Birch–Murnaghan Equation Of State

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

The third-order Birch–Murnaghan isothermal equation of state is given by $${\displaystyle P(V)=\frac{3B_0}{2}[{\left(\frac{V_0}{V}\right)}^{7/3}-{\left(\frac{V_0}{V}\right)}^{5/3}]\{1+\frac{3}{4}(B_0^{\prime }-4)[{\left(\frac{V_0}{V}\right)}^{2/3}-1]\}.}$$ 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 $${\displaystyle {B_0}^{\prime }={\left(\frac{\partial B}{\partial P}\right)}_{P=0}}$$ The expression for the equation of state is obtained by expanding the Helmholtz free energy in powers of the finite strain parameter f, defined as $${\displaystyle f=\frac{1}{2}[{\left(\frac{V_0}{V}\right)}^{2/3}-1],}$$ in the form of a series.^{[3]: 68–69} This is more evident by writing the equation in terms of f. Expanded to third order in finite strain, the equation reads,^[3]: 72 $${\displaystyle P(f)=3B_{0}f(1+2f{)}^{5/2}(1+af+{h}{i}{g}{h}{e}{r}{o}{r}{d}{e}{r}{t}{e}{r}{m}{s}),}$$ with ${\displaystyle a=\frac{3}{2}({B_0}^{\prime }-4)}$.

The internal energy, E(V), is found by integration of the pressure: $${\displaystyle E(V)=E_0+\frac{9V_0{B}_0}{16}\{{[{\left(\frac{V_0}{V}\right)}^{2/3}-1]}^3{B}_0^{\prime }+{[{\left(\frac{V_0}{V}\right)}^{2/3}-1]}^2[6-4{\left(\frac{V_0}{V}\right)}^{2/3}]\}.}$$

• Albert Francis Birch
• Francis Dominic Murnaghan
• Murnaghan equation of state

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