Under various conditions of temperature and pressure, manyreal gases behave qualitatively like an ideal gas where the gas molecules (or atoms formonatomic gas) play the role of the ideal particles. Many gases such asnitrogen,oxygen,hydrogen,noble gases, some heavier gases likecarbon dioxide and mixtures such as air, can be treated as ideal gases within reasonable tolerances[2] over a considerable parameter range aroundstandard temperature and pressure. Generally, a gas behaves more like an ideal gas at highertemperature and lowerpressure,[2] as thepotential 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. Onemole of an ideal gas has a volume of22.71095464... L (exact value based on2019 revision of the SI)[3] at standard temperature and pressure (atemperature of 273.15 K and anabsolute pressure of exactly 105Pa).[note 1]
The ideal gas model tends to fail at lower temperatures or higher pressures, where intermolecular forces and molecular size become important. It also fails for most heavy gases, such as manyrefrigerants,[2] and for gases with strong intermolecular forces, notablywater 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 aphase transition, such as to aliquid or asolid. 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 adimensionless quantity, thecompressibility factor,Z.
If the pressure of an ideal gas is reduced in athrottling 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 itsJoule–Thomson coefficient is positive or negative.)
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 classicalstatistical mechanics, and certain thermodynamic parameters such as theentropy 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 theSackur–Tetrode equation for the entropy of an ideal gas and theSaha ionization equation for a weakly ionizedplasma.
R is thegas constant, which must be expressed in units consistent with those chosen for pressure, volume and temperature. For example, inSI unitsR = 8.3145 J⋅K−1⋅mol−1 when pressure is expressed inpascals, volume in cubicmeters, and absolute temperature inkelvin.
The ideal gas law is an extension of experimentally discoveredgas laws. It can also be derived from microscopic considerations.
Realfluids at lowdensity and hightemperature 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 itcondenses from a gas into a liquid or as itdeposits from a gas into a solid. This deviation is expressed as acompressibility factor.
The other equation of state of an ideal gas must expressJoule's second law, that the internal energy of a fixed mass of ideal gas is a function only of its temperature, with. For the present purposes it is convenient to postulate an exemplary version of this law by writing:
ĉV is the dimensionless specificheat capacity at constant volume, approximately3/2 for amonatomic gas,5/2 fordiatomic gas, and 3 for non-linear molecules if we treat translations and rotations classically and ignore quantum vibrational contribution and electronic excitation. These formulas arise from application of the classicalequipartition theorem to the translational and rotational degrees of freedom.[8]
ThatU for an ideal gas depends only on temperature is a consequence of the ideal gas law, although in the general caseĉV depends on temperature and an integral is needed to computeU.
The probability distribution of particles by velocity or energy is given by theMaxwell 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.
whereS is theentropy. This quantity is generally a function of temperature due to intermolecular and intramolecular forces, but for moderate temperatures it is approximately constant. Specifically, theEquipartition Theorem predicts that the constant for a monatomic gas isĉV = 3/2 while for a diatomic gas it isĉV = 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:
Sometimes, a distinction is made between an ideal gas, whereĉV andĉP could vary with temperature, and aperfect gas, for which this is not the case.
The ratio of the constant volume and constant pressure heat capacity is theadiabatic index
For air, which is a mixture of gases that are mainly diatomic (nitrogen and oxygen), this ratio is often assumed to be 7/5, the value predicted by the classicalEquipartition Theorem for diatomic gases.
Using the results ofthermodynamics only, we can go a long way in determining the expression for theentropy of an ideal gas. This is an important step since, according to the theory ofthermodynamic potentials, if we can express the entropy as a function ofU (U is a thermodynamic potential), volumeV and the number of particlesN, 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 anexact differential, using thechain rule, the change in entropy when going from a reference state 0 to some other state with entropyS may be written as
where the reference variables may be functions of thenumber of particlesN. Using the definition of theheat capacity at constant volume for the first differential and the appropriateMaxwell relation for the second, we have
ExpressingCV in terms ofĉV as developed in the above section, differentiating the ideal gas equation of state, and integrating yields
which implies that the entropy may be expressed as
where all constants have been incorporated into the logarithm asf(N) which is some function of the particle numberN having the same dimensions asVTĉV in order that the argument of the logarithm be dimensionless. We now impose the constraint that the entropy is extensive, meaning that when the extensive parameters (V andN) are multiplied by a constant, the entropy is multiplied by the same constant. Mathematically:
From this we find an equation for the functionf(N):
Differentiating this with respect toa, settinga equal to 1, and then solving the differential equation yields
whereΦ may vary for different gases but is independent of the thermodynamic state of the gas. It has the dimensions ofVTĉV/N. Substituting into the equation for the entropy,
and using the expression for the internal energy of an ideal gas, the entropy may be written
Since this is an expression for entropy in terms ofU,V, andN, 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 – as the temperature approaches zero, the entropy approaches negative infinity, in contradiction to thethird 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 – the concept of an ideal gas breaks down at low values ofV/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 theSackur–Tetrode equation, which expresses the entropy of a monatomic (ĉV = 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 is
The chemical potential is usually referenced to the potential at some standard pressurePo so that, with:
For a mixture (j=1,2,...) of ideal gases, each at partial pressurePj, it can be shown that the chemical potentialμj will be given by the above expression with the pressureP replaced byPj.
The thermodynamic potentials for an ideal gas can now be written as functions ofT,V, andN as:
where, as before,
.
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:
In the above-mentionedSackur–Tetrode equation, the best choice of the entropy constant was found to be proportional to the quantumthermal 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 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 eitherbosons orfermions. (See thegas 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 theBoyle temperature.
The ideal Boltzmann gas yields the same results as the classical thermodynamic gas, but makes the following identification for the undetermined constantΦ:
Compressibility factor – Correction factor which describes the deviation of a real gas from ideal gas behavior
Dynamical billiards – Dynamical system abstract an ideal game of billiards, with elastic collisions off boundaries – billiard balls as a model of an ideal gas
^Until 1982, STP was defined as a temperature of 273.15 K and anabsolute pressure of exactly 1 atm. The volume of one mole of an ideal gas at this temperature and pressure is 22.413962(13) litres.[4] IUPAC recommends that the former use of this definition should be discontinued;[5] however, some textbooks still use these old values.
References
^Tuckerman, Mark E. (2010).Statistical Mechanics: Theory and Molecular Simulation (1st ed.). p. 87.ISBN978-0-19-852526-4.
^Adkins, C. J. (1983).Equilibrium Thermodynamics (3rd ed.). Cambridge, UK: Cambridge University Press. pp. 116–120.ISBN0-521-25445-0.
^Tschoegl, N. W. (2000).Fundamentals of Equilibrium and Steady-State Thermodynamics. Amsterdam: Elsevier. p. 88.ISBN0-444-50426-5.
^Attard, Phil (2012).Non-equilibrium thermodynamics and statistical mechanics : foundations and applications. Oxford University Press.ISBN9780191639760.OCLC810281588.
