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Perfect gas

Thermodynamics
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An ideal gas is a theoretical model that is widely used to describe the properties and behavior of real gases at moderate pressures and temperatures . In this model, firstly, it is assumed that the particles making up the gas do not interact with each other, that is, their sizes are negligibly small, therefore, there are no mutual collisions of particles in the volume occupied by the ideal gas. Particles of ideal gas undergo collisions only with the walls of the vessel. The second assumption: there is no long-range interaction between gas particles, for example, electrostatic or gravitational. An additional condition for elastic collisions between molecules and vessel walls within the framework of the molecular-kinetic theory leads to the thermodynamics of an ideal gas.

In various extended models of an ideal gas, it is assumed that the particles have an internal structure and extended sizes, which can be represented by particles in the form of ellipsoids or spheres connected by elastic bonds (for example, diatomic molecules). The representation of gas particles in the form of polyatomic molecules leads to the appearance of additional degrees of freedom, which makes it necessary to take into account the energy of not only translational, but also rotational-vibrational motion of particles, as well as not only central, but also off-center collisions of particles [1] .

The model is widely used to solve the problems of thermodynamics of gases and gas dynamics . For example, air at atmospheric pressure and room temperature with sufficient accuracy for practical calculations is well described by the ideal gas model.

In the case of very high pressures , the use of more accurate equations of state of real gases, for example, the semi-empirical van der Waals equation , which takes into account the attraction between the molecules, is required and their final sizes. At very high temperatures, the molecules of real gases can dissociate into their constituent atoms, or the atoms can ionize with the removal of electrons. Therefore, in cases of high pressures and / or temperatures, the equations of state of an ideal gas are applicable only with some assumptions, or are not applicable at all.

Distinguish the classical ideal gas (its properties are derived from the laws of classical mechanics and obey Maxwell-Boltzmann statistics ) , a semiclassical ideal gas [2] (for which, unlike the classical ideal gas, the law of uniform distribution of energy over the degrees of freedom [3] [4] is not satisfied) and quantum ideal gas (its properties are determined by the laws of quantum mechanics and are described by Fermi statistics - Dirac or Bose - Einstein ) .

From a thermodynamic point of view, the difference between classical and semiclassical ideal gases is as follows. The heat capacity of a classical ideal gas does not depend on temperature and is uniquely determined by the geometry of the gas molecule [5] , which thereby determines the form of the caloric equation of state of the gas. Classical ideal gases with the same geometry of molecules obey the same caloric equation of state. The heat capacity of a semiclassical ideal gas depends on temperature [6] [K 1] , and this dependence is individual for each gas; accordingly, each semiclassical ideal gas is described by its own caloric equation of state. Very often, including in this article, when the differences between the classical and semiclassical approximations do not play a role, the term “classical ideal gas” is considered as a synonym for the expression “ non-quantum ideal gas ”. In the macroscopic approach, hypothetical (non-existent) gases that obey the thermal equation of state of Clapeyron [11] [12] (Clapeyron-Mendeleev [13] [12] ) are called ideal classical and quasiclassical gases. Sometimes it is additionally indicated that the Joule law [14] [15] [16] [17] is valid for a classical ideal gas. Thermodynamics claims that the Joule law holds for any fluid with an equation of state of the formpT=f(V) {\ displaystyle {\ frac {p} {T}} = f (V)} {\ displaystyle {\ frac {p} {T}} = f (V)} orpV=f(T) {\ displaystyle pV = f (T)} {\ displaystyle pV = f (T)} wherep {\ displaystyle p} p - pressureT {\ displaystyle T} T - absolute temperature andV {\ displaystyle V} V - volume (see [18] [19] [20] ). Therefore, in defining the classical ideal gas, it is not necessary to mention the Joule law. On the other hand, if we consider this law as a generalization of experimental data, then the presentation of the macroscopic theory of the classical ideal gas requires the involvement of only the most basic information from thermodynamics.

The popularity of the “ideal gas” model in thermodynamic training courses is due to the fact that the results obtained using the Clapeyron equation are not very complex mathematical expressions and usually allow simple analytical and / or graphical analysis of the behavior of the quantities included in them. The semiclassical approximation is used to calculate the thermodynamic functions of gases from their molecular data [21] [22] [23] .

Content

  • 1 History
  • 2 Classic perfect gas
    • 2.1 Molecular-kinetic theory of an ideal gas
    • 2.2 Thermal equation of state and thermal coefficients of an ideal gas
    • 2.3 Mixture of ideal gases
    • 2.4 Perfect gas (hydroaeromechanics)
    • 2.5 Heat capacity
    • 2.6 Entropy and thermodynamic potentials
  • 3 Application of the ideal gas theory
    • 3.1 Physical meaning of gas temperature
    • 3.2 Boltzmann distribution
    • 3.3 adiabatic process
    • 3.4 Sound speed
  • 4 quantum ideal gas
    • 4.1 Fermi gas
    • 4.2 Bose gas
  • 5 Ideal gas in a gravitational field
  • 6 The limits of applicability of the ideal gas theory
  • 7 See also
  • 8 Comments
  • 9 notes
  • 10 Literature

History

 
Benoit Clapeyron

The history of the concept of ideal gas dates back to the successes of experimental physics, the beginning of which was laid in the 17th century. In 1643, Evangelista Torricelli first proved that air has weight (mass), and, together with V. Viviani , conducted an experiment on measuring atmospheric pressure using a glass tube filled with mercury sealed at one end. So the first mercury barometer was born. In 1650, the German physicist Otto von Guericke invented an air pump and in 1654 conducted the famous experiment with the Magdeburg hemispheres , which clearly confirmed the existence of atmospheric pressure. The experiments of the English physicist Robert Boyle in balancing the mercury column with compressed air pressure in 1662 led to the conclusion of the gas law, later called the Boyle – Mariotte law [24] , due to the fact that the French physicist Edm Marriott conducted a similar independent study in 1679.

