Ideal gas - Wikipedia, the free encyclopedia 頁 1 / 8

Ideal gas From Wikipedia, the free encyclopedia

An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point Thermodynamics particles. 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.

At normal conditions such as standard temperature and pressure, most real gases behave Branches qualitatively like an ideal gas. Many gases such Classical · Statistical · Chemical as air, nitrogen, oxygen, hydrogen, noble gases, Equilibrium / Non-equilibrium and some heavier gases like carbon dioxide can Thermofluids be treated like ideal gases within reasonable tolerances.[1] Generally, a gas behaves more like Laws an ideal gas at higher temperature and lower Zeroth · First · Second · Third density (i.e. lower pressure),[1] as the work Systems performed by intermolecular forces becomes less significant compared with the particles' kinetic State: energy, and the size of the molecules becomes Equation of state less significant compared to the empty space Ideal gas · Real gas between them. Phase of matter · Equilibrium Control volume · Instruments The ideal gas model tends to fail at lower temperatures or higher pressures, when Processes: intermolecular forces and molecular size become Isobaric · Isochoric · Isothermal important. It also fails for most heavy gases, such Adiabatic · Isentropic · Isenthalpic [1] as water vapor or many refrigerants. At some Quasistatic · Polytropic point of low temperature and high pressure, real Free expansion gases undergo a phase transition, such as to a Reversibility · Irreversibility liquid or a solid. The model of an ideal gas, Endoreversibility however, does not describe or allow phase transitions. These must be modeled by more Cycles: complex equations of state. Heat engines · Heat pumps The ideal gas model has been explored in both Thermal efficiency the Newtonian dynamics (as in "kinetic theory") System properties and in quantum mechanics (as a "gas in a box"). The ideal gas model has also been used to model Property diagrams the behavior of electrons in a metal (in the Drude Intensive and extensive properties model and the free electron model), and it is one State functions: of the most important models in statistical mechanics. Temperature / Entropy (intro.) † Pressure / Volume † Chemical potential / Particle no. † († Conjugate variables) Contents Vapor quality Reduced properties ■ 1 Types of ideal gas ■ 2 Classical thermodynamic ideal gas

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■ 3 Heat capacity Process functions: ■ 4 Entropy Work · Heat ■ 5 Thermodynamic potentials ■ 5.1 Multicomponent Material properties systems Specific heat capacity c = ■ 6 Speed of sound 1 ■ 7 Equation Table for an Ideal Gas Compressibility β = − ■ 8 Ideal quantum gases V ■ 8.1 Ideal Boltzmann gas 1 Thermal expansion α = ■ 8.2 Ideal Bose and Fermi gases Property database

■ 9See also Equations ■ 10 References Carnot's theorem Clausius theorem Fundamental relation Types of ideal gas Ideal gas law Maxwell relations There are three basic classes of ideal gas: Table of thermodynamic equations ■ the classical or Maxwell-Boltzmann ideal gas, Potentials ■ the ideal quantum Bose gas, composed Free energy · Free entropy of bosons, and ■ the ideal quantum Fermi gas, Internal energy U(S,V) composed of fermions. Enthalpy H(S,p) = U + pV The classical ideal gas can be separated into two Helmholtz free energy A(T,V) = U − TS types: The classical thermodynamic ideal gas and Gibbs free energy G(T,p) = H − TS the ideal quantum Boltzmann gas. Both are History and culture essentially the same, except that the classical thermodynamic ideal gas is based on classical Philosophy: statistical mechanics, and certain thermodynamic Entropy and time · Entropy and life parameters such as the entropy are only specified Brownian ratchet to within an undetermined additive constant. The Maxwell's demon ideal quantum Boltzmann gas overcomes this Heat death paradox limitation by taking the limit of the quantum Loschmidt's paradox Bose gas and quantum Fermi gas in the limit of Synergetics high temperature to specify these additive constants. The behavior of a quantum Boltzmann History: gas is the same as that of a classical ideal gas General · Heat · Entropy · Gas laws except for the specification of these constants. Perpetual motion The results of the quantum Boltzmann gas are Theories: used in a number of cases including the Sackur- Caloric theory · Vis viva Tetrode equation for the entropy of an ideal gas and the Saha ionization equation for a weakly Theory of heat ionized plasma. Mechanical equivalent of heat Motive power Publications: Classical thermodynamic ideal "An Experimental Enquiry Concerning ... Heat" gas "On the Equilibrium of Heterogeneous Substances"

