multiplied by a constant, the entropy will be multiplied by the same constant. Mathematically:

:<math>S(T,aV,aN)=a S(T,V,N).\,</math>

From this we find an equation for the function {{math|''f''(''N'')}}

:<math>af(N)=f(aN).\,</math>

Differentiating this with respect to {{mvar|a}}, setting {{mvar|a}} equal to 1, and then solving the differential equation yields {{math|''f''(''N'')}}:

:<math>f(N)=\Phi N\,</math>

where {{mvar|Φ}} may vary for different gases, but will be independent of the thermodynamic state of the gas. It will have the dimensions of {{math|''VT<sup>ĉ<sub>V</sub></sup>''/''N''}}. Substituting into the equation for the entropy:

:<math>\frac{S}{Nk} = \ln\left(\frac{VT^{\hat{c}_V}}{N\Phi}\right).\,</math>

and using the expression for the internal energy of an ideal gas, the entropy may be written:

:<math>\frac{S}{Nk} = \ln\left[\frac{V}{N}\,\left(\frac{U}{\hat{c}_V k N}\right)^{\hat{c}_V}\,\frac{1}{\Phi}\right]</math>

Since this is an expression for entropy in terms of {{mvar|U}}, {{mvar|V}}, and {{mvar|N}}, it is a fundamental equation from which all other properties of the ideal gas may be derived.

This is about as far as we can go using thermodynamics alone. Note that the above equation is flawed&nbsp;– as the temperature approaches zero, the entropy approaches negative infinity, in contradiction to the [[third law of thermodynamics]]. In the above "ideal" development, there is a critical point, not at absolute zero, at which the argument of the logarithm becomes unity, and the entropy becomes zero. This is unphysical. The above equation is a good approximation only when the argument of the logarithm is much larger than unity&nbsp;– the concept of an ideal gas breaks down at low values of {{mvar|{{sfrac|V|N}}}}. Nevertheless, there will be a "best" value of the constant in the sense that the predicted entropy is as close as possible to the actual entropy, given the flawed assumption of ideality. A quantum-mechanical derivation of this constant is developed in the derivation of the [[Sackur–Tetrode equation]] which expresses the entropy of a monatomic ({{mvar|''ĉ<sub>V</sub>''}}&nbsp;=&nbsp;{{sfrac|3|2}}) ideal gas. In the Sackur–Tetrode theory the constant depends only upon the mass of the gas particle. The Sackur–Tetrode equation also suffers from a divergent entropy at absolute zero, but is a good approximation for the entropy of a monatomic ideal gas for high enough temperatures.

An alternative way of expressing the change in entropy:

<math>\frac {\Delta S}{Nk\hat{c}_V}
= \ln\left(\frac{P}{P_0}\right)+\gamma \ln\left(\frac{V}{V_0}\right)=\ln\left(\frac{PV^\gamma}{P_0V_0^\gamma}\right) \implies PV^\gamma=const. \; \text{for isentropic process}
</math>

==Thermodynamic potentials==
{{Main|Thermodynamic potential}}
Expressing the entropy as a function of {{mvar|T}}, {{mvar|V}}, and {{mvar|N}}:

:<math>\frac{S}{kN}=\ln\left( \frac{VT^{\hat{c}_V}}{N\Phi}\right)</math>

The [[chemical potential]] of the ideal gas is calculated from the corresponding equation of state (see [[thermodynamic potential]]):

:<math>\mu=\left(\frac{\partial G}{\partial N}\right)_{T,P}</math>

where {{mvar|G}} is the [[Gibbs free energy]] and is equal to {{math|''U'' + ''PV'' − ''TS''}} so that:

:<math>\mu(T,P)=kT\left(\hat{c}_P-\ln\left(\frac{kT^{\hat{c}_P}}{P\Phi}\right)\right)</math>

The chemical potential is usually referenced to the potential at some standard pressure ''P<sup>o</sup>'' so that, with <math>\mu^o(T)=\mu(T,P^o)</math>:

:<math>\mu(T,P)=\mu^o(T)+ kT\ln\left(\frac{P}{Po}\right)</math>

For a mixture (''j''=1,2,...) of ideal gases, each at partial pressure ''P<sub>j</sub>'', it can be shown that the chemical potential ''&mu;<sub>j</sub>'' will be given by the above expression with the pressure ''P'' replaced by ''P<sub>j</sub>''.

