Ideal gas

An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces. Additionally, the constituent atoms or molecules undergo perfectly elastic collisions with the walls of the container. Real gases do not exhibit these exact properties, although the approximation is often good enough to describe real gases.

Types of ideal gases

There exist three basic types of ideal gas:

• the classical or Maxwell-Boltzmann ideal gas,
• the ideal quantum Bose gas, composed of bosons, and
• the ideal quantum Fermi gas, composed of fermions.

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 thermodyamic ideal gas is based on classical thermodynamics alone, 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.

Classical thermodynamic ideal gas

The thermodynamic properties of an ideal gas can be described by two equations: The equation of state of a classical ideal gas is given by the ideal gas law.

Failed to parse (Missing texvc executable; please see math/README to configure.): pV = N k_B T = nRT\,

The internal energy of an ideal gas is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): U = \hat{c}_V nRT = \hat{c}_V NkT

where:

• Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{c}_V

is a constant dependent on temperature (e.g. equal to 3/2 for a monatomic gas for moderate temperatures)

• U is the internal energy
• p is the pressure
• V is the volume
• n is the amount of gas (moles)
• R is the gas constant, 8.314 J·K⁻¹mol^-1
• T is the absolute temperature
• N is the number of particles
• k 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.

Heat capacity

The heat capacity at constant volume of an ideal gas is:

Failed to parse (Missing texvc executable; please see math/README to configure.): C_V = \left(\frac{\partial U}{\partial T}\right)_V = \hat{c}_V Nk

It is seen that the constant Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{c}_V

is just the dimensionless heat capacity at constant volume. It is equal to half the number of degrees of freedom per particle. For moderate temperatures a monatomic gas is just  Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{c}_V=3/2
while for a diatomic gas is Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{c}_V=5/2

. 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 an ideal gas is:

Failed to parse (Missing texvc executable; please see math/README to configure.): C_p = \left(\frac{\partial H}{\partial T}\right)_p = (\hat{c}_V+1) Nk

where Failed to parse (Missing texvc executable; please see math/README to configure.): H=U+pV

is the enthalpy of the gas. It is seen that Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{c}_p
is also a constant and that the dimensionless heat capacities are related by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{c}_p-\hat{c}_V=1

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, of which the internal energy U is one, if we can express the entropy as a function of U 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S

where:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S =\int_{T_0}^{T} \frac{C_v}{T}\,dT+\int_{V_0}^{V}\left(\frac{\partial P}{\partial T}\right)_VdV

Expressing Failed to parse (Missing texvc executable; please see math/README to configure.): C_V

in terms of Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{c}_V
as developed in the above section, differentiating the ideal gas equation of state, and integrating yields:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S = \hat{c}_VNk\ln\left(\frac{T}{T_0}\right)+Nk\ln\left(\frac{V}{V_0}\right) = Nk\ln\left(\frac{VT^{\hat{c}_v}}{f(N)}\right)

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 Failed to parse (Missing texvc executable; please see math/README to configure.): VT^{\hat{c}_v}

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta S(T,aV,aN)=a\Delta S(T,V,N)\,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): af(N)=f(aN)\,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f(N)=\phi N\,

where φ is some constant with the dimensions of Failed to parse (Missing texvc executable; please see math/README to configure.): VT^{\hat{c}_v}/N . Substituting into the equation for the change in entropy:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\Delta S}{Nk} = \ln\left(\frac{VT^{\hat{c}_v}}{N\phi}\right)\,

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{c}_p

is a constant we can express the entropy in what will prove to be a more convenient form:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{S}{kN}=\ln\left( \frac{VT^{\hat{c}_V}}{N\Phi}\right)+\hat{c}_p

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):

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu=\left(\frac{\partial G}{\partial N}\right)_{T,p}

where G is the Gibbs free energy and is equal to Failed to parse (Missing texvc executable; please see math/README to configure.): U+pV-TS

so that:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mu(T,V,N)=-kT\ln\left(\frac{VT^{\hat{c}_V}}{N\Phi}\right).

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

Failed to parse (Missing texvc executable; please see math/README to configure.): U\,		Failed to parse (Missing texvc executable; please see math/README to configure.): =\hat{c}_V NkT\,
Failed to parse (Missing texvc executable; please see math/README to configure.): A=\,	Failed to parse (Missing texvc executable; please see math/README to configure.): U-TS\,	Failed to parse (Missing texvc executable; please see math/README to configure.): =\mu N-NkT\,
Failed to parse (Missing texvc executable; please see math/README to configure.): H=\,	Failed to parse (Missing texvc executable; please see math/README to configure.): U+pV\,	Failed to parse (Missing texvc executable; please see math/README to configure.): =\hat{c}_p NkT\,
Failed to parse (Missing texvc executable; please see math/README to configure.): G=\,	Failed to parse (Missing texvc executable; please see math/README to configure.): U+pV-TS\,	Failed to parse (Missing texvc executable; please see math/README to configure.): =\mu N\,

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-specie ideal gas are:

Failed to parse (Missing texvc executable; please see math/README to configure.): U(S,V,N)=\hat{c}_V Nk\left(\frac{N\Phi\,e^{S/Nk-\hat{c}_p}}{V}\right)^{1/\hat{c}_V}

Failed to parse (Missing texvc executable; please see math/README to configure.): A(T,V,N)=-NkT\left(1+\ln\left(\frac{VT^{\hat{c}_V}}{N\Phi}\right)\right)

Failed to parse (Missing texvc executable; please see math/README to configure.): H(S,P,N)=\hat{c}_p Nk\left(\frac{p\Phi\,e^{S/Nk-\hat{c}_p}}{k}\right)^{1/\hat{c}_p}

Failed to parse (Missing texvc executable; please see math/README to configure.): G(T,P,N)=-NkT\ln\left(\frac{kT^{\hat{c}_p}}{p\Phi}\right)

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 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): S=\sum_j S_j = \sum_j N_j\,k \left[\ln\left( \frac{VT^{\hat{c}_V}}{N_j\Phi}\right)+\hat{c}_p\right]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi=\sum_j \Phi_j\,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): U=\sum_j \hat{c}_V N_jk\left(\frac{N_j\Phi\,e^{S_j/N_jk-\hat{c}_p}}{V}\right)^{1/\hat{c}_V}=\hat{c}_V NkT

where N is defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): N=\sum_j N_j\,

.

Speed of sound

Main article: Speed of sound

The speed of sound in an ideal gas is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): v_{sound} = \sqrt{\frac{\gamma R T}{M}}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \gamma \,

is the adiabatic index

Failed to parse (Missing texvc executable; please see math/README to configure.): R \,

is the universal gas constant

Failed to parse (Missing texvc executable; please see math/README to configure.): T \,

is the temperature

Failed to parse (Missing texvc executable; please see math/README to configure.): M \,

is the molar mass for the gas (in kg/mol)

Equation Table for an Ideal Gas (Failed to parse (Missing texvc executable; please see math/README to configure.): pV^m=constant )

The following table gives the values for the change in the value of some thermodynamic variables under the specified transformation.

or Failed to parse (Missing texvc executable; please see math/README to configure.): C_V\,

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.)

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi = \frac{T^{3/2}\Lambda^3}{g}

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

• Ideal gas law
• Kinetic theory
• Compressibility factor
• Dynamical billiards - billiard balls as a model of an ideal gasbg:Идеален газ

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