In 1802, the French physicist Gay-Lussac published in open press the law of volumes (called the Gay-Lussac law in Russian literature) [25] , however, Gay-Lussac himself believed that the discovery was made by Jacques Charles in an unpublished work dating back to 1787 . Regardless of them, this law was discovered in 1801 by the English physicist John Dalton . In addition, it was qualitatively described by the French scientist Guillaume Amonton at the end of the 17th century. Gay-Lussac also found that the coefficient of volume expansion is the same for all gases, despite the generally accepted view that different gases expand when heated in different ways.

Gay-Lussac (1822) [26] [27] [28] and Sadi Carnot (1824) [29] [30] [28] were the first to combine the laws of Boyle - Marriott and Charles - Dalton - Gay-Lussac in a single equation . Since, however, Gay-Lussac did not use the equation he found, but did not get acquainted with the results obtained by Carnot not in his bibliographic rarity [31] book “Reflections on the driving force of fire and machines capable of developing this force” [32] , but in exposing Carnot’s ideas in Benoit Clapeyron ’s work “A Memoir on the Driving Force of Fire” [33] , the derivation of the thermal equation of state of an ideal gas was attributed to Clapeyron [34] [30] , and the equation was called the Clapeyron equation , although this scientist himself never claimed to be authorship discussion equation [28] . Meanwhile, there is no doubt that it was Clapeyron who first understood the fruitfulness of applying the equation of state, which greatly simplified all gas-related calculations.

Experimental studies of the physical properties of real gases in those years were not quite accurate and were carried out under conditions not very different from normal (temperature 0 ℃, pressure 760 mm Hg ). It was also assumed that gas, unlike vapor , is a substance that is unchanged under any physical conditions. The first blow to these ideas was the liquefaction of chlorine in 1823. Later it turned out that real gases are superheated vapors , quite distant from the regions of condensation and critical state. Any real gas can be turned into a liquid by condensation, or by continuous changes in a single-phase state. Thus, it turned out that real gases represent one of the aggregate states of the corresponding simple bodies, and the equation of state of a simple body can be the exact equation of state of the gas. Despite this, gas laws have been preserved in thermodynamics and in its technical applications as the laws of ideal gases — limiting (practically unattainable) states of real gases [35] . The Clapeyron equation was derived under certain assumptions based on the molecular-kinetic theory of gases (by August Kroenig in 1856 [36] and Rudolf Clausius in 1857) [37] . Clausius introduced the very concept of “ideal gas” [38] (in the Russian literature of the late XIX - early XX centuries, the term “perfect gas” was used instead of the name “ideal gas [39] ).

The next important step in the formulation of the thermal equation of state of an ideal gas — the transition from an individual constant for each gas to a universal gas constant — was made by Russian engineer Ilya Alymov [40] [30] [41] , whose work, published in a publication little known among physicists and chemists, not attracted attention. The same result was obtained by Mendeleev in 1874 [39] [30] [41] . Regardless of the works of Russian scientists, [42] , Kato Guldberg (1867) [43] and (1873) [44] came to the conclusion that the product of an individual constant for each gas in the equation Clapeyron on the molecular weight of the gas should be constant for all gases.

In 1912, when deriving the Nernst constant, the principle of dividing the phase space into equal-sized cells was first applied. Subsequently, in 1925, S. Bose published an article entitled “Planck’s Law and the Hypothesis of Light Quantums”, in which he developed this idea in relation to photon gas. Einstein said about this article that “the method used here allows us to obtain the quantum theory of an ideal gas” [45] . In December of that year, Enrico Fermi developed statistics on half-integer spin particles obeying the Pauli principle , which were later called fermions [46] [47] .

In Russian literature published before the end of the 1940s, the thermal equation of state of an ideal gas was called the Clapeyron equation [48] [49] [50] [51] [52] [53] or the Clapeyron equation for 1 mole [54] . In the fundamental Russian monograph of 1948, devoted to various equations of state of gases [55] , Mendeleev - unlike Clapeyron - is not mentioned at all. The surname Mendeleev in the name of the equation we are considering appeared after the start of the "struggle against cringing the West" and the search for "Russian priorities . " It was then that in the scientific and educational literature began to use such variations of the name as the Mendeleev equation [39] [56] , the Mendeleev – Clapeyron equation [57] [58] [59] and the Clapeyron – Mendeleev equation [56] [60] [ 61] [62] .