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The thermodynamic properties of an ideal gas "Reflections on the can be described by two equations: The equation Motive Power of Fire" of state of a classical ideal gas is the ideal gas law Timelines of: Thermodynamics · Heat engines and the internal energy of an ideal gas given by: Art: Maxwell's thermodynamic surface where Education: Entropy as energy dispersal ■ P is the pressure ■ V is the volume Scientists ■ n is the amount of substance of the Daniel Bernoulli gas (in moles) Sadi Carnot ■ R is the gas constant (8.314 Benoît Paul Émile Clapeyron J·K−1mol-1) Rudolf Clausius ■ T is the absolute temperature Hermann von Helmholtz ■ U is the internal energy Constantin Carathéodory ■ is the dimensionless specific Pierre Duhem heat capacity at constant volume, ≈ Josiah Willard Gibbs 3/2 for monatomic gas, 5/2 for James Prescott Joule diatomic gas and 3 for more James Clerk Maxwell complex molecules. Julius Robert von Mayer The amount of gas in J·K−1 is William Rankine John Smeaton Georg Ernst Stahl Benjamin Thompson where William Thomson, 1st Baron Kelvin John James Waterston ■ N is the number of gas particles ■ kB is the Boltzmann constant (1.381×10−23J·K−1).

The probability distribution of particles by velocity or energy is given by the Boltzmann distribution.

The ideal gas law is an extension of experimentally discovered gas laws. Real fluids 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 condenses from a gas into a liquid or solid. The deviation is expressed as a compressibility factor. Heat capacity

The heat capacity at constant volume of nR = 1 J·K−1 of any gas, including an ideal gas is:

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This is the dimensionless heat capacity at constant volume, which is generally a function of temperature due to intermolecular forces. For moderate temperatures, the constant for a monoatomic gas is while for a diatomic gas it is . It is seen that macroscopic measurements on heat capacity provide information on the microscopic structure of the molecules.

The heat capacity at constant pressure of 1 J·K−1 ideal gas is:

where H = U + pV is the enthalpy of the gas.

Sometimes, a distinction is made between an ideal gas, where and could vary with pressure and temperature, and a perfect gas, for which this is not the case. 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 potentials, if we can express the entropy as a function of U (U is a thermodynamic potential) and the volume V, 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 S may be written as ΔS where:

where the reference variables may be functions of the number of particles 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:

Expressing CV in terms of as developed in the above section, differentiating the ideal gas equation of state, and integrating yields:

where all constants have been incorporated into the logarithm as f(N) which is some function of the particle number N having the same dimensions as 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 (V and N) are multiplied by a constant, the entropy will be multiplied by the same constant. Mathematically:

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From this we find an equation for the function f(N)

Differentiating this with respect to a, setting a equal to unity, and then solving the differential equation yields f(N):

where Φ is some constant with the dimensions of . Substituting into the equation for the change in entropy:

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 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 — the concept of an ideal gas breaks down at low values of 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. It remained for quantum mechanics to introduce a reasonable value for the value of Φ which yields the Sackur-Tetrode equation for the entropy of an ideal gas. It too suffers from a divergent entropy at absolute zero, but is a good approximation to an ideal gas over a large range of densities. Thermodynamic potentials

Main article: Thermodynamic potentials

Since the dimensionless heat capacity at constant pressure is a constant we can express the entropy in what will prove to be a more convenient form:

where Φ is now the undetermined constant. The chemical potential of the ideal gas is calculated from the corresponding equation of state (see thermodynamic potential):

where G is the Gibbs free energy and is equal to U + PV − TS so that:

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

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

Multicomponent systems

By Gibbs' theorem, the entropy of a multicomponent system is equal to the sum of the entropies of each chemical species (assuming no surface effects). The entropy of a multicomponent system will be

where the sum is over all species. Likewise, the free energies are equal to the sums of the free energies of each species so that if Φ is a thermodynamic potential then

where Φj is expressed in terms of its natural variables. For example, the internal energy will be

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where N is defined as

Ideal gasses are not found in the real world. So they are different from real gasses. There are basic assumptions made in the kinetic theory of gasses. Speed of sound

Main article: Speed of sound

The speed of sound in an ideal gas is given by

where

is the adiabatic index is the universal gas constant is the temperature is the molar mass of the gas. Equation Table for an Ideal Gas

See Table of thermodynamic equations#Equation Table for an 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 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 bosons or fermions. (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 Φ:

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where Λ is the thermal de Broglie wavelength of the gas and g is the degeneracy of states.

Ideal Bose and Fermi gases

An ideal 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 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

■ Compressibility factor ■ Dynamical billiards - billiard balls as a model of an ideal gas ■ Table of thermodynamic equations ■ Scale-free ideal gas References

1. ^ abcCengel, Yunus A.; Boles, Michael A.. Thermodynamics: An Engineering Approach (Fourth ed.). p. 89. ISBN 0-07-238332-1. Retrieved from "http://en.wikipedia.org/w/index.php?title=Ideal_gas&oldid=461933221" Categories: Ideal gas Introductory physics

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