# Ideal gas

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Specific heat capacity

$\displaystyle c=$

c
=

\displaystyle c=

 $\displaystyle T$ T \displaystyle T $\displaystyle \partial S$ ∂ S \displaystyle \partial S $\displaystyle N$ N \displaystyle N $\displaystyle \partial T$ ∂ T \displaystyle \partial T
Compressibility

$\displaystyle \beta =-$

β
=

\displaystyle \beta =-

 $\displaystyle 1$ 1 \displaystyle 1 $\displaystyle \partial V$ ∂ V \displaystyle \partial V $\displaystyle V$ V \displaystyle V $\displaystyle \partial p$ ∂ p \displaystyle \partial p
Thermal expansion

$\displaystyle \alpha =$

α
=

\displaystyle \alpha =

 $\displaystyle 1$ 1 \displaystyle 1 $\displaystyle \partial V$ ∂ V \displaystyle \partial V $\displaystyle V$ V \displaystyle V $\displaystyle \partial T$ ∂ T \displaystyle \partial T
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$\displaystyle U(S,V)$

U
(
S
,
V
)

\displaystyle U(S,V)

• Enthalpy

$\displaystyle H(S,p)=U+pV$

H
(
S
,
p
)
=
U
+
p
V

\displaystyle H(S,p)=U+pV

• Helmholtz free energy

$\displaystyle A(T,V)=U-TS$

A
(
T
,
V
)
=
U

T
S

\displaystyle A(T,V)=U-TS

• Gibbs free energy

$\displaystyle G(T,p)=H-TS$

G
(
T
,
p
)
=
H

T
S

\displaystyle G(T,p)=H-TS

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Book:Thermodynamics
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• e

An ideal gas is a theoretical gas composed of many randomly moving point particles whose only interactions are perfectly elastic collisions . 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 .

In most usual conditions (for instance at standard temperature and pressure), most real gases behave qualitatively like an ideal gas. Many gases such as nitrogen , oxygen , hydrogen , noble gases , and some heavier gases like carbon dioxide can be treated like ideal gases within reasonable tolerances. [1] Generally, a gas behaves more like an ideal gas at higher temperature and lower pressure , [1] as the potential 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. One mole of an ideal gas has a volume of 22.710947(13) litres [2] at standard temperature and pressure (a temperature of 273.15  K and an absolute pressure of exactly 105  Pa ) as defined by IUPAC since 1982. [note 1]

The ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forces and molecular size becomes important. It also fails for most heavy gases, such as many refrigerants , [1] and for gases with strong intermolecular forces, notably water vapor . At high pressures, the volume of a real gas is often considerably greater 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 a phase transition , such as to a liquid or a solid . 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 behaviour can be described by a dimensionless quantity , the compressibility factor , Z.

The ideal gas model has been explored in both the Newtonian dynamics (as in ” kinetic theory “) and in quantum mechanics (as a ” gas in a box “). The ideal gas model has also been used to model the behavior of electrons in a metal (in the Drude model and the free electron model ), and it is one of the most important models in statistical mechanics.

## Contents

• 1 Types of ideal gas
• 2 Classical thermodynamic ideal gas
• 2.1 Ideal gas law
• 2.2 Internal energy
• 2.3 Microscopic model
• 3 Heat capacity
• 4 Entropy
• 5 Thermodynamic potentials
• 6 Speed of sound
• 7 Table of ideal gas equations
• 8 Ideal quantum gases
• 8.1 Ideal Boltzmann gas
• 8.2 Ideal Bose and Fermi gases
• 10 References

## Types of ideal gas[ edit ]

There are three basic classes of ideal gas:[ citation needed ]

• 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 .

## Classical thermodynamic ideal gas[ edit ]

The classical thermodynamic properties of an ideal gas can be described by two equations of state : [5] [6]

### Ideal gas law[ edit ]

Main article: Ideal gas law

The ideal gas law is the equation of state for an ideal gas, given by:

$\displaystyle PV=nRT\,$

P
V
=
n
R
T

\displaystyle PV=nRT\,

where

• P is the pressure
• V is the volume
• n is the amount of substance of the gas (in moles )
• R is the gas constant (8.314  J · K −1· mol −1)
• T is the absolute temperature .