Classic Perfect Gas

Molecular Kinetic Theory of Ideal Gas

 
The volume of an ideal gas linearly depends on temperature at constant pressure

The properties of an ideal gas based on molecular kinetic representations are determined based on the physical model of an ideal gas, in which the following assumptions are made:

  • The size of the molecules is negligible compared to the average distance between them, so that the total volume occupied by the molecules is much smaller than the volume of the vessel [63] [64] [65] ;
  • the momentum is transmitted only during collisions, that is, the attractive forces between the molecules are not taken into account, and the repulsive forces arise only during collisions [65] ;
  • collisions of particles between themselves and with the walls of the vessel are absolutely elastic [65] ;
  • the number of molecules in the gas is large and fixed, which allows us to calculate average values ​​from a small (compared to the size of the system) volume, the system is ergodic so that the ensemble averages are equal to the time averages;
  • the gas is in thermodynamic equilibrium with the walls of the vessel and there are no additional macroscopic flows of matter. Here it should be clarified that the gradients of thermodynamic quantities can take place, such as when an external field, for example, a gravitational field, is turned on.

In this case, the gas particles move independently of each other, the gas pressure on the wall is equal to the total momentum transmitted when the particles collide with the wall area of ​​a unit area per unit time [65] , the internal energy is the sum of the energies of the gas particles [66] .

According to the equivalent macroscopic formulation, an ideal gas is a gas that simultaneously obeys the Boyle – Mariotte and Gay – Lussac law [64] [67] , that is:

pV=const⋅T,{\ displaystyle pV = \ mathrm {const} \ cdot T,}  

Wherep {\ displaystyle p}   - pressureV {\ displaystyle V}   - volumeT {\ displaystyle T}   - absolute temperature.

Thermal equation of state and thermal coefficients of an ideal gas

 
Plots of isoprocesses in an ideal constant-mass gas
 
Isotherms of an ideal gas on a p - V - T diagram

The thermal properties of the classical and semiclassical ideal gas are described by the Clapeyron equation [68] [69] [58] :

pV=mMRT,{\ displaystyle pV = {\ frac {m} {M}} RT,}  

where R is the universal gas constant (8.3144598 J ⁄ ( mol ∙ K) ), m is the mass of gas, M is its molar mass , or

pV=νRT,{\ displaystyle pV = \ nu RT,}  

where ν is the amount of gas in moles .

In the formulas of statistical physics, it is customary to use the Boltzmann constant k (1.3806 · 10 −23 J ⁄ K ), the particle massm´ {\ displaystyle {\ acute {m}}}   and the number of particles N. Statistical and thermodynamic quantities are related by the relations:

m=m´N,ν=NNA,R=kNA,kN=νR,{\ displaystyle m = {\ acute {m}} N, ~~~ \ nu = {\ frac {N} {N_ {A}}}, ~~~ R = kN_ {A}, ~~~ kN = \ nu R,}  

where N A is the Avogadro number (6.02214 · 10 23 1 ⁄ mol ). Using the notation of statistical physics, the Clapeyron equation takes the form

pV=NkT,{\ displaystyle pV = NkT,}  

or

p=nkT,{\ displaystyle p = nkT,}  

where n is the concentration of particles .

Материал, касающийся термических коэффициентов идеального газа, изложен в статье Уравнение состояния .

Смесь идеальных газов

Смесь идеальных газов тоже идеальный газ. Каждой компоненте газа соответствует своё парциальное давление и общее давление смеси есть сумма парциальных давлений компонент смеси p=pone+p2+p3{\displaystyle p=p_{1}+p_{2}+p_{3}}   … Также можно получить общее количество молей в смеси газов как сумму ν=νone+ν2+ν3{\displaystyle \nu =\nu _{1}+\nu _{2}+\nu _{3}}   … Тогда уравнение состояния для смеси идеальных газов [70]

pV=νRT.{\displaystyle pV=\nu RT.}  

Совершенный газ (гидроаэромеханика)

В отличие от термодинамики в гидроаэромеханике газ, подчиняющийся уравнению Клапейрона, называют совершенным . У совершенного газа молярные изохорная CV{\displaystyle C_{V}}   и изобарная CP{\displaystyle C_{P}}   теплоёмкости постоянны. В то же время идеальным в гидроаэромеханике называют газ, у которого отсутствуют вязкость и теплопроводность . Модель совершенного газа широко применяют при исследовании течения газов [71] .

Теплоёмкость

Определим теплоёмкость при постоянном объёме для идеального газа как

c^V=oneνRT(∂S∂T)V=oneνR(∂U∂T)V,{\displaystyle {\hat {c}}_{V}={\frac {1}{\nu R}}T\left({\frac {\partial S}{\partial T}}\right)_{V}={\frac {1}{\nu R}}\left({\frac {\partial U}{\partial T}}\right)_{V},}  

где S — энтропия . Это безразмерная теплоёмкость при постоянном объёме, которая обычно зависит от температуры из-за межмолекулярных сил. При умеренных температурах это константа: для одноатомного газа ĉ V = 3/2, для двухатомного газа и многоатомных газов с линейными молекулами это ĉ V = 5/2, а для многоатомного газа с нелинейными молекулами ĉ V = 6/2=3. Видно, что макроскопические измерения теплоемкости могут дать информацию о микроскопической структуре молекул. В отечественной учебной литературе, где понятие безразмерной теплоёмкости не получило распространения, для классического идеального газа его теплоёмкость при постоянном объёме C V полагают не зависящей от температуры и, согласно теореме о равнораспределении , равной [72] : 3 Rν /2 для всех одноатомных газов, 5 Rν /2 для всех двухатомных газов и многоатомных газов с линейными молекулами, 3 Rν для всех многоатомных газов с нелинейными молекулами. Отличие квазиклассического идеального газа от классического состоит в ином виде зависимости внутренней энергии газа от его температуры [73] . Для классического идеального газа его теплоёмкость при постоянном объёме C V не зависит от температуры (она составляет), то есть внутренняя энергия газа всегда пропорциональна его температуре; для квазиклассического идеального газа его теплоёмкость CV{\displaystyle C_{V}}   зависит от химического состава газа и температуры, то есть имеет место нелинейная зависимость внутренней энергии газа от температуры [74] .