The thermodynamic potentials for an ideal gas can now be written as functions of {{mvar|T}}, {{mvar|V}}, and {{mvar|N}} as:

:{|
|-
|<math>U\,</math>
|
|<math>=\hat{c}_V NkT\,</math>
|-
|<math>A\,</math>
|<math>=U-TS\,</math>
|<math>=\mu N-NkT\,</math>
|-
|<math>H\,</math>
|<math>=U+PV\,</math>
|<math>=\hat{c}_P NkT\,</math>
|-
|<math>G\,</math>
|<math>=U+PV-TS\,</math>
|<math>=\mu N\,</math>
|}

where, as before,
:<math>\hat{c}_P=\hat{c}_V+1</math>.

The most informative way of writing the potentials is in terms of their natural variables, since each of these equations can be used to derive all of the other thermodynamic variables of the system. In terms of their natural variables, the thermodynamic potentials of a single-species ideal gas are:

:<math>U(S,V,N)=\hat{c}_V N k \left(\frac{N\Phi}{V}\,e^{S/Nk}\right)^{1/\hat{c}_V}</math>
:<math>A(T,V,N)=NkT\left(\hat{c}_V-\ln\left(\frac{VT^{\hat{c}_V}}{N\Phi}\right)\right)</math>
:<math>H(S,P,N)=\hat{c}_P Nk\left(\frac{P\Phi}{k}\,e^{S/Nk}\right)^{1/\hat{c}_P}</math>
:<math>G(T,P,N)=NkT\left(\hat{c}_P-\ln\left(\frac{kT^{\hat{c}_P}}{P\Phi}\right)\right)</math>

In [[statistical mechanics]], the relationship between the [[Helmholtz free energy]] and the [[partition function (statistical mechanics)|partition function]] is fundamental, and is used to calculate the [[thermodynamic properties]] of matter; see [https://web.archive.org/web/20120428193950/http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configuration integral] for more details.

==Speed of sound==
The [[Speed of sound#Equations|speed of sound]] in an ideal gas is given by the Newton-Laplace formula:
:<math>c_\text{sound} = \sqrt{\frac{K_s}{\rho}}=\sqrt{\left(\frac{\partial P}{\partial \rho}\right)_{s}}, </math>

where the isentropic [[Bulk modulus]] <math>K_s=\rho \left(\frac{\partial P}{\partial \rho}\right)_{s} </math>.

For an isentropic process of an ideal gas, <math>PV^\gamma=const \Rightarrow P\propto \left(\frac{1}{V}\right)^\gamma\propto \rho ^\gamma
</math>, therefore

:<math>c_\text{sound} = \sqrt{\left(\frac{\partial P}{\partial \rho}\right)_{s}} = \sqrt{\frac{\gamma P}{\rho}}=\sqrt{\frac{\gamma R T}{M}} </math>

Here,
:{{mvar|γ}} is the [[adiabatic index]] ({{mvar|{{sfrac|ĉ<sub>P</sub>|ĉ<sub>V</sub>}}}})
:{{mvar|s}} is the [[entropy]] per particle of the gas.
:{{mvar|ρ}} is the [[mass density]] of the gas.
:{{mvar|P}} is the [[pressure]] of the gas.
:{{mvar|R}} is the [[universal gas constant]]
:{{mvar|T}} is the [[temperature]]
:{{mvar|M}} is the [[molar mass]] of the gas.

== Table of ideal gas equations ==
{{main|Table of thermodynamic equations#Ideal gas}}

==Ideal quantum gases==

In the above-mentioned [[Sackur–Tetrode equation]], the best choice of the entropy constant was found to be proportional to the quantum [[thermal wavelength]] of a particle, and the point at which the argument of the logarithm becomes zero is roughly equal to the point at which the average distance between particles becomes equal to the thermal wavelength. In fact, [[Quantum mechanics|quantum]] theory itself predicts the same thing. Any gas behaves as an ideal gas at high enough temperature and low enough density, but at the point where the Sackur–Tetrode equation begins to break down, the gas will begin to behave as a quantum gas, composed of either [[boson]]s or [[fermion]]s. (See the [[gas in a box]] article for a derivation of the ideal quantum gases, including the ideal Boltzmann gas.)

