# Birch–Murnaghan equation of state

Not to be confused with Murnaghan equation of state.

The Birch–Murnaghan isothermal equation of state, published in 1947 by Francis Birch of Harvard[1], is a relationship between the volume of a body and the pressure to which it is subjected. This equation is named after Albert Francis Birch and Francis Dominic Murnaghan. Birch proposed this equation in a publication in 1947, based on the work of Murnaghan of Johns Hopkins University published in 1944[2].

## Expressions for the equation of state

The third-order Birch–Murnaghan isothermal equation of state is given by:

${\displaystyle P(V)={\frac {3B_{0}}{2}}\left[\left({\frac {V_{0}}{V}}\right)^{\frac {7}{3}}-\left({\frac {V_{0}}{V}}\right)^{\frac {5}{3}}\right]\left\{1+{\frac {3}{4}}\left(B_{0}^{\prime }-4\right)\left[\left({\frac {V_{0}}{V}}\right)^{\frac {2}{3}}-1\right]\right\}.}$

where P is the pressure, V0 is the reference volume, V is the deformed volume, B0 is the bulk modulus, and B0' is the derivative of the bulk modulus with respect to pressure. The bulk modulus and its derivative are usually obtained from fits to experimental data and are defined as

${\displaystyle B_{0}=-V\left({\frac {\partial P}{\partial V}}\right)_{P=0}}$

and

${\displaystyle B_{0}'=\left({\frac {\partial B}{\partial P}}\right)_{P=0}}$

The expression for the equation of state is obtained by expanding the free energy f in the form of a series:

${\displaystyle f={\frac {1}{2}}\left[\left({\frac {V}{V_{0}}}\right)^{-{\frac {2}{3}}}-1\right]\,.}$

The internal energy, E(V), is found by integration of the pressure:

${\displaystyle E(V)=E_{0}+{\frac {9V_{0}B_{0}}{16}}\left\{\left[\left({\frac {V_{0}}{V}}\right)^{\frac {2}{3}}-1\right]^{3}B_{0}^{\prime }+\left[\left({\frac {V_{0}}{V}}\right)^{\frac {2}{3}}-1\right]^{2}\left[6-4\left({\frac {V_{0}}{V}}\right)^{\frac {2}{3}}\right]\right\}.}$