Sackur–Tetrode equation

Formula

The Sackur–Tetrode equation is written:

S = k N ( ln ⁡ [ V N ( 4 π m 3 h 2 U N ) 3 / 2 ] + 5 2 )

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 .

S k N = ln ⁡ [ V N Λ 3 ] + 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 Boltzmann counting"). From the definition of the thermal wavelength, this means the Sackur–Tetrode equation is only valid for

V N Λ 3 ≫ 1

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

Sackur–Tetrode constant

The Sackur–Tetrode constant, written S₀/R, is equal to S/kN evaluated at a temperature of T = 1 kelvin, at standard pressure (100 kPa or 101.325 kPa, to be specified), for one mole of an ideal gas composed of particles of mass equal to one atomic mass unit (m_u = 1.660 539 040(20)×10⁻²⁷ kg). Its 2014 CODATA recommended value is:

S₀/R = −1.151 7084(14) for p^o = 100 kPa S₀/R = −1.164 8714(14) for p^o = 101.325 kPa.

Derivation from information theoretic perspective

In addition to using the thermodynamic perspective of entropy, the tools of information theory can be used to provide an information perspective of entropy. The Sackur–Tetrode equation for entropy can be derived in information theoretic terms. The equation can be seen to consist of the sum of four entropies (missing information) due to positional uncertainty, momenta uncertainty, the quantum mechanical uncertainty principle and the indistinguishability of the particles.

Including k, the Sackur–Tetrode equation is then given as:

S k = [ N ln ⁡ V ] + [ 3 2 N ln ⁡ ( 2 π e m k T ) ] + [ − 3 N ln ⁡ h ] + [ − ln ⁡ N ! ] ≈ N ln ⁡ [ V N ( 2 π m k T h 2 ) 3 2 ] + 5 2 N

The derivation uses the Stirling's approximation, ln ⁡ N ! ≈ N ln ⁡ N − N .