Tait equation

Around 1895,[1] the original isothermal Tait equation was replaced by Tammann with an equation of the form

${\displaystyle -{\frac {1}{V}}\,{\frac {dV}{dP}}={\frac {A}{V(B+P)}}\,.}$

The temperature-dependent version of the above equation is popularly known as the Tait equation and is commonly written as[2]

${\displaystyle \beta =-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}={\frac {0.4343C}{V(B+P)}}}$

or in the integrated form

${\displaystyle V=V_{0}-C\log _{10}\left({\frac {B+P}{B+P_{0}}}\right)}$

where

• ${\displaystyle \beta }$ is the compressibility of the substance (often, water) (in units of bar⁻¹ or Pa)
• ${\displaystyle V\ }$ is the specific volume of the substance (in units of ml/g or m³/kg)
• ${\displaystyle V_{0}}$ is the specific volume at ${\displaystyle P=P_{0}}$ = 1 bar
• ${\displaystyle B\ }$ and ${\displaystyle C\ }$ are functions of temperature that are independent of pressure[2]

Specific volume as a function of pressure predicted by the Tait-Murnaghan equation of state.

Another popular isothermal equation of state that goes by the name "Tait equation"[3][4] is the Murnaghan model[5] which is sometimes expressed as

${\displaystyle {\frac {V}{V_{0}}}=\left[1+{\frac {n}{K_{0}}}\,(P-P_{0})\right]^{-1/n}}$

where ${\displaystyle V}$ is the specific volume at pressure ${\displaystyle P}$, ${\displaystyle V_{0}}$ is the specific volume at pressure ${\displaystyle P_{0}}$, ${\displaystyle K_{0}}$ is the bulk modulus at ${\displaystyle P_{0}}$, and ${\displaystyle n}$ is a material parameter.

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).[1][6] This relation has the form

${\displaystyle V(P,S,T)=V_{\infty }-K_{1}S+{\frac {\lambda }{P_{0}+K_{2}S+P}}}$

where ${\displaystyle V(P,S,T)}$ is the specific volume, ${\displaystyle P}$ is the pressure, ${\displaystyle S}$ is the salinity, ${\displaystyle T}$ is the temperature, and ${\displaystyle V_{\infty }}$ is the specific volume when ${\displaystyle P=\infty }$, and ${\displaystyle K_{1},K_{2},P_{0}}$ are parameters that can be fit to experimental data.

The Tumlirz-Tammann version of the Tait equation for fresh water, i.e., when ${\displaystyle S=0}$, is

${\displaystyle V=V_{\infty }+{\frac {\lambda }{P_{0}+P}}\,.}$

For pure water, the temperature-dependence of ${\displaystyle V_{\infty },\lambda ,P_{0}}$ are:[6]

${\displaystyle {\begin{aligned}\lambda &=1788.316+21.55053\,T-0.4695911\,T^{2}+3.096363\times 10^{-3}\,T^{3}-0.7341182\times 10^{-5}\,T^{4}\\P_{0}&=5918.499+58.05267\,T-1.1253317\,T^{2}+6.6123869\times 10^{-3}\,T^{3}-1.4661625\times 10^{-5}\,T^{4}\\V_{\infty }&=0.6980547-0.7435626\times 10^{-3}\,T+0.3704258\times 10^{-4}\,T^{2}-0.6315724\times 10^{-6}\,T^{3}\\&+0.9829576\times 10^{-8}\,T^{4}-0.1197269\times 10^{-9}\,T^{5}+0.1005461\times 10^{-11}\,T^{6}\\&-0.5437898\times 10^{-14}\,T^{7}+0.169946\times 10^{-16}\,T^{8}-0.2295063\times 10^{-19}\,T^{9}\end{aligned}}}$

In the above fits, the temperature ${\displaystyle T}$ is in degrees Celsius, ${\displaystyle P_{0}}$ is in bars, ${\displaystyle V_{\infty }}$ is in cc/gm, and ${\displaystyle \lambda }$ is in bars-cc/gm.

• Equation of state