Теплоёмкость при постоянном давлении 1/R моль идеального газа:

c^P=oneνRT(∂S∂T)P=oneνR(∂H∂T)P=c^V+one,{\displaystyle {\hat {c}}_{P}={\frac {1}{\nu R}}T\left({\frac {\partial S}{\partial T}}\right)_{P}={\frac {1}{\nu R}}\left({\frac {\partial H}{\partial T}}\right)_{P}={\hat {c}}_{V}+1,}  

где H = U + PV — энтальпия газа.

Иногда проводится различие между классическим идеальным газом, где ĉ V и ĉ P могут меняться с температурой и квазиклассическим идеальным газом, для которого это не так.

Для любого классического и квазиклассического идеального газа справедливо соотношение Майера [75] :

CP−CV=R,{\displaystyle C_{P}-C_{V}=R,}  

Where CP{\displaystyle C_{P}}   — молярная теплоёмкость при постоянном давлении.

Соотношение теплоёмкостей при постоянном объёме и постоянном давлении

γ=cPcV{\displaystyle \gamma ={\frac {c_{P}}{c_{V}}}}  

called the adiabatic exponent . For air, which is a mixture of gases, this ratio is 1.4. For the adiabatic exponent, the Resh theorem is valid [76] :

CPCV=(∂P∂V)S(∂P∂V)T.{\ displaystyle {\ frac {C_ {P}} {C_ {V}}} = {\ frac {\ left ({\ frac {\ partial P} {\ partial V}} \ right) _ {S}} { \ left ({\ frac {\ partial P} {\ partial V}} \ right) _ {T}}}.  (Resh theorem)

Entropy and Thermodynamic Potentials

Expressing C V in terms of ĉ V as shown in the previous section, differentiating the equation of state of an ideal gas and integrating, we can obtain the expression of entropy [77] :

ΔS=c^VNkln⁡(TT0)+Nkln⁡(VV0),{\ displaystyle \ Delta S = {\ hat {c}} _ {V} Nk \ ln \ left ({\ frac {T} {T_ {0}}} \ right) + Nk \ ln \ left ({\ frac {V} {V_ {0}}} \ right),}  


This expression, after a series of transformations, allows one to obtain thermodynamic potentials for an ideal gas as a function of T , V , and N in the form [78] :

U{\ displaystyle U \,}  =c^VNkT{\ displaystyle = {\ hat {c}} _ {V} NkT \,}  
A{\ displaystyle A \,}  =U-TS{\ displaystyle = U-TS \,}  =μN-NkT{\ displaystyle = \ mu N-NkT \,}  
H{\ displaystyle H \,}  =U+PV{\ displaystyle = U + PV \,}  =c^PNkT{\ displaystyle = {\ hat {c}} _ {P} NkT \,}  
G{\ displaystyle G \,}  =U+PV-TS{\ displaystyle = U + PV-TS \,}  =μN{\ displaystyle = \ mu N \,}  

where, as before,

c^P=c^V+one.{\ displaystyle {\ hat {c}} _ {P} = {\ hat {c}} _ {V} +1.}  

Application of the ideal gas theory

The physical meaning of gas temperature

 
Pressure as a process of momentum transfer of gas molecules to the walls of a vessel

In the framework of the molecular kinetic theory, the pressure of gas molecules on the wall of a vesselp=FS {\ displaystyle p = {\ frac {F} {S}}}   equal to force ratioF {\ displaystyle F}   acting on the wall from the side of the molecules to the wall areaS {\ displaystyle S}   . The force can be calculated as the ratio of the total momentumK {\ displaystyle K}   transferred to the wall during collisions of molecules duringΔt {\ displaystyle \ Delta t}   , to the duration of this interval:

p=KSΔt.{\ displaystyle p = {\ frac {K} {S \ Delta t}}. \ qquad \ qquad}   (one)

In elastic collision , the mass moleculem {\ displaystyle m}   transmits momentum to the wall

km=2mvcos⁡ϑ,{\ displaystyle k_ {m} = 2mv \ cos \ vartheta, \ qquad \ qquad}   (2)

Whereϑ {\ displaystyle \ vartheta}   - the angle between the momentum of the molecule before collision and the normal to the wall andv {\ displaystyle v}   Is the velocity of the molecule [79] . The number of wall collisions isNst=nvScos⁡ϑΔt. {\ displaystyle N _ {\ rm {st}} = nvS \ cos \ vartheta \ Delta t.}   Expression averagingK=kmNst {\ displaystyle K = k_ {m} N _ {\ rm {st}}}   at all possible angles and speeds, gives:

K=23nεkinSΔt,{\ displaystyle K = {\ frac {2} {3}} n \ varepsilon _ {\ rm {kin}} S \ Delta t, \ qquad \ qquad}   (3)

Whereεkin {\ displaystyle \ \ varepsilon _ {\ rm {kin}}}   - the average kinetic energy of the translational motion of gas molecules. Substituting (3) into (1), we find that the pressure of gas molecules on the vessel wall is determined by the formulap=23nεkin {\ displaystyle p = {\ frac {2} {3}} n \ varepsilon _ {\ rm {kin}}}   [79] substitutingp {\ displaystyle p}   to the Clapeyron equation in the formp=nkT {\ displaystyle p = nkT}   we get the expressionεkin=32kT {\ displaystyle \ \ varepsilon _ {\ rm {kin}} = {\ frac {3} {2}} kT}   , whence it follows that the gas temperature is directly proportional to the average energy of the translational motion of molecules [79] .

