Sackur-Tetrode equation

From Wikipedia, the free encyclopedia

The Sackur-Tetrode equation is an expression for the entropy of a monatomic classical ideal gas which uses quantum considerations to arrive at an exact formula. Classical thermodynamics can only give the entropy of a classical ideal gas to within a constant. The Sackur-Tetrode equation is written:

$S = k N \ln \left[ \left(\frac VN\right) \left(\frac UN \right)^{\frac 32}\right]+ {\frac 32}kN\left( {\frac 53}+ \ln\frac{4\pi m}{3h^2}\right)$

where V is the volume of the gas, N is the number of particles in the gas, U is the internal energy of the gas, k is Boltzmann's constant, m is the mass of a gas particle, h is Planck's constant and ln() is the natural logarithm. See Gibbs paradox for a derivation of the Sackur-Tetrode equation. See also the ideal gas article for the constraints placed upon the entropy of an ideal gas by thermodynamics alone.

The Sackur-Tetrode equation can also be conveniently expressed in terms of the thermal wavelength $Λ$ . Using the classical ideal gas relationship U = (3/2)NkT for a monatomic gas gives

$\frac{S}{kN} = \ln\left[\frac{V}{N\Lambda^3}\right]+\frac{5}{2}$

Note that the assumption was made that the gas is in the classical regime, and is described by Maxwell-Boltzmann statistics (with "correct counting"). From the definition of the thermal wavelength, this means the Sackur-Tetrode equation is only valid for

$\frac{V}{N\Lambda^3}\gg 1.$

and in fact, the entropy predicted by the Sackur-Tetrode equation approaches negative infinity as the temperature approaches zero.

[edit] The Sackur-Tetrode constant

The Sackur-Tetrode constant, written $S 0 / R$ , is equal to S/kN evaluated at a temperature of T = 1 kelvin, at standard atmospheric pressure (101.325 kPa), for a particle of mass equal to one atomic mass unit (m = 1.6605388628x10⁻²⁷ kg), which yields the dimensionless quantity:

$S_0/R = -1.1648678\pm 0.0000044\,$