Ideal Gas |

**In General** > s.a. gas; fluids;
states in statistical mechanics; thermodynamical
systems [including geometry of state space].

* __Idea__: A large collection of particles
with no internal structure, non-interacting except for collisions (small hard spheres).

* __Equation of state__: The ideal gas law,
obtained combining Boyle's volume-pressure law, Gay-Lussac's pressure-temperature law
and Charles' law,

*pV* = *nRT* = *Nk*_{B}*T* ,

where *R* is the molar gas constant, and *k*_{B}
the Boltzmann constant (> see constants); First
stated in 1834 by Émile Clapeyron; It does not hold at very low temperatures
or high densities, when quantum effects have to be taken into account.

* __Remark__: Most gases at room
*T* and *p* behave like ideal gases, but as *T* →
0 they can't because of quantum effects.

* __Internal energy__: If *s*
is the number of degrees of freedom (*s* = 3 for a monatomic gas),

*U* = (*s*/2)* Nk*_{B}*T* .

* __Entropy__: An explicit
expression in terms of basic constants for a monatomic gas is given by
the Sackur-Tetrode equation; The general form is given by

*S* = *C*_{V} ln(*T*/*T*_{0})
+ *Nk*_{B} ln(*V*/*V*_{0}) .

* __Consequences__: Aerosol cans get cold when used.

* __In a gravitational field__:
The equilibrium pressure of a perfect gas in a constant gravitational field
decreases exponentially with height (barometric formula).

**Second-Order Quantities** > see Compressibility;
specific heat.

**References**

@ __General__: Creaco & Kalogeropoulos MPLB(09)-a0811 [thermodynamic limit, phase space measure];
Arnaud et al Ent(13)-a1105 [assumptions behind the ideal gas law].

@ __Quantum__: Meyer IJMPC(97)qp [quantum lattice gas];
Bloch pw(04)apr [in optical lattices];
Nattermann AJP(05)apr [scaling approach];
Velenich et al JPA(08) [Brownian gas, Poissonian ground state];
Dodonov & Vieira Lopes PLA(08) [temperature increase from sudden expansion];
Pérez & Sauer AHES(10)-a1004 [Einstein's work];
Nakamura et al PRE(11)-a1105 [in an expanding cavity];
Quevedo & Zaldivar a1512 [geometrothermodynamic approach];
> s.a. gas [fermion gas, boson gas]; Susceptibility.

@ __Relativistic__: Becattini & Piccinini AP(08),
Becattini & Tinti AP(10)-a0911 [rotating];
Chakrabarti et al PhyA(10) [non-extensive statistics];
Basu & Mondal a1103 [4-velocity distribution];
Cannoni PRD(14)-a1311 [probability distribution for the relative velocity of colliding particles];
Montakhab et al PhyA(14)-a1406 [morphological phase transition];
> s.a. deformed special relativity; quantum-gravity
phenomenology; statistical-mechanical systems.

@ __Modified__: Das et al PRD(09)-a0908 [DSR, equation of state];
Chandra & Chatterjee PRD(12)-a1108 [DSR, thermodynamics];
> s.a. phenomenology of modified uncertainty relations;
thermodynamic systems [non-commutative, etc].

@ __Entropy__: Kolekar & Padmanabhan PRD(11)-a1012 [in a strong gravitational field];
Oikonomou & Bagci SHPMP(13) [monatomic gas, Clausius versus Sackur-Tetrode entropies];
> s.a. Gibbs Theorem.

@ __From particles to fluid__: Park JPCS(14)-a1310 [phase transition at finite particle number for ideal Bose gas].

@ __Related topics__: Landsberg et al AJP(94)aug,
Pantellini AJP(00)jan [in a constant **g** field];
Boozer AJP(10)jan [1D, details];
Kothawala PLB(13)-a1108 [in free fall in curved spacetime];
> s.a. Boltzmann Equation;
Maxwell-Boltzmann Distribution.

> __Online resources__:
see Wikipedia page.

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send feedback and suggestions to bombelli at olemiss.edu – modified 31 may 2018