Boltzmann distribution

 
Velocity distribution for 10 6 oxygen molecules at −100, 20, 600 degrees Celsius. The horizontal axis shows the delayed velocity, the vertical axis shows the number of molecules in each velocity range 1 m / s wide.

The equilibrium distribution of particles of a classical ideal gas by state can be obtained as follows. Using the expression for the potential energy of the gas in the gravitational field and the Clapeyron equation, the barometric formula [80] is derived and using it is found the energy distribution of gas molecules in the gravitational field. Boltzmann showed that the distribution thus obtained is valid not only in the case of a potential field of gravitational forces, but also in any potential field of forces for the totality of any identical particles in a state of chaotic thermal motion [81] . This distribution is called the Boltzmann distribution :

n¯j=ae-εjkT,{\ displaystyle {\ bar {n}} _ {j} = ae ^ {- {{\ varepsilon _ {j}} \ over {kT}}},}  

Wheren¯j {\ displaystyle {\ bar {n}} _ {j}}   - the average number of particles inj {\ displaystyle j}   state with energyεj {\ displaystyle \ varepsilon _ {j}}   , and the constanta {\ displaystyle a}   determined by the normalization condition:

∑nj=N,{\ displaystyle \ sum {n_ {j}} = N,}  

WhereN {\ displaystyle N}   Is the total number of particles.

The Boltzmann distribution is the limiting case of the Fermi – Dirac and Bose – Einstein distributions for high temperatures, and, accordingly, the classical ideal gas is the limiting case of the Fermi gas and Bose gas . This limiting case corresponds to a situation where the filling of energy levels is small and quantum effects can be neglected [82] .

Adiabatic process

 
Graph of adiabat (bold line) onp-V {\ displaystyle pV}   chart for gas.
p{\ displaystyle p}   - gas pressure;
V{\ displaystyle V}   - volume

Using the ideal gas model, it is possible to predict a change in the state parameters of the gas during an adiabatic process. We write the Clapeyron equation in this form:

pV=νRT.{\ displaystyle pV = \ nu RT.}  

Differentiating both parts, we get:

pdV+Vdp=νRdT.(one){\ displaystyle pdV + Vdp = \ nu RdT. (1)}  

According to the experimentally established Joule law (Gay-Lussac – Joule law), the internal energy of an ideal gas does not depend on the pressure or volume of the gas [15] . By the definition of molar heat capacity at a constant volume,(∂U∂T)V=CV {\ displaystyle \ left ({\ frac {\ partial U} {\ partial T}} \ right) _ {V} = C_ {V}}   [83] . Therefore, we obtain

dU=νCVdT,{\ displaystyle \ mathrm {d} U = \ nu C_ {V} \ mathrm {d} T,}  

Whereν {\ displaystyle \ nu}   - the number of moles of an ideal gas.

Given the absence of heat exchange with the environment, we have [84] :

dU=-pdV.{\ displaystyle \ mathrm {d} U = -p \, \ mathrm {d} V.}  

With this in mind, equation (1) takes the form

pdV+Vdp=-pdV⋅RCV,{\ displaystyle p \ mathrm {d} V + V \ mathrm {d} p = -p \ mathrm {d} V \ cdot {\ frac {R} {C_ {V}}},}  

further by entering the coefficientγ=one+R/CV {\ displaystyle {\ mathsf {\ gamma}} = 1 + R / C_ {V}}   , we finally obtain the Poisson equation :

p⋅Vγ=const.{\ displaystyle p \, \ cdot V ^ {\ mathsf {\ gamma}} = \ mathrm {const}.}  

For a nonrelativistic nondegenerate monatomic ideal gask=5/3 {\ displaystyle {\ mathsf {k}} = 5/3}   [85] for the diatomick=7/5 {\ displaystyle {\ mathsf {k}} = 7/5}   [85] .

Speed ​​of sound

The speed of sound in an ideal gas is determined [86]

cs=(∂P∂ρ)s=γPρ=γRTM,{\ displaystyle c_ {s} = {\ sqrt {\ left ({\ frac {\ partial P} {\ partial \ rho}} \ right) _ {s}}} = {\ sqrt {\ frac {\ gamma P } {\ rho}}} = {\ sqrt {\ frac {\ gamma RT} {M}}},}  

where γ is the adiabatic exponent ( ĉ P / ĉ V ), s is the entropy per gas particle, ρ is the gas density, P is the gas pressure, R is the universal gas constant , T is the temperature , M is the molar mass of gas. Since density fluctuations are fast, the process as a whole occurs without heat exchange, which explains the appearance of the adiabatic index in the expression for the speed of sound. For air, we take γ = 1.4, M = 28.8, T = 273 K, then c s = 330 m / s.