The ideal gas law is an extension of experimentally discovered gas laws . It can also be derived from microscopic considerations.

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 as it deposits from a gas into a solid. This deviation is expressed as a compressibility factor .

This equation is derived from

• Boyle’s law :
$\displaystyle V\propto \frac 1P$

V

1
P

\displaystyle V\propto \frac 1P

;

• Charles’s law :
$\displaystyle V\propto T$

V

T

\displaystyle V\propto T

;

• Avogadro’s law :
$\displaystyle V\propto n$

V

n

\displaystyle V\propto n

.

After combining three laws we get

$\displaystyle V\propto \frac nTP$

V

n
T

P

\displaystyle V\propto \frac nTP

That is:

$\displaystyle V=R\left(\frac nTP\right)$

V
=
R

(

n
T

P

)

\displaystyle V=R\left(\frac nTP\right)

$\displaystyle PV=nRT$

P
V
=
n
R
T

\displaystyle PV=nRT

.

### Internal energy[ edit ]

The other equation of state of an ideal gas must express Joule’s law , that the internal energy of a fixed mass of ideal gas is a function only of its temperature. For the present purposes it is convenient to postulate an exemplary version of this law by writing:

$\displaystyle U=\hat c_VnRT$

U
=

c
^

V

n
R
T

\displaystyle U=\hat c_VnRT

where

• U is the internal energy
• ĉV is the dimensionless specific heat capacity at constant volume, approximately 3/2 for a monatomic gas , 5/2 for diatomic gas, and 3 for non-linear molecules if we ignore quantum vibrational contribution.

### Microscopic model[ edit ]

In order to switch from macroscopic quantities (left hand side of the following equation) to microscopic ones (right hand side), we use

$\displaystyle nR=Nk_\mathrm B \$

n
R
=
N

k

B

\displaystyle nR=Nk_\mathrm B \

where

• N is the number of gas particles
• kB is the Boltzmann constant (1.381×10−23 J·K−1).

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

## Heat capacity[ edit ]

The heat capacity at constant volume is defined by

$\displaystyle \hat c_V=\frac 1nRT\left(\frac \partial S\partial T\right)_V=\frac 1nR\left(\frac \partial U\partial T\right)_V$

c
^

V

=

1

n
R

T

(

S

T

)

V

=

1

n
R

(

U

T

)

V

\displaystyle \hat c_V=\frac 1nRT\left(\frac \partial S\partial T\right)_V=\frac 1nR\left(\frac \partial U\partial T\right)_V

where S is the entropy . This is also true for an ideal gas. 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 monatomic gas is ĉV = 3/2 while for a diatomic gas it is ĉV = 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 1/R mole of ideal gas is:

$\displaystyle \hat c_P=\frac 1nRT\left(\frac \partial S\partial T\right)_P=\frac 1nR\left(\frac \partial H\partial T\right)_P=\hat c_V+1$

c
^

P

=

1

n
R

T

(

S

T

)

P

=

1

n
R

(

H

T

)

P

=

c
^

V

+
1

\displaystyle \hat c_P=\frac 1nRT\left(\frac \partial S\partial T\right)_P=\frac 1nR\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 ĉV and ĉP 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

$\displaystyle \gamma =\frac c_Pc_V$

γ
=

c

P

c

V

\displaystyle \gamma =\frac c_Pc_V

For air, which is a mixture of gases, this ratio is 1.4.

## Entropy[ edit ]

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:

$\displaystyle \Delta S=\int _S_0^SdS=\int _T_0^T\left(\frac \partial S\partial T\right)_V\!dT+\int _V_0^V\left(\frac \partial S\partial V\right)_T\!dV$

Δ
S
=

S

0

S

d
S
=

T

0

T

(

S

T

)

V

d
T
+

V

0

V

(

S

V

)

T

d
V

\displaystyle \Delta S=\int _S_0^SdS=\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:

$\displaystyle \Delta S=\int _T_0^T\frac C_VT\,dT+\int _V_0^V\left(\frac \partial P\partial T\right)_VdV.$

Δ
S
=

T

0

T

C

V

T

d
T
+

V

0

V

(

P

T

)

V

d
V
.

