where is the hydrostatic pressure in addition to the atmospheric one, is the volume at atmospheric pressure, is the volume under additional pressure , and are experimentally determined parameters.
A very detailed historical study on the Tait equation with the physical interpretation of the two parameters and is given in reference.[2]
The expression for the pressure in terms of the specific volume is
A highly detailed study on the Tait-Tammann equation of state with the physical interpretation of the two empirical parameters and is given in chapter 3 of reference.[2] Expressions as a function of temperature for the two empirical parameters and are given for water, seawater, helium-4, and helium-3 in the entire liquid phase up to the critical temperature . The special case of the supercooled phase of water is discussed in Appendix D of reference.[5] The case of liquid argon between the triple point temperature and 148 K is dealt with in detail in section 6 of the reference.[6]
This equation, in pressure form, can be written as
where are mass densities at , respectively.
For pure water, typical parameters are = 101,325 Pa, = 1000 kg/cu.m, = 2.15 GPa, and = 7.15.[citation needed]
Tumlirz-Tammann-Tait equation of state based on fits to experimental data on pure water.
A related equation of state that can be used to model liquids is the Tumlirz equation (sometimes called the Tammann equation and originally proposed by Tumlirz in 1909 and Tammann in 1911 for pure water).[4][10] This relation has the form
where is the specific volume, is the pressure, is the salinity, is the temperature, and is the specific volume when , and are parameters that can be fit to experimental data.
The Tumlirz–Tammann version of the Tait equation for fresh water, i.e., when , is
For pure water, the temperature-dependence of are:[10]
In the above fits, the temperature is in degrees Celsius, is in bars, is in cc/gm, and is in bars-cc/gm.
Following in particular the study of underwater explosions and more precisely the shock waves emitted, J.G. Kirkwood proposed in 1965[11] a more appropriate form of equation of state to describe high pressures (>1 kbar) by expressing the isentropic compressibility coefficient as
where represents here the entropy.
The two empirical parameters and are now function of entropy such that
is dimensionless
has the same units as
The integration leads to the following expression for the volume along the isentropic
The expression for the pressure in terms of the specific volume along the isentropic is
A highly detailed study on the Modified Tait equation of state with the physical interpretation of the two empirical parameters and is given in chapter 4 of reference.[2] Expressions as a function of entropy for the two empirical parameters and are given for water, helium-3 and helium-4.
^Tait, P. G. (1888). "Report on some of the physical properties of fresh water and of sea water". Physics and Chemistry of the Voyage of H.M.S. Challenger. Vol. II, part IV.
^Aitken, F.; Volino, F. (November 2021). "A new single equation of state to describe the dynamic viscosity and self-diffusion coefficient for all fluid phases of water from 200 to 1800 K based on a new original microscopic model". Physics of Fluids. 33 (11): 117112. arXiv:2108.10666. Bibcode:2021PhFl...33k7112A. doi:10.1063/5.0069488. S2CID237278734.
^Thompson, P. A., & Beavers, G. S. (1972). Compressible-fluid dynamics. Journal of Applied Mechanics, 39, 366.
^Kedrinskiy, V. K. (2006). Hydrodynamics of Explosion: experiments and models. Springer Science & Business Media.
^Macdonald, J. R. (1966). Some simple isothermal equations of state. Reviews of Modern Physics, 38(4), 669.
^ abFisher, F. H., and O. E. Dial Jr. Equation of state of pure water and sea water. No. MPL-U-99/67. SCRIPPS INSTITUTION OF OCEANOGRAPHY LA JOLLA CA MARINE PHYSICAL LAB, 1975. http://www.dtic.mil/dtic/tr/fulltext/u2/a017775.pdf
^Cole, R. H. (1965). Underwater Explosions. New York: Dover Publications.