Birch–Murnaghan equation of state

Template:Short description Template:Confusion The Birch–Murnaghan isothermal equation of state, published in 1947 by Albert Francis Birch of Harvard,<ref name=Birch47>Template:Cite journal</ref> 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,<ref name=Murn44>Template:Cite journal</ref> so that the equation is named in honor of both scientists.

Expressions for the equation of state

The third-order Birch–Murnaghan isothermal equation of state is given by <math display="block"> P(V)=\frac{3B_0}{2} \left[\left(\frac{V_0}{V}\right)^{7/3} - \left(\frac{V_0}{V}\right)^{5/3}\right] \left\{1+\frac{3}{4}\left(B_0^\prime-4\right) \left[\left(\frac{V_0}{V}\right)^{2/3} - 1\right]\right\}. </math> 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 <math display="block"> B_0 = -V \left(\frac{\partial P}{\partial V}\right)_{P = 0}</math> and <math display="block">B_0' = \left(\frac{\partial B}{\partial P}\right)_{P = 0}</math> 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 <math display="block">f = \frac{1}{2}\left[\left(\frac{V_0}{V}\right)^Template:2/3 - 1\right] \,,</math> in the form of a series.<ref name=Poirier>Template:Cite book</ref>Template:Rp This is more evident by writing the equation in terms of f. Expanded to third order in finite strain, the equation reads,<ref name=Poirier/>Template:Rp <math display="block"> P(f) = 3 B_0 f (1 + 2 f)^{5/2} ( 1 + a f + \mathit{higher~order ~terms})\,, </math> with <math> a = \frac{3}{2}(B_0' - 4) </math>.

The internal energy, Template:Math, is found by integration of the pressure: <math display="block"> E(V) = E_0 + \frac{9V_0B_0}{16} \left\{ \left[\left(\frac{V_0}{V}\right)^{2/3} - 1\right]^3 B_0^\prime + \left[\left(\frac{V_0}{V}\right)^{2/3} - 1\right]^2 \left[6-4\left(\frac{V_0}{V}\right)^{2/3}\right]\right\}. </math>

See also

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

References

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