Sackur–Tetrode equation

The Sackur–Tetrode equation is an expression for the entropy of a monatomic classical ideal gas which incorporates quantum considerations which give a more detailed description of its regime of validity.

The Sackur–Tetrode equation is named for Hugo Martin Tetrode^[1] (1895–1931) and Otto Sackur^[2] (1880–1914), who developed it independently as a solution of Boltzmann's gas statistics and entropy equations, at about the same time in 1912.

Formula

The Sackur–Tetrode equation is written:

$S = k N \left(\ln \left[ \frac VN \left(\frac{4\pi m}{3h^2}\frac UN\right)^{3/2}\right]+ {\frac 52}\right)$

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 $\Lambda$.

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

Note that the assumption was made that the gas is in the classical regime, and is described by Maxwell–Boltzmann statistics (with "correct Boltzmann counting"). From the definition of the thermal wavelength, this means the Tommo–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.

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 = 1.660 538 782(83)×10⁻²⁷ kg). Its 2010 CODATA recommended value is:^[3]

S₀/R = −1.151 7078(23) for p^o = 100 kPa

S₀/R = −1.164 8708(23) 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.^[4]

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

$\begin{align} \frac{S}{k} & = [ N\ln V] + \left[\frac 32 N\ln\left(2\pi e m k T\right)\right] + [ -3N\ln h] + [-\ln N!] \\ & \approx N \ln \left[\frac{V}{N} \left(\frac{2\pi m k T}{h^2}\right)^{\frac 32}\right] + \frac 52 N \end{align}$

The derivation uses the Stirling's approximation, $\ln N! \approx N \ln N - N$.

References

1. H. Tetrode (1912) "Die chemische Konstante der Gase und das elementare Wirkungsquantum" (The chemical constant of gases and the elementary quantum of action), Annalen der Physik 38: 434–442. See also: H. Tetrode (1912) "Berichtigung zu meiner Arbeit: "Die chemische Konstante der Gase und das elementare Wirkungsquantum" " (Correction to my work: "The chemical constant of gases and the elementary quantum of action"), Annalen der Physik 39: 255–256.
2. Sackur published his findings in the following series of papers:
   1. O. Sackur (1911) "Die Anwendung der kinetischen Theorie der Gase auf chemische Probleme" (The application of the kinetic theory of gases to chemical problems), Annalen der Physik, 36: 958–980.
   2. O. Sackur, "Die Bedeutung des elementaren Wirkungsquantums für die Gastheorie und die Berechnung der chemischen Konstanten" (The significance of the elementary quantum of action to gas theory and the calculation of the chemical constant), Festschrift W. Nernst zu seinem 25jährigen Doktorjubiläum gewidmet von seinen Schülern (Halle an der Salle, Germany: Wilhelm Knapp, 1912), pages 405–423.
   3. O. Sackur (1913) "Die universelle Bedeutung des sog. elementaren Wirkungsquantums" (The universal significance of the so-called elementary quantum of action), Annalen der Physik 40: 67–86.
3. CODATA2010
4. Ben-Naim, Arieh (2008). A Farewell to Entropy: Statistical Thermodynamics Based on Information. World Scientific Publishing Company. ISBN 978-981-270-706-2. Retrieved 2009-11-28.