Quantum Perfect Gas

Lowering the temperature and increasing the gas density can lead to a situation where the average distance between particles becomes comparable with the de Broglie wavelength for these particles, which leads to a transition from classical to quantum ideal gas (see Degenerate gas ). In this case, the behavior of the gas depends on the spin of the particles: in the case of a half-integer spin ( fermions ), the Fermi – Dirac statistics ( Fermi gas ) are valid, in the case of a whole spin ( bosons ), the Bose – Einstein statistics ( Bose gas ) [87] .

Fermi gas

For fermions , the Pauli principle is in effect, prohibiting two identical fermions from being in the same quantum state [88] . As a result, at absolute zero temperature, the particle momenta and, correspondingly, the pressure and energy density of the Fermi gas are nonzero and proportional to the number of particles per unit volume [82] . There is an upper limit to the energy that Fermi gas particles can have at absolute zero ( Fermi EnergyEF {\ displaystyle E_ {F}}   ) If the thermal energy of the Fermi gas particles is much lower than the Fermi energy, then this state is called a degenerate gas [89] .

Examples of Fermi gases are electron gas in metals , heavily doped and degenerate semiconductors , and degenerate electron gas in white dwarfs [89] .

Bose gas

 
The velocity distribution of rubidium atoms near absolute zero. On the left is the distribution before condensation forms, in the center after formation, on the right after the evaporation of the gaseous component and the appearance of pure condensate

Since bosons can be strictly identical to each other [90] [91] and, accordingly, the Pauli principle does not apply to them, when the temperature of the Bose gas decreases below a certain temperatureT0 {\ displaystyle T_ {0}}   bosons can transition to the lowest energy level with zero momentum, i.e., the formation of a Bose - Einstein condensate . Since the gas pressure is equal to the sum of the momenta of the particles transferred to the wall per unit time, forT<T0 {\ displaystyle T <T_ {0}}   Bose gas pressure depends only on temperature. This effect was observed experimentally in 1995 , and in 2001 the authors of the experiment were awarded the Nobel Prize [92] .

Examples of Bose gases are various types of quasiparticle gases (weak excitations) in solids and liquids , the superfluid component of helium II, the Bose – Einstein condensate of Cooper electron pairs in superconductivity . An example of an ultrarelativistic Bose gas is a photon gas ( thermal radiation ) [90] [91] . An example of a Bose gas consisting of quasiparticles is phonon gas [93] .

The ideal gas in a gravitational field

 
Pressure change in the earth's atmosphere with altitude

In general relativistic thermodynamics, with the thermal equilibrium of a gas (liquid) sphere, the intrinsic temperature, measured by a local observer, decreases when moving along the radius from the center of the sphere to its surface. This relativistic effect is small (except for the case of superstrong gravitational fields) and it is neglected at the Earth’s surface [94] .

The real effect of the gravitational field on the gas (liquid) is manifested primarily through the dependence of hydrostatic pressure on the height of the column of gas (liquid). The influence of the gravitational field on the thermodynamic properties of the system can not be taken into account when the pressure change in height is much less than the absolute value of the pressure. Without going beyond thermodynamics, J. Maxwell established [95] [96] [97] that “... in a vertical column of gas left to itself, the temperature is the same everywhere after the column has reached thermal equilibrium through thermal conductivity; in other words, gravity has no effect on the temperature distribution in the column, ”and that this conclusion is valid for any gases (liquids), that is, the equality of temperatures over the entire volume of the system is a necessary condition for equilibrium in the gravitational field [98] [99] [100 ] ] [101] . By the methods of molecular kinetic theory, the same result for gases was obtained by L. Boltzmann [102] .

The dependence of pressure on the height of the isothermal column of an ideal gas gives a barometric formula . In the simplest thermodynamic model explaining the observed non-isothermality of the Earth’s atmosphere , they consider not the equilibrium , but the stationary state of the ideal gas column, achieved by the equilibrium adiabatic process of air circulation [103] , when the heat transfer in the direction of decreasing temperature (up) is balanced by the transfer of potential energy of air molecules in the opposite direction [104] .

The limits of applicability of the ideal gas theory

 
Isotherms of real gas (schematically)

If the gas density increases, then collisions of molecules begin to play an increasingly important role and it becomes impossible to neglect the size and interaction of molecules. The behavior of such a gas is poorly described by the ideal gas model, and therefore it is called a real gas [1] . Similarly, the ideal gas model cannot be used to describe a plasma in which there is a significant interaction between individual molecules [105] . To describe real gases, various modified equations of state are used, for example, virial decomposition .

Another widely used equation is obtained if we take into account that the molecule is not infinitely small, but has a certain diameterd {\ displaystyle d}   , then the Clapeyron equation for one mole of gas will take the form [106] :

p(V-b)=RT,{\ displaystyle p (Vb) = RT,}  

the valueb {\ displaystyle b}   equal to [106] :

b=2π3Nd3,{\ displaystyle b = {\ frac {2 \ pi} {3}} Nd ^ {3},}  

WhereN {\ displaystyle N}   - the number of molecules in the gas. Taking into account additional intermolecular attraction forces ( Van der Waals forces ) will lead to a change in the equation to the Van der Waals equation [106] :

(p+aν2V2)(V-bν)=νRT.{\ displaystyle \ left (p + {\ frac {a \ nu ^ {2}} {V ^ {2}}} \ right) \ left ({V} - {b \ nu} \ right) = \ nu RT. }  

There are a number of empirical equations of state, for example, Berthelot and Clausius , which even better describe the behavior of a real gas under certain conditions [107] .