\displaystyle \Delta S=\int _T_0^T\frac C_VT\,dT+\int _V_0^V\left(\frac \partial P\partial T\right)_VdV.

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

$\displaystyle \Delta S=\hat c_VNk\ln \left(\frac TT_0\right)+Nk\ln \left(\frac VV_0\right)$

Δ
S
=

c
^

V

N
k
ln

(

T

T

0

)

+
N
k
ln

(

V

V

0

)

\displaystyle \Delta S=\hat c_VNk\ln \left(\frac TT_0\right)+Nk\ln \left(\frac VV_0\right)

which implies that the entropy may be expressed as:

$\displaystyle S=Nk\ln \left(\frac VT^\hat c_Vf(N)\right)$

S
=
N
k
ln

(

V

T

c
^

V

f
(
N
)

)

\displaystyle S=Nk\ln \left(\frac VT^\hat c_Vf(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 VTĉV 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:

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

S
(
T
,
a
V
,
a
N
)
=
a
S
(
T
,
V
,
N
)
.

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

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

$\displaystyle af(N)=f(aN).\,$

a
f
(
N
)
=
f
(
a
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):

$\displaystyle f(N)=\Phi N\,$

f
(
N
)
=
Φ
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 VTĉV/N. Substituting into the equation for the entropy:

$\displaystyle \frac SNk=\ln \left(\frac VT^\hat c_VN\Phi \right).\,$

S

N
k

=
ln

(

V

T

c
^

V

N
Φ

)

.

\displaystyle \frac SNk=\ln \left(\frac VT^\hat c_VN\Phi \right).\,

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

$\displaystyle \frac SNk=\ln \left[\frac VN\,\left(\frac U\hat c_VkN\right)^\hat c_V\,\frac 1\Phi \right]$

S

N
k

=
ln

[

V
N

(

U

c
^

V

k
N

)

c
^

V

1
Φ

]

\displaystyle \frac SNk=\ln \left[\frac VN\,\left(\frac U\hat c_VkN\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 (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.

## Thermodynamic potentials[ edit ]

Main article: Thermodynamic potential

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

$\displaystyle \frac SkN=\ln \left(\frac VT^\hat c_VN\Phi \right)$

S

k
N

=
ln

(

V

T

c
^

V

N
Φ

)

\displaystyle \frac SkN=\ln \left(\frac VT^\hat c_VN\Phi \right)

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

$\displaystyle \mu =\left(\frac \partial G\partial N\right)_T,P$

μ
=

(

G

N

)

T
,
P

\displaystyle \mu =\left(\frac \partial G\partial N\right)_T,P

where G is the Gibbs free energy and is equal to U + PVTS so that:

$\displaystyle \mu (T,V,N)=kT\left(\hat c_P-\ln \left(\frac VT^\hat c_VN\Phi \right)\right)$

μ
(
T
,
V
,
N
)
=
k
T

(

c
^

P

ln

(

V

T

c
^

V

N
Φ

)

)

\displaystyle \mu (T,V,N)=kT\left(\hat c_P-\ln \left(\frac VT^\hat c_VN\Phi \right)\right)

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

 $\displaystyle U\,$ U \displaystyle U\, $\displaystyle =\hat c_VNkT\,$ = c ^ V N k T \displaystyle =\hat c_VNkT\, $\displaystyle A\,$ A \displaystyle A\, $\displaystyle =U-TS\,$ = U − T S \displaystyle =U-TS\, $\displaystyle =\mu N-NkT\,$ = μ N − N k T \displaystyle =\mu N-NkT\, $\displaystyle H\,$ H \displaystyle H\, $\displaystyle =U+PV\,$ = U + P V \displaystyle =U+PV\, $\displaystyle =\hat c_PNkT\,$ = c ^ P N k T \displaystyle =\hat c_PNkT\, $\displaystyle G\,$ G \displaystyle G\, $\displaystyle =U+PV-TS\,$ = U + P V − T S \displaystyle =U+PV-TS\, $\displaystyle =\mu N\,$ = μ N \displaystyle =\mu N\,

where, as before,

$\displaystyle \hat c_P=\hat c_V+1$

c
^

P

=

c
^

V

+
1

\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:

