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.

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 \( \Lambda \) . Using the classical ideal gas relationship U = C*NkT (where C* is the dimensionless specific heat capacity) yields:

\( \frac{S}{kN} = \ln\left[\frac{V}{N\Lambda^3}\left(\frac{2}{3}C^*\right)^{2/3}\right]+\frac{5}{2} \)

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 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.
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 (mu = 1.660 538 782(83)×10−27 kg). Its 2006 CODATA recommended value is:[3]

S0/R = −1.151 7047(44) for po = 100 kPa
S0/R = −1.164 8677(44) for po = 101.325 kPa

Interpretation of the equation through information theory

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 physical chemist Arieh Ben-Naim rederived the Sackur–Tetrode equation for entropy in terms of information theory, and in doing so he tied in well known concepts from modern physics. He showed the equation to consist of stacking the entropy (missing information) due to four terms: positional uncertainty, momenta uncertainty, quantum mechanical uncertainty principle and the indistinguishability of the particles.[4]
References

^ 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" " (Corrrection to my work: "The chemical constant of gases and the elementary quantum of action"), Annalen der Physik 39: 255 - 256.
^ Sackur published his findings in the following series of papers:
O. Sackur (1911) "Die Anwendung der kinetischen Theorie der Gase auf chemische Probleme" (The application of the the kinetic theory of gases to chemical problems), Annalen der Physik, 36: 958 - 980.
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.
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.
^ Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80 (2): 633–730. Bibcode 2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633. Direct link to value.
^ Ben-Naim, Arieh (2008). A Farewell to Entropy: Statistical Thermodynamics Based on Information. World Scientific Publishing Company. ISBN 978-9812707062. Retrieved 2009-11-28.

Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms. There remain some unsolved problems. The class number problem is particularly important.

A set K with these properties is said to be absolutely convex.