Tait equation

Tait-Murnaghan equation of state

Another popular isothermal equation of state that goes by the name "Tait equation" is the Murnaghan model 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.

Pressure formula

This equation, in pressure form, can be written as

${\displaystyle P=\frac{K_0}{n}\left[{\left(\frac{V_0}{V}\right)}^n-1\right]+P_0=\frac{K_0}{n}\left[{\left(\frac{\rho }{{\rho }_0}\right)}^n-1\right]+P_0.}$

where ${\displaystyle \rho ,{\rho }_0}$ are mass densities at ${\displaystyle P,P_0}$, respectively. For pure water, typical parameters are ${\displaystyle P_0}$ = 101,325 Pa, ${\displaystyle {\rho }_0}$ = 1000 kg/cu.m, ${\displaystyle K_0}$ = 2.15 GPa, and ${\displaystyle n}$ = 7.15.

Note that this form of the Tate equation of state is identical to that of the Murnaghan equation of state.

Bulk modulus formula

The tangent bulk modulus predicted by the MacDonald–Tait model is

${\displaystyle K=K_0{\left(\frac{V_0}{V}\right)}^n.}$

Tumlirz–Tammann–Tait equation of state

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). This relation has the form

${\displaystyle V(P,S,T)=V_{\mathrm{\infty }}-K_{1}S+\frac{\lambda }{P_0+K_{2}S+P}}$

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 ${\displaystyle V_{\mathrm{\infty }}}$ is the specific volume when ${\displaystyle P=\mathrm{\infty }}$, 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

${\displaystyle V=V_{\mathrm{\infty }}+\frac{\lambda }{P_0+P}.}$

For pure water, the temperature-dependence of ${\displaystyle V_{\mathrm{\infty }},\lambda ,P_0}$ are:

${\displaystyle \begin{array}{cc}\lambda & =1788.316+21.55053T-0.4695911T^2+3.096363\times {10}^{-3}{T}^3-0.7341182\times {10}^{-5}{T}^4\\ P_0& =5918.499+58.05267T-1.1253317T^2+6.6123869\times {10}^{-3}{T}^3-1.4661625\times {10}^{-5}{T}^4\\ V_{\mathrm{\infty }}& =0.6980547-0.7435626\times {10}^{-3}T+0.3704258\times {10}^{-4}{T}^2-0.6315724\times {10}^{-6}{T}^3\\ & +0.9829576\times {10}^{-8}{T}^4-0.1197269\times {10}^{-9}{T}^5+0.1005461\times {10}^{-11}{T}^6\\ & -0.5437898\times {10}^{-14}{T}^7+0.169946\times {10}^{-16}{T}^8-0.2295063\times {10}^{-19}{T}^9\end{array}}$

In the above fits, the temperature ${\displaystyle T}$ is in degrees Celsius, ${\displaystyle P_0}$ is in bars, ${\displaystyle V_{\mathrm{\infty }}}$ is in cc/gm, and ${\displaystyle \lambda }$ is in bars-cc/gm.

Pressure formula

The inverse Tumlirz–Tammann–Tait relation for the pressure as a function of specific volume is

${\displaystyle P=\frac{\lambda }{V-V_{\mathrm{\infty }}}-P_0.}$

Bulk modulus formula

The Tumlirz-Tammann-Tait formula for the instantaneous tangent bulk modulus of pure water is a quadratic function of ${\displaystyle P}$ (for an alternative see )

${\displaystyle K=-V\frac{\mathrm{\partial }P}{\mathrm{\partial }V}=\frac{V\lambda }{(V-V_{\mathrm{\infty }}{)}^2}=(P_0+P)+\frac{V_{\mathrm{\infty }}}{\lambda }(P_0+P{)}^2.}$

Modified Tait equation of state

Following in particular the study of underwater explosions and more precisely the shock waves emitted, J.G. Kirkwood proposed in 1965 a more appropriate form of equation of state to describe high pressures (>1 kbar) by expressing the isentropic compressibility coefficient as

${\displaystyle -\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }P}\right)}_S=\frac{1}{n(B+P)}}$

where ${\displaystyle S}$ represents here the entropy. The two empirical parameters ${\displaystyle n}$ and ${\displaystyle B}$ are now function of entropy such that

• ${\displaystyle n}$ is dimensionless
• ${\displaystyle B}$ has the same units as ${\displaystyle P}$

The integration leads to the following expression for the volume ${\displaystyle V(P,S)}$ along the isentropic ${\displaystyle S}$

${\displaystyle \frac{V}{V_0}={\left(1+\frac{P_0}{B}\right)}^{1/n}{\left(1+\frac{P}{B}\right)}^{-1/n}}$

where ${\displaystyle V_0=V(P_0,S)}$.

Pressure formula

The expression for the pressure ${\displaystyle P(V,S)}$ in terms of the specific volume along the isentropic ${\displaystyle S}$ is

${\displaystyle P=(B+P_0){\left(\frac{V_0}{V}\right)}^n-B.}$

A highly detailed study on the Modified Tait equation of state with the physical interpretation of the two empirical parameters ${\displaystyle n}$ and ${\displaystyle B}$ is given in chapter 4 of reference. Expressions as a function of entropy for the two empirical parameters ${\displaystyle n}$ and ${\displaystyle B}$ are given for water, helium-3 and helium-4.