$\displaystyle U(S,V,N)=\hat c_VNk\left(\frac N\Phi V\,e^S/Nk\right)^1/\hat c_V$

U
(
S
,
V
,
N
)
=

c
^

V

N
k

(

N
Φ

V

e

S

/

N
k

)

1

/

c
^

V

\displaystyle U(S,V,N)=\hat c_VNk\left(\frac N\Phi V\,e^S/Nk\right)^1/\hat c_V

$\displaystyle A(T,V,N)=NkT\left(\hat c_V-\ln \left(\frac VT^\hat c_VN\Phi \right)\right)$

A
(
T
,
V
,
N
)
=
N
k
T

(

c
^

V

ln

(

V

T

c
^

V

N
Φ

)

)

\displaystyle A(T,V,N)=NkT\left(\hat c_V-\ln \left(\frac VT^\hat c_VN\Phi \right)\right)

$\displaystyle H(S,P,N)=\hat c_PNk\left(\frac P\Phi k\,e^S/Nk\right)^1/\hat c_P$

H
(
S
,
P
,
N
)
=

c
^

P

N
k

(

P
Φ

k

e

S

/

N
k

)

1

/

c
^

P

\displaystyle H(S,P,N)=\hat c_PNk\left(\frac P\Phi k\,e^S/Nk\right)^1/\hat c_P

$\displaystyle G(T,P,N)=NkT\left(\hat c_P-\ln \left(\frac kT^\hat c_PP\Phi \right)\right)$

G
(
T
,
P
,
N
)
=
N
k
T

(

c
^

P

ln

(

k

T

c
^

P

P
Φ

)

)

\displaystyle G(T,P,N)=NkT\left(\hat c_P-\ln \left(\frac kT^\hat c_PP\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.

## Speed of sound[ edit ]

Main article: Speed of sound

The speed of sound in an ideal gas is given by

$\displaystyle c_\textsound=\sqrt \left(\frac \partial P\partial \rho \right)_s=\sqrt \frac \gamma P\rho =\sqrt \frac \gamma RTM$

c

sound

=

(

P

ρ

)

s

=

γ
P

ρ

=

γ
R
T

M

\displaystyle c_\textsound=\sqrt \left(\frac \partial P\partial \rho \right)_s=\sqrt \frac \gamma P\rho =\sqrt \frac \gamma RTM

where

γ is the adiabatic index (ĉP/ĉV)
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.

## Table of ideal gas equations[ edit ]

Main article: Table of thermodynamic equations § Ideal gas

## Ideal quantum gases[ edit ]

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[ edit ]

The ideal Boltzmann gas yields the same results as the classical thermodynamic gas, but makes the following identification for the undetermined constant Φ:

$\displaystyle \Phi =\frac T^\frac 32\Lambda ^3g$

Φ
=

T

3
2

Λ

3

g

\displaystyle \Phi =\frac T^\frac 32\Lambda ^3g

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

### Ideal Bose and Fermi gases[ edit ]

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 .

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

## References[ edit ]

Notes
1. ^ Until 1982, STP was defined as a temperature of 273.15  K and an absolute pressure of exactly 1  atm . The volume of one mole of an ideal gas at this temperature and pressure is 22.413962(13) litres. [3] IUPAC recommends that the former use of this definition should be discontinued; [4] however, some textbooks still use these old values.
References
1. ^ a b c Cengel, Yunus A.; Boles, Michael A. Thermodynamics: An Engineering Approach (4th ed.). p. 89. ISBN   0-07-238332-1 .
2. ^ “CODATA Value: molar volume of ideal gas (273.15 K, 100 kPa)” . Retrieved 2017-02-07.
3. ^ “CODATA Value: molar volume of ideal gas (273.15 K, 101.325 kPa)” . Retrieved 2017-02-07.
4. ^ Calvert, J. G. (1990). “Glossary of atmospheric chemistry terms (Recommendations 1990)”. Pure and Applied Chemistry. 62 (11): 2167–2219. doi : 10.1351/pac199062112167 .
5. ^ Adkins, C. J. (1983). Equilibrium Thermodynamics (3rd ed.). Cambridge, UK: Cambridge University Press. pp. 116–120. ISBN   0-521-25445-0 .
6. ^ Tschoegl, N. W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics. Amsterdam: Elsevier. p. 88. ISBN   0-444-50426-5 .