Ideal gas

There are three basic classes 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 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.

The classical thermodynamic properties of an ideal gas can be described by two equations of state:[6][7]

The dimensionless heat capacity at constant volume is generally defined by

${\displaystyle {\hat {c}}_{V}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{V}={\frac {1}{nR}}\left({\frac {\partial U}{\partial T}}\right)_{V}}$

where 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 ĉ_V = 3/2 while for a diatomic gas it is ĉ_V = 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:

${\displaystyle {\hat {c}}_{P}={\frac {1}{nR}}T\left({\frac {\partial S}{\partial T}}\right)_{P}={\frac {1}{nR}}\left({\frac {\partial H}{\partial T}}\right)_{P}={\hat {c}}_{V}+1}$

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

Sometimes, a distinction is made between an ideal gas, where ĉ_V and ĉ_P 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

${\displaystyle \gamma ={\frac {c_{P}}{c_{V}}}}$

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 classical Equipartition Theorem for diatomic gases.

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), volume V and the number of particles 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 S may be written as ΔS where:

${\displaystyle \Delta S=\int _{S_{0}}^{S}dS=\int _{T_{0}}^{T}\left({\frac {\partial S}{\partial T}}\right)_{V}\!dT+\int _{V_{0}}^{V}\left({\frac {\partial S}{\partial V}}\right)_{T}\!dV}$

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:

${\displaystyle \Delta S=\int _{T_{0}}^{T}{\frac {C_{V}}{T}}\,dT+\int _{V_{0}}^{V}\left({\frac {\partial P}{\partial T}}\right)_{V}dV.}$

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

${\displaystyle \Delta S={\hat {c}}_{V}Nk\ln \left({\frac {T}{T_{0}}}\right)+Nk\ln \left({\frac {V}{V_{0}}}\right)}$

which implies that the entropy may be expressed as:

${\displaystyle S=Nk\ln \left({\frac {VT^{{\hat {c}}_{V}}}{f(N)}}\right)}$

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 VT^ĉ_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:

${\displaystyle S(T,aV,aN)=aS(T,V,N).}$

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

${\displaystyle af(N)=f(aN).}$

Differentiating this with respect to a, setting a equal to 1, and then solving the differential equation yields f(N):

${\displaystyle f(N)=\Phi N}$

where Φ may vary for different gases, but will be independent of the thermodynamic state of the gas. It will have the dimensions of VT^ĉ_V/N. Substituting into the equation for the entropy:

${\displaystyle {\frac {S}{Nk}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right).}$

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

${\displaystyle {\frac {S}{Nk}}=\ln \left[{\frac {V}{N}}\,\left({\frac {U}{{\hat {c}}_{V}kN}}\right)^{{\hat {c}}_{V}}\,{\frac {1}{\Phi }}\right]}$

Since this is an expression for entropy in terms of U, V, and 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 – 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. A quantum-mechanical derivation of this constant is developed in the derivation of the Sackur–Tetrode equation which expresses the entropy of a monatomic (ĉ_V = 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:

Expressing the entropy as a function of T, V, and N:

${\displaystyle {\frac {S}{kN}}=\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)}$

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

${\displaystyle \mu =\left({\frac {\partial G}{\partial N}}\right)_{T,P}}$

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

${\displaystyle \mu (T,P)=kT\left({\hat {c}}_{P}-\ln \left({\frac {kT^{{\hat {c}}_{P}}}{P\Phi }}\right)\right)}$

The chemical potential is usually referenced to the potential at some standard pressure P^o so that, with ${\displaystyle \mu ^{o}(T)=\mu (T,P^{o})}$:

${\displaystyle \mu (T,P)=\mu ^{o}(T)+kT\ln \left({\frac {P}{Po}}\right)}$

For a mixture (j=1,2,...) of ideal gases, each at partial pressure P_j, it can be shown that the chemical potential μ_j will be given by the above expression with the pressure P replaced by P_j.

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

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

where, as before,

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

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:

${\displaystyle U(S,V,N)={\hat {c}}_{V}Nk\left({\frac {N\Phi }{V}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{V}}}$

${\displaystyle A(T,V,N)=NkT\left({\hat {c}}_{V}-\ln \left({\frac {VT^{{\hat {c}}_{V}}}{N\Phi }}\right)\right)}$

${\displaystyle H(S,P,N)={\hat {c}}_{P}Nk\left({\frac {P\Phi }{k}}\,e^{S/Nk}\right)^{1/{\hat {c}}_{P}}}$

${\displaystyle G(T,P,N)=NkT\left({\hat {c}}_{P}-\ln \left({\frac {kT^{{\hat {c}}_{P}}}{P\Phi }}\right)\right)}$

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 matter; see configuration integral for more details.

The speed of sound in an ideal gas is given by the Newton-Laplace formula:

${\displaystyle c_{\text{sound}}={\sqrt {\frac {K_{s}}{\rho }}}={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}},}$

where the isentropic Bulk modulus ${\displaystyle K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}.}$

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

${\displaystyle c_{\text{sound}}={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {\gamma P}{\rho }}}={\sqrt {\frac {\gamma RT}{M}}}}$

Here,

• γ is the adiabatic index (ĉ_P/ĉ_V)
• s is the entropy per particle of the gas.
• ρ is the mass density of the gas.
• P is the pressure of the gas.
• R is the universal gas constant
• T is the temperature
• M is the molar mass of the gas.

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.

• 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
• Table of thermodynamic equations – ThermodynamicsPages displaying short descriptions with no spaces
• Scale-free ideal gas – Ideal gas with no physical scale

Notes

References