Gases tend to behave as an ideal gas over a wider range of pressures when the temperature reaches the [[Boyle temperature]].

===Ideal Boltzmann gas===

The ideal Boltzmann gas yields the same results as the classical thermodynamic gas, but makes the following identification for the undetermined constant {{mvar|Φ}}:

:<math>\Phi = \frac{T^\frac32 \Lambda^3}{g}</math>

where {{mvar|Λ}} is the [[thermal de Broglie wavelength]] of the gas and {{mvar|g}} is the [[degenerate energy level|degeneracy]] of states.

===Ideal Bose and Fermi gases===

An ideal [[Bose gas|gas of bosons]] (e.g. a [[photon gas]]) will be governed by [[Bose–Einstein statistics]] and the distribution of energy will be in the form of a [[Bose–Einstein distribution]]. An ideal [[Fermi gas|gas of fermions]] will be governed by [[Fermi–Dirac statistics]] and the distribution of energy will be in the form of a [[Fermi–Dirac distribution]].

==See also==
* {{annotated link|Compressibility factor}}
* {{annotated link|Dynamical billiards}} – billiard balls as a model of an ideal gas
* {{annotated link|Table of thermodynamic equations}}
* {{annotated link|Scale-free ideal gas}}

==References==
;Notes
{{reflist|group=note}}
;References
{{reflist|30em}}

{{Atomic models}}

[[Category:Ideal gas| ]]
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Revision as of 04:10, 22 December 2020

An Ideal Gas is a Gas that is Ideal enough to be an Ideal Gas. The state of being an ideal gas is transitive. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec eu lorem vehicula, congue enim vel, consequat dui. Donec condimentum ligula a eleifend pellentesque. Donec eget tellus a nisl sodales cursus. Vestibulum sit amet nisl placerat, viverra lorem sit amet, tincidunt dolor. Cras pretium orci at vulputate molestie. Duis vitae viverra dolor. Aenean eros neque, luctus a urna vitae, dapibus viverra risus. In id elementum nibh. Praesent eu quam imperdiet, ornare dolor et, tristique neque. Phasellus a gravida nunc. Aliquam iaculis ex convallis tellus tempus, sit amet malesuada mi convallis. Interdum et malesuada fames ac ante ipsum primis in faucibus. Mauris odio quam, condimentum sit amet sollicitudin eu, tempus ut erat. Donec finibus, metus eu vehicula dapibus, sem sem lacinia augue, non fringilla turpis purus at risus. Etiam rutrum ligula leo, ultricies pharetra risus sollicitudin a.

Integer feugiat orci mi, vitae vulputate turpis condimentum in. Quisque eu tellus at magna venenatis interdum sed eu urna. Nullam id dapibus massa, et pharetra nisi. Duis hendrerit rhoncus rhoncus. Duis iaculis orci vel ex egestas gravida. Sed hendrerit, odio eget aliquam fringilla, massa odio rutrum urna, eget eleifend ex ipsum id nisl. Lorem ipsum dolor sit amet, consectetur adipiscing elit. Suspendisse ullamcorper ante non risus tempor interdum.

Aenean ultricies est at tortor congue fermentum. Morbi ornare dolor id arcu eleifend, pretium faucibus odio tincidunt. Curabitur ultrices, orci quis consequat volutpat, odio risus semper eros, quis interdum augue nibh a eros. Proin aliquam, ligula posuere tempus tempus, dolor magna dignissim felis, eget condimentum tellus tortor sit amet est. Curabitur iaculis bibendum odio, eu fermentum quam sagittis non. Nulla lacinia dictum purus condimentum tempus. In non tellus enim.

Maecenas venenatis vestibulum molestie. Morbi a interdum tellus. Praesent pellentesque velit ut urna ornare placerat. Suspendisse id pulvinar sem. Morbi eu porta felis. Orci varius natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Sed lobortis gravida libero, a volutpat quam scelerisque et. Donec sed erat sit amet nunc lobortis elementum.

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