See also

  • Bose gas
  • Gibbs Paradox
  • Real gas
  • Equation of state
  • Fermi gas

Comments

  1. ↑ To calculate the temperature dependence of the heat capacity of gases, semiclassical statistics [7] are used (statistical physics in the semiclassical approximation [8] [9] , semiclassical formulas [10] ) , which explains the origin of the term “ semiclassical ideal gas ”.

Notes

  1. ↑ 1 2 Lyubitov Yu. N. Ideal gas // Physical Encyclopedia / Ch. ed. A.M. Prokhorov . - M .: Soviet Encyclopedia , 1990. - T. 2. - S. 98. - 704 p. - 100,000 copies. - ISBN 5-85270-061-4 .
  2. ↑ Dzhalmukhambetov A.U., Fisenko M.A., Tasks-estimates and models of physical systems, 2016 , p. 12.
  3. ↑ Smirnova N. A., Methods of statistical thermodynamics in physical chemistry, 1982 , p. 201-202.
  4. ↑ Putilov K.A., Thermodynamics, 1971 , p. 168-169.
  5. ↑ Kuznetsova E.M., Ageev E.P. , Thermodynamics in questions and answers. The First Law and Its Consequences, 2003 , p. 98-100.
  6. ↑ Kuznetsova E.M., Ageev E.P. , Thermodynamics in questions and answers. The First Law and Its Consequences, 2003 , p. one hundred.
  7. ↑ Godnev I.N. , Calculation of thermodynamic functions according to molecular data, 1956 , p. 33.
  8. ↑ Levich V.G. , Introduction to Statistical Physics, 1954 , p. 9.
  9. ↑ Levich V.G. , Course in Theoretical Physics, vol. 1, 1969 , p. 333.
  10. ↑ Putilov K.A., Thermodynamics, 1971 , p. 169.
  11. ↑ Belokon N.I., Basic principles of thermodynamics, 1968 , p. 29.
  12. ↑ 1 2 Sivukhin, Thermodynamics and Molecular Physics, 2005 , p. 34.
  13. ↑ Bazarov I.P., Thermodynamics, 2010 , p. 31.
  14. ↑ Gorshkov V.I., Kuznetsov I.A., Fundamentals of Physical Chemistry, 1993 , p. 29.
  15. ↑ 1 2 Gerasimov Ya. I. et al., Course in Physical Chemistry, vol. 1, 1970 , p. 50-51.
  16. ↑ Semenchenko V.K., Selected Heads of Theoretical Physics, 1966 , p. 74.
  17. ↑ Belokon N.I., Thermodynamics, 1954 , p. 79.
  18. ↑ Arshava N.V., State functions of thermodynamic systems and functions of thermodynamic processes, 2006 , p. 75-76.
  19. ↑ Kuznetsova E.M., Ageev E.P. , Thermodynamics in questions and answers. The First Law and Its Consequences, 2003 , p. eighteen.
  20. ↑ Kubo R., Thermodynamics, 1970 , p. 91-92.
  21. ↑ Smirnova N. A., Methods of statistical thermodynamics in physical chemistry, 1982 , p. 201-248.
  22. ↑ Putilov K.A., Thermodynamics, 1971 , p. 168-176.
  23. ↑ Godnev I.N. , Calculation of thermodynamic functions from molecular data, 1956 .
  24. ↑ Kudryavtsev, 1956 , p. 185-186.
  25. ↑ Gay-Lussac, JL Recherches sur la dilatation des gaz et des vapeurs // Annales de chimie. - 1802. - Vol. Xliii. - P. 137.
  26. ↑ Gay-Lussac, 1822 , p. 87.
  27. ↑ Yastrzhembsky A.S., 1966 , p. 25.
  28. ↑ 1 2 3 Gelfer Y.M., 1981 , p. 122.
  29. ↑ Second Law of Thermodynamics, 1934 , p. 41.
  30. ↑ 1 2 3 4 Kipnis A.Ya., 1962 .
  31. ↑ У. Томсон в 1845 году тщетно пытался достать в Париже книгу Карно. См. Кричевский И. Р. , Понятия и основы термодинамики, 1970, с. 172.
  32. ↑ Второе начало термодинамики, 1934 , с. 16—69.
  33. ↑ Clapeyron, E. Mémoire sur la puissance motrice de la chaleur (неопр.) // Journal de l'École Polytechnique . — 1834. — Т. XIV . — С. 153—190 . (фр.) Facsimile at the Bibliothèque nationale de France (pp. 153—90).
  34. ↑ Zeuner G., 1866 , p. 103.
  35. ↑ Белоконь Н. И., Термодинамика, 1954 , с. 47.
  36. ↑ Krönig, A. Grundzüge einer Theorie der Gase (неопр.) // Annalen der Physik . — 1856. — Т. 99 , № 10 . — С. 315—322 . — DOI : 10.1002/andp.18561751008 . — . (нем.) Facsimile at the Bibliothèque nationale de France (pp. 315—22).
  37. ↑ Clausius, R. Ueber die Art der Bewegung, welche wir Wärme nennen (нем.) // Annalen der Physik und Chemie : magazin. — 1857. — Bd. 176 , Nr. 3 . — S. 353—379 . — DOI : 10.1002/andp.18571760302 . — . (нем.) Facsimile at the Bibliothèque nationale de France (pp. 353—79).
  38. ↑ Клаузиус // Большая советская энциклопедия : [в 30 т.] / гл. ed. А. М. Прохоров . — 3-е изд. — М. : Советская энциклопедия, 1969—1978.
  39. ↑ 1 2 3 Голоушкин В. Н., 1951 .
  40. ↑ Алымов И., 1865 , с. 106.
  41. ↑ 1 2 Гельфер Я. М., 1981 , с. 123.
  42. ↑ Zeuner G., 1866 , p. 105.
  43. ↑ Partington JR, 1913 , p. 135.
  44. ↑ Partington JR, 1949 , p. 644.
  45. ↑ Кудрявцев, том 3, 1971 , с. 397—398.
  46. ↑ About Enrico Fermi (англ.) . Чикагский университет . Дата обращения 7 января 2012. Архивировано 10 января 2013 года.
  47. ↑ Ферми Энрико — статья из Большой советской энциклопедии . Б. М. Понтекорво .
  48. ↑ Ястржембский А. С., 1933 , с. 15.
  49. ↑ БСЭ, 1-е изд., т. 32, 1936 , с. 406—407 (стб. 812—813).
  50. ↑ Ландау Л. Д., Лифшиц Е. М., Статистическая физика, 1938 , с. 73.
  51. ↑ Рубинштейн Д. Л., 1940 , с. 12.
  52. ↑ Гухман А. А., 1947 , с. 94.
  53. ↑ Каблуков И. А. и др., 1949 , с. 28.
  54. ↑ Литвин А. М., 1947 , с. 28.
  55. ↑ Вукалович М. П., Новиков И. И., 1948 .
  56. ↑ 1 2 БСЭ, 2-е изд., т. 21, 1953 , с. 357.
  57. ↑ Карапетьянц М. Х., 1949 , с. 115.
  58. ↑ 1 2 Василевский А. С., Термодинамика и статистическая физика, 2006 , с. 41.
  59. ↑ Бурдаков В. П. и др., Термодинамика, ч. 1, 2009 , с. 38.
  60. ↑ Ястржембский А. С., 1960 , с. 24—25.
  61. ↑ БРЭ, т. 14, 2009 , с. 215 .
  62. ↑ Борщевский А. Я., Физическая химия, т. 1, 2017 , с. 12.
  63. ↑ Коган М. Н. Динамика разреженного газа // Кинетическая теория. — М. , 1967.
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  65. ↑ 1 2 3 4 Ландау Л.Д., Ахиезер А.И., 1965 , с. 169—170.
  66. ↑ Ландау Л.Д., Ахиезер А.И., 1965 , с. 183.
  67. ↑ Сивухин, 1975 , с. 35.
  68. ↑ Барилович B. A., Смирнов Ю. А., Основы технической термодинамики, 2014 , с. 12.
  69. ↑ Базаров И. П., Термодинамика, 2010 , с. 65.
  70. ↑ Сивухин, 1975 , с. 36—37.
  71. ↑ Вишневецкий Л. С. Совершенный газ // Физическая энциклопедия : [в 5 т.] / Гл. ed. A.M. Prokhorov . — М. : Большая российская энциклопедия, 1994. — Т. 4: Пойнтинга — Робертсона — Стримеры. — С. 569. — 704 с. — 40 000 экз. — ISBN 5-85270-087-8 .
  72. ↑ Кубо Р., Термодинамика, 1970 , с. 25.
  73. ↑ Алмалиев А. Н. и др., Термодинамика и статистическая физика. Статистика идеального газа, 2004 , с. 35.
  74. ↑ Смирнова Н. А., Методы статистической термодинамики в физической химии, 1982 , с. 201—248.
  75. ↑ Савельев, 2001 , с. 29—32.
  76. ↑ Толпыго К. Б., Термодинамика и статистическая физика, 1966 , с. 83.
  77. ↑ Сивухин, 1975 , с. 128.
  78. ↑ Сивухин, 1975 , с. 139—140.
  79. ↑ 1 2 3 Савельев, 2001 , с. 53—56.
  80. ↑ Савельев, 2001 , с. 41—42.
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  82. ↑ 1 2 Савельев, квантовая оптика, 2001 , с. 205—208.
  83. ↑ Ландау Л.Д., Ахиезер А.И., 1965 , с. 185.
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  85. ↑ 1 2 Адиабата // А — Ангоб. — М. : Советская энциклопедия, 1969. — ( Большая советская энциклопедия : [в 30 т.] / гл. ред. А. М. Прохоров ; 1969—1978, т. 1).
  86. ↑ Сивухин, 1975 , с. 80—81.
  87. ↑ F. Reif, 1965 , pp. 246—248.
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  90. ↑ 1 2 Einstein A. Quantentheorie des einatomigen idealen Gases (неопр.) // Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse. — 1924. — Т. 1924 . — С. 261—267 . (нем.)
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  96. ↑ Maxwell, 1871 .
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  98. ↑ Кириллин В. А. и др., Техническая термодинамика, 2008 , с. 139.
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Источник — https://ru.wikipedia.org/w/index.php?title=Идеальный_газ&oldid=101233933


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